Scientiﬁc Computation Editorial Board J.-J. Chattot, Davis, CA, USA P. Colella, Berkeley, CA, USA Weinan E, Princeton, NJ, USA R. Glowinski, Houston, TX, USA M. Holt, Berkeley, CA, USA Y. Hussaini, Tallahassee, FL, USA P. Joly, Le Chesnay, France H. B. Keller, Pasadena, CA, USA D. I. Meiron, Pasadena, CA, USA O. Pironneau, Paris, France A. Quarteroni, Lausanne, Switzerland J. Rappaz, Lausanne, Switzerland R. Rosner, Chicago, IL, USA. J. H. Seinfeld, Pasadena, CA, USA A. Szepessy, Stockholm, Sweden M. F. Wheeler, Austin, TX, USA

Pierre Sagaut

Large Eddy Simulation for Incompressible Flows An Introduction

Third Edition With a Foreword by Massimo Germano

With 99 Figures and 15 Tables

123

Prof. Dr. Pierre Sagaut LMM-UPMC/CNRS Boite 162, 4 place Jussieu 75252 Paris Cedex 05, France [emailprotected]

Title of the original French edition: Introduction à la simulation des grandes échelles pour les écoulements de ﬂuide incompressible, Mathématique & Applications. © Springer Berlin Heidelberg 1998

Library of Congress Control Number: 2005930493

ISSN 1434-8322 ISBN-10 3-540-26344-6 Third Edition Springer Berlin Heidelberg New York ISBN-13 978-3-540-26344-9 Third Edition Springer Berlin Heidelberg New York ISBN 3-540-67841-7 Second Edition Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2001, 2002, 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig, Germany Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

55/3141/YL

543210

Foreword to the Third Edition

It is with a sense of great satisfaction that I write these lines introducing the third edition of Pierre Sagaut’s account of the ﬁeld of Large Eddy Simulation for Incompressible Flows. Large Eddy Simulation has evolved into a powerful tool of central importance in the study of turbulence, and this meticulously assembled and signiﬁcantly enlarged description of the many aspects of LES will be a most welcome addition to the bookshelves of scientists and engineers in ﬂuid mechanics, LES practitioners, and students of turbulence in general. Hydrodynamic turbulence continues to be a fundamental challenge for scientists striving to understand ﬂuid motions in ﬁelds as diverse as oceanography, acoustics, meteorology and astrophysics. The challenge also has socioeconomic attributes as engineers aim at predicting ﬂows to control their features, and to improve thermo-ﬂuid equipment design. Drag reduction in external aerodynamics or convective heat transfer augmentation are well-known examples. The fundamental challenges posed by turbulence to scientists and engineers have not, in essence, changed since the appearance of the second edition of this book, a mere two years ago. What has evolved signiﬁcantly is the ﬁeld of Large Eddy Simulation (LES), including methods developed to address the closure problem associated with LES (also called the problem of subgrid-scale modeling), numerical techniques for particular applications, and more explicit accounts of the interplay between numerical techniques and subgrid modeling. The original hope for LES was that simple closures would be appropriate, such as mixing length models with a single, universally applicable model parameter. Kolmogorov’s phenomenological theory of turbulence in fact supports this hope but only if the length-scale associated with the numerical resolution of LES falls well within the ideal inertial range of turbulence, in ﬂows at very high Reynolds numbers. Typical applications of LES most often violate this requirement and the resolution length-scale is often close to some externally imposed scale of physical relevance, leading to loss of universality and the need for more advanced, and often much more complex, closure models. Fortunately, the LES modeler disposes of large amount of raw materials from which to assemble improved models. During LES, the resolved motions present rich multi-scale ﬁelds and dynamics including highly non-trivial nonlinear interactions which can be interrogated to learn about

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the local state of turbulence. This availability of dynamical information has led to the formulation of a continuously growing number of diﬀerent closure models and methodologies and associated numerical approaches, including many variations on several basic themes. In consequence, the literature on LES has increased signiﬁcantly in recent years. Just to mention a quantitative measure of this trend, in 2000 the ISI science citation index listed 164 papers published including the keywords ”large-eddy-simulation” during that year. By 2004 this number had doubled to over 320 per year. It is clear, then, that a signiﬁcantly enlarged version of Sagaut’s book, encompassing much of what has been added to the literature since the book’s second edition, is a most welcome contribution to the ﬁeld. What are the main aspects in which this third edition has been enlarged compared to the ﬁrst two? Sagaut has added signiﬁcantly new material in a number of areas. To begin, the introductory chapter is enriched with an overview of the structure of the book, including an illuminating description of three fundamental errors one incurs when attempting to solve ﬂuid mechanics’ inﬁnite-dimensional, non-linear diﬀerential equations, namely projection error, discretization error, and in the case of turbulence and LES, the physically very important resolution error. Following the chapters describing in signiﬁcant detail the relevant foundational aspects of ﬁltering in LES, Sagaut has added a new section dealing with alternative mathematical formulations of LES. These include statistical approaches that replace spatial ﬁltering with conditionally averaging the unresolved motions, and alternative model equations in which the Navier-Stokes equations are replaced with mathematically better behaved equations such as the Leray model in which the advection velocity is regularized (i.e. ﬁltered). In the chapter dealing with functional modeling approaches, in which the subgrid-scale stresses are expressed in terms of local functionals of the resolved velocity gradients, a more complete account of the various versions of the dynamic model is given, as well as extended discussions of new structurefunction and multiscale models. The chapter on structural modeling, in which the stress tensor is reconstructed based on its deﬁnition and various direct hypotheses about the small-scale velocity ﬁeld is signiﬁcantly enhanced: Closures in which full prognostic transport equations are solved for the subgridscale stress tensor are reviewed in detail, and entire new subsections have been added dealing with ﬁltered density function models, with one-dimensional turbulence mapping models, and variational multi-scale models, among others. The chapter focussing on numerical techniques contains an interesting new description of the eﬀects of pre-ﬁltering and of the various methods to perform grid reﬁnement. In the chapter on analysis and validation of LES, a new detailed account is given about methods to evaluate the subgrid-scale kinetic energy. The description of boundary and inﬂow conditions for LES is enhanced with new material dealing with one-dimensional-turbulence models near walls as well as stochastic tools to generate and modulate random ﬁelds

Foreword to the Third Edition

VII

for inlet turbulence speciﬁcation. Chapters dealing with coupling of multiresolution, multidomain, and adaptive grid reﬁnement techniques, as well as LES - RANS coupling, have been extended to include recent additions to the literature. Among others, these are areas to which Sagaut and his co-workers have made signiﬁcant research contributions. The most notable additions are two entirely new chapters at the end of the book, on the prediction of scalars using LES. Both passive scalars, for which subgrid-scale mixing is an important issue, and active scalars, of great importance to geophysical ﬂows, are treated. The geophysics literature on LES of stably and unstably stratiﬁed ﬂows is voluminous - the ﬁeld of LES in fact traces its origins to simulating atmospheric boundary layer ﬂows in the early 1970s. Sagaut summarizes this vast ﬁeld using his classiﬁcations of subgrid closures introduced earlier, and the result is a conceptually elegant and concise treatment, which will be of signiﬁcant interest to both engineering and geophysics practitioners of LES. The connection to geophysical ﬂow prediction reminds us of the importance of LES and subgrid modeling from a broader viewpoint. For the ﬁeld of large-scale numerical simulation of complex multiscale nonlinear systems is, today, at the center of scientiﬁc discussions with important societal and political dimensions. This is most visible in the discussions surrounding the trustworthiness of global change models. Among others, these include boundarylayer parameterizations that can be studied by means of LES done at smaller scales. And LES of turbulence is itself a prime example of large-scale computing applied to prediction of a multi-scale complex system, including issues surrounding the veriﬁcation of its predictive capabilities, the testing of the cumulative accuracy of individual building blocks, and interesting issues on the interplay of stochastic and deterministic aspects of the problem. Thus the book - as well as its subject - Large Eddy Simulation of Incompressible Flow, has much to oﬀer to one of the most pressing issues of our times. With this latest edition, Pierre Sagaut has fully solidiﬁed his position as the preeminent cartographer of the complex and multifaceted world of LES. By mapping out the ﬁeld in meticulous fashion, Sagaut’s work can indeed be regarded as a detailed and evolving atlas of the world of LES. And yet, it is not a tourist guide: as with any relatively young terrain in which the main routes have not yet been ﬁrmly established, what is called for is unbiased, objective, and sophisticated cartography. The cartographer describes the topography, scenery, and landmarks as they appear, without attempting to preach to the traveler which route is best. In return, the traveler is expected to bring along a certain sophistication to interpret the maps and to discern which among the many paths will most likely lead towards particular destinations of interest. The reader of this latest edition will thus be rewarded with a most solid, insightful, and up-to-date account of an important and exciting ﬁeld of research. Baltimore, January 2005

Charles Meneveau

Foreword to the Second Edition

It is a particular pleasure to present the second edition of the book on Large Eddy Simulation for Incompressible Flows written by Pierre Sagaut: two editions in two years means that the interest in the topic is strong and that a book on it was indeed required. Compared to the ﬁrst one, this second edition is a greatly enriched version, motivated both by the increasing theoretical interest in Large Eddy Simulation (LES) and the increasing numbers of applications and practical issues. A typical one is the need to decrease the computational cost, and this has motivated two entirely new chapters devoted to the coupling of LES with multiresolution multidomain techniques and to the new hybrid approaches that relate the LES procedures to the classical statistical methods based on the Reynolds Averaged Navier–Stokes equations. Not that literature on LES is scarce. There are many article reviews and conference proceedings on it, but the book by Sagaut is the ﬁrst that organizes a topic that by its peculiar nature is at the crossroads of various interests and techniques: ﬁrst of all the physics of turbulence and its diﬀerent levels of description, then the computational aspects, and ﬁnally the applications that involve a lot of diﬀerent technical ﬁelds. All that has produced, particularly during the last decade, an enormous number of publications scattered over scientiﬁc journals, technical notes, and symposium acta, and to select and classify with a systematic approach all this material is a real challenge. Also, by assuming, as the writer does, that the reader has a basic knowledge of ﬂuid mechanics and applied mathematics, it is clear that to introduce the procedures presently adopted in the large eddy simulation of turbulent ﬂows is a diﬃcult task in itself. First of all, there is no accepted universal deﬁnition of what LES really is. It seems that LES covers everything that lies between RANS, the classical statistical picture of turbulence based on the Reynolds Averaged Navier–Stokes equations, and DNS, the Direct Numerical Simulations resolved in all details, but till now there has not been a general uniﬁed theory that gradually goes from one description to the other. Moreover we should note the diﬀerent importance that the practitioners of LES attribute to the numerical and the modeling aspects. At one end the supporters of the no model way of thinking argue that the numerical scheme should and could capture by itself the resolved scales. At the other end the theoretical

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modelers try to develop new universal equations for the ﬁltered quantities. In some cases LES is regarded as a technique imposed by the present provisional inability of the computers to solve all the details. Others think that LES modeling is a contribution to the understanding of turbulence and the interactions among diﬀerent ideas are often poor. Pierre Sagaut has elaborated on this immense material with an open mind and in an exceptionally clear way. After three chapters devoted to the basic problem of the scale separation and its application to the Navier–Stokes equations, he classiﬁes the various subgrid models presently in use as functional and structural ones. The chapters devoted to this general review are of the utmost interest: obviously some selection has been done, but both the student and the professional engineer will ﬁnd there a clear unbiased exposition. After this ﬁrst part devoted to the fundamentals a second part covers many of the interdisciplinary problems created by the practical use of LES and its coupling with the numerical techniques. These subjects, very important obviously from the practical point of view, are also very rich in theoretical aspects, and one great merit of Sagaut is that he presents them always in an attractive way without reducing the exposition to a mere set of instructions. The interpretation of the numerical solutions, the validation and the comparison of LES databases, the general problem of the boundary conditions are mathematically, physically and numerically analyzed in great detail, with a principal interest in the general aspects. Two entirely new chapters are devoted to the coupling of LES with multidomain techniques, a topic in which Pierre Sagaut and his group have made important contributions, and to the new hybrid approaches RANS/LES, and ﬁnally in the last expanded chapter, enriched by new examples and beautiful ﬁgures, we have a review of the diﬀerent applications of LES in the nuclear, aeronautical, chemical and automotive ﬁelds. Both for graduate students and for scientists this book is a very important reference. People involved in the large eddy simulation of turbulent ﬂows will ﬁnd a useful introduction to the topic and a complete and systematic overview of the many diﬀerent modeling procedures. At present their number is very high and in the last chapter the author tries to draw some conclusions concerning their eﬃciency, but probably the person who is only interested in the basic question “What is the best model for LES? ” will remain a little disappointed. As remarked by the author, both the structural and the functional models have their advantages and disadvantages that make them seem complementary, and probably a mixed modeling procedure will be in the future a good compromise. But for a textbook this is not the main point. The fortunes and the misfortunes of a model are not so simple to predict, and its success is in many cases due to many particular reasons. The results are obviously the most important test, but they also have to be considered in a textbook with a certain reserve, in the higher interest of a presentation that tries as much as possible to be not only systematic but also rational.

Foreword to the Second Edition

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To write a textbook obliges one in some way or another to make judgements, and to transmit ideas, sometimes hidden in procedures that for some reason or another have not till now received interest from the various groups involved in LES and have not been explored in full detail. Pierre Sagaut has succeeded exceptionally well in doing that. One reason for the success is that the author is curious about every detail. The ﬁnal task is obviously to provide a good and systematic introduction to the beginner, as rational as a book devoted to turbulence can be, and to provide useful information for the specialist. The research has, however, its peculiarities, and this book is unambiguously written by a passionate researcher, disposed to explore every problem, to search in all models and in all proposals the germs of new potentially useful ideas. The LES procedures that mix theoretical modeling and numerical computation are often, in an inextricable way, exceptionally rich in complex problems. What about the problem of the mesh adaptation on unstructured grids for large eddy simulations? Or the problem of the comparison of the LES results with reference data? Practice shows that nearly all authors make comparisons with reference data or analyze large eddy simulation data with no processing of the data .... Pierre Sagaut has the courage to dive deep into procedures that are sometimes very diﬃcult to explore, with the enthusiasm of a genuine researcher interested in all aspects and conﬁdent about every contribution. This book now in its second edition seems really destined for a solid and durable success. Not that every aspect of LES is covered: the rapid progress of LES in compressible and reacting ﬂows will shortly, we hope, motivate further additions. Other developments will probably justify new sections. What seems, however, more important is that the basic style of this book is exceptionally valid and open to the future of a young, rapidly evolving discipline. This book is not an encyclopedia and it is not simply a monograph, it provides a framework that can be used as a text of lectures or can be used as a detailed and accurate review of modeling procedures. The references, now increased in number to nearly 500, are given not only to extend but largely to support the material presented, and in some cases the dialogue goes beyond the original paper. As such, the book is recommended as a fundamental work for people interested in LES: the graduate and postgraduate students will ﬁnd an immense number of stimulating issues, and the specialists, researchers and engineers involved in the more and more numerous ﬁelds of application of LES will ﬁnd a reasoned and systematic handbook of diﬀerent procedures. Last, but not least, the applied mathematician can ﬁnally enjoy considering the richness of challenging and attractive problems proposed as a result of the interaction among diﬀerent topics. Torino, April 2002

Massimo Germano

Foreword to the First Edition

Still today, turbulence in ﬂuids is considered as one of the most diﬃcult problems of modern physics. Yet we are quite far from the complexity of microscopic molecular physics, since we only deal with Newtonian mechanics laws applied to a continuum, in which the eﬀect of molecular ﬂuctuations has been smoothed out and is represented by molecular-viscosity coeﬃcients. Such a system has a dual behaviour of determinism in the Laplacian sense, and extreme sensitivity to initial conditions because of its very strong nonlinear character. One does not know, for instance, how to predict the critical Reynolds number of transition to turbulence in a pipe, nor how to compute precisely the drag of a car or an aircraft, even with today’s largest computers. We know, since the meteorologist Richardson,1 numerical schemes allowing us to solve in a deterministic manner the equations of motion, starting with a given initial state and with prescribed boundary conditions. They are based on momentum and energy balances. However, such a resolution requires formidable computing power, and is only possible for low Reynolds numbers. These Direct-Numerical Simulations may involve calculating the interaction of several million interacting sites. Generally, industrial, natural, or experimental conﬁgurations involve Reynolds numbers that are far too large to allow direct simulations,2 and the only possibility then is Large Eddy Simulations, where the small-scale turbulent ﬂuctuations are themselves smoothed out and modelled via eddy-viscosity and diﬀusivity assumptions. The history of large eddy simulations began in the 1960s with the famous Smagorinsky model. Smagorinsky, also a meteorologist, wanted to represent the eﬀects upon large synoptic quasi-two-dimensional atmospheric or oceanic motions3 of a three-dimensional subgrid turbulence cascading toward small scales according to mechanisms described by Richardson in 1926 and formalized by the famous mathematician Kolmogorov in 1941.4 It is interesting to note that Smagorinsky’s model was a total failure as far as the 1 2 3 4

L.F. Richardson, Weather Prediction by Numerical Process, Cambridge University Press (1922). More than 1015 modes should be necessary for a supersonic-plane wing! Subject to vigorous inverse-energy cascades. L.F. Richardson, Proc. Roy. Soc. London, Ser A, 110, pp. 709–737 (1926); A. Kolmogorov, Dokl. Akad. Nauk SSSR, 30, pp. 301–305 (1941).

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atmosphere and oceans are concerned, because it dissipates the large-scale motions too much. It was an immense success, though, with users interested in industrial-ﬂow applications, which shows that the outcomes of research are as unpredictable as turbulence itself! A little later, in the 1970s, the theoretical physicist Kraichnan5 developed the important concept of spectral eddy viscosity, which allows us to go beyond the separation-scale assumption inherent in the typical eddy-viscosity concept of Smagorinsky. From then on, the history of large eddy simulations developed, ﬁrst in the wake of two schools: Stanford–Torino, where a dynamic version of Smagorinsky’s model was developed; and Grenoble, which followed Kraichnan’s footsteps. Then researchers, including industrial researchers, all around the world became infatuated with these techniques, being aware of the limits of classical modeling methods based on the averaged equations of motion (Reynolds equations). It is a complete account of this young but very rich discipline, the large eddy simulation of turbulence, which is proposed to us by the young ONERA researcher Pierre Sagaut, in a book whose reading brings pleasure and interest. Large-Eddy Simulation for Incompressible Flows - An Introduction very wisely limits itself to the case of incompressible ﬂuids, which is a suitable starting point if one wants to avoid multiplying diﬃculties. Let us point out, however, that compressible ﬂows quite often exhibit near-incompressible properties in boundary layers, once the variation of the molecular viscosity with the temperature has been taken into account, as predicted by Morkovin in his famous hypothesis.6 Pierre Sagaut shows an impressive culture, describing exhaustively all the subgrid-modeling methods for simulating the large scales of turbulence, without hesitating to give the mathematical details needed for a proper understanding of the subject. After a general introduction, he presents and discusses the various ﬁlters used, in cases of statistically hom*ogeneous and inhom*ogeneous turbulence, and their applications to Navier–Stokes equations. He very aptly describes the representation of various tensors in Fourier space, Germano-type relations obtained by double ﬁltering, and the consequences of Galilean invariance of the equations. He then goes into the various ways of modeling isotropic turbulence. This is done ﬁrst in Fourier space, with the essential wave-vector triad idea, and a discussion of the transfer-localness concept. An excellent review of spectral-viscosity models is provided, with developments going beyond the original papers. Then he goes to physical space, with a discussion of the structure-function models and the dynamic procedures (Eulerian and Lagrangian, with energy equations and so forth). The study is then generalized to the anisotropic case. Finally, functional approaches based on Taylor series expansions are discussed, along with non-linear models, hom*ogenization techniques, and simple and dynamic mixed models. 5 6

He worked as a postdoctoral student with Einstein at Princeton. M.V. Morkovin, in M´ecanique de la Turbulence, A. Favre et al. (eds.), CNRS, pp. 367–380 (1962).

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Pierre Sagaut also discusses the importance of numerical errors, and proposes a very interesting review of the diﬀerent wall models in the boundary layers. The last chapter gives a few examples of applications carried out at ONERA and a few other French laboratories. These examples are well chosen in order of increasing complexity: isotropic turbulence, with the non-linear condensation of vorticity into the “worms” vortices discovered by Siggia;7 planar Poiseuille ﬂow with ejection of “hairpin” vortices above low-speed streaks; the round jet and its alternate pairing of vortex rings; and, ﬁnally, the backward-facing step, the unavoidable test case of computational ﬂuid dynamics. Also on the menu: beautiful visualizations of separation behind a wing at high incidence, with the shedding of superb longitudinal vortices. Completing the work are two appendices on the statistical and spectral analysis of turbulence, as well as isotropic and anisotropic EDQNM modeling. A bold explorer, Pierre Sagaut had the daring to plunge into the jungle of multiple modern techniques of large-scale simulation of turbulence. He came back from his trek with an extremely complete synthesis of all the models, giving us a very complete handbook that novices can use to start oﬀ on this enthralling adventure, while specialists can discover models diﬀerent from those they use every day. Large-Eddy Simulation for Incompressible Flows - An Introduction is a thrilling work in a somewhat austere wrapping. I very warmly recommend it to the broad public of postgraduate students, researchers, and engineers interested in ﬂuid mechanics and its applications in numerous ﬁelds such as aerodynamics, combustion, energetics, and the environment. Grenoble, March 2000

7

E.D. Siggia, J. Fluid Mech., 107, pp. 375–406 (1981).

Marcel Lesieur

Preface to the Third Edition

Working on the manuscript of the third edition of this book was a very exciting task, since a lot of new developments have been published since the second edition was printed. The large-eddy simulation (LES) technique is now recognized as a powerful tool and real applications in several engineering ﬁelds are more and more frequently found. This increasing demand for eﬃcient LES tools also sustains growing theoretical research on many aspects of LES, some of which are included in this book. Among them, it is worth noting the mathematical models of LES (the convolution ﬁlter being only one possiblity), the deﬁnition of boundary conditions, the coupling with numerical errors, and, of course, the problem of deﬁning adequate subgrid models. All these issues are discussed in more detail in this new edition. Some good news is that other monographs, which are good complements to the present book, are now available, showing that LES is a topic with a fastly growing audience. The reader interested in mathematics-oriented discussions will ﬁnd many details in the monoghaphs by Volker John (Large-Eddy Simulation of Turbulent Incompressible Flows, Springer) and Berselli, Illiescu and Layton (Mathematics of Large-Eddy Simulation of Turbulent Flows, Springer), while people looking for a subsequent description of numerical methods for LES and direct numerical simulation will enjoy the book by Bernard Geurts (Elements of Direct and Large-Eddy Simulation, Edwards). More monographs devoted to particular features of LES (implicit LES appraoches, mathematical backgrounds, etc.) are to come in the near future. My purpose while writing this third edition was still to provide the reader with an up-to-date review of existing methods, approaches and models for LES of incompressible ﬂows. All chapters of the previous edition have been updated, with the hope that this nearly exhaustive review will help interested readers avoid rediscovering old things. I would like to apologize in advance for certainly forgetting some developments. Two entirely new chapters have been added. The ﬁrst one deals with mathematical models for LES. Here, I believe that the interesting point is that the ﬁltering approach is nothing but a model for the true LES problem, and other models have been developed that seem to be at least as promising as this very popular one. The second new chapter is dedicated to the scalar equation, with both passive scalar and active scalar

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(stable/unstable stratiﬁcation eﬀects) cases being discussed. This extension illustrates the way the usual LES can be extended and how new physical mechanisms can be dealt with, but also inspires new problems. Paris, November 2004

Pierre Sagaut

Preface to the Second Edition

The astonishingly rapid development of the Large-Eddy Simulation technique during the last two or three years, both from the theoretical and applied points of view, have rendered the ﬁrst edition of this book lacunary in some ways. Three to four years ago, when I was working on the manuscript of the ﬁrst edition, coupling between LES and multiresolution/multilevel techniques was just an emerging idea. Nowadays, several applications of this approach have been succesfully developed and applied to several ﬂow conﬁgurations. Another example of interest from this exponentially growing ﬁeld is the development of hybrid RANS/LES approaches, which have been derived under many diﬀerent forms. Because these topics are promising and seem to be possible ways of enhancing the applicability of LES, I felt that they should be incorporated in a general presentation of LES. Recent developments in LES theory also deal with older topics which have been intensely revisited by reseachers: a uniﬁed theory for deconvolution and scale similarity ways of modeling have now been established; the “no model” approach, popularized as the MILES approach, is now based on a deeper theoretical analysis; a lot of attention has been paid to the problem of the deﬁnition of boundary conditions for LES; ﬁltering has been extended to Navier–Stokes equations in general coordinates and to Eulerian time–domain ﬁltering. Another important fact is that LES is now used as an engineering tool for several types of applications, mainly dealing with massively separated ﬂows in complex conﬁgurations. The growing need for unsteady, accurate simulations, more and more associated with multidisciplinary applications such as aeroacoustics, is a very powerful driver for LES, and it is certain that this technique is of great promise. For all these reasons, I accepted the opportunity to revise and to augment this book when Springer oﬀered it me. I would also like to emphasize the fruitful interactions between “traditional” LES researchers and mathematicians that have very recently been developed, yielding, for example, a better understanding of the problem of boundary conditions. Mathematical foundations for LES are under development, and will not be presented in this book, because I did not want to include specialized functional analysis discussions in the present framework.

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I am indebted to an increasing number of people, but I would like to express special thanks to all my colleagues at ONERA who worked with me on LES: Drs. E. Garnier, E. Labourasse, I. Mary, P. Qu´em´er´e and M. Terracol. All the people who provided me with material dealing with their research are also warmly acknowledged. I also would like to thank all the readers of the ﬁrst edition of this book who very kindly provided me with their remarks, comments and suggestions. Mrs. J. Ryan is once again gratefully acknowledged for her help in writing the English version. Paris, April 2002

Pierre Sagaut

Preface to the First Edition

While giving lectures dealing with Large-Eddy Simulation (LES) to students or senior scientists, I have found diﬃculties indicating published references which can serve as general and complete introductions to this technique. I have tried therefore to write a textbook which can be used by students or researchers showing theoretical and practical aspects of the Large Eddy Simulation technique, with the purpose of presenting the main theoretical problems and ways of modeling. It assumes that the reader possesses a basic knowledge of ﬂuid mechanics and applied mathematics. Introducing Large Eddy Simulation is not an easy task, since no uniﬁed and universally accepted theoretical framework exists for it. It should be remembered that the ﬁrst LES computations were carried out in the early 1960s, but the ﬁrst rigorous derivation of the LES governing equations in general coordinates was published in 1995! Many reasons can be invoked to explain this lack of a uniﬁed framework. Among them, the fact that LES stands at the crossroads of physical modeling and numerical analysis is a major point, and only a few really successful interactions between physicists, mathematicians and practitioners have been registered over the past thirty years, each community sticking to its own language and center of interest. Each of these three communities, though producing very interesting work, has not yet provided a complete theoretical framework for LES by its own means. I have tried to gather these diﬀerent contributions in this book, in an understandable form for readers having a basic background in applied mathematics. Another diﬃculty is the very large number of existing physical models, referred to as subgrid models. Most of them are only used by their creators, and appear in a very small number of publications. I made the choice to present a very large number of models, in order to give the reader a good overview of the ways explored. The distinction between functional and structural models is made in this book, in order to provide a general classiﬁcation; this was necessary to produce an integrated presentation. In order to provide a useful synthesis of forty years of LES development, I had to make several choices. Firstly, the subject is restricted to incompressible ﬂows, as the theoretical background for compressible ﬂow is less evolved. Secondly, it was necessary to make a uniﬁed presentation of a large

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number of works issued from many research groups, and very often I have had to change the original proof and to reduce it. I hope that the authors will not feel betrayed by the present work. Thirdly, several thousand journal articles and communications dealing with LES can be found, and I had to make a selection. I have deliberately chosen to present a large number of theoretical approaches and physical models to give the reader the most general view of what has been done in each ﬁeld. I think that the most important contributions are presented in this book, but I am sure that many new physical models and results dealing with theoretical aspects will appear in the near future. A typical question of people who are discovering LES is “what is the best model for LES?”. I have to say that I am convinced that this question cannot be answered nowadays, because no extensive comparisons have been carried out, and I am not even sure that the answer exists, because people do not agree on the criterion to use to deﬁne the “best” model. As a consequence, I did not try to rank the model, but gave very generally agreed conclusions on the model eﬃciency. A very important point when dealing with LES is the numerical algorithm used to solve the governing equations. It has always been recognized that numerical errors could aﬀect the quality of the solution, but new emphasis has been put on this subject during the last decade, and it seems that things are just beginning. This point appeared as a real problem to me when writing this book, because many conclusions are still controversial (e.g. the possibility of using a second-order accurate numerical scheme or an artiﬁcial diﬀusion). So I chose to mention the problems and the diﬀerent existing points of view, but avoided writing a part dealing entirely with numerical discretization and time integration, discretization errors, etc. This would have required writing a companion book on numerical methods, and that was beyond the scope of the present work. Many good textbooks on that subject already exist, and the reader should refer to them. Another point is that the analysis of the coupling of LES with typical numerical techniques, which should greatly increase the range of applications, such as Arbitrary Lagrangian–Eulerian methods, Adaptive Mesh-Reﬁnement or embedded grid techniques, is still to be developed. I am indebted to a large number of people, but I would like to express special thanks to Dr. P. Le Qu´ere, O. Daube, who gave me the opportunity to write my ﬁrst manuscript on LES, and to Prof. J.M. Ghidaglia who oﬀered me the possibility of publishing the ﬁrst version of this book (in French). I would also like to thank ONERA for helping me to write this new, augmented and translated version of the book. Mrs. J. Ryan is gratefully acknowledged for her help in writing the English version. Paris, September 2000

Pierre Sagaut

Contents

1.

2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Levels of Approximation: General . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Statement of the Scale Separation Problem . . . . . . . . . . . . . . . . 1.4 Usual Levels of Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Large-Eddy Simulation: from Practice to Theory. Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formal Introduction to Scale Separation: Band-Pass Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Deﬁnition and Properties of the Filter in the hom*ogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Characterization of Diﬀerent Approximations . . . . . . . . 2.1.4 Diﬀerential Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Three Classical Filters for Large-Eddy Simulation . . . . 2.1.6 Diﬀerential Interpretation of the Filters . . . . . . . . . . . . . 2.2 Spatial Filtering: Extension to the Inhom*ogeneous Case . . . . . 2.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Non-uniform Filtering Over an Arbitrary Domain . . . . 2.2.3 Local Spectrum of Commutation Error . . . . . . . . . . . . . . 2.3 Time Filtering: a Few Properties . . . . . . . . . . . . . . . . . . . . . . . . . Application to Navier–Stokes Equations . . . . . . . . . . . . . . . . . . 3.1 Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Formulation in Physical Space . . . . . . . . . . . . . . . . . . . . . 3.1.2 Formulation in General Coordinates . . . . . . . . . . . . . . . . 3.1.3 Formulation in Spectral Space . . . . . . . . . . . . . . . . . . . . . 3.2 Filtered Navier–Stokes Equations in Cartesian Coordinates (hom*ogeneous Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Formulation in Physical Space . . . . . . . . . . . . . . . . . . . . . 3.2.2 Formulation in Spectral Space . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 5 9

15 15 15 17 18 20 21 26 31 31 32 42 43 45 46 46 46 47 48 48 48

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3.3 Decomposition of the Non-linear Term. Associated Equations for the Conventional Approach . . . . . . . 3.3.1 Leonard’s Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Germano Consistent Decomposition . . . . . . . . . . . . . . . . 3.3.3 Germano Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Invariance Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Realizability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Extension to the Inhom*ogeneous Case for the Conventional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Second-Order Commuting Filter . . . . . . . . . . . . . . . . . . . . 3.4.2 High-Order Commuting Filters . . . . . . . . . . . . . . . . . . . . . 3.5 Filtered Navier–Stokes Equations in General Coordinates . . . . 3.5.1 Basic Form of the Filtered Equations . . . . . . . . . . . . . . . 3.5.2 Simpliﬁed Form of the Equations – Non-linear Terms Decomposition . . . . . . . . . . . . . . . . . . . 3.6 Closure Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Functional and Structural Modeling . . . . . . . . . . . . . . . . 4.

5.

49 49 59 61 64 72 74 74 77 77 77 78 78 78 79 80

Other Mathematical Models for the Large-Eddy Simulation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Ensemble-Averaged Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Yoshizawa’s Partial Statistical Average Model . . . . . . . . 4.1.2 McComb’s Conditional Mode Elimination Procedure . . 4.2 Regularized Navier–Stokes Models . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Leray’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Holm’s Navier–Stokes-α Model . . . . . . . . . . . . . . . . . . . . . 4.2.3 Ladyzenskaja’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 83 84 85 86 86 89

Functional Modeling (Isotropic Case) . . . . . . . . . . . . . . . . . . . . . 5.1 Phenomenology of Inter-Scale Interactions . . . . . . . . . . . . . . . . . 5.1.1 Local Isotropy Assumption: Consequences . . . . . . . . . . . 5.1.2 Interactions Between Resolved and Subgrid Scales . . . . 5.1.3 A View in Physical Space . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Basic Functional Modeling Hypothesis . . . . . . . . . . . . . . . . . . . . 5.3 Modeling of the Forward Energy Cascade Process . . . . . . . . . . 5.3.1 Spectral Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Physical Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Improvement of Models in the Physical Space . . . . . . . 5.3.4 Implicit Diﬀusion: the ILES Concept . . . . . . . . . . . . . . . . 5.4 Modeling the Backward Energy Cascade Process . . . . . . . . . . . 5.4.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 92 93 102 104 104 105 105 109 133 161 171 171

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XXV

5.4.2 Deterministic Statistical Models . . . . . . . . . . . . . . . . . . . . 172 5.4.3 Stochastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.

7.

Functional Modeling: Extension to Anisotropic Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Application of Anisotropic Filter to Isotropic Flow . . . . . . . . . . 6.2.1 Scalar Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Batten’s Mixed Space-Time Scalar Estimator . . . . . . . . 6.2.3 Tensorial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Application of an Isotropic Filter to a Shear Flow . . . . . . . . . . 6.3.1 Phenomenology of Inter-Scale Interactions . . . . . . . . . . . 6.3.2 Anisotropic Models: Scalar Subgrid Viscosities . . . . . . . 6.3.3 Anisotropic Models: Tensorial Subgrid Viscosities . . . . . 6.4 Remarks on Flows Submitted to Strong Rotation Eﬀects . . . . Structural Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Formal Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Models Based on Approximate Deconvolution . . . . . . . . 7.2.2 Non-linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 hom*ogenization-Technique-Based Models . . . . . . . . . . . . 7.3 Scale Similarity Hypotheses and Models Using Them . . . . . . . . 7.3.1 Scale Similarity Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Scale Similarity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 A Bridge Between Scale Similarity and Approximate Deconvolution Models. Generalized Similarity Models . 7.4 Mixed Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Examples of Mixed Models . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Diﬀerential Subgrid Stress Models . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Deardorﬀ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Fureby Diﬀerential Subgrid Stress Model . . . . . . . . . . . . 7.5.3 Velocity-Filtered-Density-Function-Based Subgrid Stress Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Link with the Subgrid Viscosity Models . . . . . . . . . . . . . 7.6 Stretched-Vortex Subgrid Stress Models . . . . . . . . . . . . . . . . . . . 7.6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 S3/S2 Alignment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 S3/ω Alignment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.4 Kinematic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Explicit Evaluation of Subgrid Scales . . . . . . . . . . . . . . . . . . . . . 7.7.1 Fractal Interpolation Procedure . . . . . . . . . . . . . . . . . . . . 7.7.2 Chaotic Map Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 187 188 191 191 193 193 198 202 208 209 209 210 210 223 228 231 231 232 236 237 237 239 243 243 244 245 248 249 249 250 250 250 251 253 254

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7.7.3 Kerstein’s ODT-Based Method . . . . . . . . . . . . . . . . . . . . . 7.7.4 Kinematic-Simulation-Based Reconstruction . . . . . . . . . 7.7.5 Velocity Filtered Density Function Approach . . . . . . . . . 7.7.6 Subgrid Scale Estimation Procedure . . . . . . . . . . . . . . . . 7.7.7 Multi-level Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Direct Identiﬁcation of Subgrid Terms . . . . . . . . . . . . . . . . . . . . . 7.8.1 Linear-Stochastic-Estimation-Based Model . . . . . . . . . . 7.8.2 Neural-Network-Based Model . . . . . . . . . . . . . . . . . . . . . . 7.9 Implicit Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.1 Local Average Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.2 Scale Residual Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257 259 260 261 263 272 274 275 275 276 278

Numerical Solution: Interpretation and Problems . . . . . . . . . 8.1 Dynamic Interpretation of the Large-Eddy Simulation . . . . . . . 8.1.1 Static and Dynamic Interpretations: Eﬀective Filter . . 8.1.2 Theoretical Analysis of the Turbulence Generated by Large-Eddy Simulation . . . . . . . . . . . . . . . 8.2 Ties Between the Filter and Computational Grid. Pre-ﬁltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Numerical Errors and Subgrid Terms . . . . . . . . . . . . . . . . . . . . . 8.3.1 Ghosal’s General Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Pre-ﬁltering Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Remarks on the Use of Artiﬁcial Dissipations . . . . . . . . 8.3.5 Remarks Concerning the Time Integration Method . . .

281 281 281

Analysis and Validation of Large-Eddy Simulation Data . . 9.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Type of Information Contained in a Large-Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Validation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Statistical Equivalency Classes of Realizations . . . . . . . 9.1.4 Ideal LES and Optimal LES . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Mathematical Analysis of Sensitivities and Uncertainties in Large-Eddy Simulation . . . . . . . . . 9.2 Correction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Filtering the Reference Data . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Evaluation of Subgrid-Scale Contribution . . . . . . . . . . . . 9.2.3 Evaluation of Subgrid-Scale Kinetic Energy . . . . . . . . . . 9.3 Practical Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

305 305

10. Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 General Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Mathematical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Physical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

323 323 323 324

8.

9.

283 288 290 290 294 297 299 303

305 306 307 310 311 313 313 314 315 318

Contents XXVII

10.2 Solid Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 A Few Wall Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Wall Models: Achievements and Problems . . . . . . . . . . . 10.3 Case of the Inﬂow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Required Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Inﬂow Condition Generation Techniques . . . . . . . . . . . . . 11. Coupling Large-Eddy Simulation with Multiresolution/Multidomain Techniques . . . . . . . . . . . . 11.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Methods with Full Overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 One-Way Coupling Algorithm . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Two-Way Coupling Algorithm . . . . . . . . . . . . . . . . . . . . . 11.2.3 FAS-like Multilevel Method . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Kravchenko et al. Method . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Methods Without Full Overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Coupling Large-Eddy Simulation with Adaptive Mesh Reﬁnement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

326 326 332 351 354 354 354

369 369 371 372 372 373 374 376 377 377 378

12. Hybrid RANS/LES Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Motivations and Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Zonal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Sharp Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Smooth Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 Zonal RANS/LES Approach as Wall Model . . . . . . . . . . 12.3 Nonlinear Disturbance Equations . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Universal Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Germano’s Hybrid Model . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Speziale’s Rescaling Method and Related Approaches . 12.4.3 Baurle’s Blending Strategy . . . . . . . . . . . . . . . . . . . . . . . . 12.4.4 Arunajatesan’s Modiﬁed Two-Equation Model . . . . . . . 12.4.5 Bush–Mani Limiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.6 Magagnato’s Two-Equation Model . . . . . . . . . . . . . . . . . . 12.5 Toward a Theoretical Status for Hybrid RANS/LES Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

383 383 384 384 385 387 388 390 391 392 393 394 396 397 398

13. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Filter Identiﬁcation. Computing the Cutoﬀ Length . . . . . . . . . 13.2 Explicit Discrete Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Uniform One-Dimensional Grid Case . . . . . . . . . . . . . . . . 13.2.2 Extension to the Multi-Dimensional Case . . . . . . . . . . . .

401 401 404 404 407

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13.2.3 Extension to the General Case. Convolution Filters . . . 407 13.2.4 High-Order Elliptic Filters . . . . . . . . . . . . . . . . . . . . . . . . . 408 13.3 Implementation of the Structure Function Models . . . . . . . . . . 408 14. Examples of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 hom*ogeneous Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Isotropic hom*ogeneous Turbulence . . . . . . . . . . . . . . . . . 14.1.2 Anisotropic hom*ogeneous Turbulence . . . . . . . . . . . . . . . 14.2 Flows Possessing a Direction of Inhom*ogeneity . . . . . . . . . . . . . 14.2.1 Time-Evolving Plane Channel . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Other Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Flows Having at Most One Direction of hom*ogeneity . . . . . . . 14.3.1 Round Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Backward Facing Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Square-Section Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.4 Other Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Large-Eddy Simulation for Nuclear Power Plants . . . . . 14.4.2 Flow in a Mixed-Flow Pump . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 Flow Around a Landing Gear Conﬁguration . . . . . . . . . 14.4.4 Flow Around a Full-Scale Car . . . . . . . . . . . . . . . . . . . . . . 14.5 Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.1 General Lessons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2 Subgrid Model Eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.3 Wall Model Eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.4 Mesh Generation for Building Blocks Flows . . . . . . . . . .

411 411 411 412 414 414 418 419 419 426 430 431 432 432 435 437 437 439 439 442 444 445

15. Coupling with Passive/Active Scalar . . . . . . . . . . . . . . . . . . . . . . 15.1 Scope of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Passive Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Dynamics of the Passive Scalar . . . . . . . . . . . . . . . . . . . . . 15.2.3 Extensions of Functional Models . . . . . . . . . . . . . . . . . . . 15.2.4 Extensions of Structural Models . . . . . . . . . . . . . . . . . . . . 15.2.5 Generalized Subgrid Modeling for Arbitrary Non-linear Functions of an Advected Scalar . . . . . . . . . . . . . . . . . . . . 15.2.6 Models for Subgrid Scalar Variance and Scalar Subgrid Mixing Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.7 A Few Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 The Active Scalar Case: Stratiﬁcation and Buoyancy Eﬀects . 15.3.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Some Insights into the Active Scalar Dynamics . . . . . . . 15.3.3 Extensions of Functional Models . . . . . . . . . . . . . . . . . . . 15.3.4 Extensions of Structural Models . . . . . . . . . . . . . . . . . . . . 15.3.5 Subgrid Kinetic Energy Estimates . . . . . . . . . . . . . . . . . .

449 449 450 450 453 461 466 468 469 472 472 472 474 481 487 490

Contents

XXIX

15.3.6 More Complex Physical Models . . . . . . . . . . . . . . . . . . . . 492 15.3.7 A Few Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 A. Statistical and Spectral Analysis of Turbulence . . . . . . . . . . . A.1 Turbulence Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Foundations of the Statistical Analysis of Turbulence . . . . . . . A.2.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Statistical Average: Deﬁnition and Properties . . . . . . . . A.2.3 Ergodicity Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.4 Decomposition of a Turbulent Field . . . . . . . . . . . . . . . . . A.2.5 Isotropic hom*ogeneous Turbulence . . . . . . . . . . . . . . . . . A.3 Introduction to Spectral Analysis of the Isotropic Turbulent Fields . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.2 Modal Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.3 Spectral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Characteristic Scales of Turbulence . . . . . . . . . . . . . . . . . . . . . . . A.5 Spectral Dynamics of Isotropic hom*ogeneous Turbulence . . . . A.5.1 Energy Cascade and Local Isotropy . . . . . . . . . . . . . . . . A.5.2 Equilibrium Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . .

495 495 495 495 496 496 498 499

B. EDQNM Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Isotropic EDQNM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Cambon’s Anisotropic EDQNM Model . . . . . . . . . . . . . . . . . . . . B.3 EDQNM Model for Isotropic Passive Scalar . . . . . . . . . . . . . . . .

507 507 509 511

499 499 501 502 504 504 504 505

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

1. Introduction

1.1 Computational Fluid Dynamics Computational Fluid Dynamics (CFD) is the study of ﬂuids in ﬂow by numerical simulation, and is a ﬁeld advancing by leaps and bounds. The basic idea is to use appropriate algorithms to ﬁnd solutions to the equations describing the ﬂuid motion. Numerical simulations are used for two types of purposes. The ﬁrst is to accompany research of a fundamental kind. By describing the basic physical mechanisms governing ﬂuid dynamics better, numerical simulation helps us understand, model, and later control these mechanisms. This kind of study requires that the numerical simulation produce data of very high accuracy, which implies that the physical model chosen to represent the behavior of the ﬂuid must be pertinent and that the algorithms used, and the way they are used by the computer system, must introduce no more than a low level of error. The quality of the data generated by the numerical simulation also depends on the level of resolution chosen. For the best possible precision, the simulation has to take into account all the space-time scales aﬀecting the ﬂow dynamics. When the range of scales is very large, as it is in turbulent ﬂows, for example, the problem becomes a stiﬀ one, in the sense that the ratio between the largest and smallest scales becomes very large. Numerical simulation is also used for another purpose: engineering analyses, where ﬂow characteristics need to be predicted in equipment design phase. Here, the goal is no longer to produce data for analyzing the ﬂow dynamics itself, but rather to predict certain of the ﬂow characteristics or, more precisely, the values of physical parameters that depend on the ﬂow, such as the stresses exerted on an immersed body, the production and propagation of acoustic waves, or the mixing of chemical species. The purpose is to reduce the cost and time needed to develop a prototype. The desired predictions may be either of the mean values of these parameters or their extremes. If the former, the characteristics of the system’s normal operating regime are determined, such as the fuel an aircraft will consume per unit of time in cruising ﬂight. The question of study here is mainly the system’s performance. When extreme parameter values are desired, the question is rather the system’s characteristics in situations that have a little probability of ever existing, i.e. in the presence of rare or critical phenomena, such

2

1. Introduction

as rotating stall in aeronautical engines. Studies like this concern system safety at operating points far from the cruising regime for which they were designed. The constraints on the quality of representation of the physical phenomena diﬀer here from what is required in fundamental studies, because what is wanted now is evidence that certain phenomena exist, rather than all the physical mechanisms at play. In theory, then, the description does not have to be as detailed as it does for fundamental studies. However, it goes without saying that the quality of the prediction improves with the richness of the physical model. The various levels of approximation going into the physical model are discussed in the following.

1.2 Levels of Approximation: General A mathematical model for describing a physical system cannot be deﬁned before we have determined the level of approximation that will be needed for obtaining the required precision on a ﬁxed set of parameters (see [307] for a fuller discussion). This set of parameters, associated with the other variables characterizing the evolution of the model, contain the necessary information for describing the system completely. The ﬁrst decision that is made concerns the scale of reality considered. That is, physical reality can be described at several levels: in terms of particle physics, atomic physics, or micro- and macroscopic descriptions of phenomena. This latter level is the one used by classical mechanics, especially continuum mechanics, which will serve as the framework for the explanations given here. A system description at a given scale can be seen as a statistical averaging of the detailed descriptions obtained at the previous (lower) level of description. In ﬂuid mechanics, which is essentially the study of systems consisting of a large number of interacting elements, the choice of a level of description, and thus a level of averaging, is fundamental. A description at the molecular level would call for a deﬁnition of a discrete system governed by Boltzmann equations, whereas the continuum paradigm would be called for in a macroscopic description corresponding to a scale of representation larger than the mean free path of the molecules. The system will then be governed by the Navier–Stokes equations, if the ﬂuid is Newtonian. After deciding on a level of reality, several other levels of approximation have to be considered in order to obtain the desired information concerning the evolution of the system: – Level of space-time resolution. This is a matter of determining the time and space scales characteristic of the system evolution. The smallest pertinent

1.3 Statement of the Scale Separation Problem

3

scale is taken as the resolution reference so as to capture all the dynamic mechanisms. The system spatial dimension (zero to three dimensions) has to be determined in addition to this. – Level of dynamic description. Here we determine the various forces exerted on the system components, and their relative importance. In the continuum mechanics framework, the most complete model is that of the Navier–Stokes equations, complemented by empirical laws for describing the dependency of the diﬀusion coeﬃcients as a function of the other variables, and the state law. This can ﬁrst be simpliﬁed by considering that the elliptic character of the ﬂow is due only to the pressure, while the other variables are considered to be parabolic, and we then refer to the parabolic Navier–Stokes equations. Other possible simpliﬁcations are, for example, Stokes equations, which account only for the pressure and diﬀusion eﬀects, and the Euler equations, which neglect the viscous mechanisms. The diﬀerent choices made at each of these levels make it possible to develop a mathematical model for describing the physical system. In all of the following, we restrict ourselves to the case of a Newtonian ﬂuid of a single species, of constant volume, isothermal, and isochoric in the absence of any external forces. The mathematical model consists of the unsteady Navier– Stokes equations. The numerical simulation then consists in ﬁnding solutions of these equations using algorithms for Partial Diﬀerential Equations. Because of the way computers are structured, the numerical data thus generated is a discrete set of degrees of freedom, and of ﬁnite dimensions. We therefore assume that the behavior of the discrete dynamical system represented by the numerical result will approximate that of the exact, continuous solution of the Navier–Stokes equations with adequate accuracy.

1.3 Statement of the Scale Separation Problem Solving the unsteady Navier–Stokes equations implies that we must take into account all the space-time scales of the solution if we want to have a result of maximum quality. The discretization has to be ﬁne enough to represent all these scales numerically. That is, the simulation is discretized in steps ∆x in space and ∆t in time that must be smaller, respectively, than the characteristic length and the characteristic time associated with the smallest dynamically active scale of the exact solution. This is equivalent to saying that the space-time resolution scale of the numerical result must be at least as ﬁne as that of the continuous problem. This solution criterion may turn out to be extremely constrictive when the solution to the exact problem contains scales of very diﬀerent sizes, which is the case for turbulent ﬂows. This is illustrated by taking the case of the simplest turbulent ﬂow, i.e. one that is statistically hom*ogeneous and isotropic (see Appendix A for a more

4

1. Introduction

precise deﬁnition). For this ﬂow, the ratio between the characteristic length of the most energetic scale, L, and that of the smallest dynamically active scale, η, is evaluated by the relation: L = O Re3/4 η

,

(1.1)

in which Re is the Reynolds number, which is a measure of the ratio of the forces of inertia and the molecular viscosity eﬀect, ν. We therefore need O Re9/4 degrees of freedom in order to be able to represent all the scales in a cubic volume of edge L. The ratio of characteristic times varies as O Re1/2 , but the use of explicit time-integration algorithm leads to a linear dependency of the time step with respect to the mesh size. So in order to calculate the evolution of the solution in a volume L3 for a duration equal to the characteristic time of the most energetic scale, we have to solve the Navier–Stokes equations numerically O Re3 times! This type of computation for large Reynolds numbers (applications in the aeronautical ﬁeld deal with Reynolds numbers of as much as 108 ) requires computer resources very much greater than currently available supercomputer capacities, and is therefore not practicable. In order to be able to compute the solution, we need to reduce the number of operations, so we no longer solve the dynamics of all the scales of the exact solution directly. To do this, we have to introduce a new, coarser level of description of the ﬂuid system. This comes down to picking out certain scales that will be represented directly in the simulation while others will not be. The non-linearity of the Navier–Stokes equations reﬂects the dynamic coupling that exists among all the scales of the solution, which implies that these scales cannot be calculated independently of each other. So if we want a quality representation of the scales that are resolved, their interactions with the scales that are not have to be considered in the simulation. This is done by introducing an additional term in the equations governing the evolution of the resolved scales, to model these interactions. Since these terms represent the action of a large number of other scales with those that are resolved (without which there would be no eﬀective gain), they reﬂect only the global or average action of these scales. They are therefore only statistical models: an individual deterministic representation of the inter-scale interactions would be equivalent to a direct numerical simulation. Such modeling oﬀers a gain only to the extent that it is universal, i.e. if it can be used in cases other than the one for which it is established. This means there exists a certain universality in the dynamic interactions the models reﬂect. This universality of the assumptions and models will be discussed all through the text.

1.4 Usual Levels of Approximation

5

1.4 Usual Levels of Approximation There are several common ways of reducing the number of degrees of freedom in the numerical solution: – By calculating the statistical average of the solution directly. This is called the Reynolds Averaged Numerical Simulation (RANS)[424], which is used mostly for engineering calculations. The exact solution u splits into the sum of its statistical average u and a ﬂuctuation u (see Appendix A): u(x, t) = u(x, t) + u (x, t) . This splitting, or “decomposition”, is illustrated by Fig. 1.1. The ﬂuctuation u is not represented directly by the numerical simulation, and is included only by way of a turbulence model. The statistical averaging operation is in practice often associated with a time averaging: 1 T u(x, t)dt . u(x, t) ≈ u(x) = lim T →∞ T 0 The mathematical model is then that of the steady Navier–Stokes equations. This averaging operation makes it possible to reduce the number of scales in the solution considerably, and therefore the number of degrees of freedom of the discrete system. The statistical character of the solution prevents a ﬁne description of the physical mechanisms, so that this approach is not usable for studies of a fundamental character, especially so when the statistical average is combined with a time average. Nor is it possible to isolate rare events. On the other hand, it is an appropriate approach for analyzing performance as long as the turbulence models are able to reﬂect the existence of the turbulent ﬂuctuation u eﬀectively.

Fig. 1.1. Decomposition of the energy spectrum of the solution associated with the Reynolds Averaged Numerical Simulation (symbolic representation).

– By calculating directly only certain low-frequency modes in time (of the order of a few hundred hertz) and the average ﬁeld. This approach goes by a number of names: Unsteady Reynolds Averaged Numerical Simula-

6

1. Introduction

tion (URANS), Semi-Deterministic Simulation (SDS), Very Large-Eddy Simulation (VLES), and sometimes Coherent Structure Capturing (CSC) [726, 44]. The ﬁeld u appears here as the sum of three contributing terms [456, 451, 240, 726]: u(x, t) = u(x) + u(x, t)c + u (x, t) . The ﬁrst term is the time average of the exact solution, the second its conditional statistical average, and the third the turbulent ﬂuctuation. This decomposition is illustrated in Fig. 1.2. The conditional average is associated with a predeﬁned class of events. When these events occur at a set time period, this is a phase average. The u(x, t)c term is interpreted as the contribution of the coherent modes to the ﬂow dynamics, while the u term, on the other hand, is supposed to represent the random part of the turbulence. The variable described by the mathematical model is now the sum u(x) + u(x, t)c , with the random part being represented by a turbulence model. It should be noted that, in the case where there exists a deterministic low-frequency forcing of the solution, the conditional average is conventionally interpreted as a phase average of the solution, for a frequency equal to that of the forcing term; but if this does not exist, the interpretation of the results is still open to debate. Since this is an unsteady approach, it contains more information than the previous one; but it still precludes a deterministic description of a particular event. It is of use for analyzing the performance characteristics of systems in which the unsteady character is forced by some external action (such as periodically pulsed ﬂows).

Fig. 1.2. Decomposition of the energy spectrum of the solution associated with the Unsteady Reynolds Averaged Numerical Simulation approach, when a predominant frequency exists (symbolic representation).

– By projecting the solution on the ad hoc function basis and retaining only a minimum number of modes, to get a dynamical system with fewer degrees of freedom. The idea here is to ﬁnd an optimum decomposition basis for representing the phenomenon, in order to minimize the number of degrees of freedom in the discrete dynamical system. There is no averaging done here, so the space-time and dynamics resolution of the numerical model is

1.4 Usual Levels of Approximation

7

still as ﬁne as that of the continuum model, but is now optimized. Several approaches are encountered in practice. The ﬁrst is to use standard basis function (Fourier modes in the spectral space or polynomials in the physical space, for example) and distribute the degrees of freedom as best possible in space and time to minimize the number of them, i.e. adapt the space-time resolution of the simulation to the nature of the solution. We thus adapt the topology of the discrete dynamical system to that of the exact solution. This approach results in the use of self-adapting grids and time steps in the physical space. It is not associated with an operation to reduce the complexity by switching to a higher level of statistical description of the system. It leads to a much less important reduction of the discrete system than those techniques based on statistical averaging, and is limited by the complexity of the continuous solution. Another approach is to use optimal basis functions, a small number of which will suﬃce for representing the ﬂow dynamics. The problem is then to determine what these basis functions are. One example is the Proper Orthogonal Decomposition (POD) mode basis, which is optimum for representing kinetic energy (see [55] for a survey). This technique allows very high data compression, and generates a dynamical system of very small dimensions (a few dozen degrees of freedom at most, in practice). The approach is very seldom used because it requires very complete information concerning the solution in order to be able to determine the base functions. The various approaches above all return complete information concerning the solutions of the exact problem, so they are perfectly suited to studies of a fundamental nature. They may not, on the other hand, be optimal in terms of reducing the complexity for certain engineering analyses that do not require such complete data. – By calculating only the low-frequency modes in space directly. This is what is done in Large-Eddy Simulation (LES). It is this approach that is discussed in the following. It is illustrated in Fig. 1.3. Typical results obtained by these three approaches are illustrated in Fig. 1.4.

Fig. 1.3. Decomposition of the energy spectrum in the solution associated with large-eddy simulation (symbolic representation).

8

1. Introduction

Fig. 1.4. Pressure spectrum inside a cavity. Top: experimental data (ideal directnumerical simulation) (courtesy of L. Jacquin, ONERA); Middle: large-eddy simulation (Courtesy of L. Larchevˆeque, ONERA); Bottom: unsteady RANS simulation (Courtesy of V. Gleize, ONERA).

1.5 Large-Eddy Simulation: from Practice to Theory. Structure of the Book

9

1.5 Large-Eddy Simulation: from Practice to Theory. Structure of the Book As mentioned above, the Large-Eddy Simulation approach relies on the definition of large and small scales. This fuzzy and empirical concept requires further discussion to become a tractable tool from both the theoretical and practical points of view. Bases for the theoretical understanding and modeling of this approach are now introduced. In practice, the Large-Eddy Simulation technique consists in solving the set of ad hoc governing equations on a computational grid which is too coarse to represent the smallest physical scales. Let ∆x and η be the computional mesh size (assumed to be uniform for the sake of simplicity) and the characteristic size of the smallest physical scales. Let u be the exact solution of the following continuous generic conservation law (the case of the Navier–Stokes equations will be extensively discussed in the core of the book) ∂u + F (u, u) = 0 (1.2) ∂t where F (·, ·) is a non-linear ﬂux function. The Large-Eddy Simulation problem consists in ﬁnding the best approximation of u on the computational grid by solving the following discrete problem δud + Fd (ud , ud ) = 0 δt

(1.3)

where ud , δ/δt and Fd (·, ·) are the discrete approximations of u, ∂/∂t and F (·, ·) on the computational grid, respectively. Thus, the question arise of deﬁning what is the best possible approximation of u, uΠ , among all discrete solutions ud associated with ∆x. Let e(u, ud ) be a measure of the diﬀerence between u and ud , which does not need to be explicitely deﬁned for the present purpose. It is just emphasized here that since Large-Eddy Simulation is used to compute turbulent ﬂows, u exhibits a chaotic behavior and therefore e(u, ud ) should rely on statistical moments of the solutions. A consistency constraint on the deﬁnition of the error functional is that it must vanish in the limit case of the Direct Numerical Simulation lim e(u, ud ) = 0

∆x−→η

(1.4)

A careful look at the problem reveals that the error can be decomposed as e(u, ud ) = eΠ (u, ud ) + ed (u, ud ) + er (u, ud ) where

(1.5)

10

1. Introduction

1. eΠ (u, ud ) is the projection error which accounts for the fact that the exact solution u is approximated using a ﬁnite number of degrees of freedom. The Nyquist theorem tells us that no scale smaller than 2∆x can be captured in the simulation. As a consequence, ud can never be strictly equal to u : (1.6) |u − ud | = 0 2. ed (u, ud ) is the discretization error which accounts for the fact that partial derivatives which appear in the continuous problem are approximated on the computational grid using Finite Diﬀrence, Finite Volume, Finite Element (or other similar) schemes. Putting the emphasis on spatial derivatives, this is expressed as Fd (u, u) = F (u, u)

(1.7)

3. er (u, ud ) is the resolution error, which accounts for the fact that, some scales of the exact solution being missing, the evaluation of the non-linear ﬂux function cannot be exact, even if the discretization error is driven to zero: (1.8) F (ud , ud ) = F (u, u)

This analysis shows that the Large-Eddy Simulation problem is very complex, since it depends explicitely on the exact solution, the computational grid and the numerical method, making each problem appearing as unique. Therefore, it is necessary to ﬁnd some mathematical models for the Large-Eddy Simulation problem which will mimic its main features, the most important one being the removal of the small scales of the exact solution. A simpliﬁed heuristic view of this problem is illustrated in Fig. 1.5, where the eﬀect of the Nyquist ﬁlter is represented. Several mathematical models have been proposed to handle the true Large-Eddy Simulation problem. The most popular one (see [216, 440, 495, 619, 627]) relies on the representation of the removal of the small scales as the result of the application of a low-pass convolution ﬁlter (in terms of wave number) to the exact solution. The deﬁnition and the properties of this ﬁltering operator are presented in Chap. 2. The application of this ﬁlter to the Navier–Stokes equations, described in Chap. 3, yields the corresponding constitutive mathematical model for the large-eddy simulation. Alternate mathematical models are detailed in Chap. 4. The second question raised by the Large-Eddy Simulation approach deals with the search for the best approximate solution uΠ ∈ {ud } that will minimize the error e(u, ud). The short analysis given above shows that the projection error, eΠ (u, ud ) cannot be avoided. Therefore, the best, ideal Large-Eddy Solution is such that e(u, ud ) = e(u, uΠ ) = eΠ (u, uΠ )

(1.9)

1.5 Large-Eddy Simulation: from Practice to Theory. Structure of the Book

11

Fig. 1.5. Schematic view of the simplest scale separation operator: grid and theoretical ﬁlters are the same, yielding a sharp cutoﬀ ﬁltering in Fourier space between the resolved and subgrid scales. The associated cutoﬀ wave number is denoted kc , which is directly computed from the cutoﬀ length ∆ in the physical space. Here, ∆ is assumed to be equal to the size of the computational mesh.

and, following relation (1.5), it is associated to ed (u, uΠ ) + er (u, uΠ ) = 0

(1.10)

The best solution in sought in practice trying to enforce the sequel relation (1.10). Two basic diﬀerent ways are identiﬁed for that purpose: – The explicit Large-Eddy Simulation approach, in which an extra forcing term, referred to as a subgrid model, is introduced in the governing equation to cancel the resolution error. Two modeling approaches are discussed here: functional modeling, based on representing kinetic energy transfers (covered in Chaps. 5 and 6), and structural modeling, which aims to reproduce the eigenvectors of the statistical correlation tensors of the subgrid modes (presented in Chap. 7). The basic assumptions and the subgrid models corresponding to each of these approaches are presented. In the hypothetical case where a perfect subgrid model could be found, expression (1.10) shows that the discretization error ed (u, ud ) must also be driven to zero to recover the ideal Large-Eddy Simulation solution. A perfect numerical method is obviously a natural candidate for that purpose, but reminding that the error measure is based on statistical moments, the much less stringent requirement that the numerical method must be neutral with respect to the error deﬁnition is suﬃcient. Chapter 8 is devoted to the theoretical

12

1. Introduction

problems related to the eﬀects of the numerical method used in the simulation. The representation of the numerical error in the form of an additional ﬁlter is introduced, along with the problem of the relative weight of the various ﬁlters used in the numerical simulation. – The implicit Large-Eddy Simulation approach, in which no extra term is introduced in the governing equations, but the numerical method is chosen such that the numerical error and the resolution error will cancel each other, yielding a direct fulﬁlment of relation (1.10). This approach is brieﬂy presented in this book in Sect. 5.3.4. The interested reader can refer to [276] for an exhaustive description. The fact that the ideal solution uΠ is associated to a non-vanishing projection error eΠ (u, uΠ ) raises the problem of the reliability of data obtained via Large-Eddy Simulation for practical purposes. Several theoretical and practical problems are met when addressing the issue of validating and exploiting Large-Eddy Simulation. The deﬁnition of the best solution being intrinsically based on the deﬁnition of the error functional (which is arbitrary), a universal answer seems to be meaningless. Questions concerning the analysis and validation of the large-eddy simulation calculations are dealt with in Chap. 9. The concept of statistically partially equivalent simulations is introduced, which is of major importance to interpret the nature of the data recovered from Large-Eddy Simulation. A short survey of available results dealing with the properties of ﬁltered Navier–Stokes solutions (ideally uΠ ) and Large-Eddy Simulation solutions (true ud ﬁelds) is presented. The discussions presented above deal with the deﬁnition of the LargeEddy Simulation problem inside the computational domain. As all diﬀerential problems, it must be supplemented with ad hoc boundary conditions to yield a well-posed problem. Thus, the new question of deﬁning discrete boundary conditions in a consistent way appears. The problem is similar to the previous one: what boundary conditions should be used to reach the best solution uΠ ? A weaker constraint is to ﬁnd boundary conditions which do not deteriorate the accuracy that could potentially be reached with the selected numerical scheme and closure. The boundary conditions used for large-eddy simulation are discussed in Chap. 10, where the main cases treated are solid walls and turbulent inﬂow conditions. In the solid wall case, the emphasis is put on the problem of deﬁning wall stress models, which are subgrid models derived for the speciﬁc purpose of taking into account the dynamics of the inner layer of turbulent boundary layers. The issue of deﬁning eﬃcient turbulent inﬂow condtions raises from the need to truncate the computational domain, which leads to the requirement of ﬁnding a way to take into account upstream turbulent ﬂuctuations in the boundary conditions. Despite the fact that it yields very signiﬁcant complexity reduction in terms of degrees of freedom with respect to Direct Numerical Simulation, Large-Eddy Simulation still requires considerable computational eﬀorts to handle realistic applications. To obtain further complexity reduction, several

1.5 Large-Eddy Simulation: from Practice to Theory. Structure of the Book

13

hybridizations of the Large-Eddy Simulation technique have been proposed. Methods for reducing the cost of Large-Eddy Simulation by coupling it with multiresolution and multidomain techniques are presented in Chap. 11. Hybrid RANS/LES approaches are presented in Chap. 12. The deﬁnition of such multiresolution methods and/or hybrid RANS/LES techniques raises many practical and theoretical issues. Among the most important ones, the emphasis is put in the dedicated chapters on the coupling strategies and the fact that the instantaneous ﬁelds can be fully discontinuous (fully meaning here that the velocity ﬁeld is not a priori continuous at the interfaces between domains with diﬀerent resolution, but also that even the number of space dimension and the number of unknwons can be diﬀerent). Practical aspects concerning the implementation of subgrid models are described in Chap. 13. Lastly, the discussion is illustrated by examples of largeeddy simulation applications for diﬀerent categories of ﬂows, in Chap. 14. Chapter 15 is devoted the the extension of concepts, methods and models presented in previous chapters to the case of a more complex physical system, in which an additional equation for a scalar is added to the Navier–Stokes equations. Two cases are considered: the passive scalar case, in which there is no feedback in the momentum equation and the new problem is restricted to closing the ﬁltered scalar equation, and the active scalar case, which corresponds to a two-way coupling between the scalar ﬁeld and the velocity ﬁeld. In the latter, the deﬁnition of subgrid models for both the velocity and the scalar is a full problem. For the sake of clarity, the discussion is limited to stably stratiﬁed ﬂows and buoyancy driven ﬂows. Combustion and two-phase ﬂows are not treated.

2. Formal Introduction to Scale Separation: Band-Pass Filtering

The idea of scale separation introduced in the preceding chapter will now be formalized on the mathematical level, to show how to handle the equations and derive the subgrid models. This chapter is devoted to the representation of the ﬁltering as a convolution product, which is the most common way to model the removal of small scales in the Larg-Eddy Simulation approach. Other deﬁnitions, such as partial statistical averaging or conditional averaging [251, 250, 465], will be presented in Chap. 4. The ﬁltering approach is ﬁrst presented in the ideal case of a ﬁlter of uniform cutoﬀ length over an inﬁnite domain (Sect. 2.1). Fundamental properties of ﬁlters and their approximation via diﬀerential operators is presented. Extensions to the cases of a bounded domain and a ﬁlter of variable cutoﬀ length are then discussed (Sect. 2.2). The chapter is closed by discussing a few properties of the Eulerian time-domain ﬁlters (Sect. 2.3).

2.1 Deﬁnition and Properties of the Filter in the hom*ogeneous Case The framework is restricted here to the case of hom*ogeneous isotropic ﬁlters, for the sake of easier analysis, and to allow a better understanding of the physics of the phenomena. The ﬁlter considered is isotropic. This means that its properties are independent of the position and orientation of the frame of reference in space, which implies that it is applied to an unbounded domain and that the cutoﬀ scale is constant and identical in all directions of space. This is the framework in which subgrid modeling developed historically. The extension to anisotropic and inhom*ogeneous1 ﬁlters, which researchers have only more recently begun to look into, is described in Sect. 2.2. 2.1.1 Deﬁnition Scales are separated by applying a scale high-pass ﬁlter, i.e. low-pass in frequency, to the exact solution. This ﬁltering is represented mathematically in 1

That is, whose characteristics, such as the mathematical form or cutoﬀ frequency, are not invariant by translation or rotation of the frame of reference in which they are deﬁned.

16

2. Formal Introduction to Filtering

physical space as a convolution product. The resolved part φ(x, t) of a spacetime variable φ(x, t) is deﬁned formally by the relation:

+∞

+∞

φ(x, t) = −∞

−∞

φ(ξ, t )G(x − ξ, t − t )dt d3 ξ

,

(2.1)

in which the convolution kernel G is characteristic of the ﬁlter used, which is associated with the cutoﬀ scales in space and time, ∆ and τ c , respectively. This relation is denoted symbolically by: φ= G φ

.

(2.2)

The dual deﬁnition in the Fourier space is obtained by multiplying the ω) of φ(x, t) by the spectrum G(k, ω) of the kernel G(x, t): spectrum φ(k, ω) = φ(k, ω)G(k, ω) , φ(k,

(2.3)

φ , φ = G

(2.4)

or, in symbolic form: where k and ω are the spatial wave number and time frequency, respectively. is the transfer function associated with the kernel G. The The function G spatial cutoﬀ length ∆ is associated with the cutoﬀ wave number kc and time τ c with the cutoﬀ frequency ωc . The unresolved part of φ(x, t), denoted φ (x, t), is deﬁned operationally as: φ (x, t) = φ(x, t) − φ(x, t) +∞ = φ(x, t) − −∞

or:

(2.5) +∞

−∞

φ(ξ, t )G(x − ξ, t − t )dt d3 ξ, (2.6)

φ = (1 − G) φ .

(2.7)

The corresponding form in spectral space is:

i.e.

ω) − φ(k, ω) = 1 − G(k, ω) , ω) φ(k, φ (k, ω) = φ(k,

(2.8)

φ . φ = (1 − G)

(2.9)

2.1 Deﬁnition and Properties of the Filter in the hom*ogeneous Case

17

2.1.2 Fundamental Properties In order to be able to manipulate the Navier–Stokes equations after applying a ﬁlter, we require that the ﬁlter verify the three following properties: 1. Conservation of constants a = a ⇐⇒

+∞

−∞

+∞

G(ξ, t )d3 ξdt = 1

.

(2.10)

−∞

2. Linearity φ+ψ =φ+ψ

.

(2.11)

This property is automatically satisﬁed, since the product of convolution veriﬁes it independently of the characteristics of the kernel G. 3. Commutation with derivation ∂φ ∂φ = , ∂s ∂s

s = x, t

.

(2.12)

Introducing the commutator [f, g] of two operators f and g applied to the dummy variable φ [f, g]φ ≡ f ◦ g(φ) − g ◦ f (φ) = f (g(φ)) − g(f (φ))

,

the relation (2.12) can be re-written symbolically ∂ G , =0 . ∂s

(2.14)

The commutator deﬁned by relation (2.13) has the properties2 : [f, g] = −[g, f ] Skew-symmetry , [f ◦ g, h] = [f, h] ◦ g + f ◦ [g, h] [f, [g, h]] + [g, [h, f ]] + [h, [f, g]] = 0

(2.13)

following (2.15)

Leibniz identity ,

(2.16)

Jacobi’s identity .

(2.17)

The ﬁlters that verify these three properties are not, in the general case, Reynolds operators (see Appendix A), i.e. φ = φ =

2

G G φ = G2 φ = φ = G φ G (1 − G) φ = 0 ,

,

(2.18) (2.19)

In the linear case, the commutator satisﬁes all the properties of the Poissonbracket operator, as deﬁned in classical mechanics.

18

2. Formal Introduction to Filtering

which is equivalent to saying that G is not a projector (excluding the trivial case of the identity application). Let us recall that an application P is deﬁned as being a projector if P ◦ P = P . Such an application is idempotent because it veriﬁes the relation ◦ ... ◦ P = P, ∀n ∈ IN+ P n ≡ P ◦ P

.

(2.20)

n times

When G is not a projector, the ﬁltering can be interpreted as a change of variable, and can be inverted, so there is no loss of information3 [243]. The kernel of the application is reduced to the null element, i.e. ker(G) = {0}. If the ﬁlter is a Reynolds operator, we get G2 = 1

,

(2.21)

or, remembering the property of conservation of constants: G=1

.

(2.22)

In the spectral space, the idempotency property implies that the transfer function takes the following form:

0 G(k, ω) = ∀k, ∀ω . (2.23) 1 therefore takes the form of a sum of Dirac funcThe convolution kernel G tions and Heaviside functions associated with non-intersecting domains. The is 1 for the modes that are constant conservation of constants implies that G in space and time. The application can no longer be inverted because its kernel ker(G) = {φ } is no longer reduced to the zero element; and consequently, the ﬁltering induces an irremediable loss of information. A ﬁlter is said to be positive if: G(x, t) > 0, ∀x and ∀t .

(2.24)

2.1.3 Characterization of Diﬀerent Approximations The various methods mentioned in the previous section for reducing the number of degrees of freedom will now be explained. We now assume that the 3

The reduction of the number of degrees of freedom comes from the fact that the new variable, i.e. the ﬁltered variable, is more regular than the original one in the sense that it contains fewer high frequencies. Its characteristic scale in space is therefore larger, which makes it possible to use a coarser solution to describe it, and therefore fewer degrees of freedom. The result is a direct numerical simulation of the smoothed variable. As in all numerical simulations, a numerical cutoﬀ is imposed by the use of a ﬁnite number of degrees of freedom. But in the case considered here the numerical cutoﬀ is assumed to occur within the dissipative range of the spectrum, so that no active scales are missing.

2.1 Deﬁnition and Properties of the Filter in the hom*ogeneous Case

19

space-time convolution kernel G(x− ξ, t− t ) in IR4 is obtained by tensorizing mono-dimensional kernels: G(x − ξ, t − t ) = G(x − ξ)Gt (t − t ) = Gt (t − t ) Gi (xi − ξi ) . (2.25) i=1,3

The Reynolds time average over a time interval T is found by taking: Gt (t − t ) =

HT , Gi (xi − ξi ) = δ(xi − ξi ), i = 1, 2, 3 , T

(2.26)

in which δ is a Dirac function and HT the Heaviside function corresponding to the interval chosen. This average is extended to the ith direction of space by letting Gi (xi − ξi ) = HL /L, in which L is the desired integration interval. The phase average corresponding to the frequency ωc is obtained by letting: t (ω) = δ(ω − ωc ), Gi (xi − ξi ) = δ(xi − ξi ), i = 1, 2, 3 . G

(2.27)

In all of the following, the emphasis will be put on the large-eddy simulation technique based on spatial ﬁltering, because it is the most employed approach, with very rare exceptions [160, 161, 603, 107]. This is expressed by: (2.28) Gt (t − t ) = δ(t − t ) . Diﬀerent forms of the kernel Gi (xi − ξi ) in common use are described in the following section. It should nonetheless be noted that, when a spatial ﬁltering is imposed, it automatically induces an implicit time ﬁltering, since the dynamics of the Navier–Stokes equations makes it possible to associate a characteristic time with each characteristic length scale. This time scale is evaluated as follows. Let ∆ be the cutoﬀ length associated with the ﬁlter, and kc = π/∆ the associated wave number. Let E(k) be the energy spectrum of the exact solution (see Appendix A for a deﬁnition). The kinetic energy associated with the wave number kc is kc E(kc ). The velocity scale vc associated with this same wave number is estimated as: vc = kc E(kc ) . (2.29) The characteristic time tc associated with the length ∆ is calculated by dimensional arguments as follows: tc = ∆/vc

.

(2.30)

The corresponding frequency is ωc = 2π/tc . The physical analysis shows that, for spectrum forms E(k) considered in the large-eddy simulation framework, vc is a monotonic decreasing function of kc (resp. monotonic increasing

20

2. Formal Introduction to Filtering

function of ∆), so that ωc is a monotonic increasing function of kc (resp. monotonic decreasing function of ∆). Suppressing the spatial scales corresponding to wave numbers higher than kc induces the disappearance of the time frequencies higher than ωc . We nonetheless assume that the description with a spatial ﬁltering alone is relevant. Eulerian time-domain ﬁltering for spatial large-eddy simulation is recovered taking (2.31) Gi (xi − ξi ) = δ(xi − ξi ) . A reasoning similar to the one given above shows that time ﬁltering induces an implicit spatial ﬁltering. 2.1.4 Diﬀerential Filters A subset of the ﬁlters deﬁned in the previous section is the set of diﬀerential ﬁlters [242, 243, 245, 248]. These ﬁlters are such that the kernel G is the Green’s function associated to an inverse linear diﬀerential operator F : φ = =

F (G φ) = F (φ) φ+θ

∂φ ∂2φ ∂φ + ∆l + ∆lm + ... , ∂t ∂xl ∂xl ∂xm

(2.32)

where θ and ∆l are some time and space scales, respectively. Diﬀerential ﬁlters can be grouped into several classes: elliptic, parabolic or hyperbolic ﬁlters. In the framework of a generalized space-time ﬁltering, Germano [242, 243, 245] recommends using a parabolic or hyperbolic time ﬁlter and an elliptic space ﬁlter, for reasons of physical consistency with the nature of the Navier–Stokes equations. It is recalled that a ﬁlter is said to be elliptic (resp. parabolic or hyperbolic) if F is an elliptic (resp. parabolic, hyperbolic) operator. Examples are given below [248]. Time Low-Pass Filter. A ﬁrst example is the time low-pass ﬁlter. The associated inverse diﬀerential relation is: φ=φ+θ

∂φ ∂t

.

(2.33)

The corresponding convolution ﬁlter is: 1 φ= θ

t − t φ(x, t ) exp − dt θ −∞

t

.

(2.34)

It is easily seen that this ﬁlter commutes with time and space derivatives. This ﬁlter is causal, because it incorporates no future information, and therefore is applicable to real-time or post-processing of the data.

2.1 Deﬁnition and Properties of the Filter in the hom*ogeneous Case

21

Helmholtz Elliptic Filter. An elliptic ﬁlter is obtained by taking: 2∂

φ=φ−∆

2

φ ∂x2l

.

(2.35)

It corresponds to a second-order elliptic operator, which depends only on space. The convolutional integral form is: |x − ξ| 1 φ(ξ, t) exp − φ= dξ . (2.36) 2 |x − ξ| ∆ 4π∆ This ﬁlter satisﬁes the three previously mentioned basic properties. Parabolic Filter. A parabolic ﬁlter is obtained taking φ = φ+θ

2 ∂φ 2∂ φ −∆ ∂t ∂x2l

,

(2.37)

yielding φ(ξ, t) (x − ξ)2 θ t − t dξdt . φ= exp − 2 − 3 )3/2 3/2 θ (t − t (4π) ∆ −∞ 4∆ (t − t ) (2.38) It is easily veriﬁed that the parabolic ﬁlter satistiﬁes the three required properties. √ θ

t

Convective and Lagrangian Filters. A convective ﬁlter is obtained by adding a convective part to the parabolic ﬁlter, leading to: φ=φ+θ

2 ∂φ ∂φ 2∂ φ + θVl −∆ ∂t ∂xl ∂x2l

,

(2.39)

where V is an arbitrary velocity ﬁeld. This ﬁlter is linear and constant preserving, but commutes with derivatives if and only if V is uniform. A Lagrangian ﬁlter is obtained when V is taken equal to u, the velocity ﬁeld. In this last case, the commutation property is obviously lost. 2.1.5 Three Classical Filters for Large-Eddy Simulation Three convolution ﬁlters are ordinarily used for performing the spatial scale separation. For a cutoﬀ length ∆, in the mono-dimensional case, these are written: – Box or top-hat ﬁlter: ⎧ ⎪ 1 ⎪ ⎨ ∆ G(x − ξ) = ⎪ ⎪ ⎩ 0

if |x − ξ| ≤ otherwise

∆ 2

,

(2.40)

22

2. Formal Introduction to Filtering

sin(k∆/2) G(k) = k∆/2

.

(2.41)

are represented in The convolution kernel G and the transfer function G Figs. 2.1 and 2.2, respectively. – Gaussian ﬁlter: G(x − ξ) =

1/2

γ π∆

exp

2

−γ|x − ξ|2

2

,

(2.42)

∆

G(k) = exp

2

−∆ k 2 4γ

,

(2.43)

in which γ is a constant generally taken to be equal to 6. The convolution are represented in Figs. 2.3 and 2.4, kernel G and the transfer function G respectively. – Spectral or sharp cutoﬀ ﬁlter: G(x − ξ) =

sin (kc (x − ξ)) π , with kc = kc (x − ξ) ∆

G(k) =

⎧ ⎨ 1

if |k| ≤ kc

⎩

otherwise

. 0

,

(2.44)

(2.45)

are represented in The convolution kernel G and the transfer function G Figs. 2.5 and 2.6, respectively. It is trivially veriﬁed that the ﬁrst two ﬁlters are positive while the sharp cutoﬀ ﬁlter is not. The top-hat ﬁlter is local in the physical space (its support is compact) and non-local in the Fourier space, inversely from the sharp cutoﬀ ﬁlter, which is local in the spectral space and non-local in the physical space. As for the Gaussian ﬁlter, it is non-local both in the spectral and physical spaces. Of all the ﬁlters presented, only the sharp cutoﬀ has the property: n ·G G · G...

= G = G , n times

and is therefore idempotent in the spectral space. Lastly, the top-hat and Gaussian ﬁlters are said to be smooth because there is a frequency overlap between the quantities u and u . Modiﬁcation of the exact solution spectrum by the ﬁltering operator is illustrated in ﬁgure 2.7.

2.1 Deﬁnition and Properties of the Filter in the hom*ogeneous Case

23

Fig. 2.1. Top-hat ﬁlter. Convolution kernel in the physical space normalized by ∆.

Fig. 2.2. Top-hat ﬁlter. Associated transfer function.

24

2. Formal Introduction to Filtering

Fig. 2.3. Gaussian ﬁlter. Convolution kernel in the physical space normalized by ∆.

Fig. 2.4. Gaussian ﬁlter. Associated transfer function.

2.1 Deﬁnition and Properties of the Filter in the hom*ogeneous Case

Fig. 2.5. Sharp cutoﬀ ﬁlter. Convolution kernel in the physical space.

Fig. 2.6. Sharp cutoﬀ ﬁlter. Associated transfer function.

25

26

2. Formal Introduction to Filtering

Fig. 2.7. Energy spectrum of the unﬁltered and ﬁltered solutions. Filters considered are a projective ﬁlter (sharp cutoﬀ ﬁlter) and a smooth ﬁlter (Gaussian ﬁlter) with the same cutoﬀ wave number kc = 500.

2.1.6 Diﬀerential Interpretation of the Filters General results. Convolution ﬁlters can be approximated as simple diﬀerential operators via a Taylor series expansion, if some additional constraints are fulﬁlled by the convolution kernel, thus yielding simpliﬁed and local ﬁltering operators. Validation of the use of Taylor series expansions in the representation of the ﬁltering operator and conditions for convergence will be discussed in the next section. We ﬁrst consider space ﬁltering and recall its deﬁnition using a convolution product: +∞

φ(x, t) = −∞

φ(ξ, t)G(x − ξ)dξ

.

(2.46)

To obtain a diﬀerential interpretation of the ﬁlter, we perform a Taylor series expansion of the φ(ξ, t) term at (x, t): φ(ξ, t) = φ(x, t) + (ξ − x)

∂ 2 φ(x, t) ∂φ(x, t) 1 + (ξ − x)2 + ... ∂x 2 ∂x2

(2.47)

Introducing this expansion into (2.46), and considering the symmetry and conservation properties of the constants of the kernel G, we get:

2.1 Deﬁnition and Properties of the Filter in the hom*ogeneous Case

27

φ(x, t)

1 ∂ 2 φ(x, t) +∞ 2 z G(z)dz + ... 2 ∂x2 −∞ 1 ∂ n φ(x, t) + z n G(z)dz + ... n! ∂xn α(l) ∂ l φ(x, t) φ(x, t) + , l! ∂xl

=

φ(x, t)

=

(2.48)

l=1,∞

where α(l) designates the lth-order moment of the convolution kernel: +∞ α(l) = (−1)l z l G(z)dz . (2.49) −∞

Assuming that the solution is 2π-periodic, the moments of the convolution kernel can be rewritten as follows [728, 607]:

l

α(l) = ∆ Ml ,

(−π−x)/∆

ξ l G(ξ)dξ

Ml =

,

(2.50)

(π−x)/∆

leading to the following expression for the ﬁltered variable φ: ∞ (−1)k

φ(x) =

k!

k=0

k

∆ Mk (x)

∂kφ (x) ∂xk

.

(2.51)

With this relation, we can interpret the ﬁltering as the application of a diﬀerential operator to the primitive variable φ. The subgrid part can also be rewritten using the following relation φ (x)

= =

=

φ(x) − φ(x) α(l) ∂ l φ(x) − l! ∂xl l=1,∞ ∞ k+1 k=1

(−1) k!

k

∆ Mk (x)

∂k φ (x) ∂xk

.

(2.52)

The ﬁltered variable φ can also be expanded using derivatives of the transˆ of the ﬁlter [607]. Assuming periodicity and diﬀerentiability fer function G of φ, we can write φ(x) =

+∞

φˆk eıkx

,

ı2 = −1 ,

(2.53)

k=−∞

and

+∞ ∂lφ (x) = (ık)l φˆk eıkx ∂xl k=−∞

.

(2.54)

28

2. Formal Introduction to Filtering

The ﬁltered ﬁeld is expanded as follows: φ(x) =

+∞

ˆ φˆk eıkx G(k)

.

(2.55)

k=−∞

The ﬁltered ﬁeld can be expressed as a Taylor series expansion in the ﬁlter width ∆: 2

φ(x, ∆) = φ(x, 0) + ∆

∆ ∂2φ ∂φ (x, 0) + (x, 0) + ... 2 ∂∆2 ∂∆

.

(2.56)

By diﬀerentiating (2.55) with respect to ∆, we obtain ∂lφ l

(x, 0) = l!al

∂∆

∂lφ (x) ∂xl

,

(2.57)

with

ˆ 1 ∂lG (0) . ıl l! ∂k l The resulting ﬁnal expression of the ﬁltered ﬁeld is al =

+∞ ∞ ˆ (k∆)l ∂ l G (0)φˆk eıkx l! ∂k l

φ(x) =

(2.58)

.

(2.59)

l=0 k=−∞

Time-domain ﬁlters deﬁned as a convolution product can be expanded in an exactly similar way, yielding +∞ φ(x, t) = φ(x, t )G(t − t )dt −∞

=

φ(x, t) +

α(l) ∂ l φ(x, t) l! ∂tl

,

(2.60)

l=1,∞

The values of the ﬁrst moments of the box and Gaussian ﬁlters are given in Table 2.1. It can be checked that the sharp cutoﬀ ﬁlter leads to a divergent series, because of its non-localness. For these two ﬁlters, we have the estimate n

α(n) = O(∆ )

(2.61)

α(n) = O(τc n )

(2.62)

for space-domain ﬁltering, and

for time-domain ﬁltering.

2.1 Deﬁnition and Properties of the Filter in the hom*ogeneous Case

29

Table 2.1. Values of the ﬁrst ﬁve non-zero moments for the box and Gaussian ﬁlters. α(n)

n=0

box Gaussian

n=2

n=4

n=6

2

4

6

∆ /12 2 ∆ /12

1 1

∆ /80 4 ∆ /48

∆ /448 6 5∆ /576

n=8 8

∆ /2304 8 35∆ /6912

For a general space–time ﬁlter, neglecting cross-derivatives of the kernel, this Taylor series expansion gives [160, 161]: φ(x, t)

+∞

= −∞

φ(ξ, t )G(x − ξ, t − t )dξdt l α(l) α(l) ∂ l φ(x, t) x ∂ φ(x, t) t + , (2.63) l! ∂xl l! ∂tl

= φ(x, t) +

l=1,∞

with

α(l) x

+∞

= −∞

and (l) αt

+∞

−∞

+∞

+∞

= −∞

−∞

l=1,∞

(ξ − x)l G(x − ξ, t − t )dξdt

,

(2.64)

(t − t)l G(x − ξ, t − t )dξdt

.

(2.65)

Conditions for Convergence of the Taylor Series Expansions. A ﬁrst analysis of the convergence properties of the Taylor series expansions discussed above was provided by Vasilyev et al. [728], and is given below. Assuming that the periodic one-dimensional ﬁeld φ does not contain wave numbers higher than kmax , one can write the following Fourier integral:

kmax

φ(x) =

−ıkx ˆ dk φ(k)e

,

(2.66)

−kmax

where time-dependence has been omitted for the sake of simplicity. The total energy of φ, Eφ , is equal to Eφ =

kmax

−kmax

2 ˆ |φ(k)| dk

.

(2.67)

The mth derivative of φ can be written as ∂mφ (x) = (−ı)m ∂xm

kmax

−kmax

−ıkx ˆ k m φ(k)e dk

.

(2.68)

30

2. Formal Introduction to Filtering

From this expression, we get the following bounds for the derivative: m kmax ∂ φ 2 ˆ ≤ |k|2m |φ(k)| dk ∂xm −kmax 1/2 1/2 kmax kmax ˆ |k|m dk |φ(k)|dk ≤ ≤

−kmax

2Eφ kmax m k 2m + 1 max

−kmax

.

(2.69)

From relations (2.69) and (2.51) we obtain the following inequalities: ∞ l ∞ (−1)l ∂ φ 1 l ∂ l φ l ∆ Ml (x) l (x) ≤ ∆ |Ml (x)| l (x) l! ∂x l! ∂x l=0 l=0 l ∞ kmax ∆ |Ml (x)| √ .(2.70) ≤ 2Eφ kmax l! 2l + 1 l=0 From this last inequality, it can easily be seen that the series (2.51) converges for any value of ∆ if the following constraint is satisﬁed: lim

l−→∞

|Ml+1 (x)| =0 . |Ml (x)|(l + 1)

(2.71)

For ﬁlters with compact support, the following criterion holds: lim

l−→∞

(kmax ∆)|Ml+1 (x)| <1 . |Ml (x)|(l + 1)

(2.72)

For symmetric ﬁlters, the analogous criterion is (kmax ∆)2 |M2l+2 (x)| <1 . l−→∞ |M2l (x)|(2l + 2)(2l + 1) lim

(2.73)

Pruett et al. [607] proved that all symmetric, non-negative4 ﬁlters satisfy relation (2.73). This proof is now presented. If the ﬁlter is non-negative, following an integral mean value theorem, there exists a value c, −2π ≤ −π − x ≤ c ≤ π − x ≤ 2π, such that 2 (−π−x)/∆ |c| 2l ξ G(ξ)dξ |M2l+2 | = (π−x)/∆ ∆ 2 |c| |M2l | , (2.74) = ∆ 4

It is recalled that the ﬁlter is said to be non-negative if G(x) ≥ 0, ∀x.

2.2 Spatial Filtering: Extension to the Inhom*ogeneous Case

whereby |M2l+2 | = |M2l |

|c| ∆

2 ≤

2π ∆

31

2 .

(2.75)

The expected result is trivially deduced from this last relation.

2.2 Spatial Filtering: Extension to the Inhom*ogeneous Case 2.2.1 General In the above explanations, it was assumed that the ﬁlter is hom*ogeneous and isotropic. These assumptions are at time too restrictive for the resulting conclusions to be usable. For example, the deﬁnition of bounded ﬂuid domains forbids the use of ﬁlters that are non-local in space, since these would no longer be deﬁned. The problem then arises of deﬁning ﬁlters near the domain boundaries. At the same time, there may be some advantage in varying the ﬁlter cutoﬀ length to adapt the structure of the solution better and thereby ensure optimum gain in terms of reducing the number of degrees of freedom in the system to be resolved. From relation (2.1), we get the following general form of the commutation error for a convolution ﬁlter G(y, ∆(x, t)) on a domain Ω [230, 260]: ∂ ∂ ∂φ , G φ = (G φ) − G

. (2.76) ∂x ∂x ∂x The ﬁrst term of the right-hand side of (2.76) can be expanded as ∂ ∂ (G φ) = G(x − ξ, ∆(x, t))φ(ξ, t)dξ (2.77) ∂x ∂x Ω ∂G ∂∆ + = G(x − ξ, ∆(x, t))φ(ξ, t)n(ξ)ds

φ ∂x ∂∆ ∂Ω ∂φ , (2.78) +G

∂x where n(ξ) is the outward unit normal vector to the boundary of Ω, ∂Ω, yielding

∂ ∂G ∂∆ , G φ = +

φ G(x − ξ, ∆(x, t))φ(ξ, t)n(ξ)ds. (2.79) ∂x ∂x ∂∆ ∂Ω

The ﬁrst term appearing in the right-hand side of relation (2.79) is due to spatial variation of the ﬁltering length, while a domain boundary generates the second one. A similar development leads to:

32

2. Formal Introduction to Filtering

∂G ∂∆ ∂ , G φ =

φ ∂t ∂t ∂∆

.

(2.80)

These error terms must be eliminated, or bounded, in order to be able to deﬁne a controlled and consistent ﬁltering process for large-eddy simulation. This is done by deriving new ﬁltering operators. Several alternatives to the classical convolution products have been proposed, which are described in the following. Franke and Frank [225] propose an extension of (2.79) to the case of a domain with moving boundaries and a uniform time depdendent ﬁlter length, i.e. ∂Ω = ∂Ω(t) and ∆ = ∆(t). Limiting the analysis to the one dimensionalcase for the sake of clarity, and taking Ω(t) = [a(t), b(t)], one ﬁrst notices that the constraint dealing with the preservation of the constant

b(t)

G(x − ξ, ∆(t))dξ = 1 ,

(2.81)

a(t)

yields the following ﬁlter conservation law ∂G d∆(t) db(t) da(t) dξ = − G(x − b(t), ∆(t)) − G(x − a(t), ∆(t)) . dt dt a(t) ∂∆ dt (2.82) The commutation errors have the following forms: ∂ , G φ(x, t) = G(x − a(t), ∆(t))φ(a(t), t) − G(x − b(t), ∆(t))φ(b(t), t) , ∂x (2.83)

b(t)

∂ , G φ(x, t) ∂t

=

ξ=b(t) dξ − G(x − ξ, ∆(t))φ(ξ, t) dt ξ=a(t) b(t) ∂G d∆(t) dξ . + φ(ξ, t) ∂∆ dt a(t)

(2.84)

2.2.2 Non-uniform Filtering Over an Arbitrary Domain This section presents the ﬁndings concerning the extension of the ﬁltering to the case where the ﬁlter cutoﬀ length varies in space and where the domain over which it applies is bounded or inﬁnite. New Deﬁnition of Filters and Properties: Mono-dimensional Case. Alternative proposals in the hom*ogeneous case. Ghosal and Moin [262] propose to deﬁne the ﬁltering of a variable φ(ξ), deﬁned over the interval

2.2 Spatial Filtering: Extension to the Inhom*ogeneous Case

33

] − ∞, +∞[, as φ(ξ) = G φ =

1 ∆

+∞

G −∞

ξ−η ∆

φ(η)dη

,

(2.85)

in which the cutoﬀ length ∆ is constant. The convolution kernel G is made to verify the following four properties: 1. Symmetry G(−ξ) = G(ξ)

.

(2.86)

We note that this property was not explicitly required before, but that it is veriﬁed by the three ﬁlters described in Sect. 2.1.5. 2. Conservation of constants +∞ a G(ξ)dξ, a = const. (2.87) −∞

3. Fast decay. G(ξ) → 0 as |ξ| → ∞ fast enough for all of its moments to be ﬁnite, i.e. +∞ G(ξ)ξ n dξ < ∞, ∀n ≥ 0 . (2.88) −∞

4. Quasi-local in physical space. G(ξ) is localized (in a sense to be speciﬁed) in the interval [−1/2, 1/2]. Extension of the Top-Hat Filter to the Inhom*ogeneous Case: Properties. Considering deﬁnition (2.85), the top-hat ﬁlter (2.40) is expressed:

1 if |ξ| ≤ 12 G(ξ) = . (2.89) 0 otherwise There are a number of ways of extending this ﬁlter to the inhom*ogeneous case. The problem posed is strictly analogous to that of extending ﬁnite volume type schemes to the case of inhom*ogeneous structured grids: the control volumes can be deﬁned directly on the computational grid or in a reference space carrying a uniform grid, after a change of variable. Two extensions of the box ﬁlter are discussed in the following, each based on a diﬀerent approach. Direct extension. If the cutoﬀ length varies in space, one solution is to say: 1 φ(ξ) = ∆+ (ξ) + ∆− (ξ)

ξ+∆+ (ξ)

φ(η)dη

,

(2.90)

ξ−∆− (ξ)

in which ∆+ (ξ) and ∆− (ξ) are positive functions and (∆+ (ξ) + ∆− (ξ)) is the cutoﬀ length at point ξ. These diﬀerent quantities are represented in Fig. 2.8.

34

2. Formal Introduction to Filtering

Fig. 2.8. Direct extension of the top-hat ﬁlter. Representation of the integration cell at point ξ.

If the domain is ﬁnite or semi- inﬁnite, the functions ∆+ (ξ) and ∆− (ξ) must decrease fast enough near the domain boundaries for the integration interval [ξ −∆− (ξ), ξ +∆+ (ξ)] to remain deﬁned. The box ﬁlter is extended intuitively here, as an average over the control cell [ξ − ∆− (ξ), ξ + ∆+ (ξ)]. This approach is similar to the ﬁnite volume techniques based on control volumes deﬁned directly on the computational grid. It is shown that this expression does not ensure the commutation property with derivation in space. Relation (2.12) becomes (the dependency of the functions ∆+ and ∆− as a function of ξ is not explicitly state, to streamline the notation): (d/dξ) (∆− + ∆+ ) d φ = G , φ dξ ∆− + ∆+ 1 d∆+ d∆− − + φ(ξ − ∆− ) φ(ξ + ∆+ ) . ∆+ + ∆− dξ dξ (2.91) The amplitude of the error committed cannot be evaluated a priori, and thus cannot be neglected. Also, when (2.90) is applied to the Navier–Stokes equations, all the terms, including the linear ones, will introduce unknown terms that will require a closure. Extension by Variable Change. SOCF. To remedy this problem, a more general alternative description than relation (2.90) is proposed by Ghosal and Moin [262]. This new deﬁnition consists of deﬁning ﬁlters that commute at the second order with the derivation in space (Second Order Commuting Filter, or SOCF). This is based on a change of variable that allows the use of a hom*ogeneous ﬁlter. The function φ is assumed to be deﬁned over a ﬁnite or inﬁnite interval [a, b]. Any regular monotonic function deﬁned over this interval can be related to a deﬁnite function over the interval [−∞, +∞] by performing the variable change: ξ = f (x) ,

(2.92)

in which f is a strictly monotonic diﬀerentiable function such that: f (a) = −∞, f (b) = +∞ .

(2.93)

2.2 Spatial Filtering: Extension to the Inhom*ogeneous Case

35

The constant cutoﬀ length ∆ deﬁned over the reference space [−∞, +∞] is associated with the variable cutoﬀ length δ(x) over the starting interval by the relation: ∆ . (2.94) δ(x) = f (x) In the case of a ﬁnite or semi-inﬁnite domain, the function f takes inﬁnite values at the bounds and the convolution kernel becomes a Dirac function. The ﬁltering of a function ψ(x) is deﬁned in the inhom*ogeneous case in three steps: 1. We perform the variable change x = f −1 (ξ), which leads to the deﬁnition of the function φ(ξ) = ψ(f −1 (ξ)). 2. The function φ(ξ) is then ﬁltered by the usual hom*ogeneous ﬁlter (2.85): f (x) − η 1 +∞ ψ(x) ≡ φ(ξ) = G φ(η)dη . (2.95) ∆ −∞ ∆ 3. The ﬁltered quantity is then re-expressed in the original space: f (x) − f (y) 1 b ψ(x) = G ψ(y)f (y)dy . ∆ a ∆

(2.96)

This new expression of the ﬁlter modiﬁes the commutation error with the spatial derivation. Using (2.95) and integrating by parts, we get: dψ dx

= +

y=b f (x) f (x) − f (y) − G ψ(y) ∆ ∆ y=a b f (x) − f (y) 1 G f (x)ψ (y)dy ∆ a ∆

.

(2.97)

The fast decay property of the kernel G makes it possible to cancel the ﬁrst term of the rigth-hand side. The commutation error is: f (x) − f (y) d 1 b G G , φ = f (y)ψ (y) dξ ∆ a ∆ f (x) × 1− dy . (2.98) f (y) In order to simplify this expression, we introduce a new variable ζ such that: f (y) = f (x) + ∆ζ . (2.99) The variable y is then re-expressed as a series as a function of ∆: 2

y = y0 (ζ) + ∆y1 (ζ) + ∆ y2 (ζ) + ...

(2.100)

36

2. Formal Introduction to Filtering

Then, combining relations (2.99) and (2.100), we get (the dependence of the functions according to the variable x is not explicitly shown, to streamline the notation): 2 ∆ζ ∆ f ζ y =x+ − + ... , (2.101) f 2f 3 which allows us to re-write relation (2.98) as:

d φ G , dξ

+∞

= −∞

f (x) G(ζ)ψ (y(ζ)) 1 − dζ f (y(ζ))

(2.102)

2

= C1 ∆ + C2 ∆ + ... ,

(2.103)

in which the coeﬃcients C1 and C2 are expressed as: C1

=

C2

=

f ψ f 2

+∞

ζG(ζ)dζ

,

(2.104)

−∞

2f f ψ + f f ψ − 3f ψ 2f 4 2

+∞

ζ 2 G(ζ)dζ

.

(2.105)

−∞

The symmetry property of the kernel G implies C1 = 0, which ensures that the ﬁlter commutation error with the spatial derivation is of the second order as a function of the cutoﬀ length ∆. The authors call this Second-Order Commuting Filter (SOCF). A study of the spectral distribution of the commutation error is available in reference [262]. Rather than detailing this analysis here, only the major results will be explained. Considering a function of the form: ψ(x) = ψk eıkx ,

ı2 = −1 ,

(2.106)

the two derivation operations are written: dψ = ıkψ, dx

dψ = ıkψ dx

.

(2.107)

The commutation error can be measured by comparing the two wave numbers k and k , the latter being such that ıkψ = ık ψ. The commutation error is zero if k = k . Algebraic manipulations lead to the relation:

+∞

k f = 1 − ı∆ 2 −∞ +∞ k f −∞

ζG(ζ) sin(k∆ζ/f )dζ .

G(ζ) cos(k∆ζ/f )dζ

(2.108)

2.2 Spatial Filtering: Extension to the Inhom*ogeneous Case

37

Using the modal decomposition (2.106), the commutation error can be expressed in diﬀerential form. The calculations lead to: f 2 d2 ψ d + O(k∆)4 (2.109) ψ = α(2) 3 ∆ G , dx dx2 f δ d2 ψ (2) 2 = −α δ + O(kδ)4 , (2.110) δ dx2 in which δ(x) is the local cutoﬀ length and α(2) the second-order moment of G, i.e. +∞ (2) α = ζ 2 G(ζ)dζ . (2.111) −∞

Van der Ven’s Filters. Commuting ﬁlters can be deﬁned with the spatial derivation at an order higher than 2, at least in the case of an inﬁnite domain. To obtain such ﬁlters, Van der Ven [725] proposes deﬁning the ﬁltering for the case of a variable cutoﬀ length δ(x) by direct extension of the form (2.85): +∞ 1 x−y φ(x) = G φ(y)dy . (2.112) δ(x) −∞ δ(x) The function G is assumed here to be class C 1 , symmetrical, and must conserve the constants. Also, the function δ(x) is also assumed to be class C 1 . This deﬁnition is achieved by linearizing the general formula (2.96) around x, that is by letting φ (y) = φ (x) and φ(x) − φ(y) = φ (x)(x − y) and including relation (2.94). This linearization operation is equivalent to considering that the function φ is linear in a neighbourhood of x containing the support of the convolution kernel. By introducing the variable change y = x − ζδ(x), the corresponding commutation error is expressed:

δ d G , (G(ζ) + ζG (ζ)) φ(x − ζδ(x))dζ φ= dx δ

.

(2.113)

To increase the order of the commutation error, we look for functions G that are solutions to the equation G + ζG = a G(n) ,

n>1 ,

(2.114)

in which a is a real and G(n) designates the nth derivative of the kernel G. For such functions, the commutation error becomes:

d G , φ dx

∂ ∂ζ

=

δ a (−1)n δ

=

aδ (x)δ(x)n−1 φ(n) (x)

G(ζ)

n φ(x − ζδ(x))dζ (2.115) ,

(2.116)

38

2. Formal Introduction to Filtering

Fig. 2.9. High-order commuting ﬁlter. Graph of the associated transfer function for diﬀerent values of the parameter m.

and is thus formally of order n − 1. Simple analysis shows that the Fourier of the solution to problem (2.114) verifying the constant contransform G servation property is of the form: −aın n G(k) = exp k . (2.117) n The symmetry property of G implies that n = 2m is even, and therefore: −a(−1)m 2m G(k) = exp k . (2.118) 2m The fast decay property is recovered for a = b(−1)m , b > 0. It can be seen that the Gaussian ﬁlter then occurs again by letting m = 1. It is important to note that this analysis is valid only for inﬁnite domains, because when the bounds of the ﬂuid domain are included they bring out additional error terms with which it is no longer possible to be sure of the order of the commutation error. The transfer function obtained for various values of the parameter m is represented in Fig. 2.9. High-Order Commuting Filters. Van der Ven’s analysis has been generalized by Vasilyev et al. [728] so as to contain previous works (SOCF and Van der Ven’s ﬁlters) as special cases. As for SOCF, the ﬁltering process is deﬁned thanks to the use of a reference space. We now consider that the physical domain [a, b] is mapped into the domain [α, β]. Ghosal and Moin used α = −∞ and β = +∞. The correspondances between the two domains are summarized in Table 2.2.

2.2 Spatial Filtering: Extension to the Inhom*ogeneous Case

39

Table 2.2. Correspondances for Vasilyev’s high-order commuting ﬁlters. [a, b]

Domain Coordinate Filter length Function

[α, β] −1

x = f (ξ) δ(x) = ∆/f (x) ψ(x)

ξ = f (x) ∆ φ(ξ) = ψ(f −1 (ξ))

Considering this new mapping, relation (2.95) is transformed as ξ−η 1 α φ(ξ) = G φ(η)dη , (2.119) ∆ β ∆ and using the change of variables (2.99), we get φ(ξ) =

ξ−α ∆ ξ−β ∆

G (ζ) φ(ξ − ∆ζ)dζ

.

(2.120)

The next step consists in performing a Taylor expansion of φ(ξ − ∆ζ) in powers of ∆:

φ(ξ − ∆ζ) =

(−1)k k k ∂ k φ ∆ ζ (ξ) k! ∂ξ k

.

(2.121)

(−1)k k (k) ∂ k φ ∆ α (ξ) k (ξ) , k! ∂ξ

(2.122)

k=0,+∞

Substituting (2.121) into (2.120), we get

φ(ξ) =

k=0,+∞

where the kth moment of the ﬁlter kernel is now deﬁned as ξ−α ∆ (k) G (ζ) ζ k dζ . α (ξ) = ξ−β ∆

(2.123)

Using the relation (2.122), the space derivative of the ﬁltered variable expressed in the physical space can be evaluated as follows: dψ (x) dx

= =

dφ (ξ) (2.124) dξ (−1)k k dα(k) ∂k φ ∂ k+1 φ f (x) (ξ) k (ξ) + α(k) (ξ) k+1 (ξ) . ∆ k! dξ ∂ξ ∂ξ

f (x)

k=0,+∞

(2.125) A similar procedure is used to evaluate the second part of the commutation error. Using (2.124) and the same change of variables, we get:

40

2. Formal Introduction to Filtering

dψ 1 (x) = dx ∆

β

G α

ξ−η ∆

dφ (η)f (f −1 (η))dη dη

,

(2.126)

with

f (f −1 (η)) =

⎛

l=1,+∞

1 ⎝ (l − 1)!

k=1,+∞

⎞l−1

k −1

k

∂lf (x) ∂xl

(−1) k k ∂ f ∆ ζ (ξ)⎠ k! ∂ξ k

,

(2.127) and dφ (η) = dη

k=0,+∞

(−1)k k k ∂ k+1 φ ∆ ζ (ξ) k! ∂ξ k+1

.

(2.128)

Making the assumptions that all the Taylor expansion series are convergent5, the commutation error in the physical space is equal to G ,

d ψ= dx

k

Ak α(k) (ξ)∆ +

k=1,+∞

Bk

k=0,+∞

dα(k) k (ξ)∆ dξ

,

(2.129)

where Ak and Bk are real non-zero coeﬃcients. It is easily seen from relation (2.129) that the commutation error is determined by the ﬁlter moments and the mapping function. The order of the commutation error can then be governed by chosing an adequate ﬁlter kernel. Vasilyev proposes to use a function G such that: α(0)

=

1

∀ξ ∈ [α, β] ,

(k)

= <

0 ∞

1 ≤ k ≤ n − 1, ∀ξ ∈ [α, β] k ≥ n, ∀ξ ∈ [α, β] .

α α(k)

(2.130) ,

(2.131) (2.132)

,

(2.133)

An immediate consequence is dα(k) (ξ) = 0, 0 ≤ k ≤ n − 1, ∀ξ ∈ [α, β] dξ leading to

G ,

d n ψ = O(∆ ) . dx

(2.134)

The commutation error can be controlled by choosing a kernel G with desired moment values. It is important noting that conditions (2.130) – (2.132) do not require that the ﬁlter kernel be symmetric. Discrete ﬁlters verifying theses properties will be discussed in Sect. 13.2. 5

Vasilyev et al. [728] show that this is always true for practical numerical simulations.

2.2 Spatial Filtering: Extension to the Inhom*ogeneous Case

41

Extension to the Multidimensional Case. SOCF Filters. SOCF ﬁlters are extensible to the three-dimensional case for ﬁnite or inﬁnite domains. Let (x1 , x2 , x3 ) be a Cartesian system, and (X1 , X2 , X3 ) the reference axis system associated with a uniform isotropic grid with a mesh size ∆. The two systems are related by the relations: X1 = H1 (x1 , x2 , x3 ),

x1 = h1 (X1 , X2 , X3 )

,

(2.135)

X2 = H2 (x1 , x2 , x3 ), X3 = H3 (x1 , x2 , x3 ),

x2 = h2 (X1 , X2 , X3 ) x3 = h3 (X1 , X2 , X3 )

, ,

(2.136) (2.137)

or, in vectorial form: X = H(x),

x = h(X),

h = H −1

.

(2.138)

The ﬁltering of a function ψ(x) is deﬁned analogously to the monodimensional case. We ﬁrst make a variable change to work in the reference coordinate system, in which a hom*ogeneous ﬁlter is applied, and then perform the inverse transformation. The three-dimensional convolution kernel is deﬁned by tensorizing hom*ogeneous mono-dimensional kernels. After making the ﬁrst change of variables, we get: Xi − Xi 1 ψ(h(X)) = 3 G (2.139) ψ(h(X ))d3 X , ∆ ∆ i=1,3 or, in the original space: 1 Hi (x) − Xi ψ(h(X ))d3 X ψ(x) = G (2.140) 3 ∆ ∆ i=1,3 Hi (x) − Hi (x ) 1 = G ψ(x )J(x )d3 x , (2.141) 3 ∆ ∆ i=1,3 where J(x) is the Jacobian of the transformation X = H(x). Analysis of the error shows that, for ﬁlters deﬁned this way, the commutation error with the derivation in space is always of the second order, i.e. ∂ψ ∂ψ 2 − = O(∆ ) ∂xk ∂xk

,

(2.142)

where the second term of the left-hand side is written: ∂ψ 1 1 Hj (x) − Xj G = 3 ∂xk ∆ ∆ ∆ Hi (x) − Xi G × Hj,k (x)ψ(h(X ))d3 X ,(2.143) ∆ i=1,3;i=j

42

2. Formal Introduction to Filtering

with the notation: Hj,k (x) =

∂Hj (x) ∂xk

.

(2.144)

Diﬀerential analysis of the commutation error is performed by considering the solutions of the form: k exp(ık · x) . ψ(x) = ψ

(2.145)

An analogous approach to the one already made in the mono-dimensional case leads to the relation: ∂2ψ ∂ 2 G , + O(k∆)4 , (2.146) ψ = −α(2) ∆ Γkmp ∂xk ∂xm ∂xp where the function Γkmp is deﬁned as: Γkmp = hm,jq (H(x))hp,q (H(x))Hj,k (x)

.

(2.147)

Van der Ven’s Filters. Van der Ven’s simpliﬁed ﬁltering naturally extends to the three-dimensional case in Cartesian coordinates by letting: xi − x 1 i φ(x) = G (2.148) φ(x )d3 x , δ i (x) δ i (x) R3 i=1,3 i=1,3

in which δ i (x) is the ﬁlter cutoﬀ length in the ith direction of space at point x. For a kernel G verifying (2.114), the commutation error is expressed: ∂δ i (x) ∂ n φ(x) ∂ G , δ i (x)n−1 φ=a ∂xj ∂xj ∂xni i=1,3

,

(2.149)

and is formally of order n − 1. High-Order Commuting Filters. Vasilyev’s ﬁlters are generalized to the multiple dimension case in the same way as SOCF. 2.2.3 Local Spectrum of Commutation Error A spectral tool for analyzing the wavenumber sprectrum of the commutation error, referred to as the local spectrum analysis, was introduced by Vasilyev and Goldstein [727]. This tool enables an accurate understanding of the impact of the commutation error on the derivatives of the ﬁltered quantities. Writing the convolution ﬁlter as +∞ x−y 1 φ(x) = G φ(y)dy , (2.150) ∆(x) −∞ ∆(x)

2.3 Time Filtering: a Few Properties

43

and introducing its local Fourier decomposition (which can be evaluated using a windowed Fourier transform on bounded domains)

+∞

φ(x) =

ıkx k∆(x) φ(k)e G dk

,

(2.151)

−∞

one can identify the coeﬃcients of the local Fourier transform of the ﬁltered quantity φ x) = G k∆(x) φ(k) . (2.152) φ(k, The commutation error can be written as +∞ d 1 d∆(x) 1 , G (φ)(x) = K(k∆(x))φ(k) eıkx dk dx 2π −∞ ∆(x) dx

(2.153)

is deﬁned as where the transfer function K dG(k) K(k) = −k dk

,

.

(2.154)

By analogy with the previous case, the local spectrum of the commutation error is deﬁned as d 1 d∆(x) K(k∆(x))φ(k) . (2.155) , G (φ)(k, x) = dx ∆(x) dx This expresion shows that the gradient of the cutoﬀ length ∆(x) aﬀects the amplitude of the commutation error, while the ﬁlter shape (more precisly the gradient of the transfer function in Fourier space) governs the spectral repartition of the error. Analyses carried out considering convolution kernels presented in Sect.2.1.5 reveal that the spectrum of commutation error is global for smooth ﬁlters like the Gaussian ﬁlter (i.e. error occurs at all scales) while it is much more local for sharp ﬁlters (i.e. the commutation error is concentrated on a narrow wavenumber range).

2.3 Time Filtering: a Few Properties We consider here continuous causal ﬁlters of the form [603]:

t

φ(x, t) = G φ(x, t) = −∞

φ(ξ, t )G(t − t )dt

,

(2.156)

where the kernel G satisﬁes immediately two of the three fundamental properties given in Sect. 2.1.2, namely the linearity constraint and the constantconservation constraint.

44

2. Formal Introduction to Filtering

Due to the causality constraint, the time-domain support of these ﬁlters is bounded, i.e. lim G(t) = 0 . (2.157) t−→−∞

Then, the following commutation properties hold: ∂ φ=0 , G , ∂xj ∂ ∂G G , = φ(x, t)G(0) −

φ ∂t ∂t

(2.158)

.

(2.159)

It is observed that the use of spatially bounded domains does not introduce any commutation error terms, the ﬁlter being independent of the position in space. Two examples [606] are the exponential ﬁlter t − t 1 t 1 t φ(ξ, t ) exp dt , (2.160) G(t) = e ∆ −→ φ(x, t) = ∆ ∆ −∞ ∆ where ∆ is the characteristic cutoﬀ time, and the Heaviside ﬁlter G(t) =

1 H(t + ∆) ∆

,

(2.161)

where H(t) is the Heaviside function, yielding 1 φ(x, t) = ∆

t

φ(ξ, t )dt

.

(2.162)

t−∆

An interesting property of Eulerian time-domain ﬁltering is that local diﬀerential expression of the ﬁlters are easily derived, whose practical implementation is easier than those of their original counterparts. Diﬀerentiating relation (2.162), on obtains the diﬀerential form of the Heaviside ﬁlter: 1 ∂φ(x, t) = φ(x, t) − φ(x, t − ∆) ∂t ∆

.

(2.163)

.

(2.164)

The diﬀerential exponential ﬁlter is expressed as 1 ∂φ(x, t) = φ(x, t) − φ(x, t − ∆) ∂t ∆

3. Application to Navier–Stokes Equations

This chapter is devoted to the derivation of the constitutive equations of the large-eddy simulation technique, which is to say the ﬁltered Navier–Stokes equations. Our interest here is in the case of an incompressible viscous Newtonian ﬂuid of uniform density and temperature. We ﬁrst describe the application of an isotropic spatial ﬁlter1 to the Navier–Stokes equations expressed in Cartesian coordinates or in general coordinates. The emphasis will be put on the Eulerian, velocity–pressure formulation.2 An important point is that these two formulations lead to diﬀerent commutation errors with the ﬁltering operator, and thus yield diﬀerent theoretical and practical problems. The main point is that, when curvilinear grids are considered, two possibilities arise for solving numerically the ﬁltered governing equations: – Conventional Approach: First the ﬁlter is applied to the Navier–Stokes equations written in Cartesian coordinates, and then the ﬁltered equations are transformed in general coordinates. Here, the ﬁlter is applied in the physical space, and the ﬁlter kernels are developed within the usual Cartesian framework (see Chap. 2). – Alternate Approach: First the Navier–Stokes equations are expressed in general coordinates, and then the ﬁlter is applied to the transformed equations. In this case, the transformed variables are ﬁltered using uniform ﬁlter kernels, leading to vanishing commutation errors. If physical variables are ﬁltered using a transformed ﬁlter kernel, some commutation errors appear and speciﬁc ﬁlters must be employed (see Chap. 2). The diﬀerences originate from the fact that the transformation in general coordinates is a nonlinear operation, yielding diﬀerent commutation errors between the two operations. 1 2

Refer to the deﬁnition given in Chap. 2. A few works dealing with the velocity–vorticity form of the Navier–Stokes equations exist [163, 485, 486]. It is important to note that the Lagrangian framework is employed in [485, 486]. Results dealing with large-eddy simulation within the framework of lattice-Boltzmann methods are presented in the review by Chen and Doolen [127].

46

3. Application to Navier–Stokes Equations

Most of the existing published works deal with the conventional approach. As a consequence, this chapter will be mostly devoted to this approach. It should be noted that this ideal framework, which implies that the ﬂuid domain is unbounded, is the one nearly all authors use because it is only in this framework that the theory on which the subgrid modeling is based can be fully developed. The commutation errors between the ﬁlter and the derivation in space are then ignored. Section 3.4 is on the application of an inhom*ogeneous ﬁlter to the basic equations written in Cartesian coordinates. We begin by deriving the ﬁltered Navier–Stokes equations following the conventional approach. The various decompositions of the nonlinear term as a function of the ﬁltered quantities are then discussed. We lastly introduce the closure problem, i.e. the representation of the unknowns as a function of the variables in the ﬁltered problem.

3.1 Navier–Stokes Equations We recall here the equations governing the evolution of an incompressible Newtonian ﬂuid, ﬁrst in the physical space, in general coordinates, and then in the spectral space. 3.1.1 Formulation in Physical Space In the physical space, the velocity ﬁeld u = (u1 , u2 , u3 ) expressed in a reference Cartesian coordinate system x = (x1 , x2 , x3 ) is a solution of the system comprising the momentum and continuity equations: ∂ui ∂ui ∂ ∂p ∂ ∂uj + (ui uj ) = − +ν + , i = 1, 2, 3 , (3.1) ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂ui =0 , ∂xi

(3.2)

in which p = P/ρ and ν are, respectively, the static pressure and the assumedly constant, uniform kinematic viscosity. To obtain a well-posed problem, we have to add initial and boundary conditions to this system. 3.1.2 Formulation in General Coordinates The incompressible Navier–Stokes equations written in general coordinates in strong conservation-law form [733] read: ∂ (J −1 ξik ui ) = 0 , ∂ξ k

(3.3)

3.1 Navier–Stokes Equations

47

Fig. 3.1. Schematic of the coordinate transformation.

∂ ∂ ∂ ∂ −1 (J ui ) + k (U k ui ) = − k (J −1 ξik p) + ν k ∂t ∂ξ ∂ξ ∂ξ

∂ui J −1 Gkl l , ∂ξ

(3.4)

where ξ k are the coordinate directions in the transformed space, ξik = ∂ξ k /∂xi , J −1 is the Jacobian of the transformation, Gkl = ξik ξil denotes the contravariant metric tensor, ui the Cartesian components of the velocity ﬁeld, and U k = J −1 ξik ui the contravariant velocity component (see Fig. 3.1). 3.1.3 Formulation in Spectral Space The dual system in spectral space is obtained by applying a Fourier transform to equations (3.1) and (3.2). By making use of the fact that the incompressibility constraint is reﬂected geometrically by the orthogonality3 of the wave (k) , deﬁned as (see Appendix A for greater detail vector k and of the mode u on the spectral analysis of turbulence): 1 (k) = u (3.5) u(x)e−ık·x d3 x, ı2 = −1 , (2π)3 the system (3.1) - (3.2) can be reduced to a single equation: ∂ + νk 2 u i (k) = Ti (k) , ∂t in which the non-linear term Ti (k) is of the form: u j (p) um (q)δ(k − p − q)d3 pd3 q Ti (k) = Mijm (k) 3

(3.6)

,

(3.7)

This orthogonality relation is demonstrated by re-writing the incompressibility constraint of the velocity ﬁeld in the spectral space as: ∂ui (k) = 0 = 0 ⇐⇒ ki u i (k) ≡ k · u ∂xi

.

48

3. Application to Navier–Stokes Equations

with:

ı Mijm (k) = − (km Pij (k) + kj Pim (k)) , (3.8) 2 in which δ is the Kronecker symbol and Pij (k) is the projection operator on the plane orthogonal to the vector k. This operator is written: Pij (k) =

ki kj δij − 2 k

.

(3.9)

3.2 Filtered Navier–Stokes Equations in Cartesian Coordinates (hom*ogeneous Case) This section describes the equations of large-eddy simulation such as they are obtained by applying a hom*ogeneous ﬁlter verifying the properties of linearity, conservation of constants, and commutation with derivation, to the Navier–Stokes equations. These are the equations that will be resolved in the numerical simulation. 3.2.1 Formulation in Physical Space In light of the commutation with derivation property, the application of a ﬁlter to equations (3.1) and (3.2) is expressed: ∂ui ∂ ∂p ∂ + (ui uj ) = − +ν ∂t ∂xj ∂xi ∂xj ∂ui =0 , ∂xi

∂ui ∂uj + ∂xj ∂xi

,

(3.10)

(3.11)

where p is the ﬁltered pressure. The ﬁltered momentum equation brings out the non-linear term ui uj which, in order for this equation to be usable, will have to be expressed as a function of u and u , which are now the only unknowns left in the problem and where: u = u − u

.

(3.12)

This decomposition is not unique, and will be discussed in the following section. 3.2.2 Formulation in Spectral Space ui (k), the momentum equation in the Using the equivalence u i (k) = G(k) spectral space obtained by multiplying equation (3.6) by the transfer function

3.3 Decomposition of the Non-linear Term. Conventional Approach

G(k) is expressed:

∂ ui (k) = G(k)T + 2νk 2 G(k) i (k) , ∂t

in which the ﬁltered non-linear term G(k)T i (k) is written: uj (p) G(k) G(k)T um (q)δ(k − p − q)d3 pd3 q i (k) = Mijm (k)

49

(3.13)

. (3.14)

The ﬁltered non-linear term (3.14) brings out the contributions of the (p) and u (q). To complete the decomposition, these modes also modes u have to be expressed as the sum of a ﬁltered part and a ﬂuctuation. This is the same problem as the one encountered when writing the equations in the physical space. This operation is described in the following section.

3.3 Decomposition of the Non-linear Term. Associated Equations for the Conventional Approach This section details the various existing decompositions of the non-linear term and the associated equations written in Cartesian coordinates. 3.3.1 Leonard’s Decomposition Expression in Physical Space. Leonard [436] expresses the non-linear term in the form of a triple summation: ui uj

=

(ui + ui )(uj + uj )

=

ui uj

ui uj +

+

uj ui

(3.15) +

ui uj

.

(3.16)

The non-linear term is now written entirely as a function of the ﬁltered quantity u and the ﬂuctuation u . We then have two versions of this [762]. The ﬁrst considers that all the terms appearing in the evolution equations of a ﬁltered quantity must themselves be ﬁltered quantities, because the simulation solution has to be the same for all the terms. The ﬁltered momentum equation is then expressed: ∂ui ∂ ∂p ∂ ∂uj ∂ui ∂τij + ui uj = − +ν + , (3.17) − ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj in which the subgrid tensor τ , grouping together all the terms that are not exclusively dependent on the large scales, is deﬁned as: τij = Cij + Rij = ui uj − ui uj

,

(3.18)

50

3. Application to Navier–Stokes Equations

where the cross-stress tensor, C, which represents the interactions between large and small scales, and the Reynolds subgrid tensor, R, which reﬂects the interactions between subgrid scales, are expressed: Cij

=

ui uj + uj ui

Rij

=

ui uj

,

(3.19)

.

(3.20)

In the following, this decomposition will be called double decomposition. The other point of view consists of considering that it must be possible to evaluate the terms directly from the ﬁltered variables. But the ui uj term cannot be calculated directly because it requires a second application of the ﬁlter. To remedy this, Leonard proposes a further decomposition: ui uj = ui uj − ui uj + ui uj = Lij + ui uj . (3.21) The new L term, called Leonard tensor, represents interactions among the large scales. Using this new decomposition, the ﬁltered momentum equation becomes: ∂ui ∂p ∂ ∂uj ∂τij ∂ui ∂ − (ui uj ) = − +ν + . (3.22) + ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj The subgrid tensor τ , which now groups all the terms that are not expressed directly from u, takes the form: τij = Lij + Cij + Rij = ui uj − ui uj

.

(3.23)

This decomposition will be designated hereafter the Leonard or triple decomposition. Equation (3.22) and the subgrid term τij deﬁned by (3.23) can be obtained directly from the Navier–Stokes equations without using the Leonard decomposition for this. It should be noted that the term ui uj is a quadratic term and that it contains frequencies that are in theory higher than each of the terms composing. So in order to represent it completely, more degrees of freedom are needed than for each of the terms ui and uj 4 . We may point out that, if the ﬁlter is a Reynolds operator, then the tensors Cij and Lij are identically zero5 and the two decompositions are then equivalent, since the subgrid tensor is reduced to the tensor Rij . 4

5

In practice, if the large-eddy simulation ﬁlter is associated with a given computational grid on which the Navier–Stokes equations are resolved, this means that the grid used for composing the ui uj product has to be twice as ﬁne (in each direction of space) as the one used to represent the velocity ﬁeld. If the product is composed on the same grid, then only the ui uj term can be calculated. It is recalled that if the ﬁlter is a Reynolds operator, then we have the three following properties (see Appendix A): u = u, u = 0, uu = u u

,

3.3 Decomposition of the Non-linear Term. Conventional Approach

51

Writing the Navier–Stokes equations (3.1) in the symbolic form ∂u + N S(u) = 0 ∂t

,

(3.24)

the ﬁltered Navier–Stokes equations are expressed G

∂u ∂u = ∂t ∂t

=

−G N S(u)

=

−N S(u) − [G , N S](u)

(3.25) ,

(3.26)

where [., .] is the commutator operator introduced in Sect. 2.1.2. We note that the subgrid tensor corresponds to the commutation error between the ﬁlter and the non-linear term. Introducing the bilinear form B(·, ·): B(ui , uj ) ≡ ui uj

,

(3.27)

we get τij = [G , B](ui , uj ) .

(3.28)

Double decomposition (3.18) leads to the following equation for the resolved kinetic energy qr2 = ui ui /2: ∂qr2 ∂t

=

∂ui ∂ui ∂ui ∂ui ui uj + τij −ν ∂xj ∂xj ∂xj ∂xj

−

∂ ∂ ∂qr2 (ui p) + ν ∂x ∂x ∂x i i i

I

II

IV

−

III

V

∂ ∂ ui ui uj − (ui τij ) ∂xj ∂xj

VI

.

(3.29)

V II

This equation shows the existence of several mechanisms exchanging kinetic energy at the resolved scales: whence Cij

Lij

.

=

ui uj + uj ui

= =

ui u j + uj u i 0 ,

= = =

ui uj − ui uj ui uj − ui uj 0 .

52

3. Application to Navier–Stokes Equations

– – – – – – –

I - production II - subgrid dissipation III - dissipation by viscous eﬀects IV - diﬀusion by pressure eﬀect V - diﬀusion by viscous eﬀects V I - diﬀusion by interaction among resolved scales V II - diﬀusion by interaction with subgrid modes. Leonard’s decomposition (3.23) can be used to obtain the similar form: ∂qr2 ∂t

∂ui ∂qr2 uj ∂ui ∂ui + τij −ν ∂xj ∂xj ∂xj ∂xj

=

−

−

∂ ∂ ∂qr2 (ui p) + ν ∂x ∂x ∂x i i i

V III

XI

+

IX

X

XII

∂ ∂ui ui uj − (ui τij ) ∂xj ∂xj

XIII

.

(3.30)

XIV

This equation diﬀers from the previous one only in the ﬁrst and sixth terms of the right-hand side, and in the deﬁnition of tensor τ : – – – – – – –

V III - advection IX - idem II X - idem III XI - idem IV XII - idem V XIII - production XIV - idem V II

The momentum equation for the small scales is obtained by subtracting the large scale equation from the unﬁltered momentum equation (3.1), making, for the double decomposition: ∂ ∂τij ∂p ∂ui + + (ui + ui )(uj + uj ) − ui uj = − ∂t ∂xj ∂xi ∂xi ∂uj ∂ui ∂ +ν + , ∂xj ∂xj ∂xi

(3.31)

and, for the triple decomposition: ∂τij ∂ui ∂ ∂p + + (ui + ui )(uj + uj ) − ui uj = − ∂t ∂xj ∂xi ∂xi ∂uj ∂ui ∂ + ν + . ∂xj ∂xj ∂xi

(3.32)

3.3 Decomposition of the Non-linear Term. Conventional Approach

53

2 The ﬁltered subgrid kinetic energy qsgs = uk uk /2 equation obtained by multiplying (3.32) by ui and ﬁltering the relation thus derived is expressed: 2 ∂qsgs ∂t

=

+

1 ∂ ∂ 2 ∂ (ui ui uj − uj ui ui ) − (puj − p uj ) qsgs uj − ∂xj 2 ∂xj ∂xj

XV XV I XV II 2 ∂qsgs ∂ ∂ + (τij ui ) ν ∂xj ∂xj ∂xj

−

−

ν

XIX

XV III

∂ui ∂ui ∂ui ∂ui ∂ui − − τij ∂xj ∂xj ∂xj ∂xj ∂xj

XX

– – – – – – –

.

(3.33)

XXI

XV - advection XV I - turbulent transport XV II - diﬀusion by pressure eﬀects XV III - diﬀusion by viscous eﬀects XIX - diﬀusion by subgrid modes XX - dissipation by viscous eﬀects XXI - subgrid dissipation. For the double decomposition, equation (3.31) leads to: 2 ∂qsgs ∂t

=

−

∂ ∂ui ∂ui ui ui uj − ui ui uj + ui uj − ui uj ∂xj ∂xj ∂xj

XXII XXIII ∂ 2 ui ∂ 2 ui ∂ puj − p uj + ν ui 2 − ui 2 ∂xj ∂xj ∂xj

−

XXIV

+

∂ui ∂ τij ui − τij , ∂xj ∂xj

XXV I

with: – – – – –

XXII - turbulent transport XXIII - production XXIV - diﬀusion by pressure eﬀects XXV - viscous eﬀects XXV I - subgrid dissipation and diﬀusion

XXV

(3.34)

54

3. Application to Navier–Stokes Equations

It is recalled that, if the ﬁlter used is not positive, the generalized subgrid 2 kinetic energy qgsgs deﬁned as the half-trace of the subgrid tensor, 1 u u + ui ui , 2 i i can admit negative values locally (see Sect. 3.3.5). If the ﬁlter is a Reynolds operator, the subgrid tensor is then reduced to the subgrid Reynolds tensor and the generalized subgrid kinetic energy is equal to the subgrid kinetic energy, i.e. 1 2 2 qsgs ≡ ui ui = qgsgs ≡ τkk /2 . (3.35) 2 Expression in Spectral Space. Both versions of the Leonard decomposition can be transcribed in the spectral space. Using the deﬁnition of the (k) as ﬂuctuation u ui (k) , (3.36) u i (k) = (1 − G(k)) 2 = τkk /2 = qgsgs

the ﬁltered non-linear term G(k)T i (k) is expressed, for the triple decomposition: G(k)T i (k) =

Mijm (k)

G(q) uj (p) G(p) um (q)δ(k − p − q)d3 pd3 q G(q) (1 − G(k)) G(p)

−

Mijm (k)

+

um (q)δ(k − p − q)d3 pd3 q × uj (p) G(k) G(p)(1 − G(q)) + G(q)(1 − G(p)) Mijm (k)

+

× uj (p) um (q)δ(k − p − q)d3 pd3 q G(k) (1 − G(q))(1 − G(p)) Mijm (k) × uj (p) um (q)δ(k − p − q)d3 pd3 q

.

(3.37)

The ﬁrst term of the right-hand side corresponds to the contribution ui uj , the second to the Leonard tensor L, the third to the cross stresses represented by the tensor C, and the fourth to the subgrid Reynolds tensor R. This is illustrated by Fig. 3.2. The double decomposition is derived by combination of the ﬁrst two terms of the right-hand side of (3.37): G(k)T i (k) =

Mijm (k)

G(q) G(k) G(p)

× uj (p) um (q)δ(k − p − q)d3 pd3 q

3.3 Decomposition of the Non-linear Term. Conventional Approach

55

Fig. 3.2. Representation of the various Leonard decomposition terms in the spectral space, when using a sharp cutoﬀ ﬁlter with a cutoﬀ frequency kc .

G(k) G(p)(1 − G(q)) + G(q)(1 − G(p))

+

Mijm (k)

+

× uj (p) um (q)δ(k − p − q)d3 pd3 q G(k) (1 − G(q))(1 − G(p)) Mijm (k) × uj (p) um (q)δ(k − p − q)d3 pd3 q

.

(3.38)

The ﬁrst term of the right-hand side corresponds to the contribution ui uj in the physical space, and the last two remain unchanged with respect to the triple decomposition. Let us note that the sum of the contributions of the cross tensor and the subgrid Reynolds tensor simpliﬁes to the form: G(q)) Cij + Rij = Mijm (k) (1 − G(p) G(k) × uj (p) um (q)δ(k − p − q)d3 pd3 q

.

(3.39)

The momentum equations corresponding to these two decompositions are found by replacing the right-hand side of equation (3.13) with the desired terms. For the double decomposition, we get:

56

3. Application to Navier–Stokes Equations

∂ ui (k) + νk 2 G(k) ∂t

= Mijm (k)

G(q) G(k) G(p)

× uj (p) um (q)δ(k − p − q)d3 pd3 q G(q)) + Mijm (k) (1 − G(p) G(k) × uj (p) um (q)δ(k − p − q)d3 pd3 q

, (3.40)

and for the triple decomposition: ∂ G(q) ui (k) = Mijm (k) G(p) + νk 2 G(k) ∂t × uj (p) um (q)δ(k − p − q)d3 pd3 q G(q) (1 − G(k)) G(p) − Mijm (k) × uj (p) um (q)δ(k − p − q)d3 pd3 q G(k) G(p)(1 − G(q)) + Mijm (k) +G(q)(1 − G(p)) × uj (p) um (q)δ(k − p − q)d3 pd3 q G(k) (1 − G(q))(1 − G(p)) + Mijm (k) × uj (p) um (q)δ(k − p − q)d3 pd3 q

.

(3.41)

For both decompositions, the momentum equation can be expressed in the symbolic form: ∂ 2 ui (k) = Tr (k) + Tsgs (k) , + νk G(k) (3.42) ∂t in which Tr (k) designates the transfer terms calculated directly from the resolved modes, and is therefore equivalent to the contribution of the ui uj term in the case of the triple decomposition, and that of the ui uj term for the double decomposition. The Tsgs (k) term designates the other non-linear terms, and therefore corresponds to the contribution of the subsidiary term such as deﬁned above. Let E(k) be the energy contained on the sphere of radius k. It is calculated as: 1 2 (k) · u ∗ (k)dS(k) , u (3.43) E(k) = k 2 where dS(k) is the surface element of the sphere, and where the asterisk designates a conjugate complex number. The kinetic energy of the resolved modes contained on this same sphere, denoted E r (k), is deﬁned by the relation

3.3 Decomposition of the Non-linear Term. Conventional Approach

57

E r (k)

1 2 u(k) · G(k) u∗ (k)dS(k) G(k) k 2 2 (k)E(k) . = G =

(3.44) (3.45)

We return to the kinetic energy of the resolved modes, qr2 = ui ui /2, by summation on all the wave numbers: ∞ E r (k)dk . (3.46) qr2 = 0

It is important to note that E r (k) is related to the energy of the resolved modes, which is generally not equal to the ﬁltered part of the kinetic energy which, for its part, is associated with the quantity denoted E(k), deﬁned as E(k) = G(k)E(k)

.

(3.47)

The identity of these two quantities is veriﬁed when the transfer func2 (k) = G(k), tion is such that G ∀k, i.e. when the ﬁlter used is a projector. The evolution equation for E r (k) is obtained by multiplying the ﬁltered mo u∗ (k), and then integrating the result on mentum equation (3.13) by k 2 G(k) the sphere of radius k. Using the double decomposition we get the following equation: 1 ∂ 2 G(q) G 2 (k)S(k|p, q)dpdq G(p) + 2νk E r (k) = ∂t 2 ∆ 1 G(q)) 2 (k)S(k|p, q)dpdq , + (1 − G(p) G 2 ∆ (3.48) and the triple decomposition: ∂ 1 2 G(q) G(k)S(k|p, G(p) q)dpdq + 2νk E r (k) = ∂t 2 ∆ 1 G(q)S(k|p, G(k)(1 − G(k)) G(p) q)dpdq − 2 1 2 (k) G(p) + G 2 ∆ × (1 − G(q)) + G(q)(1 − G(p)) S(k|p, q)dpdq 1 2 (k) (1 − G(q))(1 G + − G(p)) 2 ∆ ×S(k|p, q)dpdq in which

,

(3.49)

58

3. Application to Navier–Stokes Equations

S(k|p, q) = 16π 2 kpqMijm (k) uj (p) um (q) ui (−k)δ(k − p − q) , (3.50) !! and where the symbol ∆ designates integration over the interval |k − p| < q < k + p. Following the example of what was done for the momentum equations, the kinetic energy evolution equation for the resolved modes can be expressed in the abbreviated form ∂ 2 e + 2νk E r (k) = Tre (k) + Tsgs (k) . (3.51) ∂t e (k) represent, respectively, the energy transfers The terms Tre (k) and Tsgs of mode k with all the other modes associated with the terms that can be calculated directly from the resolved modes, and the subgrid terms.The kinetic energy conservation property for inviscid ﬂuids, i.e. in the case of zero viscosity, implies: e (k))d3 k = 0 . (Tre (k) + Tsgs

(3.52)

The momentum equations for the unresolved scales are obtained by algebraic manipulations strictly analogous to those used for obtaining the equations for the resolved scales, except that this time equation (3.6) is multiplied by (1 − G(k)) instead of G(k). These equations are written: ∂ ui (k) = (1 − G(k))T (3.53) + νk 2 (1 − G(k)) i (k) . ∂t

Calculations similar to those explained above lead to: ∂ G(q)(1 G(p) − G(k)) + νk 2 u i (k) = Mijm (k) ∂t

+

× uj (p) um (q)δ(k − p − q)d3 pd3 q (1 − G(k)) G(p)(1 − G(q)) Mijm (k) +G(q)(1 − G(p))

+

um (q)δ(k − p − q)d3 pd3 q × uj (p) (1 − G(k)) (1 − G(q))(1 − G(p)) Mijm (k) × uj (p) um (q)δ(k − p − q)d3 pd3 q

.

(3.54)

The ﬁrst term of the right-hand side represents the contribution of the interactions between large scale modes, the second the contribution of the cross interactions, and the last the interactions among the subgrid modes.

3.3 Decomposition of the Non-linear Term. Conventional Approach

59

Let E sgs be the energy contained in the subgrid modes. This energy is deﬁned as: 1 2 k E sgs (k) = (1 − G(k)) u(k) · (1 − G(k)) u∗ (k)dS(k) (3.55) 2 =

2 (k)E(k) (1 − G)

,

(3.56)

and is diﬀerent, in the general case, from the kinetic energy ﬂuctuation though the equality of these two quantities is verE (k) = (1 − G)(k)E(k), iﬁed when the ﬁlter is a Reynolds operator. Simple calculations give us the following evolution equation for E sgs (k): ∂ 1 2 2 G(q)(1 G(p) − G(k)) + 2νk E sgs (k) = S(k|p, q)dpdq ∂t 2 ∆ 1 + (1 − G(k)) G(p)(1 − G(q)) 2 ∆ +G(q)(1 − G(p)) S(k|p, q)dpdq 1 2 + (1 − G(k)) (1 − G(q))(1 − G(p)) 2 ∆ ×S(k|p, q)dpdq , (3.57) where the notation used is the same as for the kinetic energy evolution equa2 is obtained by tion of the resolved modes. The subgrid kinetic energy qsgs summation over the entire spectrum: ∞ 2 = E sgs (k)dk . (3.58) qsgs 0

3.3.2 Germano Consistent Decomposition This section presents the Germano consistent decomposition, which is a generalization of the Leonard decomposition. Deﬁnition and Properties of Generalized Central Moments. For convenience, we use [φ]G to denote the resolved part of the ﬁeld φ, deﬁned as in the ﬁrst chapter, where G is the convolution kernel, i.e.: +∞ G(x − ξ)φ(ξ)d3 ξ . (3.59) [φ]G (x) ≡ G φ(x) ≡ −∞

We deﬁne the generalized central moments with the ﬁlter G, denoted τG , as [244, 246, 247, 248]: τG (φ1 , φ2 ) = τG (φ1 , φ2 , φ3 ) = τG (φ1 , φ2 , φ3 , φ4 ) =

[φ1 φ2 ]G − [φ1 ]G [φ2 ]G

,

(3.60)

[φ1 φ2 φ3 ]G − [φ1 ]G τG (φ2 , φ3 ) − [φ2 ]G τG (φ1 , φ3 ) (3.61) −[φ3 ]G τG (φ1 , φ2 ) − [φ1 ]G [φ2 ]G [φ3 ]G , ...

(3.62)

60

3. Application to Navier–Stokes Equations

The generalized central moments thus deﬁned verify the following properties: τG (φ, ψ)

=

τG (φ, a) = τG (φ, ψ, a) = ∂τG (φ, ψ)/∂s =

τG (ψ, φ) 0,

,

(3.63)

for a = const.

,

(3.64)

0, for a = const. , τG (∂φ/∂s, ψ) + τG (φ, ∂ψ/∂s), s = x, t

(3.65) . (3.66)

If we perform the decomposition φ = φ1 + φ2 , ψ = ψ1 + ψ2 , we get: τG (ψ1 + ψ2 , φ1 + φ2 ) =

τG (ψ1 , φ1 ) + τG (ψ1 , φ2 )

+

τG (ψ2 , φ1 ) + τG (ψ2 , φ2 )

.

(3.67)

The generalized central moments also appear as the coeﬃcients of the following formal Taylor expansion [247]: [φ(a1 , ..., an )]G

= φ([a1 ]G , ..., [an ]G ) +

τG (al , am ) 2!

l,m

+

ylm

τG (al , am , ak ) ylmk + ... , 3!

(3.68)

l,m,k

with ylm =

∂ 2 φ([a1 ]G , ..., [an ]G ) , ∂[al ]G ∂[am ]G

ylmk =

∂ 3 φ([a1 ]G , ..., [an ]G ) ∂[al ]G ∂[am ]G ∂[ak ]G

,

and where the ai are generic turbulent quantities. The relation (3.68) establishes a link between the ﬁltered value of the functional φ and its unﬁltered counterpart applied to the ﬁltered variables [ai ]G . Consistent Decomposition: Associated Equations. By applying the property (3.67) to the decomposition φ = [φ]G + φ , ψ = [ψ]G + ψ , we get: τG ([φ]G + φ , [ψ]G + ψ ) = τG ([φ]G , [ψ]G ) + τG (φ , [ψ]G ) +τG ([φ]G , ψ ) + τG (φ , ψ )

.

(3.69)

This decomposition is said to be consistent because it is consistent with the deﬁnition of the generalized central moments, ensuring that all the terms in it are of the same form, which is not true of the Leonard decomposition. The various terms of the right-hand side of equation (3.69) can be interpreted as generalizations of the terms of the Leonard triple decomposition. By applying this deﬁnition to the components of the velocity ﬁelds, the subgrid tensor (3.23) appears in a double form: τG (ui , uj )

= [ui uj ]G − [ui ]G [uj ]G = Lij + Cij + Rij = Lij + Cij + Rij

,

(3.70)

3.3 Decomposition of the Non-linear Term. Conventional Approach

61

in which the tensors L, C and R are deﬁned as: Lij Cij

= =

τG ([ui ]G , [uj ]G ) , τG ([ui ]G , uj ) + τG (ui , [uj ]G ) ,

(3.71) (3.72)

Rij

=

τG (ui , uj )

(3.73)

,

and represent, respectively, the interactions between the large scales, the cross interactions, and the interactions among subgrid scales. They therefore represent tensors deﬁned by Leonard, but are not the same as them in the general case. By bringing out the generalized central moments, the ﬁltered momentum equations are written in the form: ∂ ∂[p]G ∂ ∂[uj ]G ∂[ui ]G ∂[ui ]G + ([ui ]G [uj ]G ) = − +ν + ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂τG (ui , uj ) − . (3.74) ∂xj This equation is equivalent to the one derived from the triple Leonard decomposition. Similarly, the subgrid kinetic energy evolution equation (3.33) is re-written as: 2 2 ∂qsgs ∂qsgs 1 ∂ = τG (ui , ui , uj ) + τG (p, uj ) − ν ∂t ∂xj 2 ∂xj

− ντG (∂ui /∂xj , ∂ui /∂xj ) − τG (ui , uj )

∂[ui ]G ∂xj

.

(3.75)

It is easy to check that the structure of the ﬁltered equations is, in terms of generalized central moments, independent of the ﬁlter used. This is called the ﬁltering (or averaging) invariance property. 3.3.3 Germano Identity Basic Germano Identity. Subgrid tensors corresponding to two diﬀerent ﬁltering levels can be related by an exact relation derived by Germano [246]. A sequential application of two ﬁlters, F and G, is denoted: [ui ]FG = [[ui ]F ]G = [[ui ]G ]F

,

(3.76)

or equivalently: [ui ]FG (x) =

+∞

G(x − y)d y

+∞

3

−∞

−∞

F (y − ξ)ui (ξ)d3 ξ

.

(3.77)

62

3. Application to Navier–Stokes Equations

Fig. 3.3. Illustration of the two ﬁltering levels, F and G, involved in the Germano identity. Associated cutoﬀ wave numbers are denoted kF and kG , respectively. Resolved velocity ﬁelds are uF and uG , and the associated subgrid tensors are τF and τG , respectively.

Here, [ui ]FG corresponds to the resolved ﬁeld for the double ﬁltering F G. The two ﬁltering levels are illustrated in Fig. 3.3. The subgrid tensor associated with the level F G is deﬁned as the following generalized central moment: τFG (ui , uj ) = [ui uj ]FG − [ui ]FG [uj ]FG

.

(3.78)

This expression is a trivial extension of the deﬁnition of the subgrid tensor associated with the G ﬁltering level. By deﬁnition, the subgrid tensor τG ([ui ]F , [uj ]F ) calculated from the scales resolved for the F ﬁltering level, is written: (3.79) τG ([ui ]F , [uj ]F ) = [[ui ]F [uj ]F ]G − [ui ]FG [uj ]FG . These two subgrid tensors are related by the following exact relation, called the Germano identity: τFG (ui , uj ) = [τF (ui , uj )]G + τG ([ui ]F , [uj ]F ) .

(3.80)

This relation can be interpreted physically as follows. The subgrid tensor at the F G ﬁltering level is equal to the sum of the subgrid tensor at the F level ﬁltered at the G level and the subgrid tensor at the G level calculated from the ﬁeld resolved at the F level. This relation is local in space and time and is independent of the ﬁlter used. It is interesting noting that re-writing the subgrid tensor as τF G (ui , uj ) = [F G , B](ui , uj ) ,

3.3 Decomposition of the Non-linear Term. Conventional Approach

63

where [., .] is the commutator operator (see equation (2.13)) and B(., .) the bilinear form deﬁned by relation (3.27), the Germano identity (3.80) is strictly equivalent to relation (2.16): [F G , B](ui , uj ) = [F , B] ◦ (G )(ui , uj ) + (F ) ◦ [G , B](ui , uj ). (3.81) The previous Germano identity can be referred to as the multiplicative Germano identity [251], because it is based on a sequential application of the two ﬁlters. An additive Germano identity can also be deﬁned considering that the second ﬁltering level is deﬁned by the operator (F + G) and not by the operator (G ) ◦ (F ). The equivalent relation for (3.80) is τF+G (ui , uj ) = τF (ui , uj ) + τG (ui , uj ) − ([ui ]F [uj ]G + [ui ]G [uj ]F ) . (3.82) Multilevel Germano Identity. The Multiplicative Germano Identity can be extended to the case of N ﬁltering levels, Gi , i = 1, N , with associated characteristic lengths ∆1 ≤ ∆2 ≤ ... ≤ ∆N [248, 710, 633]. n We deﬁne the nth level ﬁltered variable φ as n

φ = Gn Gn−1 ... G1 φ = G1n φ ,

(3.83)

with n Gm ≡ Gn Gn−1 ... Gm , Gnn = Id,

∀m ∈ [1, n] .

(3.84)

Let τijn = ui uj n −uni unj be the subgrid tensor associated to the nth ﬁltering level. The classical two-level Germano identity (3.80) reads τijn+1 = τijn

n+1

+ Ln+1 ij ,

Ln+1 = uni unj ij

n+1

− un+1 un+1 i j

.

(3.85)

Simple algebraic developments lead to the following relation between two ﬁltering levels n and m, with m < n: n n τijn = Lnij + Gk+1 Lkij + Gm+1 τijm , (3.86) k=m+1,n−1

resulting in a fully general mutilevel identity. Generalized Germano Identity. A more general multiplicative identity is obtained by applying an arbitrary operator L to the basic identity (3.81) [629], yielding L{[F G , B](ui , uj )]}

= L{[F , B] ◦ (G )(ui , uj ) + (F ) ◦ [G , B](ui , uj )} .

(3.87)

For linear operators, we get L{[F G , B](ui , uj )]} = +

L{[F , B] ◦ (G )(ui , uj )} L{(F ) ◦ [G , B](ui , uj )}

.

Application to the multilevel identity (3.86) is straightforward.

(3.88)

64

3. Application to Navier–Stokes Equations

3.3.4 Invariance Properties One of the basic principles of modeling in mechanics is to conserve the generic properties of the starting equations [681, 230, 260, 572, 326, 398]. We consider in the present section the analysis of some invariance/symmetry properties of the ﬁltered Navier–Stokes equations, and the resulting constraints for subgrid models. It is remembered that a diﬀerential equation will be said to be invariant under a transformation if it is left unchanged by this transformation. It is important to note that these properties are not shared by the boundary conditions. It is shown that properties of the ﬁltered Navier– Stokes equations depend on the ﬁlter used to operate the scale separation. The preservation of the symmetry properties of the original Navier–Stokes equations will then lead to the deﬁnition of speciﬁc requirements for the ﬁlter kernel6 G(x, ξ). The properties considered below are: – Galilean invariance for the spatial ﬁltering approach (p. 64). – Galilean invariance for the time-domain ﬁltering approach (p. 66). – General frame-invariance properties for the time-domain ﬁltering approach (p. 67). – Time invariance for the spatial ﬁltering approach (p. 69). – Rotation invariance for the spatial ﬁltering approach (p. 70). – Reﬂection invariance for the spatial ﬁltering approach (p. 70). – Asymptotic Material Frame Indiﬀerence for the spatial ﬁltering approach (p. 71). Galilean Invariance for Spatial ﬁlter. This section is devoted to the analysis by Speziale [681] of the preservation of the Galilean invariance property for translations of the Navier–Stokes equations, ﬁrst by applying a spatial ﬁlter, then by using the diﬀerent decompositions presented above. Let us take the Galilean transformation (translation): x• = x + V t + b,

t• = t ,

(3.89)

in which V and b are arbitrary uniform vectors in space and constant in time. If the (x, t) frame of reference is associated with an inertial frame, then so is (x• , t• ). Let u and u• be the velocity vectors expressed in the base frame of reference and the new translated one, respectively. The passage from one system to the other is deﬁned by the relations: u• ∂ ∂x•i ∂ ∂t• 6

= = =

u+V , ∂ , ∂xi ∂ ∂ − Vi ∂t ∂xi

(3.90) (3.91) .

(3.92)

We will only consider ﬁlters with constant and uniform cutoﬀ length, i.e. ∆ is independent on both space and time. Variable length ﬁlters are anisotropic or nonhom*ogeneous, and violate the following properties in the most general case.

3.3 Decomposition of the Non-linear Term. Conventional Approach

65

The proof of the invariance of the Navier–Stokes equations for the transformation (3.89) is trivial and is not reproduced here. With this property in hand, what remains to be shown in order to prove the invariance of the ﬁltered equations by such a transformation is that the ﬁltering process preserves this property. Let there be a variable φ such that φ• = φ .

(3.93)

The ﬁltering in the translated frame of reference is expressed: • φ = G(x• − x• )φ• (x• )d3 x• .

(3.94)

By using the previous relations, we get: x• − x• = (x + V t + b) − (x + V t + b) = x − x ∂x• d3 x• = i d3 x = d3 x , ∂xj and thus, by substitution, the equality: • φ = G(x − x )φ(x )d3 x = φ ,

,

(3.95) (3.96)

(3.97)

which completes the proof7 . The invariance of the Navier–Stokes equations for the transformation (3.89) implies that the sum of the subgrid terms and the convection term, calculated directly from the large scales, is also invariant, but not that each term taken individually is invariant. In the following, we study the properties of each term arising from the Leonard and Germano decompositions. The above relations imply: u• = u + V , u• = u , u• = u

,

(3.98)

which reﬂects the fact that the velocity ﬂuctuations are invariant by Galilean transformation, while the total velocity is not. In the spectral space, this corresponds to the fact that only the constant mode does not remain invariant by this type of transformation since, with the V ﬁeld being uniform, it alone is aﬀected by the change of coordinate system8 . 7 8

A suﬃcient condition is that the ﬁlter kernel appears as a function of x − x . This is expressed: V = cste =⇒ V (k) = 0 ∀k = 0 , and thus • (k) = u (k) ∀k = 0 , u • (0) + V (0) . u (0) = u

66

3. Application to Navier–Stokes Equations

In the translated frame, the Leonard tensor takes the form: L•ij

= u•i u•j − u•i u•j = Lij + Vi uj + Vj ui − (Vi uj + Vj ui ) . = Lij − Vi uj + Vj ui

(3.99) (3.100) (3.101)

So this tensor is not invariant. Similar analyses show that: • Cij , = Cij + Vi uj + Vj ui • Rij • L•ij + Cij

= =

Rij , Lij + Cij

(3.102) (3.103) (3.104)

.

The tensor C is thus not invariant in the general case, while the tensor R and the groups L+C and L+C+R are. A diﬀerence can be seen to appear here between the double and triple decompositions: the double retains groups of terms (subgrid tensor and terms computed directly) that are not individually invariant, while the groups in the triple decomposition are. The generalized central moments are invariant by construction. That is, by combining relations (3.67) and (3.63), we immediately get: • (u•i , u•j ) = τG (ui , uj ) . τG

(3.105)

This property results in all the terms in Germano’s consistent decomposition being invariant by Galilean transformation, which is all the more true for the tensors L, C and R. Galilean Invariance and Doppler Eﬀect for Time-Domain Filters. Pruett [603] extended the above analysis to the case of Eulerian time-domain ﬁltering. Using the properties of the Eulerian time-domain ﬁltering and the fact that Navier–Stokes equations are form-invariant under Galilean transformations, one can easily prove that time-domain ﬁltered Navier–Stokes equations are also form-invariant under these transformations: ∂u•j ∂u•i ∂ • • ∂ ∂u•i ∂p• + • ui uj = − • + ν • + • , (3.106) ∂t• ∂xj ∂xi ∂xj ∂x•j ∂xi ∂u•i =0 . ∂x•i

(3.107)

It was shown in the preceding section that the spatially ﬁltered part of a Galilean-invariant function is itself Galilean invariant, i.e. u• = u + V =⇒ u• = u + V ,

p• = p =⇒ p• = p

.

(3.108)

A fundamental diﬀerence between spatial- and time-domain ﬁltering is that what applies to the former does not apply to the latter. Writing the

3.3 Decomposition of the Non-linear Term. Conventional Approach

67

deﬁnition of the ﬁltered velocity in the translated frame, we have: u•i

≡ ui (t• , x• − V t• ) + Vi t• = ui (t• , x• − V t• )G(t• − t• )dt• + Vi

−∞ t

= −∞ t

=

−∞ t

= −∞

ui (t , x• − V t )G(t − t)dt + Vi ui (t , x• − V t)G(t − t)dt + Vi ui (t , x)G(t − t)dt + Vi

≡ ui (x, t) + Vi

.

(3.109)

It is seen from these equations that Eulerian temporally ﬁltered quantities experience a Doppler shift in the direction of the translational velocity V , and that equalities presented in (3.108) are recovered only when V = 0. The equations remain invariant under Galilean transformations because each term is subjected to the same shift. This Doppler shift results in a wave number-dependent frequency shift between the two frames [603], indicating that Eulerian time-domain ﬁltering may be inadequate for ﬂows in which structures are convected at very diﬀerent characteristic velocities, such as boundary layers [289]. On the contrary, free shear ﬂows (mixing layers, jets, and wakes) seem to be better adapted. General Investigation of Frame-Invariance Properties of TimeDomain Filtered Navier–Stokes Equations. Previous analysis of Galilean invariance properties of Eulerian time-domain ﬁlter was extended to the more general case of Euclidean group of transformations by Pruett et al. [606]. In the case the observer is ﬁxed in the inertial frame (referred to as case I below), the spatial coordinates in the noninertial frame vary with time, while those in the inertial frame are ﬁxed. Thus, the general change of reference frame is expressed as (3.110) x•i (t• ) = Qij [xi + Vi ] , where x•i is refers to the coordinates of a point in a frame of reference in arbitrary time-dependent motion (rotation and translation) relative to an inertial frame tied to coordinates xi , and Q = Q(t) is a time-dependent orthogonal tensor. The vector V is also time-dependent, i.e. V = V (t). The time in the new reference frame is obtained considering a shift t0 : t• = t + t0 . The velocity in the moving frame is " # . (3.111) u•i (t• , x• ) = Q˙ ij [xi + Vi ] + Qij ui + V˙ i The ﬁltering cutoﬀ ∆ being frame-invariant, it will not be explicited in the following.

68

3. Application to Navier–Stokes Equations

In the opposite case (case II) where the observer is ﬁxed in the noninertial frame, one obtains xi (t) = Qji x•j − Vi

,

(3.112)

ui (t, x) = Q˙ ji x•j + Qji u•j − V˙ i

.

(3.113)

The spatial coordinates in the inertial frame are now time-dependent, and those in the noninertial frame are ﬁxed. These two diﬀerent cases must be treated separately when analyzing the properties of the time-ﬁletered Navier–Stokes equations. Application of the Eulerian time ﬁlter in case I yields x•i (t• ) = Qij xi + Qij Vi

,

(3.114)

" # . u•i (t• , x• ) = Q˙ ij [xi + Vi ] + Qij ui + V˙ i

(3.115)

The subgrid velocity ﬁeld u• = u• − u• is then equal to ˙ ˙ ˙ ˙ u• i = Qik uk − Qik uk + Qik − Qik xk + (Qik bk ) − (Qik bk )

. (3.116)

In case II, the ﬁltered and subgrid quantities are expressed as follows xi (t) = Qji x•j − V i

,

ui (t, x) = Q˙ ji x•j + Qji u•j − V˙ i

(3.117)

,

ui = Qki u•k − Qki u•k + Q˙ kj − Q˙ kj x•k − b˙ i − b˙ i

(3.118)

.

(3.119)

A look at equations (3.115) and (3.118) show that the velocity is not frame invariant under general Euclidean transformations. The same conclusion apply for the subgrid velocity ﬁeld. A noticeable diﬀerence between time- and space-ﬁltering is that, because Q(t) is a time-dependent parameter, ﬁltered and unﬁltered velocity ﬁelds do not transform in the same manner in the time-ﬁltering approach. The Navier–Stokes equations are known to be not frame-invariant under the Euclidean group of transformation, and can be expressed as ∂u•i ∂u• ∂P • ∂ 2 u• + ∂u•k •i = − • + ν • i • + 2Ωik u•k + Ω˙ ik x•k • ∂t ∂xk ∂xi ∂xk ∂xk

,

(3.120)

3.3 Decomposition of the Non-linear Term. Conventional Approach

69

where the modiﬁed pressure P • and the rotation rate tensor are deﬁned as 1 P • = p• + Ωkl Ωln x•n x•k − Qnk V¨k• x•n 2 Ωik ≡ Q˙ il Qkl

,

(3.121)

,

(3.122)

with V¨k• ≡ Qkn V¨n . The Navier–Stokes equations under the Euclidean transformation group for an observer ﬁxed in the noninertial reference frame are obtained by ﬁrst taking the material derivative of (3.111) and applying the ﬁlter, leading to Du•i Duj = (Ω˙ ik − Ωil Ωlk )x•k + 2Ωik u•k + Qij V¨j + Qij Dt• Dt

.

(3.123)

and then inserting the following expression deduced from the Navier–Stokes equations written in the inertial frame: Qij

Duj p• ∂ 2 u• =− • +ν • i • Dt ∂xi ∂xk ∂xk

,

(3.124)

yielding 2 • • • • ∂u•i • ∂u i = − ∂P + ν ∂ ui • +Ω ˙ ik x•k − ∂τik + u + 2Ω u ik k k ∂t• ∂x•k ∂x•i ∂x•k ∂x•k ∂x•k

, (3.125)

where the ﬁltered pressure P • and the subgrid scale tensor τ • are deﬁned as 1 P • = p• + Ωkl Ωln x•n x•k − V¨k• x•k 2 • = u•i u•k − u• i u•k τik

(3.126)

.

(3.127)

A comparison of equations (3.125) and (3.120) shows that the time-ﬁltered Navier–Stokes equations do not retain the same form in the most general case, to the contrary of the spatially ﬁltered ones. The diﬀerences appear in the Coriolis terms, the centrifugal and the rotational acceleration terms. Time Invariance (Spatial Filters). A time shift of the amount t0 yields the following change of coordinates: t• = t + t 0 ,

x• = x,

u• = u

.

(3.128)

Since we are considering space dependent ﬁlters only, the ﬁltered Navier– Stokes equations are automatically time-invariant, without any restriction on

70

3. Application to Navier–Stokes Equations

the ﬁlter kernel. We have: u• = u,

u• = u

,

(3.129)

and • τik

=

τik

L•ik • Rik

= =

Lik , Rik ,

(3.131) (3.132)

• Cik

=

Cik

(3.133)

,

(3.130)

.

All the subgrid terms are invariant. Rotation Invariance (Spatial Filters). We now consider the following change of reference system: t• = t,

x• = Ax,

u• = Au

,

(3.134)

where A is the rotation matrix with AT A = AAT = Id and |A| = 1. Simple calculations similar to those shown for in the section devoted to Galilean invariance lead to the following relations: u• = Au,

u• = Au

,

(3.135)

if and only if the ﬁlter kernel G(x, ξ) satisﬁes G(A(x − ξ)) = G(x − ξ) =⇒ G(x, ξ) = G(|x − ξ|) ,

(3.136)

meaning that the ﬁlter must be spherically symmetric. The subgrid terms are transformed as: • τik L•ik

= Aim Akn τmn , = Aim Akn Lmn ,

(3.137) (3.138)

• Rik • Cik

= Aim Akn Rmn = Aim Akn Cmn

(3.139) (3.140)

, ,

and are seen to be invariant. Reﬂection Invariance (Spatial Filters). We now consider a reﬂection in the lth direction of space: t• = t; x•l = −xl ; x•i = xi , i = l; u•l = −ul ; u•i = ui , i = l

.

(3.141)

If the ﬁlter is such that G(x − ξ) = G(−x + ξ), i.e. is symmetric, then • u•l = −ul ; u•i = ui , i = l; u• l = −u l ; ui = u i , i = l

,

(3.142)

3.3 Decomposition of the Non-linear Term. Conventional Approach

71

yielding • τik L•ik

= =

βτik , βLik ,

(3.143) (3.144)

• Rik • Cik

= =

βRik βCik

(3.145) (3.146)

, ,

with β = −1 if i = l or k = l and i = l, and β = 1 otherwise. We can see that the subgrid tensor and all the terms appearing in both the double and triple decomposition are invariant. Asymptotic Material Frame Indiﬀerence (Spatial Filters). The last symmetry considered in the present section is the asymptotic material frame indiﬀerence, which is a generalization of the preceding cases. The change of frame is expressed as: ˙ t• = t, x• = A(t)x + c(t), u• = Au + d(t), d(t) = c˙ + Ax

,

(3.147)

where the rotation matrix A is such that AT A = AAT = Id , |A| = 1 and c(t) is a vector. The Navier–Stokes equations are not form-invariant under this group of transformation in the general case. Form invariance is recovered in the asymptotic limit of two-dimensional ﬂows. The resulting changes of the subgrid and resolved velocity ﬁeld are: u• = Au + d,

u• = Au

,

(3.148)

yielding

with

• τik

=

Aim τmn Akn

L•ik

=

Aim τmn Lkn − Bik

,

(3.150)

• Cik

=

Aim τmn Ckn + Bik

,

(3.151)

• Rik

=

Aim Rmn Akn

,

Bij = ui dj + uj di

.

,

(3.149)

(3.152)

These properties are subjected to the condition G(x, ξ) = G(|x − ξ|). We can see that the properties of the subgrid tensors are the same as in the case of the Galilean invariance case. Table 3.1 summarizes the results dealing with the symmetry properties.

72

3. Application to Navier–Stokes Equations

Table 3.1. Invariance properties of spatial convolution ﬁlters and subgrid tensors. Symmetry

G(x, ξ)

L

C

L+C

R

Galilean translation Time shift Rotation Reﬂection Asymptotic material indiﬀerence

G(x − ξ) G(x, ξ) G(|x − ξ|) G(x − ξ) = G(ξ − x) G(|x − ξ|)

no yes yes yes no

no yes yes yes no

yes yes yes yes yes

yes yes yes yes yes

3.3.5 Realizability Conditions A second-rank tensor τ is realizable or semi-positive deﬁnite, if the following inequalities are veriﬁed (without summation on the repeated greek indices) [746, 260]: ταα ≥ |ταβ | ≤ det(τ )

≥

0, α = 1, 2, 3 , √ ταα τββ , α, β = 1, 2, 3 ,

(3.153) (3.154)

0 .

(3.155)

These conditions can be written in several equivalent forms [260]. Some of these are listed below. 1. The quadratic form Q = xi τij xj

(3.156)

is positive semideﬁnite. 2. The three principal invariants of τ are nonnegative: I1

=

I2

=

ταβ ≥ 0

,

2 (ταα τββ − ταβ )≥0 ,

(3.157) (3.158)

α=β

I3

= det(τ ) ≥ 0 .

(3.159)

The positiveness of the ﬁlter as deﬁned by relation (2.24) is a necessary and suﬃcient condition to ensure the realizability of the subgrid tensor τ . Below, we reproduce the demonstration given by Vreman et al. [746], which is limited to the case of a spatial ﬁlter G(x−ξ) without restricting the general applicability of the result. Let us ﬁrst assume that G ≥ 0. To prove that the tensor τ is realizable at any position x of the ﬂuid domain Ω, we deﬁne the sub-domain Ωx representing the support of the application ξ → G(x − ξ). Let Fx be the space of real

3.3 Decomposition of the Non-linear Term. Conventional Approach

73

functions deﬁned on Ωx . Since G is positive, for φ, ψ ∈ Fx , the application G(x − ξ)φ(ξ)ψ(ξ)dξ

(φ, ψ)x =

(3.160)

Ωx

deﬁnes an inner product on Fx . Using the deﬁnition of the ﬁltering, the subgrid tensor can be re-written in the form:

=

ui uj (x) − ui (x)uj (x) ui uj (x) − ui (x)uj (x) − uj (x)ui (x) + ui (x)uj (x) G(x − ξ)ui (ξ)uj (ξ)d3 ξ − ui (x) G(x − ξ)uj (ξ)d3 ξ Ωx Ωx 3 uj (x) G(x − ξ)ui (ξ)d ξ − ui (x)uj (x) G(x − ξ)d3 ξ Ωx Ωx G(x − ξ) (ui (ξ) − ui (x)) (uj (ξ) − uj (x))

=

(uxi , uxj )x

τij (x) = = = −

Ωx

,

(3.161)

where the diﬀerence uxi (ξ) = ui (ξ) − ui (x) is deﬁned on Ωx . The tensor τ thus appears as a Grammian 3 × 3 matrix of inner products, and is consequently always deﬁned as semi-positive. This shows that the stated condition is suﬃcient. Let us now assume that the condition G ≥ 0 is not veriﬁed for a piecewise continuous kernel. There then exists a pair (x, y) ∈ Ω × Ω, an ∈ IR+ , > 0, and a neighbourhood V = {ξ ∈ Ω, |ξ − y| < }, such that G(x − ξ) < 0, ∀ξ ∈ V . For a function u1 deﬁned on Ω such that u1 (ξ) = 0 if ξ ∈ V et u1 (ξ) = 0 everywhere else, then the component τ11 is negative: 2

2

τ11 (x) = u21 (x) − (u1 (x)) ≤

G(x − ξ) (u1 (ξ)) d3 ξ < 0 .

(3.162)

V

The tensor τ is thus not semi-positive deﬁnite, which concludes the demonstration. The properties of the three usual analytical ﬁlter presented in Sect. 2.1.5 are summarized in Table 3.2. Table 3.2. Positiveness property of convolution ﬁlters. Filter Box Gaussian Sharp cutoﬀ

Eq. (2.40) (2.42) (2.44)

Positiveness yes yes no

74

3. Application to Navier–Stokes Equations

3.4 Extension to the Inhom*ogeneous Case for the Conventional Approach The results of the previous sections were obtained by applying isotropic hom*ogeneous ﬁlters on an unbounded domain to Navier–Stokes equations written in Cartesian coordinates. What is presented here are the equations obtained by applying non-hom*ogeneous convolution ﬁlters on bounded domains to these equations. Using the commutator (2.13), the most general form of the ﬁltered Navier–Stokes equations is: ∂ui ∂t

+

∂ui ∂ ∂p ∂ ∂uj ∂τij (ui uj ) + −ν + =− ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj ∂ ∂ − G , (ui uj ) (ui ) − G , ∂t ∂xj ∂ ∂2 − G , (3.163) (p) + ν G , (ui ) , ∂xi ∂xk ∂xk ∂ui ∂ = − G , (ui ) . ∂xi ∂xi

(3.164)

All the terms appearing in the right-hand side of equations (3.163) and (3.164) are commutation errors. The ﬁrst term of the right-hand side of the ﬁltered momentum equation is the subgrid force, and is subject to modeling. The other terms are artefacts due to the ﬁlter, and escape subgrid modeling. An interesting remark drawn from equation (3.164) is that the ﬁltered ﬁeld is not divergence-free if some commutation errors arise. An analysis of the breakdown of continuity constraint in large-eddy simulation is provided by Langford and Moser [420], which shows that for many common largeeddy simulation representations, there is no exact continuity constraint on the ﬁltered velocity ﬁeld. But for mean-preserving representations a bulk continuity constraint holds. The governing equations obtained using second-order commuting ﬁlters (SOCF), as well as the techniques proposed by Ghosal and Moin [262] and Iovenio and Tordella [343] to reduce the commutation error and Vasilyev’s high-order commuting ﬁlters [728], are presented in the following. 3.4.1 Second-Order Commuting Filter Here we propose to generalize Leonard’s approach by applying SOCF ﬁlters. The decomposition of the non-linear term considered here as an example is the triple decomposition; but the double decomposition is also usable. For convenience in writing the ﬁltered equations, we introduce the operator Di

3.4 Extension to the Inhom*ogeneous Case for the Conventional Approach

75

such that: ∂ψ = Di ψ ∂xi

.

(3.165)

According to the results of Sect. 2.2.2, the operator Di is of the form: Di =

∂2 ∂ 2 4 − α(2) ∆ Γijk 2 + O(∆ ) , ∂xi ∂xi

(3.166)

in which the term Γ is deﬁned by the relation (2.147). By applying the ﬁlter and bringing out the subgrid tensor τij = ui uj − ui uj , we get for the momentum equation: ∂ui + Dj (ui uj ) = −Di p + νDj Dj ui − Dj τij ∂t

.

(3.167)

To measure the errors, we introduce the expansion as a function of ∆: p

=

u =

2

p(0) + ∆ p(1) + ... ,

(3.168)

2

u(0) + ∆ u(1) + ...

(3.169)

The terms corresponding to the odd powers of ∆ are identically zero because of the symmetry of the convolution kernel. By substituting this decomposition in (3.167), at the ﬁrst order we get: (0) (0) (0) (0) ∂uj ∂τij ∂p(0) ∂ (0) (0) ∂ui ∂ ∂ui =− + ui uj +ν + , − ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj (3.170) (0) (0) in which τij is the subgrid term calculated from the ﬁeld u . The associated continuity equation is: (0) ∂ui =0 . (3.171) ∂xi These equations are identical to those obtained in the hom*ogeneous case, 2 but relate to a variable containing an error in O(∆ ) with respect to the exact solution. 2 To reduce the error, the problem of the term in ∆ has to be resolved, i.e. solve the equations that use the variables u(1) and p(1) . Simple expansions lead to the system: (1) (1) (1) ∂uj ∂p(1) ∂ (1) (0) ∂ ∂ui ∂ui (0) (1) = − + ui uj + ui uj +ν + ∂t ∂xj ∂xi ∂xj ∂xj ∂xi (1)

−

∂τij (1) + α(2) fi ∂xj

,

(3.172)

76

3. Application to Navier–Stokes Equations (1)

in which the coupling term fi

deﬁned as:

(0) (0)

(1) fi

(0)

=

∂ 2 (ui uj ) ∂ 2 τij ∂ 2 p(0) Γjmn + Γimn + Γjmn ∂xm ∂xn ∂xm ∂xn ∂xm ∂xn

−

ν

(0)

(0)

∂ 3 ui ∂Γkmn ∂ 2 ui − 2Γkmn ∂xk ∂xm ∂xn ∂xk ∂xm ∂xn

,

(3.173)

(1)

∂ui =0 ∂xi

.

(3.174)

By solving this second problem, we can ensure the accuracy of the solution 4 up to the order O(∆ ) . Another procedure aiming at removing the commutation error was proposed by Iovenio and Tordella [343]. It relies on an approximation of the commutation error terms up to the fourth order in terms of ∆ which is based on the use of several ﬁltering levels. Reminding that the commutation error between the ﬁltering operator and the ﬁrst-order spatial derivative can be expressed as G ,

d d∆(x) ∂ ∆ φ (x) (φ) = − dx dx ∂∆

,

(3.175)

∆

where φ denotes the ﬁltered quantity obtained applying a ﬁlter with length ∆ on the variable φ, and introducing the central second order ﬁnite-diﬀerence approximation for the gradient of the ﬁltered quantity with respect to the ﬁlter width: ∂ ∆ 1 ∆+h ∆−h φ = φ −φ (3.176) + O(h2 ) 2h ∂∆ one obtains the following explicit, two ﬁltering level approximation for the commutation error ⎞ ⎛ 2∆ d d∆(x) 1 ⎝ ∆ ∆ G , φ −φ ⎠ (φ) − dx dx 2∆

,

(3.177)

This evaluation is independent of the exact ﬁlter shape, and makes it possible to cancel the leading error term in each part of the ﬁltered Navier-Stokes equations (3.163) - (3.164). It just involves the deﬁnition of an auxiliary ﬁltering level with a cutoﬀ length equal to 2∆.

3.5 Filtered Navier–Stokes Equations in General Coordinates

77

3.4.2 High-Order Commuting Filters The use of Vasilyev’s ﬁlters (see Sect. 2.2.2) instead of SOCF yields a set of governing ﬁltered equations formally equivalent to (3.167), but with: Di =

∂ n + O(∆ ) , ∂xi

(3.178)

where the order of accuracy n is ﬁxed by the number of vanishing moments of the ﬁlter kernel. The classical ﬁltered equations, without extra-terms accounting for the commutation errors, relate to a variable containing an error n scaling as O(∆ ) with respect to the exact ﬁltered solution.

3.5 Filtered Navier–Stokes Equations in General Coordinates 3.5.1 Basic Form of the Filtered Equations Jordan [358, 359], followed by other researchers [780, 18], proposed operating the ﬁltering in the transformed plane, following the alternate approach, as deﬁned at the beginning of this chapter. Assuming that the ﬁlter width and local grid spacing are equal, the resolved and ﬁltered ﬂowﬁelds are identical. It is recalled that the ﬁltering operation is applied along the curvilinear lines: k k ψ(ξ )φ(ξ ) = G(ξ k − ξ k )ψ(ξ k )φ(ξ k )dξ k , (3.179) where ψ is a metric coeﬃcient or a group of metric coeﬃcients, φ a physical variable (velocity component, pressure), G a hom*ogeneous ﬁlter kernel, and ξ k the coordinate along the considered line. It is easily deduced from the results presented in Sect. 2.2 that the commutation error vanishes in the present case, thanks to the hom*ogeneity of the kernel: ∂G/∂∆ = 0. But it is worth noting that the error term coming from the boundary of the domain will not cancel in the general case.9 Application of the ﬁlter to the Navier–Stokes equations written in generalized coordinates (3.3) and (3.4) leads to the following set of governing equations for large-eddy simulation: ∂ (J −1 ξik ui ) = 0 , ∂ξ k ∂ −1 ∂ ∂ ∂ (J ui ) + k (U k ui ) = − k (J −1 ξik p) + ν k ∂t ∂ξ ∂ξ ∂ξ 9

This point is extensively discussed in Chap. 10.

(3.180) ∂ J −1 Gkl l (ui ) . ∂ξ (3.181)

78

3. Application to Navier–Stokes Equations

3.5.2 Simpliﬁed Form of the Equations – Non-linear Terms Decomposition It is seen that many ﬁltered nonlinear terms appear in (3.180) and (3.181) which originate from the coordinate transformation. In order to uncouple geometrical quantities, such as metrics and Jacobian, from quantities related to the ﬂow, like velocity, and to obtain a simpler problem, further assumptions are required. The metrics being computed by a ﬁnite diﬀerence approximation in practice, they can be considered as ﬁltered quantities, yielding: U k = J −1 ξjk uj J −1 ξjk uj

.

(3.182)

All the terms appearing in the ﬁltered equations can be simpliﬁed similarly. As for the conventional approach, convective nonlinear terms need to be decomposed in order to allow us to use them for practical purpose. The resulting equations are: ∂ (U k ) = 0 , (3.183) ∂ξ k ∂ −1 ∂ (J ui ) + k (U k ui ) = ∂t ∂ξ

∂ (J −1 ξik p) (3.184) ∂ξ k ∂ ∂ ∂ +ν k J −1 Gkl l (ui ) − k (σik ) , ∂ξ ∂ξ ∂ξ

−

where the contravariant counterpart of the subgrid tensor is deﬁned as σik = J −1 ξjk ui uj − J −1 ξjk uj ui = U k ui − U k ui

.

(3.185)

Taking into account the fact that the metrics are assumed to be smooth ﬁltered quantities, the contravariant subgrid tensor can be tied to the subgrid tensor deﬁned in Cartesian coordinates: σik = J −1 ξjk ui uj − J −1 ξjk uj ui = J −1 ξjk (ui uj − ui uj ) = J −1 ξjk τij . (3.186)

3.6 Closure Problem 3.6.1 Statement of the Problem As was already said in the ﬁrst chapter, large-eddy simulation is a technique for reducing the number of degrees of freedom of the solution. This is done by separating the scales in the exact solution into two categories: resolved scales and subgrid scales. The selection is made by the ﬁltering technique described above.

3.6 Closure Problem

79

The complexity of the solution is reduced by retaining only the large scales in the numerical solution process, which entails reducing the number of degrees of freedom in the solution in space and time. The information concerning the small scales is consequently lost, and none of the terms that use these scales, i.e. the terms in u in the physical space and in (1 − G) in the spectral space, can be calculated directly. They are grouped into the subgrid tensor τ . This scale selection determines the level of resolution of the mathematical model. Nonetheless, in order for the dynamics of the resolved scales to remain correct, the subgrid terms have to be taken into consideration, and thus have to be modeled. The modeling consists of approximating the coupling terms on the basis of the information contained in the resolved scales alone. The modeling problem is twofold: 1. Since the subgrid scales are lacking in the simulation, their existence is unknown and cannot be decided locally in space and time. The problem thus arises of knowing if the exact solution contains, at each point in space and time, any smaller scales than the resolution established by the ﬁlter. In order to answer this question, additional information has to be introduced, in either of two ways. The ﬁrst is to use additional assumptions derived from acquired knowledge in ﬂuid mechanics to link the existence of subgrid modes to certain properties of the resolved scales. The second way is to enrich the simulation by introducing new unknowns directly related to the subgrid modes, such as their kinetic energy, for example. 2. Once the existence of the subgrid modes is determined, their interactions with the resolved scales have to be reﬂected. The quality of the simulation will depend on the ﬁdelity with which the subgrid model reﬂects these interactions. Various modeling strategies and models that have been developed are presented in the following. An important remark, somewhat tautological, is that the modeling process should take into account the ﬁltering operator [597, 171, 604]. This can be seen by remarking that ﬁltered and subgrid ﬁelds are deﬁned by the ﬁltering operator, and that a change in the ﬁlter will automatically lead to a new deﬁnition of these quantities and modify their properties. 3.6.2 Postulates So far, we have assumed nothing concerning the type of ﬂow at hand, aside from those assumptions that allowed us to demonstrate the momentum and continuity equations. Subgrid modeling usually assumes the following hypothesis Hypothesis 3.1 If subgrid scales exist, then the ﬂow is locally (in space and time) turbulent.

80

3. Application to Navier–Stokes Equations

Consequently, the subgrid models will be built on the known properties of turbulent ﬂows. It should be noted that theories exist that use other basic hypotheses. We may mention as an example the description of suspensions in the form of a ﬂuid with modiﬁed properties [423]: the solid particles are assumed to have predeﬁned characteristics (mass, form, spatial distribution, and so forth) and have a characteristic size very much less than the ﬁlter cutoﬀ length, i.e. at the scale at which we want to describe the ﬂow dynamics directly. Their actions are taken into account globally, which means a very high saving compared with an individual description of each particle. The diﬀerent descriptions obtained by hom*ogenization techniques also enter into this framework. 3.6.3 Functional and Structural Modeling Preliminary Remarks. Before discussing the various ways of modeling the subgrid terms, we have to set some constraints in order to orient the choices [627]. The subgrid modeling must be done in compliance with two constraints: 1. Physical constraint. The model must be consistent from the viewpoint of the phenomenon being modeled, i.e.: – Conserve the basic properties of the starting equation, such as Galilean invariance and asymptotic behaviors; – Be zero wherever the exact solution exhibits no small scales corresponding to the subgrid scales; – Induce an eﬀect of the same kind (dispersive or dissipative, for example) as the modeled terms; – Not destroy the dynamics of the solve scales, and thus especially not inhibit the ﬂow driving mechanisms. 2. Numerical constraint. A subgrid model can only be thought of as included in a numerical simulation method, and must consequently: – Be of acceptable algorithmic cost, and especially be local in time and space; – Not destabilize the numerical simulation; – Be insensitive to discretization, i.e. the physical eﬀects induced theoretically by the model must not be inhibited by the discretization. Modeling Strategies. The problem of subgrid modeling consists in taking the interaction with the ﬂuctuating ﬁeld u , represented by the term ∇ · τ , into account in the evolution equation of the ﬁltered ﬁeld u. Two modeling strategies exist [627]: – Structural modeling of the subgrid term, which consists in making the best approximation of the tensor τ by constructing it from an evaluation of u or a formal series expansion. The modeling assumption therefore consists in using a relation of the form u = H(u) or τ = H(u).

3.6 Closure Problem

81

– Functional modeling, which consists in modeling the action of the subgrid terms on the quantity u and not the tensor τ itself, i.e. introducing a dissipative or dispersive term, for example, that has a similar eﬀect but not necessarily the same structure (not the same proper axes, for example). The closure hypothesis can then be expressed in the form ∇ · τ = H(u). These two modeling approaches do not require the same foreknowledge of the dynamics of the equations treated and theoretically do not oﬀer the same potential in terms of the quality of results obtained. The structural approach requires no knowledge of the nature of the interscale interaction, but does require enough knowledge of the structure of the small scales of the solution in order to be able to determine one of the relations u = H(u) or τ = H(u) to be possible, one of the two following conditions has to be met: – The dynamics of the equation being computed leads to a universal form of the small scales (and therefore to their total structural independence from the resolved motion, as all that remains to be determined is their energy level). – The dynamics of the equation induces a suﬃciently strong and simple interscale correlation for the structure of the subgrid scales to be deduced from the information contained in the resolved ﬁeld. As concerns the modeling of the inter-scale interaction by just taking its eﬀect into account, this requires no foreknowledge of the subgrid scale structure, but does require knowing the nature of the interaction [184] [383]. Moreover, in order for such an approach to be practical, the eﬀect of the small scales on the large must be universal in character, and therefore independent of the large scales of the ﬂow.

4. Other Mathematical Models for the Large-Eddy Simulation Problem

The two preceding chapters are devoted to the convolution ﬁltering mathematical model for Large-Eddy simulation. Others approaches are now described, that can be gouped in two classes: – Mathematical models which rely on a statistical average (Sect. 4.1), recovering this way some interesting features of the Reynolds-Averaged Navier– Stokes model by precluding some drawbacks of the convolution ﬁlter approach in general domains. – Models derived from regularized versions of the Navier–Stokes equations (Sect. 4.2), that were proposed to alleviate some theoretical problems dealing with the existence, the uniqueness and the regularity of the general solution of the three-dimensional, unsteady, incompressible Navier–Stokes equations. These regularized models have smooth solutions, in the sense that their gradients remain controlled, and are re-interpreted within the Large-Eddy Simulation framework as good candidates to account for the removal of small scales.

4.1 Ensemble-Averaged Models 4.1.1 Yoshizawa’s Partial Statistical Average Model Yoshizawa [791] proposes to combine scale decomposition and statistical average to deﬁne an ad hoc mathematical model for Large-Eddy Simulation, referred to as the partial statistical average procedure. Writing the generalized Fourier decomposition of a dummy variable φ(x, t) as φk (t)ψk (x) (4.1) φ(x, t) = k=1,+∞

where φk (t) and ψk (x) are the coeﬃcients of the decomposition and the basis functions, respectively, the ﬁltered part of φ(x, t) is deﬁned as φ(x, t) = φk (t)ψk (x) + φk (t)ψk (x) (4.2) k=1,kc

k=kc ,+∞

84

4. Other Mathematical Models for the Large-Eddy Simulation Problem

where · denotes a statistical average operator and kc is related to the cutoﬀ index of the decomposition. The partial statistical averaging method appears then as the restriction of the usual ensemble average to scales which correspond to modes higher than kc . The cutoﬀ length ∆ is deduced from the characteristic lengthscale associated to ψkc . Since it relies on an ensemble average operator, this procedure does not suﬀer the drawbacks of the convolution ﬁltering approach and can be applied on curvilinear grids on bounded domains in a straightforward manner. But it requires the computation of the coeﬃcients φk (t), and therefore several realizations of the ﬂow are necessary, rendering its practical implementation very expensive from the computational viewpoint. In the simple case of hom*ogeneous ﬂows, the statistical average can be transformed into a spatial average invoking the ergodic theorem (see Appendix A for a brief discussion). 4.1.2 McComb’s Conditional Mode Elimination Procedure Another procedure was proposed independently by McComb and coworkers [465], which is referred to as conditional mode elimination. These authors based their approach on the local chaos hypothesis, which states that in a fully turbulent ﬂow the small scales are more uncertain than the large ones. This assumption is compatible with Kolmogorov’s local isotropy hypothesis (see Sect. A.5.1 for a discussion) dealing with the universality of the small scales and their increasing (as a function of the wavenumber) statistical decoupling from the large ones. More precisely, McComb’s interpretation says that uncertainty in the high-wavenumber modes originates in the ampliﬁcation of some degree of uncertainty in low-wavenumber modes by the non-linear chaotic nature of turbulence. This scheme is illustrated in Fig. 4.1.

Fig. 4.1. Schematic view of the local chaos hypothesis proposed by McComb in the Fourier space. Left: several instantaneous spectra are shown, in which increasing uncertainty is observed. Right: ideal view, where wave numbers smaller than kc are strictly deterministic, while higher wave number exhibit a fully chaotic behavior.

4.2 Regularized Navier–Stokes Models

85

The scale separation with a cutoﬀ length ∆ is achieved carying out a conditional statistical average of scales smaller than ∆, φ< , based on ﬁxed realizations of scales larger than ∆, φ> . The former are assumed to be uncertain and to exhibit and inﬁnite number of diﬀerent realizations for each realization of the large scales. The ﬁltered part of φ(x, t) is then expressed as φ(x, t) = φ> (x, t) + φ< |φ> (x, t)

(4.3)

where f |g denotes the conditional statistical average of f with respect to g. As Yoshizawa’s procedure, the conditional mode elimination does not suffer the drawbacks of the ﬁltering approach. These two ensemble-average based models for Large-Eddy Simulation are equivalent in many cases.

4.2 Regularized Navier–Stokes Models The mathematical models discussed in this section were not originally proposed to represent the properties of the Large-Eddy Simulation technique. They are surrogates to the Navier–Stokes equations, which have better properties from the pure mathematical point of view: while the question of the existence, uniqueness and regulatity of the solution of the three-dimensional, unsteady and incompressible Navier–Stokes equations is still an open problem, these new models allow for a complete mathematical analysis. One of the main obstacle faced in the mathematical analysis of the Navier–Stokes equations is that it cannot yet be proven that its solutions remain smooth for arbitrarily long times. More precisely, no a priori estimates has been found which guarantees that the enstrophy remains ﬁnite everywhere in the domain ﬁlled by the ﬂuid (but it can be proven that it is bounded in the mean). The physical interpretation associated with this picture is that some very intermittent vorticity bursts can occur, injecting kinetic energy at scales much smaller than the Kolmorogov scale, resulting in quasi-inﬁnite local values of the enstrophy. Such events correspond to ﬁnite-time singularities of the solution, and violate the axiom of continuum mechanics. A large number of mathematical results dealing with these problems have been published, which will not be further discussed here. The important point is that some systems, which are very close to the Navier–Stokes equations, have been proposed. A common feature is that they are well-posed from the mathematical point of view, meaning that their solutions are proved to be regular. As a consequence, they appear as regularized systems derived from the original Navier–Stokes equations, the regularization being associated to the disappearance of singularities. From a physical point of view, these new systems do not allow the occurance of local inﬁnite gradient thanks to an extra damping of the smallest scales. This smoothing property originates their interpretation as models for Large-Eddy Simulation.

86

4. Other Mathematical Models for the Large-Eddy Simulation Problem

4.2.1 Leray’s Model The ﬁrst model was proposed by Leray in 1934, who suggested to regularize the Navier–Stokes equations as follows: ∂ui ∂uk ui ∂p ∂ 2 ui + =− +ν ∂t ∂xk ∂xi ∂xk ∂xk

(4.4)

∂ui =0 ∂xi

(4.5)

where the regularized (i.e. ﬁltered) velocity ﬁeld is deﬁned as u(x, t) = φ u(x, t)

(4.6)

where the mollifying function (i.e. the ﬁlter kernel) φ is assumed to have a compact support, to be C ∞ and to have an integral equal to one. It can be proved under these assumptions that the solution of the regularized system (4.4)–(4.5) is unique and C ∞ . A main drawback is that it does not share all the frame-invariance properties of the Navier–Stokes equations. As quoted by Geurts and Holm [258], the system proposed by Leray can be rewritten in the usual Large-Eddy Simulation framework applying the ﬁlter a second times, leading to ∂τijLeray ∂uk ui ∂p ∂ 2 ui ∂ui + =− +ν − ∂t ∂xk ∂xi ∂xk ∂xk ∂xj

(4.7)

∂ui =0 ∂xi

(4.8)

where the subgrid tensor ansatz is deﬁned as τijLeray = ui uj − ui uj

(4.9)

An important diﬀerence with the usual deﬁnition of the subgrid tensor τij is that this new tensor is not symmetric. Leray’s regularized model makes it possible to carry out a complete mathematical analysis, but suﬀers the same problem when dealing with curvilinear grids on bounded domains as the original convolution ﬁlter model described in the preceding chapters. 4.2.2 Holm’s Navier–Stokes-α Model The second regularized model presented in this chapter is the Navier–Stokesα proposed by Holm (see [221, 282, 182, 221, 258, 281]). The regularization is achieved by imposing an energy penalty which damps the scales smaller than the threshold scale α (to be interpreted as ∆ within the usual large-

4.2 Regularized Navier–Stokes Models

87

eddy simulation framework)1, while still allowing for non linear sweeping of the small scales by the largest ones. The regularization appears as a nonlinearly dispersive modiﬁcation of the convection term in the Navier–Stokes equations. The system of the Navier–Stokes-α (also referred to as the Camassa-Holm equations) can be derived in two diﬀerent ways, which are now presented. Method 1: Kelvin-ﬁltered Navier–Stokes equations. The ﬁrst way to obtain the Navier–Stokes-α model is to introduce the Kelvin-ﬁltering. The Navier–Stokes equations satisfy Kelvin’s circulation theorem d dt

$

$ u · dx = Γ (u)

(ν∇2 ) · dx

(4.10)

Γ (u)

where Γ (u) is a closed ﬂuid loop that moves with velocity u. The original set of equations is regularized by modifying the ﬂuid loop along which the circulation is integrated: instead of using a ﬂuid loop moving at velocity u, an new ﬂuid loop moving at the regularized velocity u is considered. The exact deﬁnition of u is not necessary at this point and will be given later. The new circulation relationship is d dt

$

$ u · dx = Γ (u)

(ν∇2 ) · dx

(4.11)

Γ (u)

and corresponds to the following modiﬁed momentum equation: ∂u + u · ∇u + ∇T u · u = −∇p + ν∇2 u ∂t

(4.12)

∇·u = 0 .

(4.13)

with

This set of equations describes the Kelvin-ﬁltered Navier–Stokes equations. The Navier–Stokes-α equations are recovered specifying the regularized ﬁeld u as the result of the application of the Helmholtz ﬁlter (2.35) to the original ﬁeld u: (4.14) u = (1 − α2 ∇2 )u . It can be proved that the kinetic energy Eα deﬁned as 1 Eα = 2 1

u · udx =

1 2 α2 2 2 |u| + |∇ u| dx 2 2

,

(4.15)

It can be shown that in the case of three-dimensional fully developed turbulence, the solution of the Navier–Stokes-α exhibits the usual k−5/3 behavior for scales larger than α and a k−3 behavior for scales smaller than α.

88

4. Other Mathematical Models for the Large-Eddy Simulation Problem

is bounded, showing that the ﬁltered ﬁeld u remains regular. The equation (4.12) can be rewritten under the usual form in Large-Eddy Simulation as a momentum equation for the ﬁltered velocity ﬁeld u (formally identical to (4.4)). The corresponding deﬁnition of the subgrid tensor is τijNSα = (ui uj − ui uj ) − α2

∂uk ∂uk + uj ∇2 ui ∂xi ∂xj

.

(4.16)

Method 2: Modiﬁed Leray’s Model. Guermond, Oden and Prudhomme [282] observe that the Navier–Stokes-α system can be interpreted as a frame-invariant modiﬁcation of original Leray’s regularized model. Starting from the rotational form of the Navier–Stokes equations ∂u + (∇ × u) × u = −∇π + ν∇2 u, ∂t

1 π = p + u2 2

,

(4.17)

∇·u = 0 ,

(4.18)

and regularizing it using the technique proposed by Leray, one obtains ∂u + (∇ × u) × u = −∇π + ν∇2 u, ∂t

1 π = p + u2 2

,

(4.19)

∇·u = 0 .

(4.20)

Now using the relations (∇ × u) × u = u · ∇u − (∇T u)u,

∇(u · (∇T u)) = (∇T u)u + (∇T u)u , (4.21) the following form of the regularized system is recovered ∂u + u · ∇u + (∇T u) · u = −∇π + ν∇2 u, ∂t

π = π − u · u

∇·u = 0 .

,

(4.22) (4.23)

The Navier–Stokes-α model is obatined using the Helmholtz ﬁlter (4.14). The corresponding equation for u is ∂u + u · ∇u = ∇ · T ∂t

,

(4.24)

with T = −pId + 2ν(1 − α2 ∇2 )S + 2α2 S

◦

,

(4.25)

4.2 Regularized Navier–Stokes Models

89

◦

where S is related to the Jaumann derivative of the regularized strain rate tensor: ◦

S =

∂S + u · ∇S + SΩ − ΩS, ∂t

Ω=

1 ∇u − ∇T u 2

.

(4.26)

This system is formally similar to the constitutive law of a rate-dependent incompressible ﬂuid of second grade with slightly modiﬁed dissipation, and it is frame-invariant. It is equivalent to the Leray model in which the term which is responsible for the failure in the frame preservation, i.e. α2 (∇T u∇2 u), has been removed. Therefore, the Navier–Stokes-α equations appear as a pertur2 bation of order α2 (i.e. ∆ ) of the original Leray model. 4.2.3 Ladyzenskaja’s Model Another regularized version of the Navier–Stokes equations was proposed by Ladyzenskaja and Kaniel [417, 418, 377], who introduced a non-linear modiﬁcation of the stress tensor which is expected to be more relevant than the linear relationship for Newtonian ﬂuids when velocity gradients are large. The equation for the regularized ﬁeld u is ∂u + u · ∇u = −∇p + ν∇2 u − ε∇ · T (∇u) , ∂t

(4.27)

∇·u = 0 ,

(4.28)

where ε is a strictly arbitrary constant and the non-linear stress tensor T is deﬁned as T (∇u) = νT (∇u2 )∇u

,

(4.29)

where the non-linear viscosity νT (τ ) is a positive monotonically-increasing function of τ ≥ 0 that obeys the following law for large values of τ : cτ µ ≤ νT (τ ) ≤ c τ µ ,

0 < c < c ,

µ≥

1 4

.

(4.30)

The equivalent expression for the subgrid stress tensor is τijLadyzenskaja = εTij (∇u2 ) .

(4.31)

Since T depends only on the gradient of the resolved ﬁeld u, Ladyzenskaja’s model is closed and does not require further modeling work.

5. Functional Modeling (Isotropic Case)

It would be illusory to try to describe the structure of the scales of motion and the interactions in all imaginable conﬁgurations, in light of the very large disparity of physical phenomena encountered. So we have to restrict this description to cases which by nature include scales that are too small for today’s computer facilities to solve them entirely, and which are at the same time accessible to theoretical analysis. This description will therefore be centered on the inter-scale interactions in the case of fully developed isotropic hom*ogeneous turbulence1 , which is moreover the only case accessible by theoretical analysis and is consequently the only theoretical framework used today for developing subgrid models. Attempts to extend this theory to anisotropic and/or inhom*ogeneous cases are discussed in Chap. 6. The text will mainly be oriented toward the large-eddy simulation aspects. For a detailed description of the isotropic hom*ogeneous turbulence properties, which are reviewed in Appendix A, the reader may refer to the works of Lesieur [439] and Batchelor [45].

5.1 Phenomenology of Inter-Scale Interactions It is important to note here the framework of restrictions that apply to the results we will be presenting. These results concern three-dimensional ﬂows and thus do not cover the physics of two-dimensional ﬂows (in the sense of ﬂows with two directions2 , and not two-component3 ﬂows), which have a totally diﬀerent dynamics [403, 404, 405, 438, 481]. The modeling in the two-dimensional case leads to speciﬁc models [42, 624, 625] which will not 1 2

3

That is, whose statistical properties are invariant by translation, rotation, or symmetry. These are ﬂows such that there exists a direction x for which we have the property: ∂u ≡0 . ∂x These are ﬂows such that there exists a framework in which the velocity ﬁeld has an identically zero component.

92

5. Functional Modeling (Isotropic Case)

be presented. For details on two-dimensional turbulence, the reader may also refer to [439]. 5.1.1 Local Isotropy Assumption: Consequences In the case of fully developed turbulence, Kolmogorov’s statistical description of the small scales of the ﬂow, based on the assumption of local isotropy, has been the one most used for a very long time. By introducing the idea of local isotropy, Kolmogorov assumes that the small scales belonging to the inertial range of the energy spectrum of a fully developed inhom*ogeneous turbulent ﬂow are: – Statistically isotropic, and therefore entirely characterized by a characteristic velocity and time; – Without time memory, therefore in energy equilibrium with the large scales of the ﬂow by instantaneous re-adjustment. This isotropy of the small scales implies that they are statistically independent of the large energetic scales, which are characteristic of each ﬂow and are therefore anisotropic. Experimental work [512] has shown that this assumption is not valid in shear ﬂows for all the scales belonging to the inertial range, but only for those whose size is of the order of the Kolmogorov scale. Numerical experiments [32] show that turbulent stresses are nearly isotropic for wave numbers k such that kLε > 50, where Lε is the integral dissipation length4 . These experiments have also shown that the existence of an inertial region does not depend on the local isotropy hypothesis. The causes of this persistence of the anisotropy in the inertial range due to interactions existing between the various scales of the ﬂow will be mentioned in Chap. 6. Works based on direct numerical simulations have also shown that the assumption of equilibrium between the resolved and subgrid scales may be faulted, at least temporarily, when the ﬂow is subject to unsteady forcing [594, 570, 454, 504]. This is due to the fact that the relaxation times of these two scale ranges are diﬀerent. In the case of impulsively accelerated ﬂows (plane channel, boundary layer, axisymmetric straining) the subgrid scales react more quickly than the resolved ones, and then also relax more quickly toward an equilibrium solution. The existence of a zone of the spectrum, corresponding to the higher frequencies, where the scales of motion are statistically isotropic, justiﬁes the study of the inter-modal interactions in the ideal case of isotropic hom*ogeneous turbulence. Strictly speaking, the results can be used for determining subgrid models only if the cutoﬀ associated with the ﬁlter is in this region, 4

The integral dissipation length is deﬁned as Lε =

ui ui 3/2 ε

where ε is the energy dissipation rate.

,

5.1 Phenomenology of Inter-Scale Interactions

93

because the dynamics of the unresolved scales then corresponds well to that of the isotropic hom*ogeneous turbulence. It should be noted that this last condition implies that the representation of the dynamics, while incomplete, is nonetheless very ﬁne, which theoretically limits the gain in complexity that can be expected from large-eddy simulation technique. Another point is that the local isotropy hypothesis is formulated for fully developed turbulent ﬂows at very high Reynolds numbers. As it aﬃrms the universal character of the small scales’ behavior for these ﬂows, it ensures the possibility using the large-eddy simulation technique strictly, if the ﬁlter cutoﬀ frequency is set suﬃciently high. There is no theoretical justiﬁcation, though, for applying the results of this analysis to other ﬂows, such as transitional ﬂows. 5.1.2 Interactions Between Resolved and Subgrid Scales In order to study the interactions between the resolved and subgrid scales, we adopt an isotropic ﬁlter by a cutoﬀ wave number kc . The subgrid scales are those represented by the k modes such that k ≥ kc . In the case of fully developed isotropic hom*ogeneous turbulence, the statistical description of the inter-scale interactions is reduced to that of the kinetic energy transfers. Consequently, only the information associated with the amplitude of the ﬂuctuations is conserved, and none concerning the phase is taken into account. These transfers are analyzed using several tools: – Analytical theories of turbulence, also called two-point closures, which describe triadic interactions on the basis of certain assumptions. They will therefore express the non-linear term S(k|p, q), deﬁned by relation (3.50) completely. For a description of these theories, the reader may refer to Lesieur’s book [439], and we also mention Waleﬀe’s analysis [748, 749], certain conclusions of which are presented in the following. – Direct numerical simulations, which provide a complete description of the dynamics. – Renormalization Group Theory [622, 328, 812, 809, 464, 775, 804, 805, 813, 802, 803, 810], with several variants. Typology of the Triadic Interactions. It appears from the developments (k) mode interacts only with of Sect. 3.1.3 (also see Appendix A) that the u those modes whose wave vectors p and q form a closed triangle with k. The wave vector triads (k, p, q) thus deﬁned are classiﬁed in several groups [805] which are represented in Fig. 5.1: – Local triads for which %p q& 1 ≤ max , ≤ a, a k k

a = O(1)

,

94

5. Functional Modeling (Isotropic Case)

Fig. 5.1. Diﬀerent types of triads.

which correspond to interactions among wave vectors of neighboring modules, and therefore to interactions among scales of slightly diﬀerent sizes; – Non-local triads, which are all those interactions that do not fall within the ﬁrst category, i.e. interactions among scales of widely diﬀering sizes. Here, we adopt the terminology proposed in [74], which distinguishes between two sub-classes of non-local triads, one being distant triads of interactions in which k p ∼ q or k ∼ q p. It should be noted that these terms are not unequivocal, as certain authors [439, 442] refer to these “distant” triads as being just “non-local”. By extension, a phenomenon will be called local if it involves wave vectors k and p such that 1/a ≤ p/k ≤ a, and otherwise non-local or distant. Canonical Analysis. This section presents the results from analysis of the simplest theoretical case, which we call here canonical analysis. This consists of assuming the following two hypotheses: 1. Hypothesis concerning the ﬂow. The energy spectrum E(k) of the exact solution is a Kolmogorov spectrum, i.e. E(k) = K0 ε2/3 k −5/3 ,

k ∈ [0, ∞] ,

(5.1)

where K0 is the Komogorov constant and ε the kinetic energy dissipation rate. We point out that this spectrum is not integrable since its corresponds to an inﬁnite kinetic energy. 2. Hypothesis concerning the ﬁlter. The ﬁlter is a sharp cutoﬀ type. The subgrid tensor is thus reduced to the subgrid Reynolds tensor. e In analyzing the energy transfers Tsgs (k) (see relation (3.51)) between the modes to either side of a cutoﬀ wave number kc located in the inertial range of the spectrum, Kraichnan [405] uses the Test Field Model (TFM) to bring out the existence of two spectral bands (see Fig. 5.2) for which the interactions with the small scales (p and/or q ≥ kc ) are of diﬀerent kinds.

5.1 Phenomenology of Inter-Scale Interactions

95

Fig. 5.2. Interaction regions between resolved and subgrid scales.

1. In the ﬁrst region (1 in Fig. 5.2), which corresponds to the modes such that k kc , the dominant dynamic mechanism is a random displacement of the momentum associated with k by disturbances associated with p and q. This phenomenon, analogous to the eﬀects of the molecular viscosity, entails a kinetic energy decay associated with k and, since the total kinetic energy is conserved, a resulting increase of it associated with p and q. So here it is a matter of a non-local transfer of energy associated with non-local triadic interactions. These transfers, which induce a damping of the ﬂuctuations, are associated with what Waleﬀe [748, 749] classiﬁes as type F triads (represented in Fig. 5.3).

Fig. 5.3. Non-local triad (k, p, q) of the F type according to Waleﬀe’s classiﬁcation, and the associated non-local energy transfers. The kinetic energy of the mode corresponding to the smallest wave vector k is distributed to the other two modes p and q, creating a forward energy cascade in the region where k kc .

96

5. Functional Modeling (Isotropic Case)

Subsequent analyses using the Direct Interaction Approximation (DIA) and the Eddy Damped Quasi-Normal Markovian (EDQNM) models [120, 136, 442, 443, 647] or Waleﬀe’s analyses [748, 749] have reﬁned this representation by showing the existence of two competitive mechanisms in the region where k kc . The ﬁrst region is where the energy of the large scales is drained by the small ones, as already shown by Kraichnan. The second mechanism, of much lesser intensity, is a return of energy from the small scales p and q to the large scale k. This mechanism also corresponds to a non-local energy transfer associated with non-local triadic interactions that Waleﬀe classiﬁes as type R (see Fig. 5.4). It represents a backward stochastic energy cascade associated with an energy spectrum in k 4 for very small wave numbers. This phenomenon has been predicted analytically [442] and veriﬁed by numerical experimentation [441, 120]. The analytical studies and numerical simulations show that this backward cascade process is dominant for very small wave numbers. On the average, these modes receive more energy from the subgrid modes than they give to them. 2. In the second region (region 2 in Fig. 5.2), which corresponds to the k modes such that (kc −k) kc , the mechanisms already present in region 1 persist. The energy transfer to the small scales is at the origin of the forward kinetic energy cascade. Moreover, another mechanism appears involving triads such that p or q kc , which is that the interactions between the scales of this region and the subgrid scales are much more intense than in the ﬁrst. Let us take q kc . This mechanism is a coherent straining of the small scales k and p by the shear associated with q, resulting in a wave number diﬀusion process between k and p through the cutoﬀ, with one of the structures being stretched (vortex stretching phenomenon) and the other unstretched. What we are observing here is a local energy transfer between k and p associated with non-local triadic interactions due to the type R triads (see Fig. 5.4). Waleﬀe reﬁnes the analysis of this phenomenon: a very large part of the energy is transferred locally from the intermediate wave number located just ahead of the cutoﬀ toward the larger wave number just after it, and the remaining fraction of energy is transferred to the smaller wave number. These ﬁndings have been corroborated by numerical data [120, 185, 189] and other theoretical analyses [136, 443]. e (k) (see relation (3.51)) between mode k and The energy transfers Tsgs the subgrid modes can be represented in a form analogous to molecular dissipation. To do this, by following Heisenberg (see [688] for a description of Heisenberg’s theory), we deﬁne an eﬀective viscosity νe (k|kc ), which represents the energy transfers between the k mode and the modes located beyond the kc cutoﬀ such that: e Tsgs (k) = −2νe(k|kc )k 2 E(k) .

(5.2)

5.1 Phenomenology of Inter-Scale Interactions

97

Fig. 5.4. Non-local (k, p, q) triad of the R type according to Waleﬀe’s classiﬁcation, and the associated energy transfers in the case q kc . The kinetic energy of the mode corresponding to the intermediate wave vector k is distributed locally to the largest wave vector p and non-locally to the smallest wave vector, q. The former transfer originates the intensiﬁcation of the coupling in the (kc − k) kc spectral band, while the latter originates the backward kinetic energy cascade.

It should be pointed out that this viscosity is real, i.e. νe (k|kc ) ∈ IR, and that if any information related to the phase were included, it would lead the deﬁnition of a complex term having an a priori non-zero imaginary part, which may seem to be more natural for representing a dispersive type of coupling. Such a term is obtained not by starting with the kinetic energy equation, but with the momentum equation5 . The two energy cascades, forward and backward, can be introduced separately by introducing distinct eﬀective viscosities, constructed in such a way as to ensure energy transfers equivalent to those of these cascades. We get the following two forms: νe+ (k|kc , t) = −

+ (k|kc , t) Tsgs 2 2k E(k, t)

,

(5.3)

νe− (k|kc , t) = −

− Tsgs (k|kc , t) 2 2k E(k, t)

,

(5.4)

+ − in which Tsgs (k|kc , t) (resp. Tsgs (k|kc , t)) is the energy transfer term from the k mode to the subgrid modes (resp. from the subgrid modes to the k mode). This leads to the decomposition: e (k) = Tsgs

=

+ − Tsgs (k|kc , t) + Tsgs (k|kc , t) 2 + −2k E(k, t) νe (k|kc , t) + νe− (k|kc , t)

(5.5) .

(5.6)

These two viscosities depend explicitly on the wave number k and the cutoﬀ wave vector kc , as well as the shape of the spectrum. The result of 5

This possibility is only mentioned here, because no works have been published on it to date.

98

5. Functional Modeling (Isotropic Case)

these dependencies on the ﬂow is that the viscosities are not, because they characterize the ﬂow and not the ﬂuid. They are of opposite sign: νe+ (k|kc , t) ensures a loss of energy of the resolved scales and is consequently positive, like the molecular viscosity, whereas νe− (k|kc , t), which represents an energy gain in the resolved scales, is negative. The conclusions of the theoretical analyses [405, 443] and numerical studies [120] are in agreement on the form of these two viscosities. Their behavior is presented in Fig. 5.5 in the canonical case.

Fig. 5.5. Representation of eﬀective viscosities in the canonical case. Short dashes: νe+ (k|kc , t); long dashes: −νe− (k|kc , t); solid: νe+ (k|kc , t) + νe− (k|kc , t).

We may note that these two viscosities become very high for wave numbers close to the cutoﬀ. These two eﬀective viscosities diverge as (kc − k)−2/3 as k tends toward kc . However, their sum νe (k|kc , t) remains ﬁnite and Leslie et al. [443] proposes the estimation: νe (kc |kc , t) = 5.24νe+ (0|kc , t) .

(5.7)

The interactions with the subgrid scales is therefore especially important in the dynamics of the smallest resolved scales. More precisely, Kraichnan’s theoretical analysis leads to the conclusion that about 75% of the energy transfers of a k mode occur with the modes located in the [k/2, 2k] spectral

5.1 Phenomenology of Inter-Scale Interactions

99

band6 . No transfers outside this spectral band have been observed in direct numerical simulations at low Reynolds numbers [185, 804, 805]. The diﬀerence with the theoretical analysis stems from the fact that this analysis is performed in the limit of the inﬁnite Reynolds numbers. In the limit of the very small wave numbers, we have the asymptotic behaviors: 2k 2 E(k, t)νe+ (k|kc , t) 2k 2 E(k, t)νe− (k|kc , t)

∝ k 1/3 , ∝ k4 .

(5.8) (5.9)

The eﬀective viscosity associated with the energy cascade takes the constant asymptotic value: νe+ (0|kc , t) = 0.292ε1/3kc−4/3

.

(5.10)

We put the emphasis on the fact that the eﬀective viscosity discussed here is deﬁned considering the kinetic enery transfer between unresolved and resolved modes. As quoted by McComb et al. [465], it is possible to deﬁne diﬀerent eﬀective viscosities by considering other balance equations, such as the enstrophy transfer. An important consequence is that kinetic-energybased eﬀective viscosities are eﬃcient surrogates of true transfer terms in the kinetic energy equation, but may be very bad representations of subgrid eﬀects for other physical mechanisms. Dependency According to the Filter. Leslie and Quarini [443] extended the above analysis to the case of the Gaussian ﬁlter. The spectrum considered is always of the Kolmogorov type. The Leonard term is now non-zero. The results of the analysis show very pronounced diﬀerences from the canonical analysis. Two regions of the spectrum are still distinguishable, though, with regard to the variation of the eﬀective viscosities νe+ and νe− , which are shown in Fig. 5.6: – In the ﬁrst region, where k kc , the transfer terms still observe a constant asymptotic behavior, independent of the wave number considered, as in the canonical case. The backward cascade term is negligible compared with the forward cascade term. – In the second region, on the other hand, when approaching cutoﬀ, the two transfer terms do not have divergent behavior, contrary to what is observed in the canonical case. The forward cascade term decreases monotonically and cancels out after the cutoﬀ for wave numbers more than a decade beyond it. The backward cascade term increases up to cutoﬀ and exhibits a decreasing behavior analogous to that of the forward cascade term. The maximum intensity of the backward cascade is encountered for modes just after the cutoﬀ. 6

The same local character of kinetic energy transfer is observed in nonhom*ogeneous ﬂow, such has the plane channel ﬂow [186].

100

5. Functional Modeling (Isotropic Case)

Fig. 5.6. Eﬀective viscosities in the application of a Gaussian ﬁlter to a Kolmogorov spectrum. Long dots νe+ (k|kc , t); dots: −νe− (k|kc , t); solid: νe+ (k|kc , t) + νe− (k|kc , t).

Fig. 5.7. Eﬀective viscosity corresponding to the Leonard term in the case of the application of a Gaussian ﬁlter to a Kolmogorov spectrum.

5.1 Phenomenology of Inter-Scale Interactions

101

In contrast to the sharp cutoﬀ ﬁlter used for the canonical analysis, the Gaussian ﬁlter makes it possible to deﬁne Leonard terms and non-identically zero cross terms. The eﬀective viscosity associated with these terms is shown in Fig. 5.7, where it can be seen that it is negligible for all the modes more than a decade away from the cutoﬀ. In the same way as for the backward cascade term, the maximum amplitude is observed for modes located just after the cutoﬀ. This term remains smaller than the forward and backward cascade terms for all the wave numbers. Dependency According to Spectrum Shape. The results of the canonical analysis are also dependent on the shape of the spectrum considered. The analysis is repeated for the case of the application of the sharp cutoﬀ ﬁlter to a production spectrum of the form: E(k) = As (k/kp )K0 ε2/3 k −5/3

,

(5.11)

with As (x) =

xs+5/3 1 + xs+5/3

,

(5.12)

and where kp is the wave number that corresponds to the maximum of the energy spectrum [443]. The shape of the spectrum thus deﬁned is illustrated in Fig. 5.8 for several values of the s parameter. The variation of the total eﬀective viscosity νe for diﬀerent values of the quotient kc /kp is diagrammed in Fig. 5.9. For low values of this quotient, i.e. when the cutoﬀ is located at the beginning of the inertial range, we observe

Fig. 5.8. Production spectrum for diﬀerent values of the shape parameter s.

102

5. Functional Modeling (Isotropic Case)

Fig. 5.9. Total eﬀective viscosity νe (k|kc ) in the case of the application of a sharp cutoﬀ ﬁlter to a production spectrum for diﬀerent values of the quotient kc /kp , normalized by its value at the origin.

that the viscosity may decrease at the approach to the cutoﬀ, while it is strictly increasing in the canonical case. This diﬀerence is due to the fact that the asymptotic reasoning that was applicable in the canonical case is no longer valid, because the non-localness of the triadic interactions involved relay the diﬀerence in spectrum shape to the whole of it. For higher values of this quotient, i.e. when the cutoﬀ is located suﬃciently far into the inertial range (for large values of the ratio kc /kp ), a behavior that is qualitatively similar to that observed in the canonical case is once again found7 . For kc = kp , no increase is observed in the energy transfers as k tends toward kc . The behavior approximates that observed for the canonical analysis as the ratio kp /kc decreases. 5.1.3 A View in Physical Space Analyses described in the preceding section were all performed in the Fourier space, and do not give any information about the location of the subgrid transfer in the physical space and its correlation with the resolved scale features8 . Complementary informations on the subgrid transfer in the physical space have been found by several authors using direct numerical simulation. 7 8

In practice, kc /kp =8 seems appropriate. This is a prerequisite for designing a functional subgrid model in physical space.

5.1 Phenomenology of Inter-Scale Interactions

103

Kerr et al. [383] propose to use the rotational form of the non-linear term of the momentum equation: N (x) = u(x) × ω(x) − ∇ph (x)

,

(5.13)

where ω = ∇ × u and ph the pressure term. By splitting the velocity and vorticity ﬁeld into a resolved and a subgrid contribution, we get: u × ω − u × ω = u × ω + u × ω + u × ω

I

II

III

.

(5.14)

IV

The four terms represent diﬀerent coupling mechanisms between the resolved motion and the subgrid scales: – – – –

I - exact subgrid term, II - interaction between resolved velocity and subgrid vorticity, III - interaction between subgrid velocity and resolved vorticity, IV - interaction between subgrid velocity and subgrid vorticity.

The corresponding complete non-linear terms N I , ..., N IV are built by adding the speciﬁc pressure term. The associated subgrid kinetic energy transfer terms are computed as εl = u · N l . The authors made three signiﬁcant observations for isotropic turbulence: – Subgrid kinetic energy transfer is strongly correlated with the boundaries of regions of large vorticity production (stretching), i.e. regions where ω i S ij ω j is large; – Term II, u × ω , has a correlation with subgrid non-linear term I up to 0.9. This term dominates the backward energy cascade; – Up to 90% of the subgrid kinetic energy transfer comes from term III, i.e. from the interaction of subgrid velocity with resolved vorticity. This term mostly contributes to the forward energy cascade. Additional results of Borue and Orszag [72] show that the subgrid transfer takes place in regions where the vorticity stretching term is positive or in 3 regions with negative skewness of the resolved strain rate tensor, Tr(S ). These authors also found that there is only a very poor local correlation between the subgrid transfer τij S ij and the local strain S ij S ij , where S ij is the resolved strain rate tensor. Horiuti [325, 327] decomposed the subgrid tensor into several contributions,9 and used direct numerical simulation data of isotropic turbulence to analyze their contributions. A ﬁrst remark is that the eigenvectors of the total subgrid tensor have a preferred orientation of 42◦ relative to those of S. Eigenvectors of (S ik S kj − Ω ik Ω kj ) are highly aligned with those of S, while 9

This decomposition is discussed in the section devoted to nonlinear models, p. 223.

104

5. Functional Modeling (Isotropic Case)

those of (S ik Ω kj − S ik Ω kj ) exhibit a 42◦ angle, from which stems the global observed diﬀerence. The ﬁrst term is associated with the forward energy cascade. The second one makes no contribution to the total production of subgrid kinetic energy, but is relevant to the vortex stretching and the backward energy cascade process.10 Similar results were obtained by Meneveau and coworkers [704, 703]. The role of coherent structures in interscale transfer is of major importance in shear ﬂows. Da Silva and M´etais [158] carried out an exhaustive study in the plane jet case: the most intense forward cascade events occur near these coherent structures and not randomly in space. The local equilibrium assumption is observed to hold globally but not locally as most viscous dissipation of subgrid kinetic energy takes place within coherent structure cores, while forward and backward cascade occur at diﬀerent locations. 5.1.4 Summary The diﬀerent analyses performed in the framework of fully developed isotropic turbulence show that: 1. Interactions between the small and large scales is reﬂected by two main mechanisms: – A drainage of energy from the resolved scales by the subgrid scales (forward energy cascade phenomenon); – A weak feedback of energy, proportional to k 4 to the resolved scales (backward energy cascade phenomenon). 2. The interactions between the subgrid scales and the smallest of the resolved scales depend on the ﬁlter used and on the shape of the spectrum. In certain cases, the coupling with the subgrid scales is strengthened for wave numbers close to the cutoﬀ and the energy toward the subgrid modes is intensiﬁed. 3. These cascade mechanisms are associated to speciﬁc features of the velocity and vorticity ﬁeld in physical space.

5.2 Basic Functional Modeling Hypothesis All the subgrid models entering into this category make more or less implicit use of the following hypothesis: Hypothesis 5.1 The action of the subgrid scales on the resolved scales is essentially an energetic action, so that the balance of the energy transfers alone between the two scale ranges is suﬃcient to describe the action of the subgrid scales. 10

This is an indication that the backward energy cascade is not associated with negative subgrid viscosity from the theoretical point of view.

5.3 Modeling of the Forward Energy Cascade Process

105

Using this hypothesis as a basis for modeling, then, we neglect a part of the information contained in the small scales, such as the structural information related to the anisotropy. As was seen above, the energy transfers between subgrid scales and resolved scales mainly exhibit two mechanisms: a forward energy transfer toward the subgrid scales and a backward transfer to the resolved scales which, it seems, is much weaker in intensity. All the approaches existing today for numerical simulation at high Reynolds numbers consider the energy lost by the resolved scales, while only a few rare attempts have been made to consider the backward energy cascade. Once hypothesis 5.1 is assumed, the modeling consists in modifying the diﬀerent evolution equations of the system in such a way as to integrate the desired dissipation or energy production eﬀects into them. To do this, two diﬀerent approaches can be found in today’s works: – Explicit modeling of the desired eﬀects, i.e. including them by adding additional terms to the equations: the actual subgrid models; – Implicit inclusion by the numerical scheme used, by arranging it so the truncation error induces the desired eﬀects. Let us note that while the explicit approach is what would have to be called the classical modeling approach, the implicit one appears generally only as an a posteriori interpretation of dissipative properties for certain numerical methods used.

5.3 Modeling of the Forward Energy Cascade Process This section describes the main functional models of the energy cascade mechanism. Those derived in the Fourier space, conceived for simulations based on spectral numerical methods, and models derived in the physical space, suited to the other numerical methods, are presented separately. 5.3.1 Spectral Models The models belonging to this category are all eﬀective viscosity models drawing upon the analyses of Kraichnan for the canonical case presented above. The following models are described: 1. The Chollet–Lesieur model (p. 106) which, based on the results of the canonical analysis (inertial range of the spectrum with a slope of -5/3, sharp cutoﬀ ﬁlter, no eﬀects associated with a production type spectrum) yields an analytical expression for the eﬀective viscosity as a function of the wave number considered and the cutoﬀ wave number. It will reﬂect the local eﬀects at the cutoﬀ, i.e. the increase in the energy transfer toward the subgrid scales. This model explicitly brings out a dependency

106

2.

3.

4.

5.

5. Functional Modeling (Isotropic Case)

of the eﬀective viscosity as a function of the kinetic energy at the cutoﬀ. This guarantees that, when all the modes of the exact solution are resolved, the subgrid model automatically cancels out. The fact that this information is local in frequency allows the model to consider (at least partially) the spectral disequilibrium phenomena that occur at the level of the resolved scales11 , though without relaxing the hypotheses underlying the canonical analysis. Only the amplitude of the transfers is variable, and not their pre-supposed shape. The eﬀective viscosity model (p. 107), which is a simpliﬁcation of the previous one and is based on the same assumptions. The eﬀective viscosity is then independent of the wave number and is calculated so as to ensure the same average value as the Chollet–Lesieur model. It is simpler to compute, but does not reﬂect the local eﬀects at the cutoﬀ. The dynamic spectral model (p. 107), which is an extension of the Chollet–Lesieur model for spectra having a slope diﬀerent from that of the canonical case (i.e. - 5/3). Richer information is considered here: while the Chollet–Lesieur model is based only on the energy level at the cutoﬀ, the dynamic spectral model also incorporates the spectrum slope at the cutoﬀ. With this improvement, we can cancel the subgrid model in certain cases for which the kinetic energy at cutoﬀ is non-zero but where the kinetic energy transfer to the subgrid modes is zero12 . This model also reﬂects the local eﬀects at the cutoﬀ. The other basic assumptions underlying the Chollet–Lesieur model are maintained. The Lesieur–Rogallo model (p. 108), which computes the intensity of the transfers by a dynamic procedure. This is an extension of the Chollet– Lesieur model for ﬂows in spectral disequilibrium, as modiﬁcations in the nature of the transfers to the subgrid scales can be considered. The dynamic procedure consists in including in the model information relative to the energy transfers at play with the highest resolved frequencies. The assumptions concerning the ﬁlter are not relaxed, though. Models based on the analytical theories of turbulence (p. 108), which compute the eﬀective viscosity without assuming anything about the spectrum shape of the resolved scales, are thus very general. On the other hand, the spectrum shape of the subgrid scales is assumed to be that of a canonical inertial range. These models, which are capable of including very complex physical phenomena, require very much more implementation and computation eﬀort than the previous models. The assumptions concerning the ﬁlter are the same as for the previous models.

Chollet–Lesieur Model. Subsequent to Kraichnan’s investigations, Chollet and Lesieur [136] proposed an eﬀective viscosity model using the results of the EDQNM closure on the canonical case. The full subgrid transfer term 11 12

This is by their action on the transfers between resolved scales and the variations induced on the energy level at cutoﬀ. As is the case, for example, for two-dimensional ﬂows.

5.3 Modeling of the Forward Energy Cascade Process

107

including the backward cascade is written: e (k|kc ) = −2k 2 E(k)νe (k|kc ) , Tsgs

(5.15)

in which the eﬀective viscosity νe (k|kc ) is deﬁned as the product νe (k|kc ) = νe+ (k|kc )νe∞

.

(5.16)

The constant term νe∞ , independent of k, corresponds the asymptotic value of the eﬀective viscosity for wave numbers that are small compared with the cutoﬀ wave number kc . This value is evaluated using the cutoﬀ energy E(kc ): ' −3/2

νe∞ = 0.441K0

E(kc ) kc

.

(5.17)

The function νe (k|kc ) reﬂects the variations of the eﬀective viscosity in the proximity of the cutoﬀ. The authors propose the following form, which is obtained by approximating the exact solution with a law of exponential form: (5.18) νe+ (k|kc ) = 1 + 34.59 exp(−3.03kc /k) . This form makes it possible to obtain an eﬀective viscosity that is nearly independent of k for wave numbers that are small compared with kc , with a ﬁnite increase near the cutoﬀ. There is a limited inclusion of the backward cascade with this model: the eﬀective viscosity remains strictly positive for all wave numbers, while the backward cascade is dominant for very small wave numbers, which would correspond to negative values of the eﬀective viscosity. Constant Eﬀective Viscosity Model. A simpliﬁed form of the eﬀective viscosity of (5.16) can be derived independently of the wave number k [440]. By averaging the eﬀective viscosity along k and assuming that the subgrid modes are in a state of energy balance, we get: ' 2 −3/2 E(kc ) νe (k|kc ) = νe = K0 . (5.19) 3 kc Dynamic Spectral Model. The asymptotic value of the eﬀective viscosity (5.17) has been extended to the case of spectra of slope −m by M´etais and Lesieur [514] using the EDQNM closure. For a spectrum proportional to k −m , m ≤ 3, we get: ' √ E(kc ) 5 − m −3/2 νe∞ (m) = 0.31 3 − mK0 . (5.20) m+1 kc For m > 3, the energy transfer cancels out, inducing zero eﬀective viscosity. Here, we ﬁnd a behavior similar to that of two-dimensional turbulence. Extension of this idea in physical space has been derived by Lamballais and his coworkers [422, 675].

108

5. Functional Modeling (Isotropic Case)

Lesieur–Rogallo Model. By introducing a new ﬁltering level corresponding to the wave number km < kc , Lesieur and Rogallo [441] propose a dynamic algorithm for adapting the Chollet–Lesieur model. The contribution to the transfer T (k), k < kc , corresponding to the (k, p, q) triads such that p and/or q are in the interval [km , kc ], can be computed explicitly by Fourier transforms. This contribution is denoted Tsub (k|km , kc ) and is associated with the eﬀective viscosity: νe (k|km , kc ) = −

Tsub (k|km , kc ) 2k 2 E(k)

.

(5.21)

The eﬀective viscosity corresponding to the interactions with wave numbers located beyond km is the sum: νe (k|km ) = νe (k|km , kc ) + νe (k|kc ) .

(5.22)

This relation corresponds exactly to Germano’s identity and was previously derived by the authors. The two terms νe (k|km ) and νe (k|kc ) are then modeled by the Chollet–Lesieur model. We adopt the hypothesis that when k < km , then k kc , which leads to νe+ (k|kc ) = νe+ (0). Relation (5.22) then leads to the equation: ' 4/3 km km + νe (k|km ) = νe (k|km , kc ) + νe+ (0) . (5.23) E(km ) kc The factor νe+ (0) is evaluated by considering that we have the relations νe+ (k|km ) ≈ νe+ (0),

νe (k|km , kc ) ≈ νe (0|km , kc ) ,

(5.24)

for k km , which leads to: ' νe+ (0)

= νe (0|km , kc )

( 4/3 )−1 km km 1− E(km ) kc

.

(5.25)

Models Based on Analytical Theories of Turbulence. The eﬀective viscosity models presented above are all based on an approximation of the eﬀective viscosity proﬁle obtained in the canonical case, and are therefore intrinsically linked to the underlying hypotheses, especially those concerning the shape of the energy spectrum. One way of relaxing this constraint is to compute the eﬀective viscosity directly from the computed spectrum using analytical theories of turbulence. This approach has been used by Aupoix [24], Chollet [132, 133], and Bertoglio [56, 57, 58]. More recently, following the recommendations of Leslie and Quarini, which are to model the forward and backward cascade mechanisms separately,

5.3 Modeling of the Forward Energy Cascade Process

109

Chasnov [120] in 1991 proposed an eﬀective viscosity model considering only the energy draining eﬀects, with the backward cascade being modeled separately (see Sect. 5.4). Starting with an EDQNM analysis, Chasnov proposes computing the eﬀective viscosity νe (k|kc ) as: p2 q2 3 3 (xy + z )E(q) + (xz + y )E(p) , dp dqΘkpq q p kc p−k (5.26) in which x, y and z are geometric factors associated with the (k, p, q) triads and Θkpq a relaxation time. These terms are explained in Appendix B. To compute this integral, the shape of the energy spectrum beyond the cutoﬀ kc must be known. As it is not known a priori, it must be speciﬁed elsewhere. In practice, Chasnov uses a Kolmogorov spectrum extending from the cutoﬀ to inﬁnity. To simplify the computations, the relation (5.26) is not used outside the interval [kc ≤ p ≤ 3kc ]. For wave numbers p > 3kc , the following simpliﬁed asymptotic form already proposed by Kraichnan is used: 1 νe (k|kc ) = 2 2k

∞

p

1 νe (k|kc ) = 15

∞

kc

∂E(p) dpΘkpq 5E(p) + p ∂p

.

(5.27)

5.3.2 Physical Space Models Subgrid Viscosity Concept. The forward energy cascade mechanism to the subgrid scales is modeled explicitly using the following hypothesis: Hypothesis 5.2 The energy transfer mechanism from the resolved to the subgrid scales is analogous to the molecular mechanisms represented by the diﬀusion term, in which the viscosity ν appears. This hypothesis is equivalent to assuming that the behavior of the subgrid scales is analogous to the Brownian motion superimposed on the motion of the resolved scales. In gaskinetics theory, molecular agitation draws energy from the ﬂow by way of molecular viscosity. So the energy cascade mechanism will be modeled by a term having a mathematical structure similar to that of molecular diﬀusion, but in which the molecular viscosity will be replaced by a subgrid viscosity denoted νsgs . As Boussinesq proposed, this choice of mathematical form of the subgrid model is written: −∇ · τ d = ∇ · νsgs (∇u + ∇T u)

,

(5.28)

in which τ d is the deviator of τ , i.e.: 1 τijd ≡ τij − τkk δij 3

.

(5.29)

110

5. Functional Modeling (Isotropic Case)

The complementary spherical tensor 13 τkk δij is added to the ﬁltered static pressure term and consequently requires no modeling. This decomposition is necessary since the tensor (∇u + ∇T u) has a zero trace, and we can only model a tensor that also has a zero trace. This leads to the deﬁnition of the modiﬁed pressure Π: 1 (5.30) Π = p + τkk . 3 It is important to note that the modiﬁed pressure and ﬁltered pressure p may take very diﬀerent values when the generalized subgrid kinetic energy becomes large [374]. The closure thus now consists in determining the relation: νsgs = N (u) . (5.31) The use of hypothesis (5.2) and of a model structured as above calls for a few comments. Obtaining a scalar subgrid viscosity requires the adoption of the following hypothesis: Hypothesis 5.3 A characteristic length l0 and a characteristic time t0 are suﬃcient for describing the subgrid scales. Then, by dimensional reasoning similar to Prandtl’s, we arrive at: νsgs ∝

l02 t0

.

(5.32)

Models of the form (5.28) are local in space and time, which is a necessity if they are to be used in practice. This local character, similar to that of the molecular diﬀusion terms, implies [26, 405, 813]: Hypothesis 5.4 (Scale Separation Hypothesis) There exists a total separation between the subgrid and resolved scales. A spectrum verifying this hypothesis is presented in Fig. 5.10. Using L0 and T0 to denote the characteristic scales, respectively, of the resolved ﬁeld in space and time, this hypothesis can be reformulated as: l0 1, L0

t0 1 . T0

(5.33)

This hypothesis is veriﬁed in the case of molecular viscosity. The ratio between the size of the smallest dynamically active scale, ηK , and the mean free path ξfp of the molecules of a gas is evaluated as: ξfp Ma ηK Re1/4

,

(5.34)

5.3 Modeling of the Forward Energy Cascade Process

111

Fig. 5.10. Energy spectrum corresponding to a total scale separation for cutoﬀ wave number kc .

where Ma is the Mach number, deﬁned as the ratio of the ﬂuid velocity to the speed of sound, and Re the Reynolds number [708]. In most of the cases encountered, this ratio is less than 103 , which ensures the pertinence of using a continuum model. For applications involving rareﬁed gases, this ratio can take on much higher values of the order of unity, and the Navier– Stokes equations are then no longer an adequate model for describing the ﬂuid dynamics. Filtering associated to large-eddy simulation does not introduce such a separation between resolved and subgrid scales because the turbulent energy spectrum is continuous. The characteristic scales of the smallest resolved scales are consequently very close to those of the largest subgrid scales13 . This continuity originates the existence of the spectrum region located near the cutoﬀ, in which the eﬀective viscosity varies rapidly as a function of the wave number. The result of this diﬀerence in nature with the molecular viscosity is that the subgrid viscosity is not a characteristic of the ﬂuid but of the ﬂow. Let us not that Yoshizawa [786, 788], using a re-normalization technique, has shown that the subgrid viscosity is deﬁned as a fourth-order non-local tensor in space and time, in the most general case. The use of the scale separation hypothesis therefore turns out to be indispensable for constructing local models, although it is contrary to the scale similar hypothesis of Bardina et al. [40], which is discussed in Chap. 7. It is worth noting that subgrid-viscosity based models for the forward energy cascace induce a spurious alignment of the eigenvectors for resolved strain rate tensor and subgrid-scale tensor, because they are expressed as 13

This is all the more true for smooth ﬁlters such as the Gaussian and box ﬁlters, which allow a frequency overlapping between the resolved and subgrid scales.

112

5. Functional Modeling (Isotropic Case)

τ d ∝ (∇u + ∇T u).14 Tao et al. [703, 704] and Horiuti [325] have shown that this alignment is unphysical: the eigenvectors for the subgrid tensor have a strongly preferred relative orientation of 35 to 45 degrees with the resolved strain rate eigenvectors. The modeling problem consists in determining the characteristic scales l0 and t0 . Model Types. The subgrid viscosity models can be classiﬁed in three categories according to the quantities they bring into play [26]: 1. Models based on the resolved scales (p. 113): the subgrid viscosity is evaluated using global quantities related to the resolved scales. The existence of subgrid scales at a given point in space and time will therefore be deduced from the global characteristics of the resolved scales, which requires the introduction of assumptions. 2. Models based on the energy at the cutoﬀ (p. 116): the subgrid viscosity is calculated from the energy of the highest resolved frequency. Here, it is a matter of information contained in the resolved ﬁeld, but localized in frequency and therefore theoretically more pertinent for describing the phenomena at cutoﬀ than the quantities that are global and thus not localized in frequency, which enter into the models of the previous class. The existence of subgrid scales is associated with a non-zero value of the energy at cutoﬀ15 . 3. Models based on the subgrid scales (p. 116), which use information directly related to the subgrid scales. The existence of the subgrid scales is no longer determined on the basis of assumptions concerning the characteristics of the resolved scales as it is in the previous cases, but rather directly from this additional information. These models, because they are richer, also theoretically allow a better description of these scales than the previous models. These model classes are presented in the following. All the developments are based on the analysis of the energy transfers in the canonical case. In order to be able to apply the models formulated from these analyses to more realistic ﬂows, such as the hom*ogeneous isotropic ﬂows associated with a production type spectrum, we adopt the assumption that the ﬁlter cutoﬀ fre14

15

But it must be also remembered that the purpose of these functional models is not to predict the subgrid tensor, but just to enforce the correct resolved kinetic energy balance. This reconstruction of the subgrid tensor is nothing but an a posteriori interpretation. This fact is used by Germano to derive new subgrid viscosity models [252]. This hypothesis is based on the fact that the energy spectrum E(k) of an isotropic turbulent ﬂow in spectral equilibrium corresponding to a Kolmogorov spectrum is a monotonic continuous decreasing function of the wave number k. If there exists a wave number k∗ such that E(k∗ ) = 0, then E(k) = 0, ∀k > k∗ . Also, if the energy is non-zero at the cutoﬀ, then subgrid modes exist, i.e. if E(kc ) = 0, then there exists a neighbourhood Ωkc = [kc , kc + c ], c > 0 such that E(kc ) ≥ E(k) ≥ 0 ∀k ∈ Ωkc .

5.3 Modeling of the Forward Energy Cascade Process

113

quency is located suﬃciently far into the inertial range for these analyses to remain valid (refer to Sect. 5.1.2). The use of these subgrid models for arbitrary developed turbulent ﬂows (anisotropic, inhom*ogeneous) is justiﬁed by the local isotropy hypothesis: we assume then that the cutoﬀ occurs in the scale range that veriﬁes this hypothesis. The case corresponding to an isotropic hom*ogeneous ﬂow associated with a production spectrum is represented in Fig. 5.11. Three energy ﬂuxes are deﬁned: the injection rate of turbulent kinetic energy into the ﬂow by the driving mechanisms (forcing, instabilities), denoted εI ; the kinetic energy transfer rate through the cutoﬀ, denoted ε*; and the kinetic energy dissipation rate by the viscous eﬀects, denoted ε. Models Based on the Resolved Scales. These models are of the generic form: νsgs = νsgs ∆, ε˜ , (5.35) in which ∆ is the characteristic cutoﬀ length of the ﬁlter and ε˜ the instantaneous energy ﬂux through the cutoﬀ. We implicitly adopt the assumption here, then, that the subgrid modes exist, i.e. that the exact solution is not entirely represented by the ﬁltered ﬁeld when this ﬂux is non-zero. First Method. Simple dimensional analysis shows that: 4/3

νsgs ∝ ε˜1/3 ∆

.

(5.36)

Fig. 5.11. Dynamics of the kinetic energy in the spectral space. The energy is injected at the rate εI . The transfer rate through the cutoﬀ, located wave number kc , is denoted ε˜. The dissipation rate due to viscous eﬀects is denoted ε. The local equilibrium hypothesis is expressed by the equality εI = ε˜ = ε.

114

5. Functional Modeling (Isotropic Case)

Reasoning as in the framework of Kolmogorov’s hypotheses for isotropic hom*ogeneous turbulence, for the case of an inﬁnite inertial spectrum of the form (5.37) E(k) = K0 ε2/3 k −5/3 , K0 ∼ 1.4 , in which ε is the kinetic energy dissipation rate, we get the equation: νsgs =

A 4/3 ˜ ε1/3 ∆ K0 π 4/3

,

(5.38)

in which the constant A is evaluated as A = 0.438 with the TFM model and as A = 0.441 by the EDQNM theory [26]. The angle brackets operator , designates a statistical average. This statistical averaging operation is intrinsically associated with a spatial mean by the fact of the ﬂow’s spatial hom*ogeneity and isotropy hypotheses. This notation is used in the following to symbolize the fact that the reasoning followed in the framework of isotropic hom*ogeneous turbulence applies only to the statistical averages and not to the local values in the physical space. The problem is then to evaluate the average ﬂux ˜ ε. In the isotropic hom*ogeneous case, we have: 2|S|2 = 2S ij S ij =

kc

2k 2 E(k)dk, kc =

π ∆

.

(5.39)

If the cutoﬀ kc is located far enough into the inertial range, the above relation can be expressed solely as a function of this region’s characteristic quantities. Using a spectrum of the shape (5.37), we get: 3 −4/3 2|S|2 = π 4/3 K0 ε2/3 ∆ 2

.

(5.40)

Using the hypothesis16 [447]: |S|3/2 |S|3/2

,

(5.41)

we get the equality: ε =

1 π2

3K0 2

−3/2

2

∆ 2|S|2 3/2

.

(5.42)

In order to evaluate the dissipation rate ε from the information contained in the resolved scales, we assume the following: Hypothesis 5.5 (Local Equilibrium Hypothesis) The ﬂow is in constant spectral equilibrium, so there is no accumulation of energy at any frequency and the shape of the energy spectrum remains invariant with time. 16

The error margin measured in direct numerical simulations of isotropic hom*ogeneous turbulence is of the order of 20% [507].

5.3 Modeling of the Forward Energy Cascade Process

115

This implies an instantaneous adjustment of all the scales of the solution to the turbulent kinetic energy production mechanism, and therefore equality between the production, dissipation, and energy ﬂux through the cutoﬀ: εI = ˜ ε = ε .

(5.43)

Using this equality and relations (5.38) and (5.42), we get the closure relation: 2 νsgs = C∆ 2|S|2 1/2 , (5.44) where the constant C is evaluated as: √ −1/4 A 3K0 C= √ ∼ 0.148 . 2 π K0

(5.45)

Second Method. The local equilibrium hypothesis allows: ε = ˜ ε ≡ −S ij τij = νsgs 2S ij S ij .

(5.46)

The idea is then to assume that: νsgs 2S ij S ij = νsgs 2S ij S ij .

(5.47)

By stating at the outset that the subgrid viscosity is of the form (5.44) and using relation (5.40), a new value is found for the constant C: C=

1 π

3K0 2

−3/4 ∼ 0.18 .

(5.48)

We note that the value of this constant is independent of the cutoﬀ wave number kc , but because of the way it is calculated, we can expect a dependency as a function of the spectrum shape. Alternate Form. This modeling induces a dependency as a function of the cutoﬀ length ∆ and the strain rate tensor S of the resolved velocity ﬁeld. In the isotropic hom*ogeneous case, we have the equality: 2|S|2 = ω · ω, ω = ∇ × u .

(5.49)

By substitution, we get the equivalent form [487]: 2 νsgs = C∆ ω · ω1/2

.

(5.50)

These two versions bring in the gradients of the resolved velocity ﬁeld. This poses a problem of physical consistency since the subgrid viscosity is non-zero as soon as the velocity ﬁeld exhibits spatial variations, even if it is laminar and all the scales are resolved. The hypothesis that links the existence of the subgrid modes to that of the mean ﬁeld gradients therefore prevents

116

5. Functional Modeling (Isotropic Case)

us from considering the large scale intermittency and thereby requires us to develop models which by nature can only be eﬀective for dealing with ﬂows that are completely turbulent and under-resolved everywhere17. Poor behavior can therefore be expected when treating intermittent or weakly developed turbulent ﬂows (i.e. in which the inertial range does not appear in the spectrum) due to too strong an action by the model. Models Based on the Energy at Cutoﬀ. The models of this category are based on the intrinsic hypothesis that if the energy at the cutoﬀ is non-zero, then subgrid modes exist. First Method. Using relation (5.38) and supposing that the cutoﬀ occurs within an inertial region, i.e.: E(kc ) = K0 ε2/3 kc−5/3

,

(5.51)

by substitution, we get: ' A νsgs = √ K0

E(kc ) , kc = π/∆ . kc

(5.52)

This model raises the problem of determining the energy at the cutoﬀ in the physical space, but on the other hand ensures that the subgrid viscosity will be null if the ﬂow is well resolved, i.e. if the highest-frequency mode captured by the grid is zero. This type of model thus ensures a better physical consistency than those models based on the large scales. It should be noted that it is equivalent to the spectral model of constant eﬀective viscosity. Second Method. As in the case of models based on the large scales, there is a second way of determining the model constant. By combining relations (5.46) and (5.51), we get: ' E(kc ) 2 νsgs = . (5.53) 3/2 kc 3K 0

Models Based on Subgrid Scales. Here we considers models of the form: 2 νsgs = νsgs ∆, qsgs , ε , (5.54) 2 in which qsgs is the kinetic energy of the subgrid scales and ε the kinetic energy dissipation rate18 . These models contain more information about the 17 18

In the sense that the subgrid modes exist at each point in space and at each time step. Other models are of course possible using other subgrid scale quantities like a length or time scale, but we limit ourselves here to classes of models for which practical results exist.

5.3 Modeling of the Forward Energy Cascade Process

117

subgrid modes than those belonging to the two categories described above, and thereby make it possible to do without the local equilibrium hypothesis (5.5) by introducing characteristic scales speciﬁc to the subgrid modes by 2 and ε. This capacity to handle the energy disequilibrium is way of qsgs expressed by the relation: * ε ≡ −τij S ij = ε ,

(5.55)

which should be compared with (5.43). In the case of an inertial range extending to inﬁnity beyond the cutoﬀ, we have the relation: 2 qsgs

1 ≡ ui ui = 2

∞

E(k)dk = kc

3 K0 ε2/3 kc−2/3 2

,

(5.56)

from which we deduce: ε =

kc q 2 3/2 (3K0 /2)3/2 sgs

.

(5.57)

By introducing this last equation into relation (5.38), we come to the general form: 1+α/3

2 (1−α)/2 νsgs = Cα εα/3 qsgs ∆

in which A Cα = K0 π 4/3

3K0 2

,

(5.58)

,

(5.59)

(α−1)/2 π (1−α)/3

and in which α is a real weighting parameter. Interesting forms of νsgs have been found for certain values: – For α = 1, we get νsgs =

A 4/3 ∆ ε1/3 K0 π 4/3

.

(5.60)

This form uses only the dissipation and is analogous to that of the models based on the resolved scales. If the local equilibrium hypothesis is used, these two types of models are formally equivalent. – For α = 0, we get 2 1/2 2 A νsgs = ∆ qsgs . (5.61) 3/2 3 πK 0

This model uses only the kinetic energy of the subgrid scales. As such, it is formally analogous to the deﬁnition of the diﬀusion coeﬃcient of an ideal gas in the framework of gaskinetics theory. In the case of an inertial

118

5. Functional Modeling (Isotropic Case)

spectrum extending to inﬁnity beyond the cutoﬀ, this model is strictly equivalent to the model based on the energy at cutoﬀ, since in this precise case we have the relation: 3 2 kc E(kc ) = qsgs 2

.

(5.62)

– For α = −3, we have: 2 2 4A qsgs νsgs = 3 9K0 ε

.

(5.63)

This model is formally analogous to the k−ε statistical model of turbulence for the Reynolds Averaged Navier–Stokes equations, and does not bring in the ﬁlter cutoﬀ length explicitly. 2 The closure problem consists in determining the quantities ε and qsgs . To do this, we can introduce one or more equations for the evolution of these quantities or we can deduce them from the information contained in the resolved ﬁeld. As these quantities represent the subgrid scales, we are justiﬁed in thinking that, if they are correctly evaluated, the subgrid viscosity will be negligible when the ﬂow is well resolved numerically. However, it should be noted that these models in principle require more computation than those based on the resolved scales, because they produce more information concerning the subgrid scales.

Extension to Other Spectrum Shapes. The above developments are based on a Kolmogorov spectrum, which reﬂects only the existence of a region of similarity of the real spectra. This approach can be extended to other more realistic spectrum shapes, mainly including the viscous eﬀects. Several extensions of the models based on the large scales were proposed by Voke [737] for this. The total dissipation ε can be decomposed into the sum of the dissipation associated with the large scales, denoted εr , and the dissipation associated with the subgrid scales, denoted εsgs , (see Fig. 5.12): ε = εr + εsgs .

(5.64)

These three quantities can be evaluated as: ε = εr =

2(νsgs + ν)|S|2 , kc 2 2ν|S| = 2ν k 2 E(k)dk 0

εsgs =

2νsgs |S|2 = Cs ∆

2

(5.65) ,

3/2 2|S|2

(5.66) ,

(5.67)

from which we get: 1 εr = , ε 1 + ν˜

ν˜ =

νsgs ν

.

(5.68)

5.3 Modeling of the Forward Energy Cascade Process

119

Fig. 5.12. Kinetic energy dynamics in the spectral space. The energy is injected at the rate εI . The transfer rate through the cutoﬀ located at the wave number kc is denoted ε˜. The dissipation rate in the form of heat by the viscous eﬀects associated with the scales located before and after the cutoﬀ kc are denoted εr and εsgs , respectively.

This ratio is evaluated by calculating the εr term analytically from the chosen spectrum shapes, which provides a way of then computing the subgrid viscosity νsgs . We deﬁne the three following parameters: κ=

k =k kd

ν3 ε

1/4 ,

2

∆ Re∆ =

κc =

kc kd

,

(5.69)

+ 2|S|2 ν

,

(5.70)

in which kd is the wave number associated with the Kolmogorov scale (see Appendix A), and Re∆ is the mesh-Reynolds number. Algebraic substitutions lead to: −1/2 (5.71) κ = πRe∆ (1 + ν˜)−1/4 . The spectra studied here are of the generic form: E(k) = K0 ε2/3 k −5/3 f (κ) ,

(5.72)

in which f is a damping function for large wave numbers. The following are some commonly used forms of this function:

120

5. Functional Modeling (Isotropic Case)

– Heisenberg–Chandrasekhar spectrum: (

f (κ) = 1 +

3K0 2

)−4/3

3 κ

4

.

(5.73)

– Kovasznay spectrum: f (κ) =

2 K0 4/3 1− κ 2

.

(5.74)

Note that this function cancels out for κ = (2/K0 )3/4 , which requires that the spectrum be forced to zero for wave numbers beyond this limit. – Pao spectrum: 3K0 4/3 κ f (κ) = exp − . (5.75) 2 These three spectrum shapes are graphed in Fig. 5.13. An analytical integration leads to: – For the Heisenberg–Chandrasekhar spectrum: εr = κ4/3 c ε

(

2 3K0

)−1/3

3 +

κ4c

,

(5.76)

Fig. 5.13. Graph of Heisenberg–Chandrasekhar, Kovasznay, and Pao spectra, for kd = 1000.

5.3 Modeling of the Forward Energy Cascade Process

or: νsgs = ν

⎧ ⎨ ⎩

κ−4/3 c

(

2 3K0

)1/3

3 + κ4c

−1

121

⎫ ⎬ ⎭

.

(5.77)

– For the Kovazsnay spectrum: 3 K0 4/3 εr =1− 1− κc ε 2 or:

,

(5.78)

⎧( ⎫ 3 )−1 ⎨ ⎬ K0 4/3 κc νsgs = ν 1− 1− −1 ⎩ ⎭ 2

.

(5.79)

– For the Pao spectrum: εr = 1 − exp ε or: νsgs = ν

⎧( ⎨ ⎩

1 − exp

3K0 4/3 κ 2 c

3K0 4/3 κ 2 c

3 ,

3 )−1 −1

(5.80) ⎫ ⎬ ⎭

.

(5.81)

These new estimates of the subgrid viscosity νsgs make it possible to take the viscous eﬀects into account, but requires that the spectrum shape be set a priori, as well as the value of the ratio κc between the cutoﬀ wave number kc and the wave number kd associated with the Kolmogorov scale. Inclusion of the Local Eﬀects at Cutoﬀ. The subgrid viscosity models in the physical space, such as they have been developed, do not reﬂect the increase in the coupling intensity with the subgrid modes when we consider modes near the cutoﬀ. These models are therefore analogous to that of constant eﬀective viscosity. To bring out these eﬀects in the proximity of the cutoﬀ, Chollet [134], Ferziger [217], Lesieur and M´etais [440], Deschamps [176], Borue and Orszag [68, 69, 70, 71], Winckelmans and co-workers [762, 163, 759] and Layton [427] propose introducing high-order dissipation terms that will have a strong eﬀect on the high frequencies of the resolved ﬁeld without aﬀecting the low frequencies. e can Chollet [134], when pointing out that the energy transfer term Tsgs be written in the general form e (k|kc ) = −2νe(n) (k|kc )k 2n E(k) , Tsgs

(5.82)

122

5. Functional Modeling (Isotropic Case) (n)

in which νe (k|kc ) is a hyper-viscosity, proposes modeling the subgrid term in the physical space as the sum of an ordinary subgrid viscosity model and a sixth-order dissipation. This new model is expressed: (5.83) ∇ · τ = −νsgs C1 ∇2 + C2 ∇6 u , in which C1 and C2 are constants. Ferziger proposes introducing a fourthorder dissipation by adding to the subgrid tensor τ the tensor τ (4) , deﬁned as: 2 2 ∂ ∂ (4) (4) ∂ ui (4) ∂ uj τij = νsgs + νsgs , (5.84) ∂xj ∂xk ∂xk ∂xi ∂xk ∂xk or as (4)

τij =

∂2 ∂xk ∂xk

∂ui ∂uj (4) + νsgs ∂xj ∂xi

,

(5.85)

(4)

in which the hyper-viscosity νsgs is deﬁned by dimensional arguments as 4

(4) νsgs = Cm ∆ |S| .

(5.86)

The full subgrid term that appears in the momentum equations is then written: (2) (4) , (5.87) τij = τij + τij (2)

in which τij is a subgrid viscosity model described above. A similar form is proposed by Lesieur and M´etais: after deﬁning the velocity ﬁeld u as u = ∇2p u

,

(5.88)

the two authors propose the composite form: Sij τij = −νsgs S ij + (−1)p+1 νsgs

,

(5.89)

hyper-viscosity obtained by applying a subgrid viscosity model in which νsgs to the u ﬁeld, and S the strain rate tensor computed from this same ﬁeld. The constant of the subgrid model used should be modiﬁed to verify the local equilibrium relation, which is

−τij S ij = ε . This composite form of the subgrid dissipation has been validated experimentaly by Cerutti et al. [115], who computed the spectral distribution of dissipation and the corresponding spectral viscosity from experimental data. It is worth noting that subgrid dissipations deﬁned thusly, as the sum of second- and fourth-order dissipations, are similar in form to certain numerical schemes designed for capturing strong gradients, like that of Jameson et al. [346].

5.3 Modeling of the Forward Energy Cascade Process

123

Borue and Orszag [68, 69, 70, 71] propose to eliminate the molecular and the subgrid viscosities by replacing them by a higher power of the Laplacian operator. Numerical tests show that three-dimensional inertial-range dynamics is relatively independent of the form of the hyperviscosity. It was also shown that for a given numerical resolution, hyperviscous dissipations increase the extent of the inertial range of three-dimensional turbulence by an order of magnitude. It is worth noting that this type of iterated Laplacian is commonly used for two-dimensional simulations. Borue and Orszag used a height-time iterated Laplacian to get these conclusions. Such operators are easily deﬁned when using spectral methods, but are of poor interest when dealing with ﬁnite diﬀerence of ﬁnite volume techniques. Subgrid-Viscosity Models. Various subgrid viscosity models belonging to the three categories deﬁned above will now be described. These are the following: 1. The Smagorinsky model (p. 124), which is based on the resolved scales. This model, though very simple to implant, suﬀers from the defects already mentioned for the models based on the large scales. 2. The second-order Structure Function model developed by M´etais and Lesieur (p. 124), which is an extension into physical space of the models based on the energy at cutoﬀ. Theoretically based on local frequency information, this model should be capable of treating large-scale intermittency better than the Smagorinsky model. However, the impossibility of localizing the information in both space and frequency (see discussion further on) reduces its eﬃciency. 3. The third-order Structure Function models developed by Shao (p. 126), which can be interpreted as an extension of the previous model based on the second-order structure function. The use of the Kolmogorov– Meneveau equation [510] for the ﬁltered third-order structure function enables the deﬁnition of several models which do not contain arbitrary constants and have improved potentiality for non-equilibrium ﬂows. 4. A model based on the kinetic energy of the subgrid modes (p. 128). This energy is considered as an additional variable of the problem, and is evaluated by solving an evolution equation. Since it contains information relative to the subgrid scales, it is theoretically more capable of handling large-scale intermittency than the previous model. Moreover, the local equilibrium hypothesis can be relaxed, so that the spectral nonequilibrium can be integrated better. The model requires additional hypotheses, though (modeling, boundary conditions). 5. The Yoshizawa model (p. 129), which includes an additional evolution equation for a quantity linked to a characteristic subgrid scale, by which it can be classed among models based on the subgrid scales. It has the same advantages and disadvantages as the previous model. 6. The Mixed Scale Model (p. 130), which uses information related both to the subgrid modes and to the resolved scales, though without incorpo-

124

5. Functional Modeling (Isotropic Case)

rating additional evolution equations. The subgrid scale data is deduced from that contained in the resolved scales by extrapolation in the frequency domain. This model is of intermediate level of complexity (and quality) between those based on the large scales and those that use additional variables. Smagorinsky Model. The Smagorinsky model [676] is based on the large scales. It is generally used in a local form the physical space, i.e. variable in space, in order to be more adaptable to the ﬂow being calculated. It is obtained by space and time localization of the statistical relations given in the previous section. There is no particular justiﬁcation for this local use of relations that are on average true for the whole, since they only ensure that the energy transfers through the cutoﬀ are expressed correctly on the average, and not locally. This model is expressed: 2 νsgs (x, t) = Cs ∆ (2|S(x, t)|2 )1/2

.

(5.90)

The constant theoretical value Cs is evaluated by the relations (5.45) or (5.48). It should nonetheless be noted that the value of this constant is, in practice, adjusted to improve the results. Clark et al. [143] use Cs = 0.2 for a case of isotropic hom*ogeneous turbulence, while Deardorﬀ [172] uses Cs = 0.1 for a plane channel ﬂow. Studies of shear ﬂows using experimental data yield similar evaluations (Cs 0.1 − 0.12) [503, 570, 724]. This decrease in the value of the constant with respect to its theoretical value is due to the fact that the ﬁeld gradient is now non-zero and that it contributes to the |S(x, t)| term. To enforce the local equilibrium relation, the value of the constant has to be reduced. It should be noted that this new value ensures only that the right quantity of resolved kinetic energy will be dissipated on the whole throughout the ﬁeld, but that the quality of the level of local dissipation is uncontrolled. 19 Second-Order Structure Function Model. This model is a transposition of M´etais and Lesieur’s constant eﬀective viscosity model into the physical space, and can consequently be interpreted as a model based on the energy at cutoﬀ, expressed in physical space. The authors [514] propose evaluating the energy at cutoﬀ E(kc ) by means of the second-order velocity structure 19

Canuto and Cheng [99] derived a more general expression for the constant Cs , which appears as an explicit function of the subgrid kinetic energy and the local shear: 1/2 2 qsgs Cs ∝ , 2 ε|S|∆ which is eﬀectively a decreasing function of the local shear |S|. That demonstrates the limited theoretical range of application of the usual Smagorinsky model.

5.3 Modeling of the Forward Energy Cascade Process

function. This is deﬁned as: DLL (x, r, t) =

[u(x, t) − u(x + x , t)] d3 x 2

|x |=r

.

125

(5.91)

In the case of isotropic hom*ogeneous turbulence, we have the relation: ∞ sin(kr) E(k, t) 1 − DLL (r, t) = DLL (x, r, t)d3 x = 4 dk . (5.92) kr 0 Using a Kolmogorov spectrum, the calculation of (5.92) leads to: DLL (r, t) =

9 Γ (1/3)K0ε2/3 r2/3 5

,

(5.93)

or, expressing the dissipation ε, as a function of DLL (r, t) in the expression for the Kolmogorov spectrum: E(k) =

5 DLL (r, t)r−2/3 k −5/3 9Γ (1/3)

.

(5.94)

To derive a subgrid model, we now have to evaluate the second-order structure function from the resolved scales alone. To do this, we decompose by: (5.95) DLL (r, t) = DLL (r, t) + C0 (r, t) , in which DLL (r, t) is computed from the resolved scales and C0 (r, t) corresponds to the contribution of the subgrid scales:

∞

C0 (r, t) = 4 kc

sin(kr) E(k, t) 1 − dk kr

.

(5.96)

By replacing the quantity E(k, t) in equation (5.96) by its value (5.94), we get: −2/3 r Hsf (r/∆) , (5.97) C0 (r, t) = DLL (r, t) ∆ in which Hsf is the function 20 3 2/3 Hsf (x) = + x Im {exp(ı5π/6)Γ (−5/3, ıπx)} 9Γ (1/3) 2π 2/3

. (5.98)

Once it is substituted in (5.95), this equation makes it possible to evaluate the energy at the cutoﬀ. The second-order Structure Function model takes the form: + νsgs (r) = A(r/∆)∆ DLL (r, t) ,

(5.99)

126

5. Functional Modeling (Isotropic Case)

in which A(x) =

−3/2 −1/2 2K 0 x−4/3 1 − x−2/3 Hsf (x) 3π 4/3 (9/5)Γ (1/3)

.

(5.100)

In the same way as for the Smagorinsky model, a local model in space can be had by using relation (5.94) locally in order to include the local intermittency of the turbulence. The model is then written: + νsgs (x, r) = A(r/∆)∆ DLL (x, r, t) . (5.101) In the case where r = ∆, the model takes the simpliﬁed form: + νsgs (x, ∆, t) = 0.105∆ DLL (x, ∆, t) .

(5.102)

A link can be established with the models based on the resolved scale gradients by noting that: u(x, t) − u(x + x , t) = −x · ∇u(x, t) + O(|x |2 )

.

(5.103)

This last relation shows that the function F 2 is hom*ogeneous to a norm of the resolved velocity ﬁeld gradient. If this function is evaluated in the simulation in a way similar to how the resolved strain rate tensor is computed for the Smagorinsky model, we can in theory expect the Structure Function model to suﬀer some of the same weaknesses: the information contained in the model will be local in space, therefore non-local in frequency, which induces a poor estimation of the kinetic energy at cutoﬀ and a loss of precision of the model in the treatment of large-scale intermittency and spectral nonequilibrium. Third-Order Structure Functions Models. Shao et al. [669, 153] deﬁned a more general class of structure function-based subgrid viscosities considering the Kolmogorov–Meneveau equation for the third-order velocity structure function in isotropic turbulence [510] 4 − rε = DLLL − 6GLLL 5

,

(5.104)

where DLLL is the third-order longitudinal velocity correlation of the ﬁltered ﬁeld DLLL (r) = [u(x + r) − u(x)]3 , (5.105) where · denotes the statistical average operator (which is equivalent to the integral sequence introduced in the presentation of the second-order structure function model). The two other quantities are the longitudinal velocity-stress correlation tensor GLLL (r) = u1 (x)τ11 (x + r) , (5.106)

5.3 Modeling of the Forward Energy Cascade Process

127

and the average subgrid dissipation ε = −τij S ij

.

(5.107)

Shao’s procedure consists in using relation (5.104) to compute the subgrid viscosity. To this end, he assumes that the velocity-stress correlation obeys the following scale-similarity hypothesis GLLL (r) ∝ rp

,

(5.108)

where p = −1/3 corresponds to the Kolmogorov local isotropy hypotheses. Using that assumption, one obtains the following relationship for two space increments r1 and r2 : 0.8r1 ε + DLLL (r1 ) = 0.8r2 ε + DLLL (r2 )

r1 r2

−1/3 .

(5.109)

Now introducing a subgrid viscosity model with subgrid viscosity νsgs , several possibilities exist for the evaluation of the subgrid viscosity. The three following models have been proposed by Shao and his coworkers: – The one-scale, constant subgrid viscosity model. Assuming that the subgrid viscosity is constant, one obtains τij = −2νsgs S ij ,

ε = νsgs |S|2 ,

GLLL = νsgs DLL,r

,

(5.110)

where comma separated indices denote derivatives. Inserting these relations into (5.104), an expression for the eddy viscosity is recovered: + −Sk DLL , (5.111) r νsgs = 2 0.8 D|S| − 4 LL

where the skewness of the longitudinal ﬁltered velocity increment is deﬁned as DLLL Sk = 3/2 . (5.112) DLL The M´etais-Lesieur model is recovered taking r = ∆ and using properties of isotropic turbulence to evaluate the diﬀerent terms appearing in (5.111). – The one-scale, variable subgrid viscosity model. Relaxing the previous contraint dealing with the constant appearing in the structure-function model, one obtains the following asymptotic expression + −Sk r DLL . (5.113) νsgs = 8

128

5. Functional Modeling (Isotropic Case)

The constant is observed to be self-adpative, depending on the computed value of the skewness parameter Sk . – The multiscale structure function model. The last model derived by Shao is more general and is based on the two-scale relation (5.109). Using the relation ε = νsgs |S|2 and considering two diﬀrent separation distances r1 and r2 , one obtain the following expression

νsgs

1/3 DLLL (r1 ) − rr12 D LLL (r2 ) = 4/3 −0.4|S|2 1 − rr12 r1

.

(5.114)

These expressions are made local in the physical space by using local values of the diﬀerent parameters involving the resolved velocity ﬁeld u. Model Based on the Subgrid Kinetic Energy. One model, of the form (5.61), based on the subgrid scales, was developed independently by a number of authors [318, 653, 791, 792, 525, 690, 391]. The subgrid viscosity 2 : is computed from the kinetic energy of the subgrid modes qsgs νsgs (x, t) = Cm ∆

+ 2 (x, t) , qsgs

(5.115)

where, for reference: 2 qsgs (x, t) =

1 (ui (x, t) − ui (x, t))2 2

.

(5.116)

The constant Cm is evaluated by the relation (5.61). This energy constitutes another variable of the problem and is evaluated by solving an evolution equation. This equation is obtained from the exact evolution equation (3.33), whose unknown terms are modeled according to Lilly’s proposals [447], or by a re-normalization method. The various terms are modeled as follows (refer to the work of McComb [464], for example): – The diﬀusion term is modeled by a gradient hypothesis, by stating that 2 gradient the non-linear term is proportional to the kinetic energy qsgs (Kolmogorov-Prandtl relation): 2 + ∂qsgs 1 ∂ ∂ 2 u u u + uj p = C2 ∆ qsgs . (5.117) ∂xj 2 i i j ∂xj ∂xj – The dissipation term is modeled using dimensional reasoning, by: 2 (qsgs ) ν ∂ui ∂ui ε= = C1 2 ∂xj ∂xj ∆

3/2

.

(5.118)

5.3 Modeling of the Forward Energy Cascade Process

129

The resulting evolution equation is: 2 2 ∂qsgs ∂uj qsgs + = ∂t ∂xj

3/2

2 qsgs −τij S ij − C1 ∆ II

I

+

∂ C2 ∂xj

III

2 2 + ∂ 2 qsgs ∂q sgs 2 ∆ qsgs +ν , (5.119) ∂xj ∂xj ∂xj

IV

V

in which C1 and C2 are two positive constants and the various terms represent: – – – – –

I - advection by the resolved modes, II - production by the resolved modes, III - turbulent dissipation, IV - turbulent diﬀusion, V - viscous dissipation.

Using an analytical theory of turbulence, Yoshizawa [791, 792] and Horiuti [318] propose C1 = 1 and C2 = 0.1. Yoshizawa Model. The ﬁlter cutoﬀ length, ∆, is the only length scale used in deriving models based on the large scales, as this derivation has been explained above. The characteristic length associated with the subgrid scales, denoted ∆f , is assumed to be proportional to this length, and the developments of Sect. 5.3.2 show that: ∆f = Cs ∆ .

(5.120)

The variations in the structure of the subgrid modes cannot be included by setting a constant value for the coeﬃcient Cs , as is done in the case of the Smagorinsky model, for example. To remedy this, Yoshizawa [787, 790] proposes diﬀerentiating these two characteristic scales and introducing an additional evolution equation to evaluate ∆f . This length can be evaluated 2 from the dissipation ε and the subgrid kinetic energy qsgs by the relation: ∆f = C1

2 3/2 2 3/2 2 2 5/2 (qsgs ) (qsgs ) Dqsgs (qsgs ) Dε + C2 − C 3 ε ε2 Dt ε3 Dt

,

(5.121)

in which D/Dt is the material derivative associated with the resolved velocity ﬁeld. The values of the constants appearing in equation (5.121) can be determined by an analysis conducted with the TSDIA technique [790]: C1 = 1.84, C2 = 4.95 et C3 = 2.91. We now express the proportionality relation between the two lengths as: ∆f = (1 + r(x, t))∆ . (5.122)

130

5. Functional Modeling (Isotropic Case)

By evaluating the subgrid kinetic energy as: 2/3 2 qsgs = ∆ε/C1

,

(5.123)

relations (5.121) and (5.122) lead to: 2/3 −4/3 Dε

r = C4 ∆

ε

,

Dt

(5.124)

with C4 = 0.04. Using the local equilibrium hypothesis, we get: ε = −τij S ij C5 ∆2f |S|3

,

(5.125)

in which C5 = 6.52.10−3. This deﬁnition completes the calculation of the factor r and the length ∆f . This variation of the characteristic length ∆f can be re-interpreted as a variation of the constant in the Smagorinsky model: 2 −2 D|S| −2 ∂ −2 ∂|S| + Cb ∆ |S| Cs = Cs0 1 − Ca |S| |S| Dt ∂xj ∂xj

. (5.126)

The constants Cs0 , Ca and Cb are evaluated at 0.16, 1.8, and 0.047, respectively, by Yoshizawa [790] and Murakami [554]. In practice, Cb is taken to be equal to zero and the constant Cs is bounded in order to ensure the stability of the simulation: 0.1 ≤ Cs ≤ 0.27. Morinishi and Kobayashi [541] recommend using the values Ca = 32 and Cs0 = 0.1. Mixed Scale Model. Ta Phuoc Loc and Sagaut [627, 626] deﬁned models having a triple dependency on the large and small structures of the resolved ﬁeld as a function of the cutoﬀ length. These models, which make up the one-parameter Mixed Scale Model family, are derived by taking a weighted geometric average of the models based on the large scales and those based on the energy at cutoﬀ: νsgs (α)(x, t) = Cm |F(u(x, t))|α (qc2 )

1−α 2

(x, t) ∆

1+α

,

(5.127)

with F (u(x, t)) = S(x, t) or ∇ × u(x, t) .

(5.128)

It should be noted that localized versions of the models are used here, so that any ﬂows that do not verify the spatial hom*ogeneity property can be processed better. The kinetic energy qc2 can be evaluated using any method presented in Sect. 9.2.3. In the original formulation of the model, it is evaluated in the physical space by the formula: qc2 (x, t) =

1 (ui (x, t)) (ui (x, t)) 2

.

(5.129)

5.3 Modeling of the Forward Energy Cascade Process

131

* is the resolved Fig. 5.14. Spectral subdivisions for double sharp-cutoﬀ ﬁltering. u ﬁeld in the sense of the test ﬁlter, (u) the test ﬁeld, and u the unresolved scales in the sense of the initial ﬁlter.

The test ﬁeld (u) represents the high-frequency part of the resolved velocity ﬁeld, deﬁned using a second ﬁlter, referred to as the test ﬁlter, des* > ∆ ignated by the tilde symbol and associated with the cutoﬀ length ∆ (see Fig. 5.14): * . (u) = u − u (5.130) The resulting model can be interpreted in two ways: – As a model based on the kinetic energy of the subgrid scales, i.e. the second form of the models based on the subgrid scales in Sect. 5.3.2, if we use Bardina’s hypothesis of scale similarity (described in Chap. 7), which allows us to set: 2 , (5.131) qc2 qsgs 2 is the kinetic energy of the subgrid scales. This assumption in which qsgs can be reﬁned in the framework of the canonical analysis. Assuming that the two cutoﬀs occur in the inertial range of the spectrum, we get:

qc2 =

kc

E(k)dk = kc

3 −2/3 K0 ε2/3 kc − kc −2/3 2

,

(5.132)

* respectively. in which kc and kc are wave numbers associated with ∆ and ∆,

132

5. Functional Modeling (Isotropic Case)

We then deﬁne the relation: ( 2 , β= qc2 = βqsgs

kc kc

)

−2/3 −1

.

(5.133)

It can be seen that the approximation is exact if β = 1, i.e. if: 1 kc = √ kc 8

.

(5.134)

This approximation is also used by Bardina et al. [40] and Yoshizawa et al. [794] to derive models based on the subgrid kinetic energy without using any additional transport equation. – As a model based on the energy at cutoﬀ, and therefore as a generalization of the spectral model of constant eﬀective viscosity into the physical space. That is, using the same assumptions as before, we get: 3 βkc E(kc ) . 2 √ Here, the approximation is exact if kc = kc / 8. qc2 =

(5.135)

It is important to note that the Mixed Scale Model makes no use of any commutation property between the test ﬁlter and the derivation operators. Also, we note that for α ∈ [0, 1] the subgrid viscosity νsgs (α) is always deﬁned, whereas the model appears in the form of a quotient for other values of α can then raise problems of numerical stability once it is discretized, because the denominator may cancel out. The model constant can be evaluated theoretically by analytical theories of turbulence in the canonical case. Resuming the results of Sect. 5.3.2, we get: Cm = Cq1−α Cs2α

,

(5.136)

in which Cs ∼ 0.18 or Cs ∼ 0.148 and Cq ∼ 0.20. Some other particular cases of the Mixed Scale Model can be found. Wong and Lilly [766], Carati [103] and Tsubokura [717] proposed using α = −1, yielding a model independent of the cutoﬀ length ∆. Yoshizawa et al. [793] used α = 0, but introduced an exponential damping term in order to enforce a satisfactory asymptotic near-wall behavior (see p. 159): ( νsgs =

0.03(qc2 )1/2

∆ 1 − exp −21

qc2 2

∆ |S|2

) .

(5.137)

Mathematical analysis in the case α = 0 was provided by Iliescu and Layton [342] and Layton and Lewandowski [430].

5.3 Modeling of the Forward Energy Cascade Process

133

5.3.3 Improvement of Models in the Physical Space Statement of the Problem. Experience shows that the various models yield good results when they are applied to hom*ogeneous turbulent ﬂows and that the cutoﬀ is placed suﬃciently far into the inertial range of the spectrum, i.e. when a large part of the total kinetic energy is contained in the resolved scales20 . In other cases, as in transitional ﬂows, highly anisotropic ﬂows, highly under-resolved ﬂows, or those in high energetic disequilibrium, the subgrid models behave much less satisfactorily. Aside from the problem stemming from numerical errors, there are mainly two reasons for this: 1. The characteristics of these ﬂows does not correspond to the hypotheses on which the models are derived, which means that the models are at fault. We then have two possibilities: deriving models from new physical hypotheses or adjusting existing ones, more or less empirically. The ﬁrst choice is theoretically more exact, but there is a lack of descriptions of turbulence for frameworks other than that of isotropic hom*ogeneous turbulence. Still, a few attempts have been made to consider the anisotropy appearing in this category. These are discussed in Chap. 6. The other solution, if the physics of the models is put to fault, consists in reducing their importance, i.e. increasing the cutoﬀ frequency to capture a larger part of the ﬂow physics directly. This means increasing the number of degrees of freedom and striking a compromise between the grid enrichment techniques and subgrid modeling eﬀorts. 2. Deriving models based on the energy at cutoﬀ or the subgrid scales (with no additional evolution equation) for simulations in the physical space runs up against Gabor-Heisenberg’s generalized principle of uncertainty [204, 627], which stipulates that the precision of the information cannot be improved in space and in frequency at the same time. This is illustrated by Fig. 5.15. Very good frequency localization implies high non-localization in space, which reduces the possibilities of taking the intermittency21 into account and precludes the treatment of highly inhom*ogeneous ﬂows. Inversely, very good localization of the information in space prevents any good spectral resolution, which leads to high errors, e.g. in computing the energy at the cutoﬀ. Yet this frequency localization is very important, since it alone can be used to detect the presence of the subgrid scales. It is important to recall here that large-eddy simulation is based on a selection in frequency of modes making up the exact 20

21

Certain authors estimate this share to be between 80% and 90% [119]. Another criterion sometimes mentioned is that the cutoﬀ scale should be of the order of Taylor’s microscale. Bagget et al. [32] propose to deﬁne the cutoﬀ length in such a way that the subgrid scales will be quasi-isotropic, leading to ∆ ≈ Lε /10, where Lε is the integral dissipation length. Direct numerical simulations and experimental data show that the true subgrid dissipation and its surrogates do not have the same scaling laws [114, 510].

134

5. Functional Modeling (Isotropic Case)

Fig. 5.15. Representation of the resolution in the space-frequency plane. The spat ial resolution ∆ is associated with frequency resolution ∆k . Gabor-Heisenberg’s uncertainty principle stipulates that the product ∆ × ∆k remains constant, i.e. that the area of the gray domain keeps the same value (from [204], courtesy of F. Ducros).

solution. Problems arise here, induced by the localization of statistical average relations that are exact, as this localization may correspond to a statistical average. Two solutions may be considered: developing an acceptable compromise between the precision in space and frequency, or enriching the information contained in the simulation, which is done either by adding variables to it as in the case of models based on the kinetic energy of the subgrid modes, or by assuming further hypotheses when deriving models. In the following, we present techniques developed to improve the simulation results, though without modifying the structure of the subgrid models deeply. The purpose of all these modiﬁcations is to adapt the subgrid model better to the local state of the ﬂow and remedy the lack of frequency localization of the information.

5.3 Modeling of the Forward Energy Cascade Process

135

We will be describing: 1. Dynamic procedures for computing subgrid model constants (p. 137). These constants are computed in such a way as to reduce an a priori estimate of the error committed with the model considered, locally in space and time, in the least squares sense. This estimation is made using the Germano identity, and requires the use of an analytical ﬁlter. It should be noted that the dynamic procedures do not change the model in the sense that its form (e.g. subgrid viscosity) remains the same. All that is done here is to minimize a norm of the error associated with the form of the model considered. The errors committed intrinsically22 by adopting an a priori form of the subgrid tensor are not modiﬁed. These procedures, while theoretically very attractive, do pose problems of numerical stability and can induce non-negligible extra computational costs. This variation of the constant at each point and each time step makes it possible to minimize the error locally for each degree of freedom, while determining a constant value oﬀers only the less eﬃcient possibility of an overall minimization. This is illustrated by the above discussion of the constant in the Smagorinsky model. 2. Dynamic procedures that are not directly based on the Germano identity (p. 152): the multilevel dynamic procedure by Terracol and Sagaut and the multiscale structure function method by Shao. These procedures have basically the same capability to monitor the constant in the subgrid model as the previous dynamic procedures. They also involve the deﬁnition of a test ﬁlter level, and are both based on considerations dealing with the dissipation scaling as a function of the resolution. Like the procedures based on the Germano identity, they are based on the implicit assumption that some degree of self-similarity exists in the computed ﬂow. 3. Structural sensors (p. 154), which condition the existence of the subgrid scales to the veriﬁcation of certain constraints by the highest frequencies of the resolved scales. More precisely, we consider here that the subgrid scales exist if the highest resolved frequencies verify topological properties that are expected in the case of isotropic hom*ogeneous turbulence. When these criteria are veriﬁed, we adopt the hypothesis that the highest resolved frequencies have a dynamics close to that of the scales contained in the inertial range. On the basis of energy spectrum continuity (see the note of page p. 112), we then deduce that unresolved scales exist, and the subgrid model is then used, but is otherwise canceled. 4. The accentuation technique (p. 156), which consists in artiﬁcially increasing the contribution of the highest resolved frequencies when evaluating 22

For example, the subgrid viscosity models described above all induce a linear dependency between the subgrid tensor and the resolved-scale tensor: d = −νsgs S ij τij

.

136

5. Functional Modeling (Isotropic Case)

the subgrid viscosity. This technique allows a better frequency localization of the information included in the model, and therefore a better treatment of the intermittence phenomena, as the model is sensitive only to the higher resolved frequencies. This result is obtained by applying a frequency high-pass ﬁlter to the resolved ﬁeld. 5. The damping functions for the near-wall region (p. 159), by which certain modiﬁcations in the turbulence dynamics and characteristic scales of the subgrid modes in the boundary layers can be taken into account. These functions are established in such a way as to cancel the subgrid viscosity models in the near-wall region so that they will not inhibit the driving mechanisms occurring in this area. These models are of limited generality as they presuppose a particular form of the ﬂow dynamics in the region considered. They also require that the relative position of each point with respect to the solid wall be known, which can raise problems in practice such as when using multidomain techniques or when several surfaces exist. And lastly, they constitute only an amplitude correction of the subgrid viscosity models for the forward energy cascade: they are not able to include any changes in the form of this mechanism, or the emergence of new mechanisms. The three “generalist” techniques (dynamic procedure, structural sensor, accentuation) for adapting the subgrid viscosity models are all based on extracting a test ﬁeld from the resolved scales by applying a test ﬁlter to these scales. This ﬁeld corresponds to the highest frequencies catured by the simulation, so we can see that all these techniques are based on a frequency localization of the information contained in the subgrid models. The loss of localness in space is reﬂected by the fact that the number of neighbors involved in the subgrid model computation is increased by using the test ﬁlter. Dynamic Procedures for Computing the Constants. Dynamic Models. Many dynamic procedures have been proposed to evaluate the parameters in the subgrid models. The following methods are presented 1. The original method proposed by Germano, and its modiﬁcation proposed by Lilly to improve its robustness (p. 137). Its recent improvements for complex kinetic energy spectrum shapes are also discussed. 2. The Lagrangian dynamic procedure (p. 144), which is well suited for fully non-hom*ogeneous ﬂows. 3. The constrained localized dynamic procedure (p. 146), which relax some strong assumptions used in the Germano–Lilly approach. q 4. The approximate localized dynamic procedure (p. 148), which is a simpliﬁcation of the constrained localized dynamic procedure that do nor requires to solve an integral problem to compute the dynamic constant. 5. The generalized dynamic procedures (p. 149), which aim at optimizing the approximation of the subgrid acceleration and make it possible to account for discretization errors.

5.3 Modeling of the Forward Energy Cascade Process

137

6. The dynamic inverse procedure (p. 150), which is designed to improve the dynamic procedure when the cutoﬀ is located at the very begining of the inertial range of the kinetic energy spectrum. 7. The Taylor series expansion based dynamic procedure (p. 151), which results in a diﬀerential expression for the dynamic constant, the test ﬁlter being replaced by its diﬀerential approximation. 8. The dynamic procedure based on dimensional parameters (p. 151), which yields a very simple expression. Germano–Lilly Dynamic Procedure. In order to adapt the models better to the local structure of the ﬂow, Germano et al. [253] proposed an algorithm for adapting the Smagorinsky model by automatically adjusting the constant at each point in space and at each time step. This procedure, described below, is applicable to any model that makes explicit use of an arbitrary constant Cd , such that the constant now becomes time- and space-dependent: Cd becomes Cd (x, t). The dynamic procedure is based on the multiplicative Germano identity (3.80) , now written in the form: Lij = Tij − τ˜ij

,

(5.138)

in which τij

≡

Tij

≡

Lij + Cij + Rij = ui uj − ui uj *i u*j , u/ i uj − u

Lij

≡

*i u*j u/ i uj − u

,

,

(5.139) (5.140) (5.141)

in which the tilde symbol tilde designates the test ﬁlter. The tensors τ and T are the subgrid tensors corresponding, respectively, to the ﬁrst and second ﬁltering levels. The latter ﬁltering level is associated with the characteristic * with ∆ * > ∆. Numerical tests show that an optimal value is ∆ * = 2∆. length ∆, The tensor L can be computed directly from the resolved ﬁeld. We then assume that the two subgrid tensors τ and T can be modeled by the same constant Cd for both ﬁltering levels. Formally, this is expressed: 1 τij − τkk δij 3 1 Tij − Tkk δij 3

= Cd βij

,

(5.142)

= Cd αij

,

(5.143)

in which the tensors α and β designate the deviators of the subgrid tensors obtained using the subgrid model deprived of its constant. It is important noting that the use of the same subgrid model with the same constant is equivalent to a scale-invariance assumption on both the subgrid ﬂuxes and the ﬁlter, to be discussed in the following.

138

5. Functional Modeling (Isotropic Case)

Table 5.1. Examples of subgrid model kernels for the dynamic procedure. Model (5.90) (5.102) (5.127)

βij

αij * 2 |S| *S * −2∆ + ij * (∆) * F * S * −2∆

2

−2∆ |S|S ij + −2∆ F (∆)S ij −2∆

1+α

|F(u)|α (qc2 )

1−α 2

S ij

* −2∆

1+α

ij

* α (˜ |F(u)| qc2 )

1−α 2

* S ij

Some examples of subgrid model kernels for αij and βij are given in Table 5.1. Introducing the above two formulas in the relation (5.138), we get23 : 1 Lij − Lkk δij ≡ Ldij = Cd αij − C/ d βij 3

.

(5.144)

We cannot use this equation directly for determining the constant Cd because the second term uses the constant only through a ﬁltered product [621]. In order to continue modeling, we need to make the approximation: * C/ d βij = Cd βij

,

(5.145)

which is equivalent to considering that Cd is constant over an interval at least equal to the test ﬁlter cutoﬀ length. The parameter Cd will thus be computed in such a way as to minimize the error committed24 , which is evaluated using the residual Eij : 1 Eij = Lij − Lkk δij − Cd αij + Cd β*ij 3

.

(5.146)

This deﬁnition consists of six independent relations, which in theory makes it possible to determine six values of the constant25 . In order to conserve a single relation and thereby determine a single value of the constant, Germano et al. propose contracting the relation (5.146) with the resolved strain rate tensor. The value sought for the constant is a solution of the problem: ∂Eij S ij =0 . (5.147) ∂Cd 23

24 25

It is important to note that, for the present case, the tensor Lij is replaced by its deviatoric part Ldij , because we are dealing with a zero-trace subgrid viscosity modeling. Meneveau and Katz [505] propose to use the dynamic procedure to rank the subgrid models, the best one being associated with the lowest value of the residual. Which would lead to the deﬁnition of a tensorial subgrid viscosity model.

5.3 Modeling of the Forward Energy Cascade Process

139

This method can be eﬃcient, but does raise the problem of indetermination when the tensor S ij cancels out. To remedy this problem, Lilly [448] proposes calculating the constant Cd by a least-squares method, by which the constant Cd now becomes a solution of the problem: ∂Eij Eij =0 , ∂Cd or Cd = in which

mij Ldij mkl mkl

mij = αij − β*ij

(5.148)

,

(5.149)

.

(5.150)

The constant Cd thus computed has the following properties: – It can take negative values, so the model can have an anti-dissipative eﬀect locally. This is a characteristic that is often interpreted as a modeling of the backward energy cascade mechanism. This point is detailed in Sect. 5.4. – It is not bounded, since it appears in the form of a fraction whose denominator can cancel out26 . These two properties have important practical consequences on the numerical solution because they are both potentially destructive of the stability of the simulation. Numerical tests have shown that the constant can remain negative over long time intervals, causing an exponential growth in the high frequency ﬂuctuations of the resolved ﬁeld. The constant therefore needs an ad hoc process to ensure the model’s good numerical properties. There are a number of diﬀerent ways of performing this process on the constant: statistical average in the directions of statistical hom*ogeneity [253, 779], in time or local in space [799]; limitation using arbitrary bounds [799] (clipping); or by a combination of these methods [779, 799]. Let us note that the averaging procedures can be deﬁned in two non-equivalent ways [801]: by averaging the denominator and numerator separately, which is denoted symbolically: Cd =

mij Ldij mkl mkl

,

or by averaging the quotient, i.e. on the constant itself: 0 1 mij Ldij Cd = Cd = . mkl mkl 26

(5.151)

(5.152)

This problem is linked to the implementation of the model in the simulation. In the continuous case, if the denominator tends toward zero, then the numerator cancels out too. These are calculation errors that lead to a problem of division by zero.

140

5. Functional Modeling (Isotropic Case)

The ensemble average can be performed over hom*ogeneous directions of the simulation (if they exist) or over diﬀerent realizations, i.e. over several statistically equivalent simulations carried out in parallel [102, 108]. The time average process is expressed: Cd (x, (n + 1)∆t) = a1 Cd (x, (n + 1)∆t) + (1 − a1 )Cd (x, n∆t) ,

(5.153)

in which ∆t is the time step used for the simulation and a1 ≤ 1 a constant. Lastly, the constant clipping process is intended to ensure that the following two conditions are veriﬁed: ν + νsgs ≥ 0

,

(5.154)

Cd ≤ Cmax

.

(5.155)

The ﬁrst condition ensures that the total resolved dissipation ε = νS ij S ij − τij S ij remains positive or zero. The second establishes an upper bound. In practice, Cmax is of the order of the theoretical value of the Smagorinsky constant, i.e. Cmax (0.2)2 . The models in which the constant is computed by this procedure are called “dynamic” because they automatically adapt to the local state of the ﬂow. When the Smagorinsky model is coupled with this procedure, it is habitually called the dynamic model, because this combination was the ﬁrst to be put to the test and is still the one most extensively used among the dynamic models. The dynamic character of the constant Cd is illustrated in Fig. 5.16, which displays the time history of the square root of the dynamic constant in freely decaying isotropic turbulence. It is observed that during the ﬁrst stage of the computation the constant is smaller than the theoretical value of the Smagorinsky constant Cd ∼ 0.18 given by equation (5.48), because the spectrum is not fully developed. In the second stage, when a self-similar state is reached, the theoretical value is automatically recovered. The use of the same value of the constant for the subgrid model at the two ﬁltering levels appearing in equation (5.138) implicitely relies on the two following self-similarity assumptions: – The two cutoﬀ wave numbers are located in the inertial range of the kinetic energy spectrum; – The ﬁlter kernels associated to the two ﬁltering levels are themselves selfsimilar. These two constraints are not automatically satisﬁed, and the validity of the dynamic procedure for computing the constant requires a careful analysis. Meneveau and Lund [507] propose an extension of the dynamic procedure for a cutoﬀ located in the viscous range of the spectrum. Writing the constant

5.3 Modeling of the Forward Energy Cascade Process

141

Fig. 5.16. Time history of the square root of the dynamic constant in large-eddy simulation of freely decaying isotropic turbulence (dynamic Smagorinsky model). Courtesy of E. Garnier, ONERA.

of the subgrid-scale model C as an explicit function of the ﬁlter characteristic length, the Germano–Lilly procedure leads to * =C C(∆) = C(∆) d

.

(5.156)

Let η be the Kolmogorov length scale. It was said in the introduction that the ﬂow is fully resolved if ∆ = η. Therefore, the dynamic procedure is consistent if, and only if lim Cd = C(η) = 0

.

(5.157)

∆→η

Numerical experiments carried out by the two authors show that the Germano–Lilly procedure is not consistent, because it returns the value of the constant associated to the test ﬁlter level * , Cd = C(∆)

(5.158)

* lim Cd = C(rη) = 0, r = ∆/∆ .

(5.159)

yielding ∆→η

Numerical tests also showed that taking the limit r → 1 or computing the two values C(∆) and C(r∆) using least-square-error minimization without

142

5. Functional Modeling (Isotropic Case)

assuming them to be equal yield inconsistent or ill-behaved solutions. A solution is to use prior knowledge to compute the dynamic constant. A robust algorithm is obtained by rewriting equation (5.146) as follows: Eij = Ldij − C(∆) f (∆, r)αij − β*ij , (5.160) where f (∆, r) = C(r∆)/C(∆) is evaluated by calculations similar to those of Voke (see page 118). A simple analytical ﬁtting is obtained in the case r = 2: f (∆, 2) ≈ max(100, 10−x), x = 3.23(Re−0.92 − Re−0.92 ) 2∆ ∆

,

(5.161)

where the mesh-Reynolds numbers are evaluated as (see equation (5.70)): Re∆ =

2 2 * ∆ |S| 4∆ |S| , Re2∆ = ν ν

.

Other cases can be considered where the similarity hypothesis between the subgrid stresses at diﬀerent resolution levels may be violated, leading to diﬀerent values of the constant [601]. Among them: – The case of a very coarse resolution, with a cutoﬀ located at the very beginning of the inertial range or in the production range. – The case of a turbulence undergoing rapid strains, where a transition length ∆T ∝ S −3/2 ε1/2 appears. Here, S and ε are the strain magnitude and the dissipation rate, respectively. Dimensional arguments show that, roughly speaking, scales larger than ∆T are rapidly distorted but have no time to adjust dynamically, while scales smaller than ∆T can relax faster via nonlinear interactions. For each of these cases, scale dependence of the model near the critical length scale is expected, which leads to a possible loss of eﬃciency of the classical Germano–Lilly dynamic procedure. A more general dynamic procedure, which does not rely on the assumption of scale similarity or location of the cutoﬀ in the dissipation range, was proposed by Port´e-Agel et al. [601]. This new scale-dependent dynamic procedure is obtained by considering a third ﬁltering level (i.e. a second test-ﬁltering * Filtered variables at > ∆. level) with a characteristic cutoﬀ length scale ∆ this new level are denoted by a caret. leads to Writing the Germano identity between level ∆ and level ∆ 1 j = C(∆)γ ui u Qij − Qkk δij ≡ ui uj − ij − C(∆)βij 3

,

(5.162)

and where γij and βij denote the expression of the subgrid model at levels ∆ ∆, respectively. By taking ∆)γ − β , (5.163) nij = Λ(∆, ij ij

5.3 Modeling of the Forward Energy Cascade Process

with ∆) = Λ(∆,

C(∆) C(∆)

,

143

(5.164)

we obtain the following value for the constant at level ∆: C(∆) =

nij Qij nij nij

.

(5.165)

By now considering relation (5.149), which expresses the Germano identity between the ﬁrst two ﬁltering levels, where mij is now written as * ∆)α − β2 , (5.166) mij = Λ(∆, ij ij and by equating the values of C(∆) obtained using the two test-ﬁltering levels, we obtain the following relation: (Lij mij )(nij nij ) − (Qij nij )(mij mij ) = 0 ,

(5.167)

* ∆) and Λ(∆, ∆). In order to obtain a closed which has two unknowns, Λ(∆, system, some additional assumptions are needed. It is proposed in [601] to assume a power-law scaling of the dynamic constant, C(x) ∝ xp , leading to C(a∆) = C(∆)ar

.

(5.168)

For this power-law behavior, the function Λ(., .) does not depend on the scales but only on the ratio of the scales, i.e. Λ(x, y) = (x/y)r . Using this simpliﬁcation, (5.167) appears as a ﬁfth-order polynomial in C(∆). The dynamic constant is taken equal to the largest root. We now consider the problem of the ﬁlter self-similarity. Let G1 and G2 be the ﬁlter kernels associated with the ﬁrst and second ﬁltering level. For * = ∆ . We assume the sake of simplicity, we use the notations ∆ = ∆1 and ∆ 2 that the ﬁlter kernels are rewritten in a form such that: |x − ξ| u(ξ)dξ , (5.169) u(x) = G1 u(x) = G1 ∆1 |x − ξ| * u(x) = G2 u(x) = G2 u(ξ)dξ . (5.170) ∆2 We also introduce the test ﬁlter Gt , which is deﬁned such that * = G2 u = Gt u = Gt G1 u . u The ﬁlters G1 and G2 are self-similar if and only if y 1 G1 (y) = d G2 , r = ∆2 /∆1 . r r

(5.171)

(5.172)

144

5. Functional Modeling (Isotropic Case)

Hence, the two ﬁlters must have identical shapes and may only diﬀer by their associated characteristic length. The problem is that in practice only Gt is known, and the self-similarity property might not be a priori veriﬁed. Carati and Vanden Eijnden [104] show that the interpretation of the resolved ﬁeld is fully determined by the choice of the test ﬁlter Gt , and that the use of the same model for the two levels of ﬁltering is fully justiﬁed. This is demonstrated by re-interpreting previous ﬁlters in the following way. Let us consider an inﬁnite set of self-similar ﬁlters {Fn ≡ F (ln )} deﬁned as x 1 (5.173) Fn (x) = n F , ln = r n l0 , r ln where F , r > 1 and l0 are the ﬁlter kernel, an arbitrary parameter and a reference length, respectively. Let us introduce a second set {Fn∗ ≡ F ∗ (ln∗ )} deﬁned by (5.174) Fn∗ ≡ Fn Fn−1 ... F−∞ . For positive kernel F , we get the following properties: – The length ln∗ obeys the same geometrical law as ln : ∗ ln∗ = rln−1 ,

r and ln∗ = √ ln r2 − 1

.

(5.175)

– {Fn∗ } constitute a set of self-similar ﬁlters. Using these two set of ﬁlters, the classical ﬁlters involved in the dynamic procedure can be deﬁned as self-similar ﬁlters: Gt (∆t ) = G1 (∆1 ) =

Fn (ln ) , ∗ ∗ Fn−1 (ln−1 ) ,

(5.176) (5.177)

G2 (∆2 ) =

Fn∗ (ln∗ ) .

(5.178)

For any test-ﬁlter Gt and any value of r, the ﬁrst ﬁlter operator can be constructed explicitly: G1 = Gt (∆t /r) Gt (∆t /r2 ) ... Gt (∆t /r∞ ) .

(5.179)

This relation shows that for any test ﬁlter of the form (5.176), the two ﬁltering operators can be rewritten as self-similar ones, justifying the use of the same model at all the ﬁltering levels. Lagrangian Dynamic Procedure. The constant regularization procedures based on averages in the hom*ogeneous directions have the drawback of not being usable in complex conﬁgurations, which are totally inhom*ogeneous. One technique for remedying this problem is to take this average along the ﬂuid particle trajectories. This new procedure [508], called the dynamic Lagrangian procedure, has the advantage of being applicable in all conﬁgurations.

5.3 Modeling of the Forward Energy Cascade Process

145

The trajectory of a ﬂuid particle located at position x at time t is, for times t previous to t, denoted as:

z(t ) = x −

t

u[z(t ), t ]dt

.

(5.180)

t

The residual (5.146) is written in the following Lagrangian form: Eij (z, t ) = Lij (z, t ) − Cd (x, t)mij (z, t ) .

(5.181)

We see that the value of the constant is ﬁxed at point x at time t, which is equivalent to the same linearization operation as for the Germano–Lilly procedure. The value of the constant that should be used for computing the subgrid model at x at time t is determined by minimizing the error along the ﬂuid particle trajectories. Here too, we reduce to a well-posed problem by deﬁning a scalar residual Elag , which is deﬁned as the weighted integral along the trajectories of the residual proposed by Lilly: Elag =

t

−∞

Eij (z(t ), t )Eij (z(t ), t )W (t − t )dt

,

(5.182)

in which the weighting function W (t−t ) is introduced to control the memory eﬀect. The constant is a solution of the problem: ∂Elag = ∂Cd

t

−∞

2Eij (z(t ), t )

∂Eij (z(t ), t ) W (t − t )dt = 0 , ∂Cd

or: Cd (x, t) =

JLM JMM

,

(5.183)

(5.184)

in which JLM (x, t) =

t

−∞

JMM (x, t) =

t

−∞

Lij mij (z(t ), t )W (t − t )dt

,

(5.185)

mij mij (z(t ), t )W (t − t )dt

.

(5.186)

These expressions are non-local in time, which makes them unusable for the simulation, because they require that the entire history of the simulation be kept in memory, which exceeds the storage capacities of today’s supercomputers. To remedy this, we choose a fast-decay memory function W : W (t − t ) =

1 Tlag

t − t exp − Tlag

,

(5.187)

146

5. Functional Modeling (Isotropic Case)

in which Tlag is the Lagrangian correlation time. With the memory function in this form, we can get the following equations: ∂JLM ∂JLM 1 DJLM ≡ + ui = (Lij mij − JLM ) Dt ∂t ∂xi Tlag ∂JMM ∂JMM DJMM 1 ≡ + ui = (mij mij − JMM ) Dt ∂t ∂xi Tlag

,

(5.188)

,

(5.189)

the solution of which can be used to compute the subgrid model constant at each point and at each time step. The correlation time Tlag is estimated by tests in isotropic hom*ogeneous turbulence at: Tlag (x, t) = 1.5 ∆ (JMM JLM )

−1/8

,

(5.190)

which comes down to considering that the correlation time is reduced in the high-shear regions where JMM is large, and in those regions where the non-linear transfers are high, i.e. where JLM is large. This procedure does not guarantee that the constant will be positive, and must therefore be coupled with a regularization procedure. Meneveau et al. [508] recommend a clipping procedure. Solving equations (5.188) and (5.189) yields a large amount of additional numerical work, resulting in a very expensive subgrid model. To alleviate this problem, the solution to these two equations may be approximated using the following Lagrangian tracking technique [596]: JLM (x, n∆t) = +

a Lij (x, n∆t)mij (x, n∆t) (1 − a)JLM (x − ∆tu(x, n∆t), (n − 1)∆t) , (5.191)

JMM (x, n∆t) = +

a mij (x, n∆t)mij (x, n∆t) (1 − a)JMM (x − ∆tu(x, n∆t), (n − 1)∆t) , (5.192)

where a=

∆t/Tlag 1 + ∆t/Tlag

.

(5.193)

This new procedure requires only the storage of the two parameters JLM and JMM at the previous time step and the use of an interpolation procedure. The authors indicate that a linear interpolation is acceptable. Constrained Localized Dynamic Procedure. Another generalization of the Germano–Lilly dynamic procedure was proposed for inhom*ogeneous cases by Ghosal et al. [261]. This new procedure is based on the idea of minimizing an integral problem rather than a local one in space, as is done in the Germano–Lilly procedure, which avoids the need to linearize the constant

5.3 Modeling of the Forward Energy Cascade Process

147

when applying the test ﬁlter. We now look for the constant Cd that will minimize the function F [Cd ], with F [Cd ] = Eij (x)Eij (x)d3 x , (5.194) in which Eij is deﬁned from relation (5.144) and not by (5.146) as was the case for the previously explained versions of the dynamic procedure. The constant sought is such that the variation of F [Cd ] is zero: δF [Cd ] = 2 Eij (x)δEij (x)d3 x = 0 , (5.195) or, by replacing Eij with its value: δCd d3 x = 0 . −αij Eij δCd + Eij βij/

(5.196)

Expressing the convolution product associated with the test ﬁlter, we get: 3 −αij Eij + βij Eij (y)G(x − y)d y δCd (x)d3 x = 0 , (5.197) from which we deduce the following Euler-Lagrange equation: −αij Eij + βij Eij (y)G(x − y)d3 y = 0 .

(5.198)

This equation can be re-written in the form of a Fredholm’s integral equation of the second kind for the constant Cd : f (x) = Cd (x) − K(x, y)Cd (y)d3 y , (5.199) where f (x) =

1 αkl (x)αkl (x)

αij (x)Lij (x) − βij (x) Lij (y)G(x − y)d3 y

,

(5.200) K(x, y) =

KA (x, y) + KA (y, x) + KS (x, y) αkl (x)αkl (x)

,

(5.201)

and KA (x, y) = αij (x)βij (y)G(x − y) , KS (x, y) =βij (x)βij (y) G(z − x)G(z − y)d3 z

(5.202) .

(5.203)

148

5. Functional Modeling (Isotropic Case)

This new formulation raises no problems concerning the linearization of the constant, but does not solve the instability problems stemming from the negative values it may take. This procedure is called the localized dynamic procedure. To remedy the instability problem, the authors propose constraining the constant to remain positive. The constant Cd (x) is then expressed as the square of a new real variable ξ(x). Replacing the constant with its decomposition as a function of ξ, the Euler-Lagrange equation (5.198) becomes: 3 (5.204) −αij Eij + βij Eij (y)G(x − y)d y ξ(x) = 0 . This equality is true if either of the factors is zero, i.e. if ξ(x) = 0 or if the relation (5.198) is veriﬁed, which is denoted symbolically Cd (x) = G[Cd (x)]. In the ﬁrst case, the constant is also zero. To make sure it remains positive, the constant is computed by an iterative procedure: ⎧ ⎨ G[Cd(n) (x)] if G[Cd(n) (x)] ≥ 0 (n+1) Cd (x) = , (5.205) ⎩ 0 otherwise

in which G[Cd (x)] = f (x) −

K(x, y)Cd (y)d3 y

.

(5.206)

This completes the description of the constrained localized dynamic procedure. It is applicable to all conﬁgurations and ensures that the subgrid model constant remains positive. This solution is denoted symbolically: Cd (x) = f (x) + K(x, y)Cd (y)d3 y , (5.207) +

in which + designates the positive part. Approximate Localized Dynamic Procedure. The localized dynamic procedure decribed in the preceding paragraph makes it possible to regularize the dynamic procedure in fully non-hom*ogeneous ﬂows, and removes the mathematical inconsistency of the Germano–Lilly procedure. But it requires to solve an integral equation, and thus induces a signiﬁcant overhead. To alleviate this problem, Piomelli and Liu [596] propose an Approximate Localized Dynamic Procedure, which is not based on a variational approach but on a time extrapolation process. Equation (5.144) is recast in the form ∗ Ldij = Cd αij − C/ d βij

,

(5.208)

where Cd∗ is an estimate of the dynamic constant Cd , which is assumed to be known. Writing the new formulation of the residual Eij , the dynamic

5.3 Modeling of the Forward Energy Cascade Process

149

constant is now evaluated as Cd =

∗ αij (Ldij + C/ d βij ) αij αij

.

(5.209)

The authors propose to evaluate the estimate Cd∗ by a time extrapolation: Cd∗

=

(n−1) Cd

(n−1) ∂Cd + ∆t + ... , ∂t

(5.210)

where the superscript (n − 1) is related to the value of the variable at the (n − 1)th time step, and ∆t is the value of the time step. In practice, Piomelli and Liu consider ﬁrst- and second-order extrapolation schemes. The resulting dynamic procedure is fully local, and does not induce large extra computational eﬀort as the original localized procedure does. Numerical experiments carried out by these authors demonstrate that it still requires clipping to yield a well-behaved algorithm. Generalized Dynamic Procedure. It is also possible to derive a dynamic procedure using the generalized Germano identity (3.87) [629]. We assume that the operator L appearing in equation (3.88) is linear, and that there exists a linear operator L such that L(a N ) = aL(N ) + L (a, N ) ,

(5.211)

where a is a scalar real function and N an arbitrary second rank tensor. The computation of the dynamic constant Cd is now based on the minimization of the residual Eij Eij = L(Ldij ) − Cd L(mij ) , (5.212) where Ldij and mij are deﬁned by equations (5.144) and (5.150). A leastsquare minimizations yields: Cd =

L(Ldij )L(mij ) L(mij )L(mij )

.

(5.213)

The reduction of the residual obtained using this new relation with respect to the classical one is analyzed by evaluating the diﬀerence: − L(Eij ) , δEij = Eij

(5.214)

is given by relation (5.212) and Eij by (5.146). Inserting the where Eij two dynamic constants Cd and Cd , deﬁned respectively by relations (5.213) and (5.149), we get:

δEij = (Cd − Cd )L(mij ) + L (Cd , mij )

.

(5.215)

150

5. Functional Modeling (Isotropic Case)

An obvious example for the linear operator L is the divergence operator. The associated L is the gradient operator. An alternative consisting in minimizing a diﬀerent form of the residual has been proposed by Morinishi and Vasilyev [542, 544] and Mossi [550]: Eij

=

L(Ldij ) − L(Cd mij )

=

L(Ldij )

(5.216)

− Cd L(mij ) − L (Cd , mij )

.

(5.217)

The use of this new form of the residual generally requires solving a differential equation, and then yields a more complex procedure than the form (5.212). These two procedures theoretically more accurate results than the classical one, because they provide reduce the error committed on the subgrid force term itself, rather than on the subgrid tensor. They also take into account for the numerical error associated to the discrete form of L. Dynamic Inverse Procedure. We have already seen that the use of the dynamic procedure may induce some problems if the cutoﬀ is not located in the inertial range of the spectra, but in the viscous one. A similar problem arises if the cutoﬀ wave number associated to the test ﬁlter occurs at the very beginning of the inertial range, or in the production range of the spectrum. In order to compensate inaccuracies arising from the use of a large ﬁlter length associated with the test ﬁlter, Kuerten et al. [415] developed a new approach, referred to as the Dynamic Inverse Procedure. It relies on the idea that if a dynamic procedure is developed involving only length scales comparable to the basic ﬁlter length, self-similarity properties will be preserved and consistent modeling may result. Such a procedure is obtained by deﬁning the ﬁrst ﬁltering operator G and the second one F by G = H −1 ◦ L, F = H

,

(5.218)

where L is the classical ﬁlter level and H an explicit test ﬁlter, whose inverse H −1 is assumed to be known explicitly. Inserting these deﬁnitions into the Germano identity (3.80), we get a direct evaluation of the subgrid tensor τ : [F G , B](ui , uj ) = [L , B](ui , uj ) = ui uj − ui uj ≡ τij

(5.219) (5.220) (5.221)

= [H , B] ◦ (H −1 L )(ui , uj ) +(H ) ◦ [H −1 L , B](ui , uj ) .

(5.222)

This new identity can be recast in a form similar to the original Germano identity (5.223) Lij = τij − H Tij ,

5.3 Modeling of the Forward Energy Cascade Process

151

with Lij = H ((H −1 ui )(H −1 uj )) − ui uj

,

Tij = H −1 ui uj − (H −1 ui )(H −1 uj ) . The term Lij is explicitly known in practice, and does not require any modeling. Using the same notation as in the section dedicated to the Germano–Lilly procedure, we get, for the Smagorinsky model: τij Tij

=

2

Cd βij , βij = −2∆ |S|S ij

,

(5.224)

,

(5.225)

2

=

|S| S Cd αij , αij = −2∆ ij

and S are the characteristic length and the strain rate tensor assowhere ∆ ciated to the H −1 ◦ L ﬁltering level, respectively. Building the residual Eij as Eij = Lij − Cd (βij − H αij ) = Lij − Cd mij

,

(5.226)

the least-square-error minimization procedure yields: Cd =

Lij mij mij mij

.

(5.227)

Since the In this new procedure, the two lengths involved are ∆ and ∆. ≤ ∆, ensurlatter is associated to an inverse ﬁltering operator, we get ∆ ing that the dynamic procedure will not bring in lengths associated to the production range of the spectrum. In practice, this procedure is observed to suﬀer the same stability problems than the Germano–Lilly procedure, and needs to be used together with a stabilization procedure (averaging, clipping, etc.). Taylor Series Expansion Based Dynamic Models. The dynamic procedures presented above rely on the use of a discrete test ﬁlter. Chester et al. [129] proposed a new formulation for the dynamic procedure based on the diﬀerential approximation of the test ﬁlter. All quantities apprearing at the test ﬁlter level can therefore be rewritten as sums and products of partial derivatives of the resolved velocity ﬁeld, leading to a new expression of dynamic constants which involves only higher-order derivatives of the velocity ﬁeld. Dynamic Procedure with Dimensional Constants. The dynamic procedures described in the preceding paragraphs are designed to ﬁnd the best values of non-dimensional constants in subgrid scale models. Wong and Lilly [766], followed by Carati and his co-workers [103] propose to extend this procedure to evaluate dimensional parameters which appear in some models. They applied

152

5. Functional Modeling (Isotropic Case)

this idea to the so-called Kolmogov formulation for the subgrid viscosity: νsgs = C∆

4/3 1/3

ε

4/3

= Cε ∆

,

(5.228)

where the parameter Cε = Cε1/3 has the dimension of the cubic root of the subgrid dissipation rate ε. Introducing this closure at both grid and test ﬁltering levels, one obtains (the tilde symbol is related to the test ﬁlter level): τij

=

4/3

−2Cε ∆

4/3

Tij

=

* −2Cε ∆

S ij

(5.229)

* S ij

(5.230)

leading to the following expression of the residual 4/3 * 4/3 S * + C ∆/ Eij = Ldij − Cε ∆ S ij ij ε

.

(5.231)

This expression can be used to generate integral expressions for Cε . A very 4/3 simple local deﬁnition is recovered further assuming that Cε ∆ is almost * Using the additional property that constant over distance of the order of ∆. the test ﬁlter perfectly commutes with spatial derivatives, relation (5.231) simpliﬁes as 4/3 * 4/3 S * Eij = Ldij − 2Cε ∆ − ∆ . (5.232) ij The least-square optimization method therefore yields the following formula for the dynamic Cε : Cε =

* Ldij S 1 ij 4/3 * * 4/3 * S ij S ij 2 ∆ −∆

.

(5.233)

As the original Germano-Lilly procedure for non-dimensional parameters, this procedure suﬀers some numerical instability problems and must therefore be regularized using clipping and/or averaging. Dynamic Procedures Without the Germano Identity. Multilevel Procedure by Terracol and Sagaut. This method proposed by Terracol and Sagaut [709] relies on the hypothesis that the computed resolved kinetic energy spectrum obeys a power-law like E(k) = E0 k α

,

(5.234)

where α is the scaling parameter. It is worth noting that Barenblatt [41] suggests that both E0 and α might be Reynolds-number dependent. A more

5.3 Modeling of the Forward Energy Cascade Process

153

accurate expression for the kinetic energy spectrum is E(k) = K0 ε2/3 k −5/3 (kΛ)ζ

,

(5.235)

where K0 = 1.4 is the Kolmogorov constant, Λ a length scale and ζ an intermittency factor. Under this assumption, the mean subgrid dissipation rate across a cutoﬀ wave number kc , ε(kc ), scales like ε(kc ) = ε0 kcγ ,

γ=

3 3α + 5 = ζ 2 2

.

(5.236)

where ε0 is a kc -independent parameter. It is observed that in the Kolmogorov case (α = −5/3), one obtains γ = 0, leading to a constant dissipation rate. Let us now introduce a set of cutoﬀ wave numbers kn , with k1 > k2 > ... The following recursive law is straithgforwardly derived from (5.236) ε(kn ) γ = Rn,n+1 , ε(kn+1 )

Rn,n+1 =

kn kn+1

,

(5.237)

leading to the following two-level evaluation of the parameter γ: γ=

log(ε(kn )/ε(kn+1 )) log(Rn,n+1 )

.

(5.238)

Now introducing a generic subgrid model for the nth cutoﬀ level n

τijn = Cfij (un , ∆ ) ,

(5.239)

where C is the constant of the model to be dynamically computed, un the n resolved ﬁeld at the considered cutoﬀ level and ∆ ≡ π/kn the current cutoﬀ length, the dissipation rate can also be expressed as n

n

n

ε(kn ) = −τijn S ij = −Cfij (un , ∆ )S ij

,

(5.240)

n

where S is the resolved strain rate at level n. Equation (5.237) shows that the ratio ε(kn )/ε(kn+1 ) is independent of the model constant C. Using this property, Terracol and Sagaut propose to introduce two test ﬁlter levels k2 and k3 (k1 being the grid ﬁlter level where the equations must be closed, i.e. 1 ∆ = ∆ ). The intermittency factor γ is then computed using relation (5.238), and one obtains the following evaluation for the subgrid dissipation rate at the grid ﬁlter level: γ ε (k2 ) , (5.241) ε(k1 ) = R1,2 where ε (k2 ) is evaluated using a reliable approximation of the subgrid tensor to close the sequence (in practice, a scale-similarity model is used in Ref. [709]). The corresponding value of C for the model at the grid level is then deduced from (5.240): γ C = R1,2

ε (k2 ) −fij (u, ∆)S ij

.

(5.242)

154

5. Functional Modeling (Isotropic Case)

Multiscale Method Based on the Kolmogorov-Meneveau Equation. Another procedure was developed by Shao [669, 153] starting from the KolmogorovMeneveau equation for ﬁltered third-order velocity structure function: 4 − rε = DLLL − 6GLLL 5

,

(5.243)

where DLLL is the third-order longitudinal velocity correlation of the ﬁltered ﬁeld, GLLL (r) the longitudinal velocity-stress correlation tensor and ε = −τij S ij the average subgrid dissipation (see p. 126 for additional details). Now assuming that the following self-similarity law is valid GLLL (r) ∝ rp

,

(5.244)

where p = −1/3 corresponds to the Kolmogorov local isotropy hypotheses, one obtains the following relationship for two space increments r1 and r2 : 0.8r1 ε + DLLL (r1 ) = 0.8r2 ε + DLLL (r2 )

r1 r2

−1/3 .

(5.245)

Now introducing the same generic subgrid closure as for the Terracol– Sagaut procedure τij = Cfij (u, ∆)

,

(5.246)

and inserting it into (5.245) to evaluate ε = −τij S ij , taking r1 = ∆ and r2 > r1 , the dynamic value of the constant C is

C=

−1/3

DLLL (r2 ) − DLLL (∆) −1/3 0.8fij (u, ∆)S ij ∆ − r∆2 r2 ∆ r2

.

(5.247)

The only ﬁxed parameter in the Shao procedure is the scaling parameter p in (5.244). This parameter can be computed dynamically introducing a third space increment r3 , leading to the deﬁnition of a dynamic procedure with the same properties as the one proposed by Terracol and Sagaut. The proposal of Shao can also be extended to subgrid models with several adjustable constant by introducing an additional space increment for each new constant and solving a linear algebra problem. Structural Sensors. Selective Models. In order to improve the prediction of intermittent phenomena, we introduce a sensor based on structural information. This is done by incorporating a selection function in the model, based on the local angular ﬂuctuations of the vorticity, developed by David [166, 440].

5.3 Modeling of the Forward Energy Cascade Process

155

The idea here is to modulate the subgrid model in such a way as to apply it only when the assumptions underlying the modeling are veriﬁed, i.e. when all the scales of the exact solution are not resolved and the ﬂow is of the fully developed turbulence type. The problem therefore consists in determining if these two hypotheses are veriﬁed at each point and each time step. David’s structural sensor tests the second hypothesis. To do this, we assume that, if the ﬂow is turbulent and developed, the highest resolved frequencies have certain characteristics speciﬁc to isotropic hom*ogeneous turbulence, and particularly structural properties. So the properties speciﬁc to isotropic hom*ogeneous turbulence need to be identiﬁed. David, taking direct numerical simulations as a base, observed that the probability density function of the local angular ﬂuctuation of the vorticity vector exhibit a peak around the value of 20o . Consequently, he proposes identifying the ﬂow as being locally under-resolved and turbulent at those points for which the local angular ﬂuctuations of the vorticity vector corresponding to the highest resolved frequencies are greater than or equal to a threshold value θ0 . The selection criterion will therefore be based on an estimation of the angle θ between the instantaneous vorticity vector ω and the local average ˜ (see Fig. 5.17), which is computed by applying a test ﬁlter vortcity vector ω to the vorticity vector. The angle θ is given by the following relation: ˜ ω(x) × ω(x) θ(x) = arcsin . (5.248) ˜ ω(x).ω(x) We deﬁne a selection function to damp the subgrid model when the angle θ is less than a threshold angle θ0 . In the original version developed by David, the selection function fθ0 is a Boolean operator:

1 if θ ≥ θ0 fθ0 (θ) = . (5.249) 0 otherwise

Fig. 5.17. Local angular ﬂuctuation of the vorticity vector.

156

5. Functional Modeling (Isotropic Case)

This function is discontinuous, which may pose problems in the numerical solution. One variant of it that exhibits no discontinuity for the threshold value is deﬁned as follows [636]:

1 if θ ≥ θ0 fθ0 (θ) = , (5.250) r(θ)n otherwise in which θ0 is the chosen threshold value and r the function: r(θ) =

tan2 (θ/2) tan2 (θ0 /2)

,

(5.251)

where the exponent n is positive. In practice, it is taken to be equal to 2. Considering the fact that we can express the angle θ as a function of the ˜ and the norm norms of the vorticity vector ω, the average vorticity vector ω, ˜ by the relation: ω of the ﬂuctuating vorticity vector deﬁned as ω = ω − ω, ˜ 2 + ω 2 − 2˜ ωω cos θ ω = ω 2

,

and the trigonometric relation: tan2 (θ/2) =

1 − cos θ 1 + cos θ

,

the quantity tan2 (θ/2) is estimated using the relation: 2˜ ωω − ω ˜ 2 − ω2 + ω 2˜ ωω + ω ˜ 2 + ω2 − ω2 2

tan2 (θ/2) =

.

(5.252)

The selection function is used as a multiplicative factor of the subgrid viscosity, leading to the deﬁnition of selective models: νsgs = νsgs (x, t)fθ0 (θ(x)) ,

(5.253)

in which νsgs is calculated by an arbitrary subgrid viscosity model. It should be noted that, in order to keep the same average subgrid viscosity value over the entire ﬂuid domain, the constant that appears in the subgrid model has to be multiplied by a factor of 1.65. This factor is evaluated on the basis of isotropic hom*ogeneous turbulence simulations. Accentuation Technique. Filtered Models. Accentuation Technique. Since large-eddy simulation is based on a frequency selection, improving the subgrid models in the physical space requires a better diagnostic concerning the spectral distribution of the energy in the calculated solution. More precisely, what we want to do here is to determine if the exact solution is entirely resolved, in which case the subgrid model should be reduced to zero, or if there exist subgrid scales that have to be taken into account by means of a model. When models expressed in the physical space

5.3 Modeling of the Forward Energy Cascade Process

157

do not use additional variables, they suﬀer from imprecision due to GaborHeisenberg’s principle of uncertainty already mentioned above, because the contribution of the low frequencies precludes any precise determination of the energy at the cutoﬀ. Let us recall that, if this energy is zero, the exact solution is completely represented and, if otherwise, then subgrid modes exist. In order to be able to detect the existence of the subgrid modes better, Ducros [204, 205] proposes an accentuation technique which consists in applying the subgrid models to a modiﬁed velocity ﬁeld obtained by applying a frequency high-pass ﬁlter to the resolved velocity ﬁeld. This ﬁlter, denoted HPn , is deﬁned recursively as: HP1 (u) n

HP (u)

2

∆ ∇2 u = HP(HP

, n−1

(5.254) (u))

.

(5.255)

We note that the application of this ﬁlter in the discrete case results in a loss of localness in the physical space, which is in conformity with GaborHeisenberg’s principle of uncertainty. We use EHPn (k) to denote the energy spectrum of the ﬁeld thus obtained. This spectrum is related to the initial spectrum E(k) of the resolved scales by: E HPn (k) = THPn (k)E(k) ,

(5.256)

in which THPn (k) is a transfer function which Ducros evaluates in the form: γn k . (5.257) THPn (k) = bn kc Here, b and γ are positive constants that depend on the discrete ﬁlter used in the numerical simulation27 . The shape of the spectrum obtained by the transfer function to a Kolmogorov spectrum is graphed in Fig. 5.18 for several values of the parameter n. This type of ﬁlter modiﬁes the spectrum of the initial solution by emphasizing the contribution of the highest frequencies. The resulting ﬁeld therefore represents mainly the high frequencies of the initial ﬁeld and serves to compute the subgrid model. To remain consistent, the subgrid model has to be modiﬁed. Such models are called ﬁltered models. The case of the Structure Function model is given as an example. Filtered versions of the Smagorinsky and Mixed Scale models have been developped by Sagaut, Comte and Ducros [628]. Filtered Second-Order Structure Function Model. We deﬁne the second-order structure function of the ﬁltered ﬁeld: HPn 2 DLL (x, r, t) = [HPn (u)(x, t) − HPn (u)(x + x , t)] d3 x , (5.258) |x |=r

27

For a Laplacian type ﬁlter discretized by second-order accurate ﬁnite diﬀerence scheme iterated three times (n = 3), Ducros ﬁnds b3 = 64, 000 and 3γ = 9.16.

158

5. Functional Modeling (Isotropic Case)

Fig. 5.18. Energy spectrum of the accentuation solution for diﬀerent values of the parameter n (b = γ = 1, kc = 1000).

for which the statistical average over the entire ﬂuid domain, denoted HPn D LL (r, t), is related to the kinetic energy spectrum by the relation:

HPn

kc

DLL (r, t) = 4 0

sin(k∆) E HPn (k) 1 − dk k∆

.

(5.259)

According to the theorem of averages, there exists a wave number k∗ ∈ [0, kc ] such that: HPn

E

HPn

D (r, t) πLL (k∗ ) = 4(π/kc ) (1 − sin(ξ)/ξ) dξ

.

(5.260)

Using a Kolmogorov spectrum, we can state the equality: E(kc ) −5/3 kc

=

E HPn (k∗ ) k∗

.

(5.261)

Considering this last relation, along with (5.256) and (5.257), the subgrid viscosity models based on the energy at cutoﬀ are expressed: −3/2

2 K0 νsgs = 3 kc1/2

'

k∗ kc

5/3−γn

1 E HPn (k∗ ) , bn

(5.262)

5.3 Modeling of the Forward Energy Cascade Process

in which:

k∗ kc

−5/3+γn =

1

π 0

π −5/3+γn

ξ −5/3+γn (1 − sin(ξ)/ξ) dξ π (1 − sin(ξ)/ξ) dξ

.

159

(5.263)

By localizing these relations in the physical space, we deduce the ﬁltered structure function model: HPn 1/2 1/2 −3/2 DLL (x, ∆, t) ∆ π γn 2 K0 νsgs (x, ∆, t) = 1/2 π 3 π 4/3 2 bn −5/3+γn ξ (1 − sin(ξ)/ξ) dξ 0 + HPn = C (n) ∆ DLL (x, ∆, t) . (5.264) The values of the constant C (n) are given in the following Table: In practice, Ducros recommends using n = 3. Table 5.2. Values of the Structure Function model constant for diﬀerent iterations of the high-pass ﬁlter. n C

(n)

1

2

3

4

0.0637

0.020

0.0043

0.000841

1.57 · 10−4

Damping Functions for the Near-wall Region. The presence of a solid wall modiﬁes the turbulence dynamics in several ways, which are discussed in Chap. 10. The only fact concerning us here is that the presence of a wall inhibits the growth of the small scales. This phenomenon implies that the characteristic mixing length of the subgrid modes ∆f has to be reduced in the near-surface region, which corresponds to a reduction in the intensity of the subgrid viscosity. To represent the dynamics in the near-wall region correctly, it is important to make sure that the subgrid models verify the good properties in this region. In the case of a canonical boundary layer (see Chap. 10), the statistical asymptotic behavior of the velocity components and subgrid tensions can be determined analytically. Let u be the main velocity component in the x direction, v the transverse component in the y direction, and w the velocity component normal to the wall, in the z direction. Using the incompressibility constraint, a Taylor series expansion of the velocity component in the region very near the wall yields: u ∝ z, v ∝ z, w ∝ z 2 , τ11 ∝ z 2 , τ22 ∝ z 2 , τ13 ∝ z 3 , τ12 ∝ z 2 ,

τ33 ∝ z 4 , τ23 ∝ z 3 .

(5.265)

(5.266)

160

5. Functional Modeling (Isotropic Case)

Experience shows that it is important to reproduce the behavior of the component τ13 in order to ensure the quality of the simulation results. It is generally assumed that the most important stress in the near-wall region is τ13 , because it is directly linked to the mean turbulence production term, P , which is evaluated as du P ∝ τ13 . (5.267) dz Thus, it is expected that subgrid-viscosity models will be such that νsgs

du ∝ τ13 ∝ z 3 dz

.

(5.268)

We deduce the following law from relations (5.265) and (5.268): νsgs ∝ z 3

.

(5.269)

We verify that the subgrid-viscosity models based on the large scales alone do not verify this asymptotic behavior. This is understood by looking at the wall value of the subgrid viscosity associated with the mean velocity ﬁeld u. A second-order Taylor series expansion of some zero-equation subgrid viscosity models presented in the preceding section yields: 2 ∂u (Smagorinsky) , νsgs |w ∝ ∆|w ∂z w ∂u (2nd order Structure Function) , νsgs |w ∝ ∆|w ∆z1 ∂z w ∂u 1/2 ∂ 2 u 1/2 3/2 νsgs |w ∝ ∆|w ∆z1 (Mixed Scale) , ∂z w ∂z 2 w (5.270) where the w subscript denotes values taken at the wall, and ∆z1 is the distance to the wall at which the model is evaluated. Because ∂u/∂z is not zero at the wall,28 , we have the following asymptotic scalings of the modeled subgrid viscosity at solid walls: νsgs |w νsgs |w

= O(∆|2w ) (Smagorinsky) , = O(∆|w ∆z1 ) (2nd order Structure Function) ,

νsgs |w

= 0

(Mixed Scale) . (5.271)

In practice, the Mixed Scale model can predict a zero subgrid viscosity at the wall if the computational grid is ﬁne enough to make it possible to 28

Or, equivalently, the skin friction is not zero.

5.3 Modeling of the Forward Energy Cascade Process

161

evaluate correctly the second-order wall–normal velocity derivative, i.e. if at least three grid points are located within the region where the mean velocity proﬁle obeys a linear law. Consequently, the ﬁrst two models must be modiﬁed in the near-wall region in order to enforce a correct asymptotic behavior of the subgrid terms in that region. This is done by introducing damping functions. The usual relation: ∆f = C∆

,

(5.272)

is replaced by: ∆f = C∆fw (z) ,

(5.273)

in which fw (z) is the damping function and z the distance to the wall. From Van Driest’s results, we deﬁne: fw (z) = 1 − exp (−zuτ /25ν)

,

(5.274)

in which the friction velocity uτ is deﬁned in Sect. 10.2.1. Piomelli et al. [600] propose the alternate form: 1/2 fw (z) = 1 − exp −(zuτ /25ν)3

.

(5.275)

From this last form we can get a correct asymptotic behavior of the sub3 grid viscosity, i.e. a decrease in z + in the near-wall region, contrary to the Van Driest function. Experience shows that we can avoid recourse to these functions by using a dynamic procedure, a ﬁltered model, a selective model, or the Yoshizawa model. It is worth noting that subgrid viscosity models can be designed, which automatically follow the correct behavior in the near-wall region. An example is the WALE model, developed by Nicoud and Ducros [567]. 5.3.4 Implicit Diﬀusion: the ILES Concept Large-eddy simulation approaches using a numerical viscosity with no explicit modeling are all based implicitly on the hypothesis: Hypothesis 5.6 The action of subgrid scales on the resolved scales is equivalent to a strictly dissipative action. This approach is referred to as Implicit Large-Eddy Simulation (ILES). Simulations belonging to this category use dissipation terms introduced either in the framework of upwind schemes for the convection or explicit artiﬁcial dissipation term, or by the use of implicit [716] or explicit [210] frequency lowpass ﬁlters. The approach most used is doubtless the use of upwind schemes for the convective term. The diﬀusive term introduced then varies both in degree and order, depending on the scheme used (QUICK [437], Godunov [776], PPM [145], TVD [150], FCT [66], MPDATA [489, 488], among others) and

162

5. Functional Modeling (Isotropic Case)

the dissipation induced can in certain cases be very close29 to that introduced by a physical model [275]. Let us note that most of the schemes introduce dissipations of the second and/or fourth order and, in so doing, are very close to subgrid models. This point is discussed more precisely in Chap. 8. This approach is widely used in cases where the other modeling approaches become diﬃcult for one of the two following reasons: – The dynamic mechanisms escape the physical modeling because they are unknown or too complex to be modeled exactly and explicitly, which is true when complex thermodynamic mechanisms, for example, interact strongly with the hydrodynamic mechanisms (e.g. in cases of combustion [135] or shock/turbulence interaction [435]). – Explicit modeling oﬀers no a priori guarantee of certain realizability constraints related to the quantities studied (such as the temperature [125] or molar concentrations of pollutants [474]). This point is illustrated in Fig. 5.19, which displays the probability density function of a passive scalar computed by Large-Eddy Simulation with diﬀerent numerical schemes for the convection term. In cases belonging to one of these two classes, the error committed by using an implicit viscosity may in theory have no more harmful consequence on the quality of the result obtained than that which would be introduced by using an explicit model based on inadequate physical considerations. This approach is used essentially for dealing with very complex conﬁgurations or those harboring numerical diﬃculties, because it allows the use of robust numerical methods. Nonetheless, high-resolution simulations of ﬂows are beginning to make their appearance [756, 602, 776, 274]. A large number of stabilized numerical methods have been used for largeeddy simulation, but only a few of them have been designed for this speciﬁc purpose or more simply have been analyzed in that sense. A few general approaches for designing stabilized methods which mimic functional subgrid modeling are discussed below: 1. The MILES (Monotone Integrated Large Eddy Simulation) approach within the framework of ﬂux-limiting ﬁnite volume methods, as discussed by Grinstein and Fureby (p. 163). 2. The adaptive ﬂux reconstruction technique within the framework of nonlimited ﬁnite volume methods, proposed by (p. 165). 3. Finite element schemes with embedded subgrid stabilization (p. 166). 4. The use of Spectral Vanishing Viscosities (p. 169) which are well suited for numerical methods with spectral-like accuracy. 5. The high-order ﬁltering technique (p. 170), originally developed within the ﬁnite-diﬀerence framework, and wich is equivalent to some approximate deconvolution based structural models. 29

In the sense where these dissipations are localized at the same points and are of the same order of magnitude.

5.3 Modeling of the Forward Energy Cascade Process

163

Fig. 5.19. Probability of the density function of the temperature (modeled as a passive scalar) in a channel ﬂow obtained via Large-Eddy Simulation. Vertical lines denote physical bounds. It is observed that the simulation carried out with centered fourth-order accurate scheme admits non-physical values, while the use of the stabilized schemes to solve the passive scalar equation cures this problem. Courtesy of F. Chatelain (CEA).

MILES Approach. A theoretical analysis of the MILES approach within the framework of ﬂux-limiting ﬁnite volume discretizations has been carried out by Fureby and Grinstein [273, 228, 231, 274, 229], which puts the emphasis on the existing relationship between leading numerical error terms and tensorial subgrid viscosities. Deﬁning a control cell Ω of face-normal unit vector n, the convective ﬂuxes are usually discretized using Green’s theorem ∇ · (u ⊗ u)dΩ = (u · n)udS , (5.276) Ω

∂Ω

where ⊗ denotes the tensorial product and ∂Ω is the boundary of Ω. The associated discrete relation is (u · n)udS ≈ FfC (u) , (5.277) ∂Ω

f

where f are the faces of Ω and the discrete ﬂux function is expressed as FfC (u) = ((u · dA)u)f

,

(5.278)

164

5. Functional Modeling (Isotropic Case)

where dA is the face-area vector of face f of ∂Ω, and ()f is the integrated value on face f. For ﬂux-limiting methods, the numerical ﬂux is decomposed as the weighted sum of a high-order ﬂux function FfH that works well in smooth regions and a low-order ﬂux function FfL : 3 4 , (5.279) FfC (u) = FfH (u) + (1 − Γ (u)) FfH (u) − FfL (u) where Γ (u) is the ﬂux limiter.30 Fureby and Grinstein analyzed the leading error term using the following assumptions: (i) time integration is performed using a three-point backward scheme, (ii) the high-order ﬂux functions use ﬁrst-order functional reconstruction, and (iii) the low-order ﬂux functions use upwind diﬀerencing. Retaining the leading dissipative error term, the continuous equivalent formulation for the discretized ﬂuxes is ∇ · (u ⊗ u) − ∇ · (u ⊗ r + r ⊗ u + r ⊗ r) ,

exact

(5.280)

dissipative error

1 u·d r = β(∇u)d, β = (1 − Γ (u))sgn , (5.281) 2 |d| where d is the topology vector connecting neighboring control volumes. Comparison of the error term in (5.280) and the usual subgrid term appearing in ﬁltered Navier–Stokes equations (3.17) yields the following identiﬁcation for the MILES subgrid tensor: with

τMILES

= −(u ⊗ r + r ⊗ u + r ⊗ r) 3 4 = − β (u ⊗ d)∇T u + ∇u(u ⊗ d)T

I

+ β 2 (∇u)d ⊗ (∇u)d

.

(5.282)

II

Term I appears as a general subgrid-viscosity model with a tensorial diﬀusivity β(u ⊗ d), while term II mimics the Leonard tensor, leading to the deﬁnition of an implicit√mixed model (see Sect. 7.4).31 A scalar-valued measure of the viscosity is 2/8|u|∆MILES , where the characteristic length associated with the grid is ∆MILES = tr[(∇T d)(d ⊗ d)(∇d)]. The authors remarked that these error terms are invariant under the Galilean group of transformations, but are not frame indiﬀerent. Realizability and non-negative dissipation of subgrid kinetic energy may be enforced for some choice of the limiter. 30

31

Many ﬂux limiters can be found in the literature: minmod, superbee, FCT limiter, ... The reader is referred to specialized reference books [307] for a detailed discussion of these functions. MILES can also be interpreted as an implicit deconvolution model, using the analogy discussed in Sect. 7.3.3.

5.3 Modeling of the Forward Energy Cascade Process

165

Adaptive Flux Reconstruction. Adams [2] established a theoretical bridge between high-order adaptive ﬂux reconstruction used in certain ﬁnite-volume schemes and the use of a subgrid viscosity. In the simpliﬁed case of the following one-dimensional conservation law ∂u ∂F (u) + =0 ∂t ∂x

,

(5.283)

the ﬁnite volume technique leads to the computation of the cell-averaged variable: xj+1/2 uj = u(ξ)dξ , (5.284) xj−1/2

with ∆x = xj+1/2 − xj−1/2 the cell spacing of the jth cell. High-order ﬁnite volume methods rely on the reconstruction of the unﬁltered value uj+1/2 on both sides of the cell face xj+1/2 on each cell j. This is achieved by deﬁltering the variable u and deﬁning a high-order polynomial interpolant. The deﬁltering step is similar to the deconvolution approach, whose related results are presented in Sect. 7.2.1 and will not be repeated here. Sticking to Adams’ demonstration, a second-order deconvolution is employed: uj = uj −

∆x2 ∂ 2 uj 24 ∂x2

.

(5.285)

Following the WENO (Weightest Essentially Non-Oscillatory) concept [350], a hierarchical family of left-hand-side interpolants of increasing order is: P +,(0) (x)j +,(1) (x) Pj +,(2) Pj (x)

= uj

,

(5.286)

=

+,(0) Pj (x)

=

+,(1) Pj

+

α+ 1,1 (x

−

(1) xj )∆j

,

(5.287)

(2) (2) + , (5.288) +(x−xj )(x−xj+1 ) α+ 1,2 ∆j−1 +α2,2 ∆j

... = ... +,(k)

(p)

where Pj (x) is the kth-order interpolant, ∆j the divided diﬀerence of degree p of the variable, and α+ m,n some weigthing parameters. Right-handsides are deﬁned in the same way, except that P −,(0) (x)j = uj+1 and weights are noted α− m,n . A kth-order interpolation is obtained under the following constraints: m

α± m,n = 1,

α± m,n > 0,

n = 1, .., k − 1

.

(5.289)

166

5. Functional Modeling (Isotropic Case)

Applying this procedure to (5.283), the numerical convection term can be expressed as 4 ∂F (u) 1 3 ≈ fj+1/2 (xj+1/2 ) − fj−1/2 (xj−1/2 ) ∂x ∆x

,

(5.290)

where fj±1/2 is a numerical ﬂux function. Adams’ analysis is based on the local Lax–Friedrichs ﬂux: 3 4 − − fj+1/2 (x) = f (Pj+ (x)) + f (Pj+1 (x)) −βj+1/2 (Pj+1 (x)−Pj+ (x))

, (5.291)

with βj+1/2 = max |f (u)| . uj ,uj+1

(5.292)

In the simpliﬁed case of the Burgers equation, i.e. F (u) = u2 /2, the leading error term is:

E

1 γ1 ∂ 3 uj ∂uj ∂ 2 uj 2 − + = ∆x uj 8 16 ∂x3 ∂x ∂x2 1 ∂ 3 uj + (βj+1/2 δ1 − βj−1/2 δ2 )∆x2 8 ∂x3 2 ∂ uj +(βj+1/2 − βj−1/2 )γ2 ∆x , ∂x2

(5.293)

where the coeﬃcients are deﬁned as + − − + + − − γ1 = (α+ 1,2 + α2,2 + α1,2 + α2,2 ) , γ2 = (α1,2 + α2,2 − α1,2 − α2,2 ) , (5.294) + δ1 = (α− 2,2 − α2,2 ),

+ δ2 = (α− 1,2 − α1,2 ) .

(5.295)

Subgrid-viscosity models can be recovered by chosing adequately the values of the constants appearing in (5.293). The Smagorinsky model with length scale ∆ = ∆x and constant CS is obtained by taking: γ1 = 2, γ2 = CS , δ1 = δ2 = 0, βj±1/2 = ±|uj+1/2 − uj−1/2 | .

(5.296)

Variational Schemes with Embedded Subgrid Stabilization. We now present ﬁnite element methods with some built-in subgrid stabilization [280, 79, 144, 299, 623, 331, 332, 336]. The presentation will be limited to the main ideas for a simple linear advection–diﬀusion equation. The reader is referred to original articles for detailed mathematical results and extension to Navier– Stokes equations. These methods are all based on the variational formulation

5.3 Modeling of the Forward Energy Cascade Process

167

of the problem. For a passive scalar φ, we have: Ω

∂φ ψdV + ∂t

∂φ u ψdV Ω ∂x

= =

∂2φ ν ψdV + f ψdV 2 Ω ∂x Ω ∂φ ∂ψ −ν dV + f ψdV Ω ∂x ∂x Ω

(5.297) ,

where Ω is the ﬂuid domain, ψ a weighting function, u the advection velocity and f a source term. Boundary terms are assumed to vanish for the sake of simplicity. Let L be the time-dependent advection–diﬀusion operator: L=

∂ ∂ ∂ +u −ν 2 ∂t ∂x ∂x

.

(5.298)

Using this operator, (5.297) can be recast under the symbolic compact form (ψ, Lφ)Ω = (L∗ ψ, φ)Ω = a(ψ, φ) = (ψ, f )Ω , (5.299) where (., .)Ω is a scalar product, a(., .) the bilinear form deduced from the preceding equations and L∗ the adjoint operator: L∗ = −

∂ ∂ ∂ −u −ν 2 ∂t ∂x ∂x

.

(5.300)

We now split the trial and weighting functions as the sum of a resolved and a subgrid function, i.e. φ = φ + φ and ψ = ψ + ψ . Inserting these decompositions into relation (5.299), we obtain a(ψ, φ) = a(ψ + ψ , φ + φ ) = (ψ + ψ , f )Ω

,

(5.301)

and, assuming that ψ and ψ are linearly independent, we get the two following subproblems: a(ψ, φ) + a(ψ, φ ) = (ψ, f )Ω

,

(5.302)

and a(ψ , φ) + a(ψ , φ ) = (ψ , f )Ω

,

(5.303)

a(ψ, φ) + (L∗ ψ, φ )Ω = (ψ, f )Ω

,

(5.304)

or, equivalently,

and (ψ , Lφ)Ω + (ψ , Lφ )Ω = (ψ , f )Ω

.

(5.305)

A ﬁrst solution consists of discretizing (5.304) and (5.305) using standard shape functions for the resolved scales and oscillatory bubble functions for the

168

5. Functional Modeling (Isotropic Case)

Fig. 5.20. Schematic of the embedded subgrid stabilization approach: linear ﬁnite element shape functions in one dimension plus typical bubbles

subgrid scales (see Fig. 5.20). The resulting method is a two-scale method, with embedded subgrid stabilization. It is important to note that degrees of freedom associated with bubble functions are eliminated by static condensation, i.e. are expressed as functions of the resolved scales, and do not require the solution of additional evolution equations. Hughes and Stewart [336] proposed regularizing (5.304), which governs the motion of resolved scales, yielding a(ψ, φ) + (L∗ ψ, M (Lφ − f ))Ω = (ψ, f )Ω

,

(5.306)

where (Lφ − f ) is the residual of the resolved scales and M an operator originating from an elliptic regularization, which can be evaluated using bubble functions. Usual stabilized methods also rely on the regularization of (5.304) without considering the subgrid scale equation. A general form of the stabilized problem is (5.307) a(ψ, φ) + (ILψ, τstab (Lφ − f ))Ω = (ψ, f )Ω , where, typically, IL is a diﬀerential operator and τstab is an algebraic operator which approximates the integral operator (−M ) of (5.306). Classical examples are: – Standard Galerkin method, which does not introduce any stabilizing term: IL = 0

.

(5.308)

5.3 Modeling of the Forward Energy Cascade Process

169

– Streamwise upwind Petrov–Galerkin method, which introduces a stabilizing term based on the advection operator: IL = u

∂ ∂x

.

(5.309)

– Galerkin/least-squares method, which extends the preceding method by including the whole diﬀerential operator: IL = L .

(5.310)

– Subgrid Stabilization (Bubbles), which recovers the method of Hughes: IL = −L∗

.

(5.311)

The amount of numerical dissipation is governed by the parameter τstab , which can assume either tensorial or scalar expression. Many deﬁnitions can be found, most of them yielding |τstab | ∝ ∆x2 , which is the right scale for a subgrid dissipation. Results dealing with the tuning of this parameter for turbulent ﬂow simulations are almost nonexistent. Spectral Vanishing Viscosities. Karniadakis et al. [379, 395] propose to adapt the Tadmor spectral viscosity [700] for large-eddy simulation purpose. This approach will be presented using a simpliﬁed non-linear conservation law for the sake of clarity. Considering the model Burgers equation ∂ ∂u + ∂t ∂x

u2 2

=0

,

(5.312)

which can develop singularities, Tadmor proposes to regularize it for numerical purpose as ∂ u2 ∂u ∂ ∂u + = Q , (5.313) ∂t ∂x 2 ∂x ∂x where , Q and u are the articial viscosity parameter, the artiﬁcial viscosity kernel and the regularized solution (interpreted as the resolved ﬁeld in largeeddy simulation), respectively. The original formulation of the regularization proposed by Tadmor is expressed in the spectral space as ∂ ∂x

∂u Q = − ∂x

u k eikx k 2 Q(k)

,

(5.314)

M≤|k|≤N

where k is the wave number, N the number of Fourier modes, and M the wave number above which the artiﬁcial viscosity is activated. Several forms for the viscosity kernel have been suggested, among which the continuous kernel of

170

5. Functional Modeling (Isotropic Case)

Maday [475] for pseudo-spectral methods based on Legendre polynomials: (k − N )2 Q(k) = exp − , (k − M )2

k>M

,

(5.315)

√ with M 5 N and ∼ 1/N . A major diﬀerence with spectral functional model in Fourier space is that Tadmor-type regularizations vanish at low wave numbers, while functional subgrid viscosities don’t. The extension of the method to multidimensional curvilinear grids has been extensivly studied by Pasquetti and Xu [773, 582, 581]. Karniadakis et al. [395] further modify this model by deﬁning a dynamic version of the artiﬁcial viscosity in which the parameter is tuned regarding the local state of the ﬂow. To this end, the regularization term is redeﬁned as ∂ 2 Qu , (5.316) c(x, t)Q ∂x2 where the self-adaptive amplitude parameter c(x, t) can be computed considering either the gradient of the solution c(x, t) =

κ |∇u| N ∇u∞

,

(5.317)

where κ is an adjustable arbitrary parameter, or the strain tensor c(x, t) =

|S| S∞

.

(5.318)

To prevent a too high dissipation near solid walls, Kirby and Karniadakis multiply Q by a damping function 2 g(y ) = tan−1 π +

2ky + π

+ 2 y 1 − exp − C

,

(5.319)

where y is he distance to the wall, the superscript + refers to quantities expressed in wall units and C is a parameter. High-Order Filtered Methods. Visbal and Rizetta deﬁne [734] another procedure to perform large-eddy simulation based on numerical stabilization without explicit physical subgrid model. Their procedure is based on the application, at the end of each time, of an high-order low-pass ﬁlter to the solution. This stabilizing procedure and the associated corresponding method originating in the works by Visbal and Gaitonde is discussed in references given therein. In practice, they use a symmetric compact ﬁnite diﬀerence ﬁlter with the followng properties:

5.4 Modeling the Backward Energy Cascade Process

1. 2. 3. 4.

it it it it

171

is non-dispersive, i.e. it is strictly dissipative, does not amplify any waves, preserves constant functions, completely eliminates the odd-even mode.

A tenth-order compact ﬁlter is observed to yield satisfactory results in simple cases (decaying isotropic turbulence). It is important to notice that this procedure is formally equivalent to the ﬁltering-form of the full deconvolution procedure proposed by Mathew et al. [499] (see p. 220 for details). Therefore, this implicit procedure can be completely rewritten within the structural modeling framework. Other authors [63] develop similar strategies, based on the use of very-high order accurate ﬁnite diﬀerence schemes.

5.4 Modeling the Backward Energy Cascade Process 5.4.1 Preliminary Remarks The above models reﬂect only the forward cascade process, i.e. the dominant average eﬀect of the subgrid scales. The second energy transfer mechanism, the backward energy cascade, is much less often taken into account in simulations. We may mention two reasons for this. Firstly, the intensity of this return is very weak compared with that of the forward cascade toward the small scales (at least on the average in the isotropic hom*ogeneous case) and its role in the ﬂow dynamics is still very poorly understood. Secondly, modeling it requires the addition of an energy source term to the equations being computed, which is potentially a generator of numerical problems. Two methods are used for modeling the backward energy cascade: – Adding a stochastic forcing term constructed from random variables and the information contained in the resolved ﬁeld. This approach makes it possible to include a random character of the subgrid scales, and each simulation can be considered a particular realization. The space-time correlations characteristic of the scales originating the backward cascade cannot be represented by this approach, though, which limits its physical representativeness. – Modifying the viscosity associated with the forward cascade mechanism deﬁned in the previous section, so as to take the energy injected at the large scales into account. The backward cascade is then represented by a negative viscosity, which is added to that of the cascade model. This approach is statistical and deterministic, and also subject to caution because it is not based on a physical description of the backward cascade phenomenon and, in particular, possesses no spectral distribution in k 4 predicted by the analytical theories like EDQNM (see also footnote p. 104). Its advantage resides mainly in the fact that it allows a reduction of the total dissipation

172

5. Functional Modeling (Isotropic Case)

of the simulation, which is generally too high. Certain dynamic procedures for automatically computing the constants can generate negative values of them, inducing an energy injection in the resolved ﬁeld. This property is sometimes interpreted as the capacity of the dynamic procedure to reﬂect the backward cascade process. This approach can therefore be classed in the category of statistical deterministic backward cascade models. Representing the backward cascade by way of a negative viscosity is controversial because the theoretical analyses, such as by the EDQNM model, distinguish very clearly between the cascade and backward cascade terms, both in their intensity and in their mathematical form [443, 442]. This representation is therefore to be linked to other statistical deterministic descriptions of the backward cascade, which take into account only an average reduction of the eﬀective viscosity, such as the Chollet–Lesieur eﬀective viscosity spectral model. The main backward cascade models belonging to these two categories are described in the following. 5.4.2 Deterministic Statistical Models This section describes the deterministic models for the backward cascade. These models, which are based on a modiﬁcation of the subgrid viscosity associated with the forward cascade process, are: 1. The spectral model based on the theories of turbulence proposed by Chasnov (p. 172). A negative subgrid viscosity is computed directly from the EDQNM theory. No hypothesis is adopted concerning the spectrum shape of the resolved scales, so that the spectral disequilibrium mechanisms can be taken into account at the level of these scales, but the spectrum shape of the subgrid scales is set arbitrarily. Also, the ﬁlter is assumed to be of the sharp cutoﬀ type. 2. The dynamic model with an equation for the subgrid kinetic energy (p. 173), to make sure this energy remains positive. This ensures that the backward cascade process is represented physically, in the sense that a limited quantity of energy can be restored to the resolved scales by the subgrid modes. However, this approach does not allow a correct representation of the spectral distribution of the backward cascade. Only the quantity of restored energy is controlled. Chasnov’s Spectral Model. Chasnov [120] adds a model for the backward cascade, also based on an EDQNM analysis, to the forward cascade model already described (see Sect. 5.3.1). The backward cascade process is represented deterministically by a negative eﬀective viscosity term νe− (k|kc ), which is of the form: νe− (k|kc , t) = −

F − (k|kc , t) 2k 2 E(k, t)

.

(5.320)

5.4 Modeling the Backward Energy Cascade Process

173

The stochastic forcing term is computed as: ∞ p k3 F − (k|kc , t) = dp dqΘkpq (1 − 2x2 z 2 − xyz)E(q, t)E(p, t), (5.321) pq kc p−k in which x, y, and z are geometric factors associated with the triad (k, p, q), and Θkpq is a relaxation time described in Appendix B. As is done when computing the draining term (see Chasnov’s eﬀective viscosity model in Sect. 5.3.1), we assume that the spectrum takes the Kolmogorov form beyond the cutoﬀ kc . To simplify the computations, formula (5.321) is not used for wave numbers kc ≤ p ≤ 3kc . For the other wave numbers, we use the asymptotic form 14 4 ∞ E 2 (p, t) k F − (k|kc , t) = dpΘkpp (t) . (5.322) 15 p2 kc This expression complete Chasnov’s spectral subgrid model which, though quite close to the Kraichnan type eﬀective viscosity models, makes it possible to take into account the backward cascade eﬀects that are dominant for very small wave numbers. Localized Dynamic Model with Energy Equation. The Germano–Lilly dynamic procedure and the localized dynamic procedure lead to the deﬁnition of subgrid models that raise numerical stability problems because the model constant can take negative values over long time intervals, leading to exponential growth of the disturbances. This excessive duration of the dynamic constant in the negative state corresponds to too large a return of kinetic energy toward the large scales [101]. This phenomenon can be interpreted as a violation of the spectrum realizability constraint: when the backward cascade is over-estimated, a negative kinetic energy is implicitly deﬁned in the subgrid scales. A simple idea for limiting the backward cascade consists in guaranteeing spectrum realizability32 . The subgrid scales cannot then restore more energy than they contain. To verify this constraint, local information is needed on the subgrid kinetic energy, which naturally means deﬁning this as an additional variable in the simulation. A localized dynamic model including an energy equation is proposed by Ghosal et al. [261]. Similar models have been proposed independently by Ronchi et al. [621, 511], Wong [765] and Kim and Menon [391, 392, 583]. The subgrid model used is based on the kinetic energy of the subgrid modes. Using the same notation as in Sect. (5.3.3), we get: + * Q2 S * , (5.323) αij = −2∆ sgs ij + 2 S , (5.324) βij = −2∆ qsgs ij 32

The spectrum E(k) is said to be realizable if E(k) ≥ 0, ∀k.

174

5. Functional Modeling (Isotropic Case)

2 in which the energies Q2sgs and qsgs are deﬁned as:

Q2sgs =

1 / * * 1 ui ui − ui ui = Tii 2 2

1 1 (ui ui − ui ui ) = τii 2 2 Germano’s identity (5.138) is written: 2 qsgs =

1 2 Q2sgs = q/ sgs + Lii 2

, .

.

(5.325) (5.326)

(5.327)

2 The model is completed by calculating qsgs by means of an additional evolution equation. We use the equation already used by Schumann, Horiuti, and Yoshizawa, among others (see Sect. 5.3.1): 2 2 ∂uj qsgs ∂qsgs + ∂t ∂xj

3/2

=

2 (qsgs ) −τij S ij − C1 ∆ 2 2 + ∂ 2 qsgs ∂qsgs ∂ 2 ∆ qsgs , (5.328) +ν +C2 ∂xj ∂xj ∂xj ∂xj

in which the constants C1 and C2 are computed by a constrained localized dynamic procedure described above. The dynamic constant Cd is computed by a localized dynamic procedure. 2 This model ensures that the kinetic energy qsgs will remain positive, i.e. that the subgrid scale spectrum will be realizable. This property ensures that the dynamic constant cannot remain negative too long and thereby destabilize the simulation. However, ﬁner analysis shows that the realizability conditions concerning the subgrid tensor τ (see Sect. 3.3.5) are veriﬁed only on the condition: + + 2 2 qsgs qsgs ≤ Cd ≤ , (5.329) − 3∆|sγ | 3∆sα where sα and sγ are, respectively, the largest and smallest eigenvalues of the strain rate tensor S. The model proposed therefore does not ensure the realizability of the subgrid tensor. The two constants C1 and C2 are computed using an extension of the constrained localized dynamic procedure. To do this, we express the kinetic energy Q2sgs evolution equation as: ∂* uj Q2sgs ∂Q2sgs + ∂t ∂xj

2

3/2

* − C (Qsgs ) = −Tij S ij 1 * ∆ 2 + ∂Q ∂ 2 Q2sgs ∂ sgs Q2sgs . (5.330) +ν + C2 ∂xj ∂xj ∂xj ∂xj

5.4 Modeling the Backward Energy Cascade Process

175

One variant of the Germano’s relation relates the subgrid kinetic energy ﬂux fj to its analog at the level of the test ﬁlter Fj : 2 / 2/ *j (p + qsgs + ui ui /2) − uj (p + qsgs + ui ui /2) , Fj − f*j = Zj ≡ u

(5.331)

in which p is the resolved pressure. To determine the constant C2 , we substitute in this relation the modeled ﬂuxes: 2 + ∂qsgs 2 fj = C2 ∆ qsgs , (5.332) ∂xj 2 + * Q2 ∂Qsgs Fj = C2 ∆ sgs ∂xj

,

(5.333)

which leads to: Zj = Xj C2 − Y/ j C2 in which

,

2 + * Q2 ∂Qsgs Xj = ∆ sgs ∂xj 2 + ∂qsgs 2 Yj = ∆ qsgs ∂xj

(5.334)

,

(5.335)

.

(5.336)

Using the same method as was explained for the localized dynamic procedure, the constant C2 is evaluated by minimizing the quantity:

Zj − Xj C2 + Y/ j C2

Zj − Xj C2 + Y/ j C2

.

(5.337)

By analogy with the preceding developments, the solution is obtained in the form: 3 C2 (x) = fC2 (x) + KC2 (x, y)C2 (y)d y , (5.338) +

in which: fC2 (x) =

1 Xj (x)Xj (x)

Xj (x)Zj (x) − Yj (x) Zj (y)G(x − y)d3 y

,

(5.339) KC2 (x, y) =

C2 KA (x, y)

C2 KA (y, x)

+ − Xj (x)Xj (x)

KSC2 (x, y)

,

(5.340)

176

5. Functional Modeling (Isotropic Case)

in which C2 KA (x, y) = Xj (x)Yj (y)G(x − y) ,

(5.341)

KSC2 (x, y)

G(z − x)G(z − y)d3 z

= Yj (x)Yj (y)

.

(5.342)

This completes the computation of constant C2 . To determine the constant C1 , we substitute (5.327) in (5.330) and get: 2 2 2 ∂* ∂ q/ uj q/ ∂ 2 q/ ∂Fj sgs sgs sgs + = −E +ν ∂t ∂xj ∂xj ∂xj ∂xj

,

(5.343)

in which E is deﬁned as: 3/2

2 * + C1 (Qsgs ) E = Tij S ij * ∆

1 1 ∂ 2 Lii + −ν 2 ∂xj ∂xj 2

*j Lii ∂Lii ∂u + ∂t ∂xj

. (5.344)

Applying the test ﬁlter to relation (5.328), we get: 2 2 ∂ q/ ∂* uj q/ sgs sgs / + = −τij S ij − ∂t ∂xj

C1

/ 2 )3/2 (qsgs ∆

+

2 ∂ 2 q/ ∂ f*j sgs +ν ∂xj ∂xj ∂xj

. (5.345)

2 By eliminating the term ∂ q/ sgs /∂t between relations (5.343) and (5.345), then replacing the quantity Fj − f*j by its expression (5.331) and the quantity Tij by its value as provided by the Germano identity, we get:

/1 χ = φC1 − ψC

,

(5.346)

in which ∂ρj 1 1 ∂ 2 Lii / * −L S * χ = τij S ij − τ*ij S − Dt Lii + ν ij ij ij + ∂xj 2 2 ∂xj ∂xj φ = (Q2sgs )

(5.347)

* , /∆

(5.348)

/∆ ,

(5.349)

3/2

3/2

2 ψ = (qsgs )

,

and *j (p + / ui ui /2) − uj (p +/ ui ui /2) . ρj = u

(5.350)

5.4 Modeling the Backward Energy Cascade Process

177

*j ∂/∂xj . The The symbol Dt designates the material derivative ∂/∂t + u constant C1 is computed by minimizing the quantity

/1 χ − φC1 + ψC

/1 χ − φC1 + ψC

,

(5.351)

by a constrained localized dynamic procedure, which is written: C1 (x) = fC1 (x) + KC1 (x, y)C1 (y)d3 y

,

(5.352)

+

in which fC1 (x) =

1 φ(x)φ(x)

KC1 (x, y) =

φ(x)χ(x) − ψ(x)

χ(y)G(x − y)d3 y

C1 C1 KA (x, y) + KA (y, x) − KSC1 (x, y) φ(x)φ(x)

,

,

(5.353)

(5.354)

in which C1 KA (x, y) = φ(x)ψ(y)G(x − y) ,

(5.355)

KSC1 (x, y) = ψ(x)ψ(y)

G(z − x)G(z − y)d3 z

,

(5.356)

which completes the computation of the constant C1 . The version by Menon et al. [391, 392, 583], also extensively used by Davidson and his coworkers [576, 577] is much simpler as far as the practical implementation is addressed. This simpliﬁed formulation is deﬁned as follows:

C1 =

* ∆ε test (Q2sgs )3/2

and C2 = Cd =

,

* 1 Ldij S ij * S * 2S ij

(5.357)

,

(5.358)

ij

where Q2sgs is computed using relation (5.327) and the dissipation at the test ﬁlter level is evaluated using a scale-similarity hypothesis, yielding ( εtest = (ν + νsgs )

∂u/ i ∂ui ∂xj ∂xj

−

∂* ui ∂ * ui ∂xj ∂xj

) .

(5.359)

178

5. Functional Modeling (Isotropic Case)

Since it is strictly local in the sense that no integral problem is involved, this new formulation is much less demanding than the previous one in terms of computational eﬀort. Another simpliﬁed local one-equation dynamic model was proposed by Fureby [231], which is deﬁned by the following relations: C1 = where

ζm mm

,

(5.360)

/ 2 3/2 3/2 2 qsgs Qsgs − m= * ∆ ∆ * − ∂ ζ = τij/ S ij − Tij S ij ∂t

1 Lkk 2

−

∂ ∂xj

,

1 *j Lkk u 2

(5.361) .

(5.362)

The remaining parameter is computed as follows: Ldij Mij Mij Mij

,

(5.363)

1 αij − β2 ij 2

.

(5.364)

C2 = Cd = where Mij =

A more complex model is proposed by Krajnovic and Davidson [406], who use a linear-combination model (see Sect. 7.4) to close both the momentum equations and the prognostic equation for the subgrid kinetic energy. 5.4.3 Stochastic Models Models belonging to this category are based on introducing a random forcing term into the momentum equations. It should be noted that this random character does not reﬂect the space-time correlation scales of the subgrid ﬂuctuations, which limits the physical validity of this approach and can raise numerical stability problems. It does, however, obtain forcing term formulations at low algorithmic cost. The models described here are: 1. Bertoglio’s model in the spectral space (p. 179). The forcing term is constructed using a stochastic process, which is designed in order to induce the desired backward energy ﬂux and to possess a ﬁnite correlation time scale. This is the only random model for the backward cascade derived in the spectral space.

5.4 Modeling the Backward Energy Cascade Process

179

2. Leith’s model (p. 180). The forcing term is represented by an acceleration vector deriving from a vector potential, whose amplitude is evaluated by simple dimensional arguments. The backward cascade is completely decoupled from forward cascade here: there is no control on the realizability of the subgrid scales. 3. Mason–Thomson model (p. 182), which can be considered as an improvement of the preceding model. The evaluations of the vector potential amplitude and subgrid viscosity modeling the forward cascade are coupled, so as to ensure that the local equilibrium hypothesis is veriﬁed. This ensures that the subgrid kinetic energy remains positive. 4. Schumann model (p. 183), in which the backward cascade is represented not as a force deriving from a vector potential but rather as the divergence of a tensor constructed from a random solenoidal velocity ﬁeld whose kinetic energy is equal to the subgrid kinetic energy. 5. Stochastic dynamic model (p. 184), which makes it possible to calculate the subgrid viscosity and a random forcing term simultaneously and dynamically. This coupling guarantees that the subgrid scales are realizable, but at the cost of a considerable increase in the algorithmic complexity of the model. Bertoglio Model. Bertoglio and Mathieu [57, 58] propose a spectral stochastic subgrid model based on the EDQNM analysis. This model appears as a new source term fi (k, t) in the ﬁltered mometum equations, and is evaluated as a stochastic process. The following constraints are enforced: – – – –

f must not modify the velocity ﬁeld incompressibility, i.e. ki fi (k, t) = 0; f will have a Gaussian probability density function; The correlation time of f , noted tf , is ﬁnite; f must induce the desired eﬀect on the statistical second-order moments of the resolved velocity ﬁeld: ∗

fi (k, t) uj (k, t) + fj (k, t) ui (k, t) = Tij− (k, t)

2π L

3 ,

(5.365)

where Tij− (k, t) is the exact backward transfer term appearing in the vari∗

ui (k, t) uj (k, t) and L the size of the computational ation equation for domain in physical space. Assuming that the response function of the simulated ﬁeld is isotropic and independent of f , and that the time correlations exhibit an exponential decay, we get the following velocity-independent relation: fi (k, t)fj∗ (k, t) + fi∗ (k, t)fj (k, t) = Tij− (k, t)

2π L

3

1 1 + θ(k, t) tf

, (5.366)

180

5. Functional Modeling (Isotropic Case)

where θ(k, t) is a relaxation time evaluated from the resolved scales. We now have to compute the stochastic variable fi . The authors propose the following algorithm, which is based on three random variables a, b and c: ' + ∆t ∆t (n+1) (n) (n) (n+1) = 1− exp(ı2πa(n+1) ) f1 + h11 β11 f1 tf tf ' + ∆t (n) (n+1) + h22 β12 exp(ı2πc(n+1) ) , (5.367) tf ' + ∆t ∆t (n+1) (n) (n) (n+1) = 1− exp(ı2πb(n+1) ) f2 f2 + h22 β22 tf tf ' + ∆t (n) (n+1) exp(ı2πc(n+1) ) , (5.368) + h11 β21 tf where the superscript (n) denotes the value at the nth time step, ∆t is the value of the time step, and hij (k, t) = fi (k, t)fj∗ (k, t). Moreover, we get the complementary set of equations, which close the system: 1 (n+1) 2 (n+1) (n) tf (n) (n+1) 2 − h22 (β12 ) − h11 ) (h11 (β11 ) = (n) ∆t h 11

(n+1) 2

=

(n+1) (n+1) β12

=

(n+1) 2

=

(β22

)

β12

(β21

)

∆t +2 − , (5.369) tf 1 (n+1) (n) tf (n) (n+1) 2 − h − h ) (β ) (h 22 22 11 21 (n) ∆t h22 ∆t +2 − , (5.370) tf 1 (n+1) (n) tf + − h12 ) (h12 ∆t (n) (n) h11 h11 ∆t (n) +h12 2 − , (5.371) tf (n+1) 2

(β12

)

,

(5.372)

which completes the description of the model. The resulting random force satisﬁes all the cited constraints, but it requires the foreknowledge of the hij tensor. This tensor is evaluated using the EDQNM theory, which requires the spectrum of the subgrid scales to be known. To alleviate this problem, arbitrary form of the spectrum can be employed. Leith Model. A stochastic backward cascade model expressed in the physical space was derived by Leith in 1990 [434]. This model takes the form of a random forcing term that is added to the momentum equations. This term

5.4 Modeling the Backward Energy Cascade Process

181

is computed at each point in space and each time step with the introduction of a vector potential φb for the acceleration, in the form of a white isotropic noise in space and time. The random forcing term with null divergence f b is deduced from this vector potential. We ﬁrst assume that the space and time auto-correlation scales of the subgrid modes are small compared with the cutoﬀ lengths in space ∆ and in time ∆t associated with the ﬁlter33 . This way, the subgrid modes appear to be de-correlated in space and time. The correlation at two points and two times of the vector potential φb is then expressed: φbi (x, t)φbk (x , t ) = σ(x, t)δ(x − x )δ(t − t )δik

,

(5.373)

in which σ is the variance. This is computed as: σ(x, t) =

1 3

dt

d3 x φbk (x, t)φbk (x , t )

.

(5.374)

Simple dimensional reasoning shows that: 7

σ(x, t) ≈ |S|3 ∆

.

(5.375)

Also, as the vector potential appears as a white noise in space and time at the ﬁxed resolution level, the integral (5.374) is written: σ(x, t) =

1 b 3 φ (x, t)φbk (x, t)∆ ∆t . 3 k

(5.376)

Considering relations (5.375) and (5.376), we get: 4

φbk (x, t)φbk (x, t) ≈ |S|3 ∆

1 ∆t

.

(5.377)

The shape proposed for the kth component of the vector potential is: 2

φbk = Cb |S|3/2 ∆ ∆t−1/2 g

,

(5.378)

in which Cb is a constant of the order of unity, ∆t the simulation time cutoﬀ length (i.e. the time step), and g the random Gaussian variable of zero average and variance equal to unity. The vector f b is then computed by taking the rotational of the vector potential, which guarantees that it is solenoidal. In practice, Leith sets the value of the constant Cb at 0.4 and applies a spatial ﬁlter with a cutoﬀ length of 2∆, so as to ensure better algorithm stability. 33

We again ﬁnd here a total scale separation hypothesis that is not veriﬁed in reality.

182

5. Functional Modeling (Isotropic Case)

Mason–Thomson Model. A similar model is proposed by Mason and Thomson [498]. The diﬀerence from the Leith model resides in the scaling of the vector potential. By calling ∆f and ∆ the characteristic lengths of the subgrid scales and spatial ﬁlter, respectively, the variants of the resolved stresses due to the subgrid ﬂuctuations is, if ∆f ∆, of the order of (∆f /∆)3 u4e , in which ue is the characteristic subgrid velocity. The amplitude a of the ﬂuctuations in the gradients of the stresses is: 3/2

a≈

∆f ∆

5/2

u2e

,

(5.379)

which is also the amplitude of the associated acceleration. The corresponding kinetic energy variation rate of the resolved scales, qr2 , is estimated as: ∂qr2 ∆3 ≈ a2 te ≈ f5 u4e te ∂t ∆

,

(5.380)

in which te is the characteristic time of the subgrid scales. As te ≈ ∆f /ue and the dissipation rate is evaluated by dimensional arguments as ε ≈ u3e /∆f , we can say: ∆5 ∂qr2 = Cb f5 ε . (5.381) ∂t ∆ The ratio ∆f /∆ is evaluated as the ratio of the subgrid scale mixing length to the ﬁlter cutoﬀ length, and is thus equal to the constant of the subgrid viscosity models discussed in Sect. 5.3.2. Previous developments have shown that this constant is not unequivocally determinate, but that it is close to 0.2. The constant Cb is evaluated at 1.4 by an EDQNM analysis. The dissipation rate that appears in equation (5.381) is evaluated in light of the backward cascade. The local subgrid scale equilibrium hypothesis is expressed by: ∆5 (5.382) −τij S ij = ε + Cb f5 ε , ∆ in which τij is the subgrid tensor. The term on the left represents the subgrid kinetic energy production, the ﬁrst term in the right-hand side the dissipation, and the last term the energy loss to the resolved scales by the backward cascade. The dissipation rate is evaluated using this last relation: ε=

−τij S ij 1 + (∆f /∆)5

,

(5.383)

which completes the computation of the right-hand side of equation (5.381), with the tensor τij being evaluated using a subgrid viscosity model.

5.4 Modeling the Backward Energy Cascade Process

183

This equation can be re-written as: ∂qr2 = σa2 ∆t , ∂t

(5.384)

in which σa2 is the sum of the variances of the acceleration component amplitudes. From the equality of the two relations (5.381) and (5.384), we can say: ∆5 ε . (5.385) σa2 = Cb f5 ∆ ∆t The vector potential scaling factor a and σa2 are related by: ∆t a = σa2 . te

(5.386)

To complete the model, we now have to evaluate the ratio of the subgrid scale characteristic time to the time resolution scale. This is done simply by evaluating the characteristic time te from the subgrid viscosity νsgs computed by the model used, to reﬂect the cascade: te =

∆2f νsgs

,

(5.387)

which completes the description of the model, since the rest of the procedure is the same as what Leith deﬁned. Schumann Model. Schumann proposed a stochastic model for subgrid tensor ﬂuctuations that originate the backward cascade of kinetic energy [654]. The subgrid tensor τ is represented as the sum of a turbulent viscosity model and a stochastic part Rst : 2 2 st τij = νsgs S ij + qsgs δij + Rij 3

.

(5.388)

st are zero: The average random stresses Rij st =0 . Rij

(5.389)

They are deﬁned as: st Rij

2 2 = γm vi vj − qsgs δij 3

,

(5.390)

in which γm is a parameter and vi a random velocity. From dimensional arguments, we can deﬁne this as: 2 2qsgs gi , vi = (5.391) 3

184

5. Functional Modeling (Isotropic Case)

in which gi is a white random number in space and has a characteristic correlation time τv : gi = 0 , gi (x, t)gj (x , t ) = δij δ(x − x ) exp(|t − t |/τv ) .

(5.392) (5.393)

The vi ﬁeld is made solenoidal by applying a projection step. We note that the time scale τv is such that: + 2 /∆ ≈ 1 . (5.394) τv qsgs The parameter γm determines the portion of random stresses that generate the backward cascade. Assuming that only the scales belonging to the interval [kc , nkc ] are active, for a spectrum of slope of −m we get: 2 γm =

nkc

k −2m dk

kc∞ k

−2m

= 1 − n1−2m

.

(5.395)

dk

kc

For n = 2 and m = 5/3, we get γm = 0.90. The subgrid kinetic energy 2 qsgs is evaluated from the subgrid viscosity model. Stochastic Localized Dynamic Model. A localized dynamic procedure including a stochastic forcing term was proposed by Carati et al. [101]. The contribution of the subgrid terms in the momentum equation appears here as the sum of a subgrid viscosity model, denoted Cd βij using the notation of Sect. 5.3.3, which models the energy cascade, and a forcing term denoted f : ∂τij ∂Cd βij = + fi ∂xj ∂xj

.

(5.396)

The βij term can be computed using any subgrid viscosity model. The force f is chosen in the form of a white noise in time with null divergence in space. The correlation of this term at two points in space and two times is therefore expressed: fi (x, t)fj (x , t ) = A2 (x, t)Hij (x − x )δ(t − t ) .

(5.397)

The statistical average here is an average over all the realizations of f conditioned by a given velocity ﬁeld u(x, t). The factor A2 is such that Hii (0) = 1. Since a stochastic term has been introduced into the subgrid model, the residual Eij on which the dynamic procedure for computing the constant Cd is founded also possesses a stochastic nature. This property will therefore be shared by the dynamically computed constant, which is not acceptable. To ﬁnd the original properties of the dynamic constant, we take a statistical average of the residual, denoted Eij , which gets rid of the

5.4 Modeling the Backward Energy Cascade Process

185

random terms. The constant of the subgrid viscosity model is computed by a localized dynamic procedure based on the statistical average of the residual, which is written: . (5.398) Eij = Lij + C/ d βij − Cd αij The amplitude of the random forcing term can also be computed dynamically. To bring out the non-zero contribution of the stochastic term in the statistical average, we base this new procedure on the resolved kinetic energy *i /2. The evolution equation of *i u balance at the level of the test ﬁlter Q2r = u this quantity is obtained in two diﬀerent forms (only the pertinent terms are detailed, the others are symbolized): ∂Q2r *i ∂ (Cd αij + P δij ) + EF = ... − u ∂t ∂xj

,

(5.399)

∂Q2r *i ∂ C/ = ... − u *δij + Ef* . d βij + Lij + p ∂t ∂xj

(5.400)

The pressure terms P and p are in equilibrium with the velocity ﬁelds * and u, respectively. The quantities EF and E * are the backward cascade u f energy injections associated, respectively, with the forcing term F computed * computed directly at the level of the test ﬁlter, and with the forcing term f at the ﬁrst level and then ﬁltered. The diﬀerence between equations (5.399) and (5.400) leads to: (5.401) Z ≡ EF − Ef* − g = 0 , in which the fully known term g is of the form: *i g=u

∂ Cd αij + P δij − C/ *δij d βij − Lij − p ∂xj

.

(5.402)

The quantity Z plays a role for the kinetic energy that is analogous to the residual Eij for the momentum. Minimizing the quantity Z=

Z2

(5.403)

can thus serve as a basis for deﬁning a dynamic procedure for evaluating the stochastic forcing. To go any further, the shape of the f term has to be speciﬁed. To simplify the use, we assume that the correlation length of f is small compared with the cutoﬀ length ∆. The function f thus appears as de-correlated in space, which is reﬂected by: 1 (5.404) Ef = A2 (x, t) . 2

186

5. Functional Modeling (Isotropic Case)

In order to be able to calculate Ef dynamically, we assume that the backward cascade is of equal intensity at the two ﬁltering levels considered, i.e. Ef = EF .

(5.405)

* scale, we assume: Also, since f is de-correlated at the ∆ Ef* Ef = EF ,

(5.406)

which makes it possible to change relation (5.401) to become Z = EF − g

.

(5.407)

We now choose f in the form: fi = Pij (Aej ) ,

(5.408)

in which ej is a random isotropic Gaussian function, A a dimensioned constant that will play the same role as the subgrid viscosity model constant, and Pij the projection operator on a space of zero divergence. We have the relations: ei (x, t) = 0 , (5.409) 1 δij δ(t − t )δ(x − x ) . 3 Considering (5.408), (5.410) and (5.404), we get: ei (x, t)ei (x , t ) =

Ef =

1 2 1 A = A2 2 3

.

(5.410)

(5.411)

The computation of the model is completed by evaluating the constant A by a constrained localized dynamic procedure based on minimizing the functional (5.403), which can be re-written in the form: Z[A] =

A2 −g 3

2 .

(5.412)

6. Functional Modeling: Extension to Anisotropic Cases

6.1 Statement of the Problem The developments of the previous chapters are all conducted in the isotropic framework, which implies that both the ﬁlter used and the ﬂow are isotropic. They can be extended to anisotropic or inhom*ogeneous cases only by localizing the statistical relations in space and time and introducing heuristic procedures for adjusting the models. But when large-eddy simulation is applied to inhom*ogeneous ﬂows, we very often have to use anisotropic grids, which correspond to using a anisotropic ﬁlter. So there are two factors contributing to the violation of the hypotheses underlying the models presented so far: ﬁlter anisotropy (respectively inhom*ogeneity) and ﬂow anisotropy (respectively inhom*ogeneity). This chapter is devoted to extensions of the modeling to anisotropic cases. Two situations are considered: application of a anisotropic hom*ogeneous ﬁlter to an isotropic hom*ogeneous turbulent ﬂow (Sect. 6.2), and application of an isotropic ﬁlter to an anisotropic ﬂow (Sect. 6.3).

6.2 Application of Anisotropic Filter to Isotropic Flow The ﬁlters considered in the following are anisotropic in the sense that the ﬁlter cutoﬀ length is diﬀerent in each direction of space. The diﬀerent types of anisotropy possible for Cartesian ﬁltering cells are represented in Fig. 6.1. In order to use an anisotropic ﬁlter to describe an isotropic ﬂow, we are ﬁrst required to modify the subgrid models, because theoretical work and numerical experiments have shown that the resolved ﬁelds and the subgrid thus deﬁned are anisotropic [368]. For example, for a mesh cell with an aspect ratio ∆2 /∆1 = 8, ∆3 /∆1 = 4, the subgrid stresses will diﬀer from their values obtained with an isotropic ﬁlter by about ten percent. It is very important to note, though, that this anisotropy is an artifact due to the ﬁlter but that the dynamic of the subgrid scales still corresponds that of isotropic hom*ogeneous turbulence. On the functional modeling level, the problem is in determining the characteristic length that has to be used to compute the model.

188

6. Functional Modeling: Extension to Anisotropic Cases

Fig. 6.1. Diﬀerent types of ﬁltering cells. Isotropic cell (on the left): ∆1 = ∆2 = ∆3 ; pancake-type anisotropic cell (center): ∆1 ∆2 ≈ ∆3 ; cigar-type anisotropic cell (right): ∆1 ≈ ∆2 ∆3 .

Two approaches are available: – The ﬁrst consists in deﬁning a single length scale for representing the ﬁlter. This lets us keep models analogous to those deﬁned in the isotropic case, using for example scalar subgrid viscosities for representing the forward cascade process. This involves only a minor modiﬁcation of the subgrid models since only the computation of the characteristic cutoﬀ scale is modiﬁed. But it should be noted that such an approach can in theory be valid only for cases of low anisotropy, for which the diﬀerent cutoﬀ lengths are of the same order of magnitude. – The second approach is based on the introduction of several characteristic length scales in the model. This sometimes yields major modiﬁcations in the isotropic models, such as the deﬁnition of tensorial subgrid viscosities to represent the forward cascade process. In theory, this approach takes the ﬁlter anisotropy better into account, but complicates the modeling stage. 6.2.1 Scalar Models These models are all of the generic form ∆ = ∆(∆1 , ∆2 , ∆3 ). We present here: 1. Deardorﬀ’s original model and its variants (p. 189). These forms are empirical and have no theoretical basis. All we do is simply to show that they are consistent with the isotropic case, i.e. ∆ = ∆1 when ∆1 = ∆2 = ∆3 . 2. The model of Scotti et al. (p. 189), which is based on a theoretical analysis considering a Kolmogorov spectrum with an anisotropic hom*ogeneous ﬁlter. This model makes a complex evaluation possible of the ﬁlter cutoﬀ length, but is limited to the case of Cartesian ﬁltering cells.

6.2 Application of Anisotropic Filter to Isotropic Flow

189

Deardorﬀ ’s Proposal. The method most widely used today is without doubt the one proposed by Deardorﬀ [172], which consists in evaluating the ﬁlter cutoﬀ length as the cube root of the volume VΩ of the ﬁltering cell Ω. Or, in the Cartesian case: 1/3 ∆(x) = ∆1 (x)∆2 (x)∆3 (x)

,

(6.1)

in which ∆i (x) is the ﬁlter cutoﬀ length in the ith direction of space at position x. Extensions of Deardorﬀ ’s Proposal. Simple extensions of deﬁnition (6.1) are often used, but are limited to the case of Cartesian ﬁltering cells: ∆(x) =

+ 2 2 2 (∆1 (x) + ∆2 (x) + ∆3 (x))/3 ,

∆(x) = max ∆1 (x), ∆2 (x), ∆3 (x)

.

(6.2) (6.3)

Another way to compute the characteristic length on a strongly nonuniform mesh, which prevents the occurrence of large values of subgrid viscosity, was proposed by Arad [16]. It relies on the use of the harmonic mean of the usual length scales: 1/3 ∆(x) = ∆ˆ1 (x)∆ˆ2 (x)∆ˆ3 (x) with

−1/ω ˆi (x) = (∆i (x))−ω + (∆M )−ω ∆ i

,

,

i = 1, 2, 3 ,

(6.4)

(6.5)

M

where ∆i is a prescribed bound for ∆i (x), and ω > 0. Proposal of Scotti et al. More recently, Scotti, Meneveau, and Lilly [664] proposed a new deﬁnition of ∆ based on an improved estimate of the dissipation rate ε in the anisotropic case. The ﬁlter is assumed to be anisotropic but hom*ogeneous, i.e. the cutoﬀ length is constant in each direction of space. We deﬁne ∆max = max(∆1 , ∆2 , ∆3 ). Aspect ratios of less than unity, constructed from the other two cutoﬀ lengths with respect to ∆max , are denoted a1 and a2 1 . The form physically sought for the anisotropy correction is: ∆ = ∆iso f (a1 , a2 )

,

(6.6)

in which ∆iso is Deardorﬀ’s isotropic evaluation computed by relation (6.1). 1

For example, by taking ∆max = ∆1 , we get a1 = ∆2 /∆1 and a2 = ∆3 /∆1 .

190

6. Functional Modeling: Extension to Anisotropic Cases

Using the approximation: 2

ε = ∆ 2S ij S ij 3/2

,

(6.7)

and the following equality, which is valid for a Kolmogorov spectrum, K0 2 −5/3 3 |G(k)| k d k , (6.8) S ij S ij = ε2/3 2π where G(k) is the kernel of the anisotropic ﬁlter considered, after calculation we get: −3/4 K0 2 −5/3 3 |G(k)| k ∆= d k . (6.9) 2π Considering a sharp cutoﬀ ﬁlter, we get the following approximate relation by integrating equation (6.9): 4 [(ln a1 )2 − ln a1 ln a2 + (ln a2 )2 ] . f (a1 , a2 ) = cosh (6.10) 27 It is interesting to note that the dynamic procedure (see Sect. 5.3.3) for the computation of the Smagorinsky constant can be interpreted as an implicit way to compute the f (a1 , a2 ) function [663]. Introducing the subgrid mixing length ∆f , the Smagorinsky model reads νsgs

= = =

∆2f |S|

(6.11)

2 Cd ∆iso |S|

(CS ∆iso f (a1 , a2 )) |S| , 2

(6.12) (6.13)

where Cd is the value of the constant computed using a dynamic procedure, and CS the theoretical value of the Smagorinsky constant evaluated through the canonical analysis. A trivial identiﬁcation leads to: f (a1 , a2 ) = Cd /CS . (6.14) This interpretation is meaningful for positive values of the dynamic constant. A variant can be derived by using the anisotropy measure f (a1 , a2 ) instead of the isotropic one inside the dynamic procedure (see equation (6.12)), yielding new deﬁnitions of the tensors αij and βij appearing in the dynamic procedure (see Table 5.1). Taking the Smagorinsky model as an example, we get: * f (* *S * a2 ))2 |S| βij = −2(∆iso f (a1 , a2 ))2 |S|S ij , αij = −2(∆ iso a1 , * ij

, (6.15)

where f (a1 , a2 ) and f (* a1 , * a2 ) are the anisotropy measures associated to the ﬁrst and second ﬁltering levels, respectively. The corresponding formulation of the f function is now: f (a1 , a2 ) = Cd /(CS ∆iso ) . (6.16)

6.2 Application of Anisotropic Filter to Isotropic Flow

191

6.2.2 Batten’s Mixed Space-Time Scalar Estimator It was shown in Sect. 2.1.3 that spatial ﬁltering induces a time ﬁltering. In a reciprocal manner, enforcing a time-frequency cutoﬀ leads to the deﬁnition of an intrinsic spatial cutoﬀ length. To account for that phenomenon, Batten [49] deﬁnes the cutoﬀ length as + 2 ∆t ∆(x) = 2 max ∆1 (x), ∆2 (x), ∆3 (x), qsgs

,

(6.17)

2 where ∆t and qsgs are the time step of the simulation and the subgrid kinetic energy, respectively. The subgrid kinetic energy can be computed solving an prognostic transport equation (p. 128) or one of the methods discussed in Sect. 9.2.3.

6.2.3 Tensorial Models The tensorial models presented in the following are constructed empirically, with no physical basis. They are justiﬁed only by intuition and only for highly anisotropic ﬁltering cells of the cigar type, for example (see Fig. 6.1). Representing the ﬁlter by a single and unique characteristic length is no longer relevant. The ﬁlter’s characteristic scales and their inclusion in the subgrid viscosity model are determined intuitively. Two such models are described: 1. The model of Bardina et al. (p. 191), which describes the geometry of the ﬁltering cell by means of six characteristic lengths calculated from the inertia tensor of the ﬁltering cell. This approach is completely general and is a applicable to all possible types of ﬁltering cells (Cartesian, curvilinear, and other), but entrains a high complexiﬁcation in the subgrid models. 2. The model of Zahrai et al. (p. 192), which is applicable only to Cartesian cells and is simple to include in the subgrid viscosity models. Proposal of Bardina et al. Deﬁnition of a Characteristic Tensor. These authors [39] propose replacing the isotropic scalar evaluation of the cutoﬀ length associated with the grid by an anisotropic tensorial evaluation linked directly to the ﬁltering cell geometry: V (x) = (∆1 (x)∆2 (x)∆3 (x)). To do this, we introduce the moments of the inertia tensor I associated at each point x: Iij (x) =

1 V (x)

xi xj dV

.

(6.18)

V

Since the components of the inertia tensor are hom*ogeneous at the square of a length, the tensor of characteristic lengths is obtained by taking the square root of them. In the case of a pancake ﬁltering cell aligned with the

192

6. Functional Modeling: Extension to Anisotropic Cases

axes of the Cartesian coordinate system, we get the diagonal matrix: ⎛

2

∆ 2⎜ 1 Iij = ⎝ 0 3 0

0 2 ∆2 0

⎞ 0 ⎟ 0 ⎠ 2 ∆3

.

(6.19)

Application to the Smagorinsky Model. As we model only the anisotropic part of the subgrid tensor, the tensor I is decomposed into the sum of a spherical term I i and an anisotropic term I d : d Iij = I i δij + (Iij − I i δij ) = I i δij + Iij

,

(6.20)

with Ii =

1 1 2 2 2 Ikk = (∆1 + ∆2 + ∆3 ) . 3 3

(6.21)

Modifying the usual Smagorinsky model, the authors ﬁnally propose the following anisotropic tensorial model for deviator of the subgrid tensor τ : 1 τij − τkk δij 3

= + +

C1 I i |S|S ij 1 C2 |S| Iik S kj + Ijk S ki − Ilk S kl δij 3 1 |S| , C3 i Iik Ijl S kl − Imk Iml S kl δij I 3

(6.22)

in which C1 , C2 and C3 are constants to be evaluated. Proposal of Zahrai et al. Principle. Zahrai et al. [795] proposed conserving the isotropic evaluation of the dissipation rate determined by Deardorﬀ and further considering that this quantity is constant over each mesh cell: 2/3 2S ij S ij 3/2 ε = ∆1 (x)∆2 (x)∆3 (x)

.

(6.23)

On the other hand, when deriving the subgrid model, we consider that the ﬁlter’s characteristic length in each direction is equal to the cutoﬀ length in that direction. This procedure calls for the deﬁnition of a tensorial model for the subgrid viscosity. Application to the Smagorinsky model. In the case of the Smagorinsky model, we get for component k: (νsgs )k = C1 (∆1 ∆2 ∆3 )2/9 (∆k )4/3 2S ij S ij 3/2 where C1 is a constant.

,

(6.24)

6.3 Application of an Isotropic Filter to a Shear Flow

193

6.3 Application of an Isotropic Filter to a Shear Flow We will now be examining the inclusion of subgrid scale anisotropy in the functional models. The ﬁrst part of this section presents theoretical results concerning subgrid scale anisotropy and the interaction mechanisms between the large and small scales in this case. These results are obtained either by the EDQNM theory or by asymptotic analysis of the triadic interactions. The second part of the section describes the modiﬁcations that have been proposed for functional type subgrid models. Only models for the forward energy cascade will be presented, because no model for the backward cascade has yet been proposed in the anisotropic case. 6.3.1 Phenomenology of Inter-Scale Interactions Anisotropic EDQNM Analysis. Aupoix [24] proposes a basic analysis of the eﬀects of anisotropy in the hom*ogeneous case using Cambon’s anisotropic EDQNM model. The essential details of this model are given in Appendix B. The velocity ﬁeld u is decomposed as usual into average part u and a ﬂuctuating part u : u = u + u

.

(6.25)

To study anisotropic hom*ogeneous ﬂows, we deﬁne the spectral tensor Φij (k) = u∗ uj (k) i (k)

,

(6.26)

which is related to the double correlations in the physical space by the relation: ui uj (x) = Φij (k)d3 k . (6.27) Starting with the Navier–Stokes equations, we obtain the evolution equation (see Appendices A and B):

∂ 2 + 2νk Φij (k) + ∂t

∂ui ∂uj Φjl (k) + Φil (k) ∂xl ∂xl ∂ul (ki Φjm (k) + kj Φmi (k)) ∂xm

−

2

−

∂ul ∂ (kl Φij (k)) ∂xm ∂km

=

Pil (k)Tlj (k) + Pjl (k)Tli∗ (k) ,

(6.28)

194

6. Functional Modeling: Extension to Anisotropic Cases

where

ui (k)ul (p)uj (−k − p)d3 p

Tij (k) = kl and

Pij (k) =

ki kj δij − 2 k

,

(6.29)

,

(6.30)

and where the * designates the complex conjugate number. We then simplify the equations by integrating the tensor Φ on spheres of radius k=cste: φij (k) =

Φij (k)dA(k) ,

(6.31)

and obtain the evolution equations:

∂ 2 + 2νk φij (k) = ∂t +

−

∂ui ∂uj φjl (k) − φil (k) ∂xk ∂xl

l nl Pijl (k) + Sij (k) + Pijnl (k) + Sij (k)

, (6.32)

where the terms P l , S l , P nl and S nl are the linear pressure, linear transfer, non-linear pressure, and non-linear transfer contributions, respectively. The linear terms are associated with the action of the average velocity gradient, and the non-linear terms with the action of the turbulence on itself. The expression of these terms and their closure by the anisotropic EDQNM approximation are given in Appendix B. Using these relations, Aupoix derives an expression for the interaction between the modes corresponding to wave numbers greater than a given cutoﬀ wave number kc (i.e. the small or subgrid scales) and those associated with small wave numbers such that k ≤ kc (i.e. the large or resolved scales). To obtain a simple expression for the coupling among the diﬀerent scales by the non-linear terms P nl and S nl , we adopt the hypothesis that there exists a total separation of scales (in the sense deﬁned in Sect. 5.3.2) between the subgrid and resolved modes, so that we can obtain the following two asymptotic forms: Pijnl (k) = +

∞

E 2 (p)Hij (p) Θ0pp [10 + a(p)] dp p2 kc ∞ 16 2 ∂ k E(k) Θ0pp (a(p) + 3)p (E(p)Hij (p)) 105 ∂p kc ∂a(p) +E(p)Hij (p) 5 {a(p) + 3} + p dp , (6.33) ∂p

−

32 4 k 175

6.3 Application of an Isotropic Filter to a Shear Flow nl Sij (k) =

− −

195

∞ E 2 (p) 14 1 8 δij + 2Hij (p) + a(p)Hij (p) dp 2k 4 Θ0pp 2 p 15 3 25 kc ∞ ∂E(p) 1 2k 2 φij (k) Θ0pp 5E(p) + p dp 15 kc ∂p 7 ∞ 2 ∂ 2 Θ0pp 2k E(k) 5E(p)Hij (p) + p (E(p)Hij (p)) 15 ∂p kc 7

8 8 (a(p) + 3) + a(p) dp , (6.34) +E(p)Hij (p) 15 25

where E(k) is the energy spectrum, deﬁned as: E(k) =

1 φll (k) 2

,

(6.35)

and Hij (k) the anisotropy spectrum: Hij (k) =

φij (k) 1 − δij 2E(k) 3

.

(6.36)

It is easily veriﬁed that, in the isotropic case, Hij cancels out by construction. The function a(k) is a structural parameter that represents the anisotropic distribution on the sphere of radius k, and Θkpq the characteristic relaxation time evaluated by the EDQNM hypotheses. The expression of this term is given in Appendix B. These equations can be simpliﬁed by using the asymptotic value of the structural parameter a(k). By taking a(k) = −4.5, we get: ∞ E 2 (p) 28 368 nl nl 4 δij − Hij (p) dp Θ0pp 2 Pij (k) + Sij (k) = k p 45 175 kc ∞ 1 ∂E(p) 2 − 2k φij (k) Θ0pp 5E(p) + p dp 15 kc ∂p ∞ 1052 E(p)Hij (p) Θ0pp + k 2 E(k) 525 kc 52 ∂ (E(p)Hij (p)) dp . − (6.37) 105 ∂p From this equation, it can be seen that the anisotropy of the small scales takes on a certain importance. In a case where the anisotropic spectrum has the same (resp. opposite) sign for the small scales as it does for the large, the term in k 4 constitutes a return of energy that has the eﬀect of a return toward isotropy (resp. departure from isotropy), and the term in k 2 E(k) represents an backward energy cascade associated with an increasing anisotropy (resp. a return to isotropy). Lastly, the term in k 2 φij (k) is a term of isotropic drainage of energy to the large scales by the small, and represents here the energy cascade phenomenon modeled by the isotropic subgrid models.

196

6. Functional Modeling: Extension to Anisotropic Cases

Asymptotic Analysis of Triadic Interactions. Another analysis of interscale interactions in the isotropic case is the asymptotic analysis of triadic interactions [74, 785]. (k) is written in the symbolic The evolution equation of the Fourier mode u form: (k) ∂u ˙ ˙ ˙ = u(k) = [u(k)] , (6.38) nl + [u(k)] vis ∂t ˙ ˙ where [u(k)] nl and [u(k)] vis represent, respectively, the non-linear terms associated with the convection and pressure, and the linear term associated with the viscous eﬀects, deﬁned as: ˙ [u(k)] nl = −i

(p)⊥k (k · u (k − p)) u

,

(6.39)

p

with u i (p)⊥k

ki kj = δij − 2 u j (p) , k

(6.40)

2 ˙ (k) . [u(k)] vis = −νk u

(6.41)

(k) · u ∗ (k), is of the The evolution equation of the modal energy, e(k) = u form: ∂e(k) (k) · u˙ ∗ (k) + cc = [e(k)] =u ˙ ˙ , (6.42) nl + [e(k)] vis ∂t with [e(k)] ˙ nl = −i

3

4 ∗ (k) · u (p) [k · u (k − p)] + cc u

,

(6.43)

p 2 [e(k)] ˙ vis = −2νk e(k) ,

(6.44)

where the symbol cc designates the complex conjugate number of the term that precedes it. The non-linear energy transfer term brings in three wave vectors (k, p, q = k−p) and is consequently a linear sum of non-linear triadic interactions. We recall (see Sect. 5.1.2) that the interactions can be classiﬁed into various categories ranging from local interactions, for which the norms of the three wave vectors are similar (i.e. k ∼ p ∼ q), to distant interactions for which the norm of one of the wave vectors is very small compared with the other two (for example k p ∼ q). The local interactions therefore correspond to the inter-scale interactions of the same size and the distant interactions to the interactions between a large scale and two small scales.

6.3 Application of an Isotropic Filter to a Shear Flow

197

Also, any interaction that introduces a (k, p, q) triad that does not verify the relation k ∼ p ∼ q is called a non-local interaction. In the following, we will be analyzing an isolated distant triadic interaction associated with three modes: k, p and q. We adopt the conﬁguration k p ∼ q and assume that k is large scale located in the energetic portion of the spectrum. An asymptotic analysis shows that: ˙ [u(k)] nl = O(δ)

,

(6.45)

∗ 3 4 ˙ ∗ (k) + O(δ) (q) p · u [u(p)] nl = −i u

,

(6.46)

∗ 3 4 (p) p · u ∗ (k) + O(δ) ˙ [u(q)] nl = −i u

,

(6.47)

where δ is the small parameter deﬁned as δ=

k 1 p

.

The corresponding energy transfer analysis leads to the following relations: [e(k)] ˙ nl = O(δ)

,

(q) [p · u (k)] + cc} + O(δ) ˙ u(p) · u [e(p)] ˙ nl = − [e(q)] nl = i {

(6.48) .

(6.49)

Several remarks can be made: – The interaction between large and small scales persists in the limit of the inﬁnite Reynolds numbers. Consistently with the Kolmogorov hypotheses, these interactions occur with no energy transfer between the large and small scales. Numerical simulations have shown that the energy transfers are negligible between two modes separated by more than two decades. (p) and u (q) is directly pro– The variation rate of the high frequencies u (k). This implies portional to the amplitude of the low-frequency mode u that the strength of the coupling with the low-frequency modes increases with the energy of the modes. Moreover, complementary analysis shows that, for modes whose wavelength is of the order of the Taylor micro-scale λ deﬁned as (see Appendix A): 8 9 9 u 2 (6.50) λ=9 9 2 , : ∂u ∂x

198

6. Functional Modeling: Extension to Anisotropic Cases

the ratio between the energy transfers due to the distant interactions and those due to the local interactions vary as: [e(k ˙ λ )]distant 11/6 ∼ Reλ [e(k ˙ λ )]local

,

(6.51)

where Reλ is the Reynolds number referenced to the Taylor micro-scale and the velocity ﬂuctuation u . This relation shows that the coupling increases with the Reynolds number, with the result that an anisotropic distribution of the energy at the low frequencies creates an anisotropic forcing of the high frequencies, leading to a deviation from isotropy of these high frequencies. A competitive mechanism exists that has an isotropy reduction eﬀect at the small scales. This is the energy cascade associated with non-local triadic interactions that do not enter into the asymptotic limit of the distant interactions. For a wave vector of norm k, the ratio of the characteristic times τ (k)cascade and τ (k)distant , associated respectively with the energy transfer of the cascade mechanism and that due to the distant interactions, is evaluated as: τ (k)cascade ∼ constant × (k/kinjection)11/6 , (6.52) τ (k)distant where kinjection is the mode in which the energy injection occurs in the spectrum. So we see that the distant interactions are much faster than the energy cascade. Also, the ﬁrst eﬀect of a sudden imposition of large scale anisotropy will be to anisotropize the small scales, followed by competition between the two mechanisms. The dominance of one of the two depends on a number of factors, such as the separation between the k and kinjection scales, or the intensity and coherence of the anisotropy at the large scale. Numerical simulations [785] performed in the framework of hom*ogeneous turbulence have shown a persistence of anisotropy at the small scales. However, it should be noted that this anisotropy is detected only on statistical moments of the velocity ﬁeld of order three or more, with ﬁrst- and secondorder moments being isotropic. 6.3.2 Anisotropic Models: Scalar Subgrid Viscosities The subgrid viscosity models presented in this section have been designed to alleviate the problem observed with basic subgrid viscosities, i.e. to prevent the occurance of too high dissipation levels in shear ﬂows, which are known to have disastrous eﬀects in near-wall regions.2 The models presented below are: 2

Problems encountered in free shear ﬂows are usually less important, since large scales are often driven by inviscid instabilities, while the existence of a critical Reynolds number may lead to relaminarization if the subgrid viscosity is too high.

6.3 Application of an Isotropic Filter to a Shear Flow

199

1. WALE model by Nicoud and Ducros (p. 199), which is built to recover the expected asymptotic behavior in the near-wall region in equilibrium turbulent boundary layers on ﬁne grids, without any additional damping function. 2. Casalino–Jacob Weighted Gradient Model (p. 199), which is based on a modiﬁcation of the of the Smagorinsky constant to make it sensitive to the mean shear stresses, rendering it more local in terms of wave number. 3. Models based on the idea of separating the ﬁeld into an isotropic part and inhom*ogeneous part (p. 200), in order to be able to isolate the contribution of the mean ﬁeld in the computation of the subgrid viscosity, for models based on the large scales, and thereby better localize the information contained in these models by frequency. This technique, however, is applicable only to ﬂows whose mean velocity proﬁle is known or can be computed on the ﬂy. Wall-Adapting Local Eddy-Viscosity Model. It has been seen before (p. 159) that most subgrid viscosity models do not exhibit the correct behavior in the vicinity of solid walls in equilibrium boundary layers on ﬁne grids, resulting in a too high damping of ﬂuctuations in that region and to a wrong prediction of the skin friction. The common way to alleviate this problem is to add a damping function, which requires the distance to the wall and the skin friction as input parameters, leading to complex implementation issues. Another possibility is to use self-adpative models, which involve a larger algorithmic complexity. An elegant solution to solve the near-wall region problem on ﬁne grids is proposed by Nicoud and Ducros [567], who found a combination of resolved velocity spatial derivatives that exhibits the expected asymptotic behavior 3 νsgs ∝ z + , where z + is the distance to the wall expressed in wall units. The subgrid viscosity is deﬁned as νsgs

d d 3/2 Sij Sij = (Cw ∆) 5/2 d d 5/4 S ij S ij + Sij Sij 2

,

(6.53)

with Cw = 0.55 − 0.60 and d = S ik S kj + Ω ik Ω kj − Sij

1 S mn S mn − Ω mn Ω mn δij 3

.

(6.54)

This model also possesses the interesting property that the subgrid viscosity vanishes when the ﬂow is two-dimensional, in agreement with the physical analysis. Weighted Gradient Subgrid Viscosity Model. Casalino, Boudet and Jacob [112] introduced a modiﬁcation in the evaluation of the Smagorinsky constant to render it sensitive to resolved gradients, with the purpose of recovering a better accuracy in shear ﬂows.

200

6. Functional Modeling: Extension to Anisotropic Cases

The weighted gradient subgrid viscosity model is written as νsgs = (C(x, t)∆)2 |S ij | ,

(6.55)

where the self-adaptive constant is equal to C(x, t) = γCS

∗ ∗ Sij Sij

m/4 ,

S ij S ij

(6.56)

where CS = 0.18 is the conventional Smagorinsky constant, γ and m are free parameters and the weighted strain tensor S ∗ is deﬁned as (without summation over repeated greek indices) ∗ Sαβ = Wαβ S αβ

.

(6.57)

The weighting matrix coeﬃcients are inversely proportional to the third moment of corresponding strain rate tensor coeﬃcients: ; < 3 S αβ ; Wαβ = (6.58) 3 <−1 . = S αβ α,β=1,3 The bracket operator is related to a local average. Numerical tests have shown that (m, γ) = (1, 3) and (3, 10) yield satisfactory results in free shear ﬂows, the last values yielding the recovery of the theoretical behavior of the subgrid viscosity in the vicinity of solid walls on ﬁne meshes. Models Based on a Splitting Technique. Subgrid viscosity models are mostly developed in the framework of the hypotheses of the canonical analysis, i.e. for hom*ogeneous turbulent ﬂows. Experience shows that the performance of these models declines when they are used in an inhom*ogeneous framework, which corresponds to a non-uniform average ﬂow. One simple idea initially proposed by Schumann [653] is to split the velocity ﬁeld into inhom*ogeneous and isotropic parts and to compute a speciﬁc subgrid term for each of these parts. In practice, Schumann proposes an anisotropic subgrid viscosity model for dealing with ﬂows whose average gradient is non-zero, and in particular any ﬂow regions close to solid walls. The model is obtained by splitting the deviator part of the subgrid tensor τ d into one locally isotropic part and one inhom*ogeneous: a τijd = −2νsgs S ij − S ij − 2νsgs S ij , (6.59) where the angle brackets . designate an statistical average, which in practice is a spatial average in the directions of hom*ogeneity in the solution. The coa eﬃcients νsgs and νsgs are the scalar subgrid viscosities representing a locally

6.3 Application of an Isotropic Filter to a Shear Flow

201

isotropic turbulence and an inhom*ogeneous turbulence, respectively. Moin and Kim [537] and Horiuti [319] give the following deﬁnitions: 2 + νsgs = C1 ∆ 2 S ij − S ij S ij − S ij , (6.60) + 2 a νsgs = C2 ∆z 2S ij S ij , (6.61) where C1 and C2 are two constants. Horiuti recommends C1 = 0.1 and C2 = 0.254, while Moin and Kim use C1 = C2 = 0.254. The isotropic part is a function of the ﬂuctuation of the viscosity gradients, so as to make sure that the extra-diagonal components thus predicted for the subgrid tensor cancel out on the average over time. This is consistent with the isotropic hypothesis. The two characteristic lengths ∆ and ∆z represent the cutoﬀ lengths for the two types of structures, and are evaluated as: ∆(z) = (∆1 ∆2 ∆3 )1/3 (1 − exp(zuτ /Aν)) ∆z (z) = ∆3 (1 − exp([zuτ /Aν]2 ))

,

,

(6.62) (6.63)

where z is the distance to the solid wall, ∆3 the cutoﬀ length in the direction normal to the surface, and uτ the friction velocity at the surface (see Sect. 10.2.1). The constant A is taken to be equal to 25. This model was initially designed for the case of a plane channel ﬂow. It requires being able to compute the statistical average of the velocity ﬁeld, and thus can be extended only to sheared ﬂows exhibiting at least one direction of hom*ogeneity, or requires the use of several statistically equivalent simulations to perform the ensemble average [102, 108]. Sullivan et al. [698] propose a variant of it that incorporates an anisotropy factor (so that the model constant can be varied to represent the ﬁeld anisotropy better): a S ij τijd = −2νsgs γS ij − 2νsgs

.

(6.64)

a The authors propose computing the viscosity νsgs as before. The νsgs term, on the other hand, is now calculated by a model with one evolution equation for the subgrid kinetic energy (see equation (5.119) in Chap. 5). Only the subgrid kinetic energy production by the isotropic part is included, which is equivalent to replacing the II term in equation (5.119) with . (6.65) 2νsgs γ S ij − S ij S ij − S ij

The authors evaluate the anisotropy factor from the shearing rates of the large and small scales. The average per plane of ﬂuctuation hom*ogeneity of the resolved strain rate tensor, calculated by + S = 2 S ij − S ij S ij − S ij , (6.66)

202

6. Functional Modeling: Extension to Anisotropic Cases

is used for evaluating the shear of the small scales. The shear of the large scales is estimated as + S = 2S ij S ij . (6.67) The isotropy factor is evaluated as: γ=

S

S + S

.

(6.68)

6.3.3 Anisotropic Models: Tensorial Subgrid Viscosities Here we describe the main models proposed in the anisotropic framework. Except for Aupoix’s spectral model, none of these take explicit account of the backward cascade mechanism. They are: 1. Aupoix’s spectral model (p. 203), which is based on the anisotropic EDQNM analysis. The interaction terms are evaluated by adopting a preset shape of the energy spectra and subgrid mode anisotropy. This model, which requires a great deal of computation, has the advantage of including all the coupling mechanisms between large and small scales. 2. Horiuti’s model (p. 204), which is based on an evaluation of the anisotropy tensor of the subgrid modes from the equivalent tensor constructed from the highest frequencies in the resolved ﬁeld. This tensor is then used to modulate the subgrid viscosity empirically in each direction of space. This is equivalent to considering several characteristic velocity scales for representing the subgrid modes. This model can only modulate the subgrid dissipation diﬀerently for each velocity component and each direction of space, but does not include the more complex anisotropic transfer mechanisms through the cutoﬀ. 3. The model of Carati and Cabot (p. 205), who propose a general form of the subgrid viscosity in the form of a fourth-rank tensor. The components of this tensor are determined on the basis of symmetry relations. However, this model is a applicable only when the ﬂow statistically exhibits an axial symmetry, which restricts its ﬁeld of validity. 4. The model of Abba et al. (p. 207) which, as in the previous example, considers the subgrid viscosity in the form of a fourth-rank tensor. The model is based on the choice of a local adapted reference system for representing the subgrid modes, and which is chosen empirically when the ﬂow possesses no obvious symmetries. 5. The model proposed by Carati (p. 207), which is based on the evaluation of the statistical anisotropy tensor. The requirement (mean velocity proﬁle) is the same as in the Schumann splitted model.

6.3 Application of an Isotropic Filter to a Shear Flow

203

Aupoix Spectral Model. In order to take the anisotropy of the subgrid scales into account, Aupoix [24] proposes adopting preset shapes of the energy spectra and anisotropy so that the relations stemming from the previously described EDQNM analysis of anisotropy can be used. Aupoix proposes the following model for the energy spectrum: E(k) = K0 ε2/3 k −5/3 exp {f (k/kd )}

,

(6.69)

where

" % &# f (x) = exp −3.5x2 1 − exp 6x + 1.2 − 196x2 − 33.6x + 1.4532

.

(6.70) This spectrum is illustrated in Fig. 6.2. The anisotropy spectrum is modeled by: k ∂E(k) Hij (k) = bij 5 + E(k) ∂k > ?) ( −2/3 k k − 1 H (|F(u)|) −1 , × 1+H kmax kmax (6.71) where F (u) = ∇ × u, kmax is the wave number corresponding to the energy spectrum maximum, and H the Heaviside function deﬁned by:

0 if x ≤ 0 H(x) = , 1 otherwise

Fig. 6.2. Aupoix spectrum (kd = 1000).

204

6. Functional Modeling: Extension to Anisotropic Cases

and where bij is the anisotropy tensor deﬁned as: bij =

ui uj 1 − δij 2 qsgs 3

.

(6.72)

Horiuti’s Model. Horiuti [322] proposes extending the Smagorinsky model to the isotropic case by choosing a diﬀerent velocity scale for characterizing each component of the subgrid tensor. Starting with an ordinary dimensional analysis, the subgrid viscosity νsgs 2 is expressed as a function of the subgrid kinetic energy qsgs and the dissipation rate ε: 2 2 (qsgs ) νsgs = C1 . (6.73) ε To make a better adjustment of the dissipation induced by the subgrid model to the local state of the ﬂow, Horiuti proposes replacing equation (6.73) by: 2 2 (qsgs ) νsgs = C1 Υ , (6.74) ε in which Υ is a dimensionless parameter whose function is to regulate the dissipation rate as a function of the anisotropy of the resolved ﬁeld. The proposed form for Υ is: 3E s Υ = 2 , (6.75) 2qsgs where E s is the square of a characteristic velocity scale of the subgrid modes. For example, near solid walls, Horiuti proposes using the ﬂuctuation of the velocity component normal to the wall, which makes it possible for the model to cancel out automatically. To generalize this approach, we associate a chars with each subgrid stress τij . acteristic velocity Eij In practice, the author proposes evaluating these characteristic velocities by the scale similarity hypothesis by means of a test ﬁlter indicated by a tilde: s ˜i )(uj − u ˜j ) , = (ui − u Eij

(6.76)

which makes it possible to deﬁne a tensorial parameter Υij as: ˜i )(uj − u ˜j ) 3(ui − u Υij = = 2 ˜ l=1,3 (ul − ul )

.

(6.77)

This tensorial parameter characterizes the anisotropy of the test ﬁeld ˜ and can be considered as an approximation of the anisotropy tensor (u − u)

6.3 Application of an Isotropic Filter to a Shear Flow

205

associated with this velocity ﬁeld (to within a coeﬃcient of 1/3 δij ). Using a model based on the large scales, Horiuti derives the tensorial subgrid viscosity νeij : 2 νeij = C1 ∆ |F(u)| Υij , (6.78) with F (u) = ∇ × u,

or

∇u + ∇T u

,

where the constant C1 is evaluated as it is for the scalar models. He proposes a model of the general form for the subgrid tensor τ : τij = δij where K=

2 2 K+ P 3 3

− νeil

1 ˜l )2 , (ul − u 2

∂uj ∂ui − νejl ∂xl ∂xl

P = νelm

l=1,3

∂um ∂xl

,

(6.79)

.

It is important to note that this is a model for the entire subgrid tensor and not for its deviatoric part alone, as is the case for the isotropic models. Carati and Cabot Model. Carati and Cabot [100] propose a tensorial anisotropic extension of the subgrid viscosity models. Generally, the deviator τ d of the subgrid tensor τ is modeled as: (1)

(2)

τijd = νijkl S kl + νijkl Ω kl

,

(6.80)

where the tensors S and Ω are deﬁned as: S=

1 ∇u + ∇T u , 2

1 ∇u − ∇T u 2

Ω=

.

The two viscosities ν (1) and ν (2) are fourth-rank tensors theoretically deﬁned by 81 independent coeﬃcients. However, the properties of the tensors τ d , S and Ω make it possible to reduce the number of these parameters. The tensors τ d and S are symmetrical and have zero trace, which entails: νijkl

(1)

= νjikl

(1)

,

(1) νijkl

(1) νjilk

,

=

νiikl

(1)

= 0

,

(1) νijkk

= 0

.

206

6. Functional Modeling: Extension to Anisotropic Cases

The tensor ν (1) therefore contains 25 independent coeﬃcients. By a similar analysis, we can say: (2)

= νjikl

νijkl

(2)

= −νjilk

(2) νiikl

= 0

νijkl

(2)

,

(2)

,

. (6.81)

The tensor ν (2) therefore contains 15 independent coeﬃcients, which raises the number of coeﬃcients to be determined to 40. Further reductions can be made using the symmetry properties of the ﬂow. For the case of symmetry about the axis deﬁned by the vector n = (n1 , n2 , n3 ), the authors show that the model takes a reduced form that now uses only four coeﬃcients, C1 , ..., C4 : τijd

= −

2 −2C1 S ij − 2C2 ni sj + si nj − sk nk δij 3 1 C3 ni nj − n2 δij sk nk − 2C4 (r i nj + ni r j ) 3

,

(6.82)

where si = S ik nk and r i = Ω ik nk . Adopting the additional hypothesis that the tensors ν (1) and ν (2) verify the Onsager symmetry relations for the covariant vector n and the contravariant vector p: (1)

νklij (n) ,

(2)

νklij (n) ,

νijkl (p) =

(1)

νklij (−p)

,

(2) νijkl (p)

(2) νklij (−p)

,

νijkl (n) = νijkl (n) =

=

(1) (2) (1)

(6.83) we get the following reduced form:

⊥

τijd = −2ν1 S ij − 2ν2 n2 S ij

,

(6.84)

where ν1 and ν2 are two scalar viscosities and

S ij =

1 1 (ni sj + si nj ) − 2 sk nk δij , n2 3n

⊥

S ij = S ij − S ij

.

Carati then proposes determining the two parameters ν1 and ν2 by an ordinary dynamic procedure.

6.3 Application of an Isotropic Filter to a Shear Flow

207

Model of Abba et al. Another tensor formulation was proposed by Abba et al. [1]. These authors propose deﬁning the subgrid viscosity in the form of the fourth-rank tensor denoted νijkl . This tensor is deﬁned as the product of a scalar isotropic subgrid viscosity νiso and an fourth-rank tensor denoted C, whose components are dimensionless constants which will play the role of the scalar constants ordinarily used. The tensor subgrid viscosity νijkl thus deﬁned is expressed: ⎞ ⎛ νijkl = Cijkl νiso = ⎝ Cαβ aiα ajβ akα alβ ⎠ νiso , (6.85) α,β

where aiα designates the ith component of the unit vector aα (α = 1, 2, 3), Cαβ is a symmetrical 3 × 3 matrix that replaces the scalar Smagorinsky constant. The three vectors aα are arbitrary and have to be deﬁned as a function of some foreknowledge of the ﬂow topology and its symmetries. When this information is not known, the authors propose using the local framework deﬁned by the following three vectors: a1 =

u , u

a3 =

∇(|u|2 ) × u , |∇(|u|2 ) × u|

a2 = a3 × a1

.

(6.86)

The authors apply this modiﬁcation to the Smagorinsky model. The scalar viscosity is thus evaluated by the formula: 2

νiso = ∆ |S| .

(6.87)

The subgrid tensor deviator is then modeled as: 2 2 2 τijd = −2 Cijkl ∆ |S|S kl + δij ∆ Cmmkl |S|S kl 3

.

(6.88)

k,l

The model constants are then evaluated by means of a dynamic procedure. Models Based on a Splitting Technique. An anisotropic tensorial subgrid viscosity model is proposed by Carati et al. [103]. The resulting form of the subgrid stress tensor is τij = νscalar γik γjl S kl

,

(6.89)

where νscalar plays the role of a scalar subgrid viscosity to be computed using an arbitrary functional model and γij = 3

ui uj uk uk

,

(6.90)

where u (x, t) = u(x, t) − u(x, t) is the local instantaneous ﬂuctuation of the resolved ﬁeld around its statistical mean value. In practice, the mean ﬂow values can be obtained performing statistical averages over hom*ogeneous directions of the ﬂow, over several realizations computed in parallel, or by performing a steady RANS computation.

208

6. Functional Modeling: Extension to Anisotropic Cases

6.4 Remarks on Flows Submitted to Strong Rotation Eﬀects All developements presented above deal with shear ﬂows. Rotation is known to also lead to isotropy breakdown and to very complex changes in the interscale dynamics [97]: – The turbulent kinetic energy dissipation rate is observed to diminish, due to a scrambling in the non-linear triadic interactions. These phenomena have been extensively described using the weak wave turbulence framework and advanced EDQNM closures. – The Kolmogorov spectrum is no longer valid when strong rotational eﬀects are present. – Anisotropy is seen to rise, even starting from initially isotropic state. This trend is observed on all statistical moments, including Reynolds stresses, integral length scales, ... Spectral analysis reveals that the induced anisotropy escapes the classical description in the physical space, and that special representations in the Fourier space must be used to describe it acuurately. – Very complex coupling of rotation with strain is observed, leading to a very large class of dynamical regimes. The question thus arises of the validity of the subgrid models in the presence of dominant rotational eﬀects, since they do not account for this change in the interscale transfers. In fact, none of the subgrid model presented above is able to account for rotation eﬀects. But numerical experiments [183, 702, 557, 577, 152, 398, 422, 576, 596, 777, 215] show that good results in ﬂows driven by strong rotation are obtained using models that are local in terms of wave numbers on ﬁne grids: dynamic models (and other selfadaptive models), approximate deconvolution models, ... The explanation for this success is that rotation eﬀects are already taken into account by resolved scales, and that local subgrid models will be built on scales that are already modiﬁed by the rotation.

7. Structural Modeling

7.1 Introduction and Motivations This chapter describes some of the family of structural models. As has already been said, these are established with no prior knowledge of the nature of the interactions between the subgrid scales and those that make up the resolved ﬁeld. These models can be grouped into several categories: – Those derived by formal series expansions (Sect. 7.2). These models make no use of any foreknowledge of the physics of the ﬂows, and are based only on series expansions of the various terms that appear in the ﬁltered Navier–Stokes equations. This group of model encompasses models based on deconvolution procedures, nonlinear models and those based on the hom*ogenization technique. – Those that use the physical hypothesis of scale similarity (Sect. 7.3). These models are based on the scale similarity hypothesis, which establishes a correspondence between the statistical structure of the ﬂow at diﬀerent ﬁltering levels. Despite the fact that these models are formally equivalent to deconvolution-type models, I chose to present them in a separate section, because they were originally derived on the grounds of physical assumptions rather than on mathematical considerations. The link between the two classes of model is explicitly discussed in Sect. 7.3.3. – The mixed models, which are based on linear combinations of the functional and structural types, are presented in Sect. 7.4. These models have historically been developed within the framework of the scale-similarity hypothesis, but recent developments dealing with the deconvolution approach show that they are a natural part of deconvolution-based subgrid models. Here again, I chose to ﬁt these to the classical presentation, in order to allow the reader to establish the link with published references more easily. The theoretical equivalence with full deconvolution models is discussed at the end of this section. – Those based on transport equations for the subgrid tensor components (Sect. 7.5). These models, though they require no information concerning the way the subgrid modes act on the resolved scales, require a very

210

–

–

–

–

7. Structural Modeling

complex level of modeling since all the unknown terms in the transport equations for the subgrid tensor components have to be evaluated. Those constructed from deterministic models for the subgrid structures (Sect. 7.6). They assume that preferential directions of alignments are known for the subgrid structures. Those based on an explicit reconstruction of the subgrid velocity ﬂuctuations on an auxiliary grid (Sect. 7.7). These models are the only ones which aim at reconstructing the subgrid motion directly. They can be interpreted as solutions for the full deconvolution problem, as deﬁned in Sect. 7.2. The main diﬀerence with deconvolution-like models is that they require the deﬁnition of a ﬁner auxiliary grid, on which the solution of the hard deconvolution problem is explicitly reconstructed. Those based on a direct identiﬁcation of subgrid terms using advanced mathematical tools, such as linear stochastic estimation or neural networks (Sect. 7.8). Those based on speciﬁc numerical algorithms, whose errors are designed to mimic the subgrid forces (Sect. 7.9).

7.2 Formal Series Expansions The structural models presented in this section belong to one of the three following families: 1. Models based on approximate deconvolution (Sect. 7.2.1). They rely on an attempt to recover, at least partially, the original unﬁltered velocity ﬁeld by inverting the ﬁltering operator. The full deconvolution being impossible to compute in practice, only approximate deconvolution is used. 2. Nonlinear models (Sect. 7.2.2), which rely on the formal derivation of a surrogate of the subgrid stress tensor τ as a function of the gradient of the resolved velocity ﬁeld. 3. Models based on the hom*ogenization technique (Sect. 7.2.3). 7.2.1 Models Based on Approximate Deconvolution General Statement of the Deconvolution Problem. The deconvolution approach, also sometimes referred to as the deﬁltering approach, aims at reconstructing the unﬁltered ﬁeld from the ﬁltered one. The subgrid modes are no longer modeled, but reconstructed using an ad hoc mathematical procedure [181]. We recall that, writing the Navier–Stokes equations symbolically as ∂u + N S(u) = 0 ∂t

,

(7.1)

7.2 Formal Series Expansions

211

we get the following for the ﬁltered ﬁeld evolution equation (see Chap. 3) ∂u + N S(u) = [N S, G ](u) ∂t

,

(7.2)

where G is the ﬁlter kernel, and [·, ·] is the commutator operator. The exact subgrid term, which corresponds to the right-hand side of relation (7.2), appears as a function of the exact nonﬁltered ﬁeld u. This ﬁeld being unknown during the computation, the idea here is to approximate it using a deconvolution procedure: −1 u ≈ u• ≡ G−1 l u = Gl G u ,

(7.3)

is an lth-order approximate inverse of the ﬁlter G where G−1 l l

G−1 l G = Id + O(∆ ) . The subgrid term is then approximated as [N S, G ](u) [N S, G ](u• ) = [N S, G ](G−1 l u) ,

(7.4)

achieving the description of the procedure. Combining the right-hand side and the left-hand side of the resulting equation, we get: ∂u + G N S(G−1 l u) = 0 ∂t

.

(7.5)

The nonlinear term appears as (G ) ◦ (N S) ◦ (G−1 l ), i.e. as the sequential application of: (i) the approximate deconvolution operator, (ii) the Navier– Stokes operator, and (iii) a regularization operator, referred to as primary regularization in the parlance of Adams [691, 4, 692, 693, 694]. It is worth noting that the eﬃciency of the present strategy will be conditioned by our capability of ﬁnding the approximate inverse operator, the two others being a priori known. The deconvolution procedure, in the general presentation given above, calls for several important remarks: 1. It is eﬃcient for invertible ﬁlters only, i.e. non-projective ﬁlters. Projective ﬁlters induce an irreversible loss of information, which cannot be recovered (see Sect. 2.1.2). For smooth ﬁlters, the unﬁltered ﬁeld can theoretically be reconstructed [106, 807]. 2. In practice, the grid ﬁlter is always present, because of the ﬁnite number of degrees of freedom used to compute the solution. As a consequence of the Nyquist theorem, a projective ﬁlter with space and time cutoﬀs equal to 2∆x and 2∆t, respectively, is always present. Here, ∆x is the mesh size of the computational grid used for the large-eddy simulation, and ∆t the time step employed for the numerical time integration. As a consequence, the ﬁlter to be considered in a practical simulation, referred to as

212

7. Structural Modeling

the eﬀective ﬁlter, is a combination of the convolution ﬁlter with cutoﬀ length scale ∆ and the projective grid ﬁlter. The latter can be modeled as a sharp cutoﬀ ﬁlter with cutoﬀ wave number kc = π/∆x. A complete discussion about the eﬀective ﬁlter is given in Sect. 8.2. This implies that the deconvolution procedure cannot reconstruct structures smaller than 2∆x, and, following the terminology of Adams, two problems can be identiﬁed in the deconvolution approach: a) The soft deconvolution problem, which corresponds to the reconstruction of the unﬁltered ﬁeld for wave numbers k ∈ [0, π/∆x]. The corresponding reconstruction procedures described below are: – Procedures relying on an iterative reconstruction of the inverse of the ﬁltering operator (p. 212); – Procedures based on a truncated Taylor series expansion of the ﬁltering operator (p. 213). b) The hard deconvolution problem: solving the soft deconvolution problem does not suﬃce for closing the reconstruction problem, because interactions with scales smaller than 2∆x are not taken into account. In order to alleviate this problem, the soft deconvolution procedure must be supplemented with a secondary procedure, sometimes referred to as secondary regularization. These scales being deﬁnitively lost, they must be modeled (and not reconstructed) using a functional model. All the models presented in Chap. 5 can be used. The speciﬁc penalization procedure developed by Stolz and Adams and other techniques are discussed in a dedicated section (p. 218). These two processes are illustrated in Fig. 7.1. Solving the Soft Deconvolution Problem: Iterative Deconvolution. Adams et al. [691, 692, 4, 694, 693, 743, 744, 742] developed an iterative deconvolution procedure based on the Van Cittert method. If the ﬁlter kernel G has an inverse G−1 , the latter can be obtained using the following expansion: G−1

= (Id − (Id − G))−1 , = (Id − G)p ,

(7.6) (7.7)

p=0,∞

yielding the following reconstruction for the deﬁltered variable φ: φ = φ + (φ − φ) + (φ − 2φ + φ) + ... ,

(7.8)

φ = (φ − φ) + (φ − 2φ + φ) + ...

(7.9)

or equivalently The series are known to be convergent if Id − G < 1. A practical model is obtained by truncating the expansion at a given power. Stolz and Adams [691] recommend using a ﬁfth-order (p = 5) expansion.

7.2 Formal Series Expansions

213

Fig. 7.1. Schematic of the full deconvolution problem. Top: soft deconvolution problem; Bottom: ideal solution of the full (soft+hard) deconvolution problem.

Other possible reconstruction techniques, such as the Tikhonov regularization, the singular value decomposition or the conjugate gradient method, are discussed in reference [4]. Solving the Soft Deconvolution Problem: Truncated Taylor SeriesExpansion-Based Deconvolution Procedures. The deconvolution procedures presented in this section are all based on the representation of the ﬁltering operator in the form of a Taylor series expansion (see Sect. 2.1.6). We recall the general form of such an expansion for the dummy variable φ: φ(x) =

∞ (−1)k k=0

k!

k

∆ Mk (x)

∂kφ (x) ∂xk

,

where Mk is the kth-order moment of the ﬁlter kernel G.

(7.10)

214

7. Structural Modeling

The common idea shared by all these models is to approximate the ﬁltering operator by truncating the Taylor series expansion, deﬁning a low-order diﬀerential operator: φ(x) ≈

N (−1)k k=0

k!

k

∆ Mk (x)

∂kφ (x) ∂xk

,

(7.11)

with N ≤ 4 in practice. The accuracy of the reconstruction obviously depends on the convergence rate of the Taylor series expansion for the ﬁltering operator. Pruett et al. [607] proved that it is fastly converging for some typical ﬁlters, such as Gaussian and top-hat ﬁlters. The unﬁltered ﬁeld φ can formally be expressed as the solution of the following inverse problem: φ(x) ≈

N (−1)k k=0

∂k ∆ Mk (x) k k! ∂x k

−1 φ(x)

.

(7.12)

The last step consists of inverting this operator, ∆ being considered as a small parameter. Two possibilities arise at this stage of the soft deconvolution procedure. The ﬁrst one consists of using an implicit method to solve (7.12). This procedure is rarely used, because of its algorithmic cost. The second one, considered by a large number of authors, relies on the use of an explicit approximation of the solution of (7.12), which is obtained using once again a Taylor series expansion. Recalling that for a small parameter we have (7.13) (1 + )−1 = 1 − + 2 − 3 + 4 − ... , and assuming that ∆ can play the role of a small parameter, we get the following approximate inverse relation, which is valid for symmetric ﬁlters: ∂2 1 2 (7.14) φ(x) ≈ Id − ∆ M2 (x) 2 + ... φ(x) . 2 ∂x This last form can be computed immediately from the resolved ﬁeld. Limiting the expansion to the second order, the subgrid part is expressed as: φ (x)

∂ 2 φ(x) 1 2 4 ∆ M2 + O(∆ ) 2 2 ∂x 1 2 ∂2 2 = ∆ M2 2 φ + O(∆ ) 2 ∂x 2 = φ (x) + O(∆ ) . =

(7.15)

This can be used to express all the contributions as a function of the resolved ﬁeld, with second-order accuracy. The various terms of the Leonard

7.2 Formal Series Expansions

215

decomposition are approximated to the second order as: ∂2 1 2 4 ∆ M2 2 (ui uj ) + O(∆ ) , (7.16) 2 ∂x ∂2 ∂2 1 2 4 Cij ≡ ui uj + uj ui = − ∆ M2 ui 2 uj + uj 2 ui + O(∆ ) . (7.17) 2 ∂x ∂x Lij ≡ ui uj − ui uj =

The combination of these two terms leads to: 2

Lij + Cij = ∆ M2

∂ui ∂uj 4 + O(∆ ) . ∂x ∂x

(7.18)

As for the subgrid Reynolds tensor, it appears only as a fourth-order term: Rij ≡ ui uj =

1 2 2 ∂ 2 ui ∂ 2 uj 6 ∆ M2 + O(∆ ) , 4 ∂x2 ∂x2

(7.19)

so that it disappears in a second-order expansion of the full subgrid tensor. The resulting model (7.18) is referred to as the gradient model, tensordiﬀusivity model or Clark’s model. It can also be rewritten using undivided diﬀerences, leading to the deﬁnition of the increment model proposed by Brun and Friedrich [82]. In practice, this approach is used only to derive models for the tensors L and C, which escape functional modeling [98, 143, 160, 161, 459]. Certain authors also use these evaluations to neglect these tensors when the numerical scheme produces errors of the same order, which is the case for second-order accurate schemes. Finer analysis allows a better evaluation of the order of magnitude of the subgrid tensor. By using a subgrid viscosity model, i.e. τij = −2νsgs S ij

,

(7.20)

and using the local equilibrium hypothesis: ε = −τij S ij = νsgs |S|2

,

(7.21)

the amplitude of the subgrid tensor can be evaluated as: |τij | ≈ νsgs |S| ≈

√ ενsgs

.

(7.22)

By basing the computation of the subgrid viscosity on the subgrid kinetic energy: + νsgs ≈ ∆

2 qsgs

,

(7.23)

and computing this energy from a Kolmogorov spectrum: + 2 qsgs

1/2

∞

=

E(k)dk kc

216

7. Structural Modeling

∞

∝

k

−5/3

1/2 dk

kc −1/3

∝ (kc )

1/3

∝ ∆

,

we get for the subgrid viscosity: 1/3

νsgs ∝ ∆ ∆

4/3

=∆

.

(7.24)

The order of magnitude of the corresponding subgrid tensor is: |τij | ∝

√

2/3

ενsgs ∝ ∆

.

(7.25)

This estimation is clearly diﬀerent from those given previously, and shows that the subgrid tensor is theoretically dominant compared with the terms 2 in ∆ . This last evaluation is usually interpreted as being that of the subgrid Reynolds tensor Rij , while the estimations of the tensors Cij and Lij given above are generally considered to be correct. A generalized expansion for the whole subgrid tensor is proposed by Carati et al. [105]. These authors have proved that the diﬀerential expansion φψ =

∞

Clm (G)

l,m=0

∂lφ ∂mψ ∂xl ∂xm

,

(7.26)

where φ(x) and ψ(x) are two C ∞ real functions, and Clm (G) are some real coeﬃcients which depend explicitly on the ﬁlter, is valid for all the ﬁlter kernels G such that: G(−i(φ + ψ)) ∈ IR, G(−iφ)G(−iψ)

i2 = −1 .

(7.27)

This is particularly true of all symmetric kernels. The resulting general form of the gradient model deduced from (7.26) is: τij = ui uj − ui uj =

l,m=0,∞;(l,m)=(0,0)

Clm (G)

∂ l ui ∂ m uj ∂xl ∂xm

.

(7.28)

The use of the Gaussian ﬁlter yields the following simpliﬁed form: 2 m ∆ ∂ m ui ∂ m uj τij = . (7.29) 16 ∂xm ∂xm m=0,∞ By retaining only the ﬁrst term in (7.29), one recovers the gradient model (7.18).

7.2 Formal Series Expansions

217

Expression (7.28) can be recast in a general tensor-diﬀusivity form, as demonstrated by Adams and Leonard [3]. These authors have shown that, for discontinuous, but otherwise smooth, functions ψ and φ the series (7.26) can be summed up, leading to φψ = φ ψ +

(∆/4)2 ∂φ ∂ψ R(ζψ )R(ζφ ) 2 ∂x ∂x

,

(7.30)

where a non-dimensional eddy-diﬀusivity R(ζ) is introduced. It is deﬁned by the following relation: 1 ζ R (ζ) = − 2 2 2

G−1 (ζ)

2

2

G(ζ )dζ

.

(7.31)

A consistency constraint is R(ζ) −→ 1 for ζ −→ 0. This constraint is satisﬁed for the top-hat ﬁlter, but is violated for the Gaussian ﬁlter. Numerical experiments show that this form is ill-conditionned and leads to unstable results, and must be supplemented by a secondary regularization. To this end, most of the authors explicitly add another dissipative term to the momentum equation (see p. 218). Layton et al. [341, 206, 234] proposed regularizing the gradient model by applying a smoothing operator. By choosing a convolution ﬁlter with a kernel G2 , the resulting model, referred to as the rational approximation model, is written as 2 ∆ ∂ui ∂uj . (7.32) τij = G2

16 ∂x ∂x Iliescu et al. [341] discuss both explicit and implicit implementation of (7.32), which was applied to channel ﬂows [219, 340, 339]. The need for a secondary regularization to get a stable computation using the gradient-like models may be understood by analysing their dissipative properties. The link with functional models of subgrid viscosity type is established by looking at the corresponding subgrid force term appearing in the momentum equations [761]. In three dimensions, we have: 2

∆ M2 2

∂ ∂xj

∂ui ∂uj ∂xk ∂xk

2

= ∆ M2 S jk

∂ ∂ ui ∂xj ∂xk

,

(7.33)

showing that ∆ M2 S jk plays the role of a tensorial subgrid viscosity. Since the tensor S jk is not positive-deﬁnite, antidissipation occurs along stretching directions, which are associated with negative eigenvalues. Full three-dimensional expressions up to the fourth-order terms of the subgrid tensor have been derived by several authors [81, 607], but these are very cumbersome and will not be presented here.

218

7. Structural Modeling

Taylor series expansion has also been used by several authors [81, 717] to derive an equivalent diﬀerential expression for the test ﬁlter and some tensorial quantities appearing in Germano’s dynamic procedure for evaluation of the constants (see p. 137). A worthy remark can be made dealing with the evaluation of the constants appearing in the models derived from truncated Taylor series expansions, which are considered as truncated polynomials in ∆. These parameters are commonly taken equal to the one appearing in the full (untruncated) expansions, and do not correspond to the coeﬃcients of the best truncated polynomial approximation of the full expansions. Considering the best approximation, diﬀerent coeﬃcients are usually found, which may lead to different properties with respect to the preservation of symmetries of the ﬁltered Navier–Stokes equations discussed in Sect. 3.3.4. Connection between Taylor Expansion-Based Deconvolution and Iterative Deconvolution. Stolz, Adams and Kleiser [693] have demonstrated the practical equivalence between the gradient-type models (or tensor diﬀusivity models) and the use of the Van Cittert iterative technique for the soft deconvolution problem. In the particular case of the Gaussian ﬁlter, an exact fourth-order deﬁltered variable is 2

4

∆ ∂2φ ∆ ∂4φ 6

φ=φ+ + + O(∆ ) . (7.34) φ≈ 24 ∂x2 1152 ∂x4 The use of this relation yields the following expression for the approximated subgrid tensor: G−1 l

ui uj − ui uj

≈

−1 (G−1 l ui )(Gl uj ) − ui uj 2

4

∆ ∂ui ∂uj ∆ ∂ 2 ui ∂ 2 uj 6 + = + O(∆ ) . (7.35) 24 ∂x ∂x 288 ∂x2 ∂x2 It is observed that each iterative deconvolution procedure yields the deﬁnition of a particular tensor-diﬀusivity model. Practical diﬀerences may arise from the discretization of the continuous derivate operators. Solving the Hard Deconvolution Problem: Secondary Regularization. The secondary regularization is needed to obtain stable numerical simulation. This can be seen by two diﬀerent ways: 1. The soft deconvolution procedure is restricted to the reconstruction of scales which are resolvable on the considered computational grid. Interactions with unresolved scales need to be taken into account from a theoretical point of view. 2. It has been demonstrated (at least in the particular case of the tensordiﬀusivity model) that negative dissipation can occur, yielding possible numerical troubles. The net drain of resolved kinetic energy by unresolved scales needs to be taken into account (see Chap. 5 for a detailed discussion).

7.2 Formal Series Expansions

219

In practice, this secondary regularization is achieved by adding a dissipative term to the deﬁltered equations. Starting from relation (7.5), we arrive at the following formal evolution equation: ∂u + G N S(G−1 l u) = S(u, ∆) , ∂t

(7.36)

where the dissipative source term S(u, ∆) can a priori be considered as a function of u and ∆. The Stolz–Adams Penalty Term. In order to account for kinetic energy transfer with scales which are not recovered by the deconvolution procedure, Stolz and Adams [691, 692, 694, 4] introduced a relaxation term, yielding S(u, ∆) = −χ(Id − G−1 l G) u ,

(7.37)

where χ is an empirical relaxation time. Since (Id − G−1

G) is positive l semideﬁnite, this relaxation term is purely dissipative. The use of this term can be interpreted as applying a second ﬁltering operator (G−1 l ) ◦ (G ) to u every 1/χ∆t time steps, ∆t being the time step selected to perform the numerical time integration. It can also be interpreted as a penalization of the ﬁltered solution. An important point is that this relaxation regularization is not equivalent to the use of subgrid viscosity type dissipation, since the associated spectral distributions of the dissipation are very diﬀerent. The relaxation time χ can be empiricaly chosen or dynamically evaluated. Numerical simulations show that the results may not be very sensitive to the value of this parameter. Stolz et al. [692, 693] reported no important changes for the channel ﬂow case in the range 12.5uτ /h ≤ χ ≤ 100uτ /h, where uτ is the skin friction and h the channel height. This relaxation time can also be evaluated dynamically in order to enforce a constant kinetic energy of the resolved subﬁlter modes, i.e. of the modes which are ﬁltered out of the exact solution by the convolution ﬁlter but which are resolved on the computational grid. These modes correspond to the spectral band [π/∆, π/∆x]. A measure of this energy is 2 1 (Id − G−1 . (7.38) EHF = l G) u 2 The equilibrium hypothesis can be expressed as ∂EHF =0 . ∂t

(7.39)

By combining relations (7.36) and (7.37), we get 43 4 ∂EHF 3 −1 −1 = (Id − G−1 l G) u · −G N S(Gl u) − χ(Id − Gl G) u , ∂t (7.40)

220

7. Structural Modeling

leading to 4 3 4 3 −1 (Id − G−1 l G) u · G N S(Gl u) 4 3 4 χ = −3 −1 (Id − G−1 l G) u · (Id − Gl G) u

.

(7.41)

Other Possibilities. The secondary regularization is achieved by many authors using subgrid-viscosity models, which provide the desired drain of resolved kinetic energy. All the subgrid-viscosity models can be used. The most employed ones are the Smagorinsky model and the dynamic Smagorinsky model. But it is worth noting that the secondary regularization can be achieved by means of the Implicit Large-Eddy Simulation approach (Sect. 5.3.4). As an example, Pasquetti and Xu used the Spectral Vanishing Viscosity approach in Refs. [582, 581]. Full Deconvolution Model Examples. As seen at the beginning of this section, the deconvolution approach necessitates the use of two models. An a priori inﬁnite number of combinations between models for the soft and the hard deconvolution problems can be deﬁned. Some examples are listed in Table 7.1. These two-part models for the full deconvolution problem can also be recast within the framework of mixed modeling. This point is discussed in Sect. 7.4. Table 7.1. Examples of solutions to the Full Deconvolution Problem. Ref.

Soft Deconvolution

Hard Deconvolution

[4, 694, 693, 692, 691] [759] [143, 759] [763, 764, 12] [341]

Van Cittert (7.8) Van Cittert (7.8) tensor-diﬀusivity (7.18) tensor-diﬀusivity (7.18) rational model (7.32)

relaxation (7.37) Smagorinsky (5.90) Smagorinsky (5.90) dynamic Smagorinsky Smagorinsky (5.90)

Single-Step Filtering Implementation of the Full Approximate Deconvolution Model. The practical implementation of the full approximate deconvolution model has been shown by Mathew et al. [499] to simplify as the application of a single ﬁltering operator to the solution at the end of each time step of the numerical computation. Observing that relation (7.5) can be recast as • ∂u + N S(u• ) = 0 , (7.42) G

∂t under the assumption that u• is suﬃcient close to u, one obtains the following relation: ∂u• ∂u = G

. (7.43) ∂t ∂t

7.2 Formal Series Expansions

221

Therefore, the basic implementation requires three steps to obtain the solution at time (n+ 1)∆t (noticed u(n+ 1) below) starting from the solution at the previous time step, u(n) 1. Evaluation of the approximate unﬁltered ﬁeld at time n∆t : u• (n) = G−1 l u(n)

.

2. Time advancement of the approximate unﬁltered solution : u• (n + 1) = u• (n) + ∆t

∂u• + O(∆t2 ) . ∂t

3. Restriction of the new approximate unﬁltered solution : u(n + 1) = G u• (n + 1) . Looking at this sequence, Mathew observes that the ﬁrst and third steps can be combined in a unique one, leading to the following two-step algorithm 1. Time advancement of the approximate unﬁltered solution : u◦ = u• (n) + ∆t

∂u• + O(∆t2 ) . ∂t

2. Restriction of the new approximate unﬁltered solution : ◦ ◦ u• (n + 1) = (G−1 l G) u = Ql u

.

It is seen that the sole ﬁlter involved in practice is the ﬁlter Ql , which is an lth order perturbation of the identity whose eﬀect is mostly concentrated on high resolved wavenumbers. This two-step method accounts for the soft deconvolution model only. The hard deconvolution problem, or equivalently the secondary regularization, is reintroduced within the same framework using the fact that it is equivalent (at least in the case of the penalty term proposed by Stolz and Adams) to the application of the ﬁlter (G−1 l )◦(G ) = Ql to the solution every 1/χ∆t time steps. The full deconvolution problem can thus be implemented in a simple two-step procedure: 1. Time advancement of the approximate unﬁltered solution: u◦ = u• (n) + ∆t

∂u• + O(∆t2 ) . ∂t

2. Restriction of the new approximate unﬁltered solution : u• (n + 1) = G u◦

.

where the basic value G = Ql is changed into G = Ql Ql = Q2l every 1/χ∆t time steps.

222

7. Structural Modeling

Toward Higher-Order Deconvolution Models. The approximate deconvolution approach presented above can be seen as the lowest-order member of a general class of deconvolution approaches. This fact is emphasized by Mathew and his coworkers [499], who carried out the development at the next order. Restricting for the sake of simplicity the discussion to the formal, one-dimensional scalar conservation law : ∂u ∂f (u) + =0 , ∂t ∂x

(7.44)

the deconvolution approach can be written as ∂u ∂f (u) ∂f (u) ∂f (u) + = −G

=R . ∂t ∂x ∂x ∂x

(7.45)

The low-order approximate deconvolution procedure presented above corresponds to the following closure R R1 =

∂f (u• ) ∂f (u) −G

∂x ∂x

.

(7.46)

The higher-order method is derived writing the remainder in this closure relation as ∂f (u• ) ∂f (u) R = R1 + R2 , R2 = G

− , (7.47) ∂x ∂x and ﬁnding a computable approximation for R2 . Such an expansion is found introducing the Taylor series expansion ∂f ∂ • • 2 R2 = G

(u − u) + O(u − u) . (7.48) ∂x ∂u u An estimation of the leading term of this expansion is ∂f ∂ −1 (G G − Id) u R2 = G

∂x ∂u u l which can be approximated as ∂f ∂ −1 • (G

G − Id)

u R2 = G

∂x ∂u u=u• l

,

(7.49)

.

(7.50)

The resulting higher-order formulation of the approximate deconvolution approach is ∂f (u• ) ∂ ∂u ∂f −1 • +G

=G

(G

G − Id)

u . (7.51) ∂t ∂x ∂x ∂u u=u• l It is worth noticing that this reﬁned modeling makes a term appearing in the right hand side of equation (7.51) that is similar to the empirical penalty term for secondary regularization (7.37) introduced by Stolz and Adams.

7.2 Formal Series Expansions

223

7.2.2 Non-linear Models There are a number of ways of deriving nonlinear models: Horiuti [321], Speziale [683], Yoshizawa [788], and Wong [765] start with an expansion in a small parameter, while Lund and Novikov [462] use the mathematical properties of the tensors considered. It is this last approach that will be described ﬁrst, because it is the one that best reveals the diﬀerence with the functional models. Kosovic’s simpliﬁed model [401] and Wong’s dynamic model [765] are then described. Generic Model of Lund and Novikov. We assume that the deviator of the subgrid tensor can be expressed as a function of the resolved velocity ﬁeld gradients (and not the velocity ﬁeld itself, to ensure the Galilean invariance property), the unit tensor, and the square of the cutoﬀ length ∆: 1 2 τij − τkk δij ≡ τijd = F (S ij , Ω ij , δij , ∆ ) 3

.

(7.52)

The isotropic part of τ is not taken into account, and is integrated in the pressure term because S and Ω have zero traces. To simplify the expansions in the following, we use the reduced notation: S Ω = S ik Ω kj ,

2

tr(S Ω ) = S ij Ω jk Ω ki

.

The most general form for relation (7.52) is a polynomial of inﬁnite dea1 a2 a3 a4 gree of tensors whose terms are of the form S Ω S Ω ..., where the ai are positive integers. Each terms in the series is multiplied by a coeﬃcient, which is itself a function of the invariants of S and Ω. This series can be reduced to a ﬁnite number of linearly independent terms by the Cayley- Hamilton theorem. Since the tensor τ d is symmetrical, we retain only the symmetrical terms here. The computations lead to the deﬁnition of eleven tensors, m1 , ..., m11 , with which I1 , ..., I6 are associated: m1 m3 m5 m7 m9 m11

= S, m2 2 = Ω , m4 2 2 = S Ω − ΩS , m6 2

2

= S Ω + Ω S, m8 2 2 = S Ω S − S Ω S, m10 2 2 2 2 = Ω S Ω − Ω S Ω, I1 I3 I5

= = =

2

tr(S ), 3 tr(S ), 2 2 tr(S Ω ),

I2 I4 I6

where Id designates the identity tensor.

= = = = =

2

S , S Ω − Ω S, Id, 2

2

Ω S Ω − Ω S Ω, 2 2 2 2 S Ω +Ω S ,

(7.53)

2

= tr(Ω ), 2 = tr(S Ω ), 2 2 = tr(S Ω S Ω),

(7.54)

224

7. Structural Modeling

These tensors are independent in the sense that none can be decomposed into a linear sum of the ten others, if the coeﬃcients are constrained to appear as polynomials of the six invariants deﬁned above. If we relax this last constraint by considering the polynomial quotients of the invariants too, then only six of the eleven tensors are linearly independent. The tensors deﬁned above are no longer linearly independent in two cases: when the tensor S has a double eigenvalue and when two components of the vorticity disappear when expressed in the speciﬁc reference of S. The ﬁrst case corresponds to an axisymmetrical shear and the second to a situation where the rotation is about a single axis aligned with one of the eigenvectors of S. Assuming that neither of these conditions is veriﬁed, six of the terms of (7.53) are suﬃcient for representing the tensor τ , and ﬁve for representing its deviator part, which is consistent with the fact that a second-order symmetrical tensor with zero trace has only ﬁve degrees of freedom in the third dimension. We then obtain the generic polynomial form: τd

2

2

2

2

2

= C1 ∆ |S|S + C2 ∆ (S )d + C3 ∆ (Ω )d 2

2

+ C4 ∆ (S Ω − Ω S) + C5 ∆

1 2 2 (S Ω − S Ω ) , |S|

(7.55)

where the Ci , i = 1, 5 are constants to be determined. This type of model is analogous in form to the non-linear statistical turbulence models [682, 683]. Numerical experiments performed by the authors on cases of isotropic hom*ogeneous turbulence have shown that this modeling, while yielding good results, is very costly. Also, computing the diﬀerent constants raises problems because their dependence as a function of the tensor invariants involved is complex. Meneveau et al. [509] attempted to compute these components by statistical techniques, but achieved no signiﬁcant improvement over the linear model in the prediction of the subgrid tensor eigenvectors. A priori tests carried out by Horiuti [325] have shown that the (ΩS − SΩ) term is responsible for a signiﬁcant improvement of the correlation coeﬃcient with the true subgrid tensor. We note that the ﬁrst term of the expansion corresponds to subgrid viscosity models for the forward energy cascade based on large scales, which makes it possible to interpret this type of expansion as a sequence of departures from symmetry: the isotropic part of the tensor is represented by a spherical tensor, and the ﬁrst term represents a ﬁrst departure from symmetry but prevents the inclusion of the inequality of the normal subgrid stresses1 . The anisotropy of the normal stresses is included by the following terms, which therefore represent a new departure from symmetry. 1

This is true for all modeling of the form τ = (V ⊗ V ) in which V is an arbitrary vector. It is trivially veriﬁed that the tensor (V ⊗ V ) admits only a single nonzero eigenvalue λ = (V12 + V22 + V32 ), while the subgrid tensor in the most general case has three distinct eigenvalues.

7.2 Formal Series Expansions

225

Kosovic’s Simpliﬁed Non-Linear Model. In order to reduce the algorithmic cost of the subgrid model, Kosovic [401] proposes neglecting certain terms in the generic model presented above. After neglecting the high-order terms on the basis of an analysis of their orders of magnitude, the author proposes the following model: 1 2 2 1/2 τij = −(Cs ∆) 2(2|S| ) S ij + C1 S ik S kj − S mn S mn δij 3 4 , (7.56) + C2 S ik Ω kj − Ω ik S kj where Cs is the constant of the subgrid viscosity model based on the large scales (see Sect. 5.3.2) and C1 and C2 two constants to be determined. After computation, the local equilibrium hypothesis is expressed: ε = =

−τij S ij

# " (Cs ∆)2 2 (2|S|2 )1/2 S ij S ij + C1 S ik S kj S ji .

(7.57)

In the framework of the canonical case (isotropic turbulence, inﬁnite inertial range, sharp cutoﬀ ﬁlter), we get (see [45]): S ij S ij = =

2 30 ∂u1 4 ∂x1 3 K0 ε2/3 kc4/3 4

,

(7.58)

3 105 ∂u1 S ik S kj S ji = 8 ∂x1 3/2 105 1 = − εkc2 S(kc ) K0 8 10

,

(7.59)

where coeﬃcient S(kc ) is deﬁned as: 3 2 ∂u1 ∂u1 / 3/2 S(kc ) = − ∂x1 ∂x1

.

(7.60)

Substituting these expressions in relation (7.57) yields: ε = (Cs ∆)

2

3/2 3 7 K0 C1 S(kc ) kc2 ε 1− √ 2 960

.

(7.61)

This relation provides a way of relating the constants Cs and C1 and thereby computing C1 once Cs is determined by reasoning similar to that explained in the chapter on functional models. The asymptotic value of S(kc ) is evaluated by theory and experimental observation at between 0.4 and 0.8,

226

7. Structural Modeling

as kc → ∞. The constant C2 cannot be determined this way, since the contribution of the anti-symmetrical of the velocity gradient to the energy transfer is null2 . On the basis of simple examples of anisotropic hom*ogeneous turbulence, Kosovic proposes: (7.62) C2 ≈ C1 , which completes the description of the model. Dynamic Non-Linear Model. Kosovic’s approach uses some hypotheses intrinsic to the subgrid modes, for example the existence of a theoretical the spectrum shape and the local equilibrium hypothesis. To relax these constraints, Wong [765] proposes computing the constants of the non-linear models by means of a dynamic procedure. To do this, the author proposes a model of the form (we use the same notation here as in the description of the dynamic model with one equation for the kinetic energy, in Sect. 5.4.2): τij =

+ 2 2 2 S −C N qsgs δij − 2C1 ∆ qsgs ij 2 ij 3

,

(7.63)

2 where C1 and C2 are constants and qsgs the subgrid kinetic energy, and

1 1 N ij = S ik S kj − S mn S mn δij + S˙ ij − S˙ mm δij 3 3

,

(7.64)

where S˙ ij is the Oldroyd3 derivative of S ij : ∂ui DS ij ∂uj − S˙ ij = S kj − S ki Dt ∂xk ∂xk

,

(7.65)

where D/Dt is the material derivative associated with the velocity ﬁeld u. The isotropic part of this model is based on the kinetic energy of the subgrid modes (see Sect. 5.3.2). Usually, we introduce a test ﬁlter symbolized by * Using the same model, the a tilde, the cutoﬀ length of which is denoted ∆. subgrid tensor corresponding to the test ﬁlter is expressed: + 2 * Q2 S * * , Tij = Q2sgs δij − 2C1 ∆ sgs ij − C2 H ij 3

2

This is because we have the relation Ω ij S ij ≡ 0

3

(7.66)

,

since the tensors Ω and S are anti-symmetrical and symmetrical, respectively. This derivative responds to the principle of objectivity, i.e. it is invariant if the frame of reference in which the motion is observed is changed.

7.2 Formal Series Expansions

227

where Q2sgs is the subgrid kinetic energy corresponding to the test ﬁlter, * the tensor analogous to N , constructed from the velocity ﬁeld u. * and H ij

ij

Using the two expressions (7.63) and (7.66), the Germano identity (5.138) is expressed: Lij

=

Tij − τ*ij 2 2 2 2 (Q − q/ sgs )δij + 2C1 ∆Aij + C2 ∆ Bij 3 sgs

in which Aij = S ij

+ + * ∆ * 2 − qsgs S Q2sgs ij ∆

* − Bij = N ij

* ∆ ∆

2

* H ij

,

,

(7.67)

(7.68)

.

(7.69)

We then deﬁne the residual Eij : 2 2 2 Eij = Lij − (Q2sgs − q/ sgs )δij + 2C1 ∆Aij + C2 ∆ Bij 3

.

(7.70)

The two constants C1 and C2 are then computed in such a way as to minimize the scalar residual Eij Eij , i.e. ∂Eij Eij ∂Eij Eij = =0 ∂C1 ∂C2

.

(7.71)

A simultaneous evaluation of these two parameters leads to: 2∆C1 ≈

2

∆ C2 ≈

Lmn (Amn Bpq Bpq − Bmn Apq Bpq ) Akl Akl Bij Bij − (Aij Bij )2

,

(7.72)

Lmn (Bmn Apq Apq − Amn Apq Bpq ) Akl Akl Bij Bij − (Aij Bij )2

.

(7.73)

2 and Q2sgs are obtained by solving the corresponding The quantities qsgs evolution equations, which are described in the chapter on functional models. This completes computation of the subgrid model. One variant that does not require the use of additional evolution equations is derived using a model based on the gradient of the resolved scales instead of one based on the subgrid kinetic energy, to describe the isotropic term. The subgrid tensor deviator is now modeled as:

1 2 τij − τkk δij = −2C1 ∆ |S|S ij − C2 N ij 3

.

(7.74)

228

7. Structural Modeling 2

The two parameters computed by the dynamic procedure are now ∆ C1 2 and ∆ C2 . The expressions obtained are identical in form to relations (7.72) and (7.73), where the tensor Aij is deﬁned as: *S * Aij = |S|S ij − |S| ij

* ∆ ∆

2 .

(7.75)

7.2.3 hom*ogenization-Technique-Based Models General Description. The theory of hom*ogenization is a two-scale expansion technique originally developed in structural mechanics to model inhom*ogeneous materials with a periodic microstructure. If the slow (i.e. large) scale and the rapid (i.e. small) scale are very diﬀerent (i.e. if the microstructure is very ﬁne compared with the large scale variations of the material), the composite material can be represented by an hom*ogeneous material whose characteristics can be computed theoretically through the two-scale expansion and an averaging step. Using the scale separation assumption between the resolved scales of motion and the subgrid scales, the hom*ogenization was introduced by Perrier and Pironneau [588] to obtain an theoretical evaluation for the subgrid viscosity. This approach was resurrected twenty years later by Persson, Fureby and Svanstedt [589] who derived a new hom*ogenizationbased tensorial subgrid viscosity model and performed the ﬁrst simulations with this class of models. The hom*ogenization approach, which consists in solving the evolution equations of the ﬁltered ﬁeld separately from those of the subgrid modes, is based on the assumption that the cutoﬀ is located within the inertial range at each point. The resolved ﬁeld u and the subgrid ﬁeld u are computed on two diﬀerent grids by a coupling algorithm. In all of the following, we adopt the hypothesis that u = 0. The subgrid modes u are then represented by a random process v δ , which depends on the dissipation ε, and the viscosity ν, and which is transported by the resolved ﬁeld u. This modeling is denoted symbolically: x − ut t δ u = v ε, , (7.76) , 2 δ δ in which δ −1 is the largest wave number in the inertial range and δ −2 the highest frequency considered. As the inertial range is assumed to extend to the high wave numbers, δ is taken as small parameter. Let uδ be the solution to the problem: ∂uδi ∂(uδi + viδ )(uδj + vjδ ) ∂ 2 viδ ∂ 2 uδi ∂pδ ∂viδ + +ν −ν =− − ∂t ∂xj ∂xk ∂xk ∂xi ∂t ∂xk ∂xk

. (7.77)

7.2 Formal Series Expansions

229

If v δ is close to u , then uδ is close to u. More precisely, we have: uδ = u + δu1 + δ 2 u2 + ...

(7.78)

The lowest-order terms in the ﬁltered expansion yields a hom*ogenized problem for u which involves a subgrid stress tensor v δ · ∇u1 + u1 · ∇v δ . The hom*ogenization technique relies on the computation of u1 and the deﬁnition of stochastic model for v δ . A modeling of this kind, while satisfactory on the theoretical level, is not so in practice because the function v δ oscillates very quickly in space and time, and the number of degrees of freedom needed in the discrete system to describe its variations remains very high. To reduce the size of the discrete system signiﬁcantly, other hypotheses are needed, leading to the deﬁnition of simpliﬁed models which are described in the following. The description of the models is limited to the minimum amount of details for the sake of simplicity. The reader is referred to Ref. [589] for a detailed presentation of the two-scale expansion of the Navier–Stokes equations. Perrier–Pironneau Models. The ﬁrst simpliﬁcation introduced by Perrier and Pironneau [588] consists in choosing the random process in the form: v δ (x, t) =

1 v (x, t, x , t ) δ

,

(7.79)

in the space and time scales x and t , respectively, of the subgrid modes are deﬁned as: x − ut t , t = 2 . (7.80) x = δ δ The new variable v (x, t, x , t ) oscillates slowly and can thus be represented with fewer degrees of freedom. Assuming that v is periodical depending on the variables x and t on a domain Ωv = Z×]0, T [, and that the average of v is null over this domain4 , it is demonstrated that the subgrid tensor is expressed in the form: τ = B∇u

,

(7.81)

where the term B∇u is computed by taking the average on the cell of periodicity Ωv of the term (v · ∇u1 + u1 · ∇v), where u1 is the a solution on this cell of the problem: ∂u1 − ν∇2x u1 + v · ∇x u1 + u1 · ∇x v = ∇q − v · ∇u − u · ∇v ∂t ∇x · u1 = 0 , 4

, (7.82) (7.83)

This is equivalent to considering that v (x, t, x , t ) is statistically hom*ogeneous and isotropic, which is theoretically justiﬁable by the physical hypothesis of local isotropy.

230

7. Structural Modeling

where ∇x designates the gradient with respect to the x variables and q the Lagrange multiplier that enforces the constraint (7.83). This model, though simpler, is still diﬃcult to use because the variable (x − ut) is diﬃcult to manipulate. So other simpliﬁcations are needed. To arrive at a usable model, the authors propose neglecting the transport of the random variable by the ﬁltered ﬁeld in the ﬁeld’s evolution equation. This way, the random variable can be chosen in the form: v δ (x, t) =

1 v (x, t, x , t ) δ

,

(7.84)

with x =

x δ

,

(7.85)

and where the time t is deﬁned as before. Assuming that v is periodic along x and t on the domain Ωv and has an average of zero over this interval, the subgrid term takes the form: τ = A∇u

,

(7.86)

where A is a deﬁnite positive tensor such that the term A∇u is equal to the average of the term (v ⊗ u1 ) over Ωv , in which u1 is a solution on Ωv of the problem: ∂u1 − ν∇2x u1 + v · ∇x u1 = ∇q + v · ∇u , ∂t ∇x · u1 = 0

.

(7.87) (7.88)

Persson Tensorial Subgrid Viscosity. Persson, Fureby and Svanstedt [589, 227] followed a similar procedure to derive a fourth-rank tensorial subgrid viscosity. They use the following equation for u1 as a starting point (the rapid scale system coordinates is the same as in the second model of Perrier and Pironneau): ∂u1 − ν∇2x u1 + ∇x · (u1 ⊗ v + v ⊗ u1 ) − ∇x q = ∇ · (v ⊗ u) ∂t ∇x · u1 = 0

.

, (7.89)

(7.90)

Based on this system, the authors derived the following expression for the subgrid tensor: ∂uk τij = Aijkl , (7.91) ∂xl

7.3 Scale Similarity Hypotheses and Models Using Them

231

where the anisotropic fourth-rank tensorial subgrid viscosity is deﬁned as Ahjkl =

2 K0 C1 qsgs ∆ *hjkl A 11/3 (2π) ν

,

(7.92)

where the constant parameters are K0 = 1.4 (Kolmogorov constant) and 2 C1 = 1.05. The subgrid kinetic energy qsgs is evaluated solving an additional evolution equation deduced from (5.119) that includes the new deﬁnition of the subgrid viscosity as a tensorial quantity instead of a scalar one. The nondimensional parameter is given by the following spectral summation over the wave vectors m ( *k )2 + (ml /∆ *l )2 (mk /∆ −17/3 *hjkl = − A 1− Rm 2 *m R m∈Z Z *k )2 (ml /∆ *l )2 (2 − δkl ) (mk /∆ (δhl δjk + δjl δhk ) + *4 R m ) *h )2 (mk /∆ *k )2 (1 − δjl ) (mh /∆ +2 , (7.93) *4 R m

where *m R

8 2 9 9 mn =: , *n ∆ n=1,3

Rm =

δ * Rm ∆

,

(7.94)

where the scalar cutoﬀ length on a Cartesian grid is deﬁned as ∆ = (∆1 ∆2 ∆3 )1/3 , and δ is the expansion parameter. The cell aspect ratio are *l = ∆l /∆. deﬁned as ∆

7.3 Scale Similarity Hypotheses and Models Using Them 7.3.1 Scale Similarity Hypotheses Basic Hypothesis. The scale similarity hypothesis such as proposed by Bardina et al. [39, 40] consists in assuming that the statistical structure of the tensors constructed on the basis of the subgrid scales is similar to that of their equivalents evaluated on the basis of the smallest resolved scales. The spectrum of the solution based on this hypothesis is therefore broken down into three bands: the largest resolved scales, the smallest resolved scales (i.e. the test ﬁeld), and the unresolved scales (see Fig. 5.14). This statistical consistency can be interpreted in two complementary ways. The ﬁrst uses the energy cascade idea. That is, the unresolved scales

232

7. Structural Modeling

and the smallest resolved scales have a common history due to their interactions with the largest resolved scales. The classical representation of the cascade has it that the eﬀect of the largest resolved scales is exerted on the smallest resolved scales, which in turn inﬂuence the subgrid scales, which are therefore indirectly forced by the largest resolved scales, but similarly to the smallest. The second interpretation is based on the idea of coherent structures. These structures have a non-local frequency signature5 , i.e. they have a contribution on the three spectral bands considered. Scale similarity is therefore associated with the fact that certain structures appear in each of the three bands, inducing a strong correlation of the ﬁeld among the various levels of decomposition. Extended Hypothesis. This hypothesis was generalized by Liu et al. [455] (see [506] for a more complete discussion) to a spectrum split into an arbitrary number of bands, as illustrated in Fig. 7.2. The scale similarity hypothesis is then re-formulated for two consecutive spectrum bands, with the consistent forcing being associated with the low frequency band closest to those considered. Thus the speciﬁc elements of the tensors constructed from the velocity ﬁeld un and their analogous elements constructed from un+1 are assumed to be the same. This hypothesis has been successfully veriﬁed in experiments in the case of a jet turbulence [455] and plane wake turbulence [570]. Liu et al. have also demonstrated that scale similarity persists during rapid straining [454]. 7.3.2 Scale Similarity Models This section presents the structural models constructed on the basis of the scale similarity hypothesis. All of them make use of a frequency extrapolation technique: the subgrid tensor is a approximated by an analogous tensor computed from the highest resolved frequencies. The following are described: 1. Bardina’s model (p. 233) in which the subgrid tensor is computed by applying the analytical ﬁlter a second time and thereby evaluating the ﬂuctuation of the resolved scales. This model is therefore inoperative when the ﬁlter is idempotent, because this ﬂuctuation is then null. 2. Filtered Bardina model (p. 234), which is an improvement on the previous one. By construction, the subgrid tensor is a ﬁltered quantity, which results in the application of a convolution product and is therefore nonlocal in the sense that it incorporates all the information contained in 5

This is due to the fact that the variations of the velocity components associated with a vortex cannot be represented by a monochromatic wave. For example, the Lamb-Oseen vortex tangential velocity radial distribution is: 2 q 1 − e−r , Uθ = r where r is the distance to the center and q the maximum velocity.

7.3 Scale Similarity Hypotheses and Models Using Them

233

Fig. 7.2. Spectral decomposition based on the extended scale similarity hypothesis.

the support of the ﬁlter convolution kernel. It is proposed in this model to recover this non-local character by applying the ﬁlter to the modeled subgrid tensor. 3. Liu–Meneveau–Katz model (p. 234), which generalizes the Bardina model to the use of two consecutive ﬁlters of diﬀerent shapes and cutoﬀ frequencies, for computing the small scale ﬂuctuations. This model can therefore be used for any type of ﬁlter. 4. The dynamic similarity model (p. 236), which can be used to compute the intensity of the modeled subgrid stresses by a dynamic procedure, whereas in the previous cases this intensity is prescribed by hypotheses on the form of the energy spectrum. Bardina Model. Starting with the hypothesis, Bardina, Ferziger, and Reynolds [40] proposed modeling the C and R terms of the Leonard decomposition by a second application of the ﬁlter that was used to separate the scales. We furthermore have the approximation: φψ φ ψ

,

(7.95)

which allows us to say: Rij = (ui − ui )(uj − uj ) ,

(7.96)

Cij = (ui − ui )uj + (uj − uj )ui

,

(7.97)

or Rij + Cij = (ui uj − ui uj )

.

(7.98)

234

7. Structural Modeling

Adding Leonard’s term, which is computed directly from the resolved scales, we get: τij = Lij + Rij + Cij = (ui uj − ui uj )

.

(7.99)

This can be re-written in another using the generalized central moments proposed by Germano [244]: τij = τG ([ui ]G , [uj ]G ) ≡ Lij

.

(7.100)

The subgrid tensor is therefore approximated by the generalized central moment of the ﬁltered ﬁeld deﬁned like the tensor Lij in the Germano decomposition (see Sect. 3.3.2). Experience shows that this model is not eﬀective when the ﬁlter is a Reynolds operator, because the contribution thus computed then cancels out. Contrary to the subgrid viscosity models, this one does not induce an alignment of the proper axis system of the subgrid tensor on those of the strain rate tensor. Tests performed on databases generated by direct numerical simulation have shown that this model leads to a very good level of correlation with the true subgrid tensor, including when the ﬂow is anisotropic [320]. Despite its very good level of correlation6, experience shows that this model is only slightly dissipative and that it underestimates the energy cascade. It does, on the other hand, include the backward cascade mechanism. Filtered Bardina Model. The Bardina model (7.99) is local in space in the sense that it appears as a product of local values. This local character is in contradiction with the non-local nature of the subgrid tensor, so that each component appears in the form of a convolution product. To remedy this problem, Horiuti [323] and Layton and his colleagues [429, 428] propose the ﬁltered Bardina model: τij = (ui uj − ui uj ) = Lij

.

(7.101)

With this additional ﬁltering operation, we recover the non-local character of the subgrid tensor. Liu–Meneveau–Katz Model. The Bardina model uses a second application of the same ﬁlter, and therefore a single cutoﬀ scale. This model is generalized to the case of two cutoﬀ levels as [455]: *i * τij = Cl (u/ uj ) = Cl Lm i uj − u ij

,

(7.102)

where the tensor Lm ij is now deﬁned by two diﬀerent levels of ﬁltering. The test ﬁlter cutoﬀ length designated by the tilde is larger than that of the ﬁrst level. 6

The correlation coeﬃcient at the scalar level is generally higher than 0.8.

7.3 Scale Similarity Hypotheses and Models Using Them

235

The constant Cl can be evaluated theoretically to ensure that the average value of the modeled generalized subgrid kinetic energy is equal to its exact counterpart [148]. This leads to the relation: Cl =

uk uk − uk uk *k *k u u/ k uk − u

.

(7.103)

Let F(k) and G(k) be transfer functions associated with the grid ﬁlter and test ﬁlter, respectively, and let E(k) be the energy spectrum of the exact solution. Relation (7.103) can then be re-written as: !∞ (1 − F 2 (k))E(k)dk . (7.104) Cl = ! ∞ 0 2 (k))F2 (k)E(k)dk (1 − G 0

Evaluations made using experimental data come to Cl 1 [570, 455]7 . Shah and Ferziger [668] propose extending this model to the case of nonsymmetric ﬁlters. To control the amplitude of the backward cascade induced by the model, especially near solid walls, Liu et al. [455] propose the modiﬁed form: τij = Cl f (ILS )Lm ij

,

(7.105)

where the dimensionless invariant ILS , deﬁned as Lm lk S lk ILS = m Llk Lm S lk S lk lk

(7.106)

measures the alignment of the proper axes of the tensors Lm and S. As the kinetic energy dissipated by the subgrid model is expressed ε = −τij S ij

,

(7.107)

we get, using model (7.105): ε = −Cl f (ILS )ILS

.

(7.108)

The backward energy cascade is modulated by controlling the sign and amplitude of the product f (ILS )ILS . The authors considered a number of choices. The ﬁrst is:

1 if ILS ≥ 0 f (ILS ) = . (7.109) 0 otherwise This solution makes it possible to cancel out the representation of the backward cascade completely by forcing the model to be strictly dissipative. 7

The initial value of 0.45 ± 0.15 given in [455] does not take the backward cascade into account.

236

7. Structural Modeling

One drawback to this is that the function f is discontinuous, which can generate numerical problems. A second solution that is continuous consists in taking:

ILS if ILS ≥ 0 f (ILS ) = . (7.110) 0 otherwise One last positive, continuous, upper-bounded solution is of the form:

f (ILS ) =

2 (1 − exp(−γILS )) 0

if ILS ≥ 0 otherwise

,

(7.111)

in which γ = 10. Dynamic Similarity Model. A dynamic version of the Liu–Meneveau– Katz model (7.102) was also proposed [455] for which the constant Cl will no longer be set arbitrarily. To compute this model, we introduce a third level of ﬁltering identiﬁed by .. The Q analogous to tensor Lm for this new level of ﬁltering is expressed: *j − u * *i u *i uj ) . Qij = (u

(7.112)

The Germano–Lilly dynamic procedure, based here on the diﬀerence: m Mij = f (IQS )Qij − f (I/ LS )Lij

,

(7.113)

where IQS =

* Qmn S mn * |Q| |S|

Cl =

Lm lk Mlk Mpq Mpq

,

(7.114)

yields: .

(7.115)

7.3.3 A Bridge Between Scale Similarity and Approximate Deconvolution Models. Generalized Similarity Models The Bardina model can be interpreted as a particular case of the approximate soft deconvolution based models described in Sect. 7.2. Using the second order diﬀerential approximation φ=φ+

α(2) ∂ 2 φ 2 ∂x2

,

(7.116)

the Bardina model (7.99) is strictly equivalent to the second order gradient model given by relations (7.18) and (7.19).

7.4 Mixed Modeling

237

It can also be derived using the Van Cittert deconvolution procedure: a zeroth-order truncation in (7.7) is used to recover relation (7.95), while a ﬁrst-order expansion is employed to derive (7.96). The Bardina model then appears as a low-order formal expansion model for the subgrid tensor. Generalized scale similarity models can then be deﬁned using higher-order truncations for the formal expansion [254]. They are formulated as −1 −1

uj ) − (G−1 τij = (G−1 l ui )(Gl l u)i − (Gl u)j

,

(7.117)

where G−1 l designates the approximate soft deconvolution operator, deﬁned in Sect. 7.2.1. The same key idea can be extended to close the Navier–Stokes on a truncated wavelet basis: this was achieved by Hoﬀman for several simpliﬁed systems [315, 311, 309, 308].

7.4 Mixed Modeling 7.4.1 Motivations The structural models based on the scale similarity idea and the soft deconvolution models/techniques on the one hand, and the functional models on the other hand, each have their advantages and disadvantages that make them seem complementary: – The functional models, generally, correctly take into account the level of the energy transfers between the resolved scales and the subgrid modes. However, their prediction of the subgrid tensor structure, i.e. its eigenvectors, is very poor. – The models based on the scale-similarity hypothesis or an approximate deconvolution procedure generally predict well the structure of the subgrid tensor better (and then are able to capture anisotropic eﬀects and disequilibrium), but are less eﬃcient for dealing with the level of the energy transfers. It is also observed that these models yield a poor prediction of the subgrid vorticity production [491], a fact coherent with their underdissipative character. Dubois et al [202] also observed that these two parts yield diﬀerent correlations with reference data. Tests have shown that mixed models are able to capture disequilibrium and anisotropy eﬀects [8, 454, 504, 506, 594, 12]. Shao et al. [670] propose a splitting of the kinetic energy transfer across the cutoﬀ that enlights the role of each one of these two model classes. These authors combine the classical large-eddy simulation convolution ﬁlter to the

238

7. Structural Modeling

ensemble average, yielding the following decompositions: u = = =

u + ue

(7.118)

u + u u + u + ue + u

(7.119) (7.120)

.

Using this hybrid decomposition, the subgrid tensor splits into τij = τijrapid + τijslow

,

(7.121)

with e e e τijslow = ue i uj − u i u j

τijrapid

=

,

(7.122)

e ui uj − ui uj + ue i uj − u i uj e +ue j ui − u j ui .

(7.123)

These two parts can be analysed as follows: – The rapid part explicitly depends on the mean ﬂow. This contribution arises only if the convolution ﬁlter is applied in directions where the mean ﬂow gradients are non-zero. It is referred to as rapid because the time scale of its response to variations of the mean ﬂow is small. Numerical experiments show that this part plays an important role when the turbulence is in a desiquilibrium state when: (i) production of kinetic energy is much larger than dissipation or (ii) the ﬁlter length is of the same order as the integral scale of turbulence. Subgrid stresses anisotropy is observed to be due to the interaction of this rapid part and the mean shear. Numerical simulations have shown that the rapid part escapes the functional modeling, but scale-similarity models and soft deconvolution models succeed in representing anisotropic energy transfer (both forward and backward cascades) associated to the rapid part. – The slow part is always present in large-eddy simulation, because it does not depend on the mean ﬂow gradients. It corresponds to the subgrid tensor analyzed through the previously described canonical analysis. It is referred to as slow because its relaxation time is long with respect to rapid part. Numerical tests show that subgrid viscosity model correctly capture the associated kinetic energy transfer. One simple idea for generating subgrid models possessing good qualities on both the structural and energy levels is to combine a functional with a structural model, making what is called mixed models. This is generally done by combining a subgrid viscosity model for representing the energy cascade mechanism with a scale similarity. The stochastic backward cascade models are usually not included because the structural models are capable of including this phenomenon.

7.4 Mixed Modeling

239

The resulting form is 1 1 τij − τkk δij = −2νsgs S ij + (Lij − Lkk δij ) , 3 3

(7.124)

where νsgs is the subgrid viscosity (evaluated using one of the previously described model), and Lij the evaluation obtained using one of the structural model8 . Another argument for using a mixed model originates from the splitting of the full deconvolution model as the sum of the soft deconvolution model and the hard deconvolution model (see Sect. 7.2.1): the scale-similarity models are formally equivalent to truncated Taylor-series-expansion-based soft deconvolution models, and thus are not able to account for interactions between resolved scales and scales smaller than the mesh size on which the equations are solved. Consequently, they must be supplemented by another model, which will play the role of the secondary regularization within the framework of the deconvolution approach. Examples of such models are described in the following. 7.4.2 Examples of Mixed Models We present several examples of mixed models here: 1. The Smagorinsky–Bardina model (p. 240), for which the respective weights of each of the contributions are preset. This model is limited by the hypotheses underlying each of the two parts constituting it: the subgrid viscosity is still based on arguments of the inﬁnite inertial range type. Experience shows, though, that combining the two models reduces the importance of the constraints associated with these underlying hypotheses, which improves the results. 2. A one-parameter mixed model whose subgrid viscosity is computed by a dynamic procedure of the Germano–Lilly type (p. 240). With this procedure, the respective weights of the structural and functional parts of the model can be modiﬁed, so that the subgrid viscosity model is now computed as a complement to the scale similarity model, which allows a better control of the dissipation induced. It can be said, though, that this procedure innately prefers the structural part. 3. The general form of N -parameter dynamic mixed model, as derived by Sagaut et al. (p. 241). This procedure is an extension of the previous one: the weights of the diﬀerent parts of the model are dynamically computed, resulting in a possibly better approximation of the true subgrid stresses. The case of two-parameter dynamic mixed model is emphasized. 8

Only scale-similarity models or approximate deconvolution models are used in practice to derive mixed models, because they are very easy to implement.

240

7. Structural Modeling

Other mixed models have already been presented within the framework of the deconvolution approach (see Sect. 7.2.1, p. 220), which will not be repeated in the present section. Application of the results dealing with the dynamic evaluation of the constants presented below to the Taylor series expansion based deconvolution models is straightforward. Mixed Smagorinsky–Bardina Model. The ﬁrst example is proposed by Bardina et al. [40] in the form of a linear combination of the Smagorinsky model (5.90) and the scale similarity model (7.99). The subgrid tensor deviator is then written: 1 1 τij − τkk δij = 3 2 in which

1 −2νsgs S ij + Lij − Lkk δij 3

Lij = ui uj − ui uj

,

(7.125)

,

(7.126)

and 2

νsgs = Cs ∆ |S| .

(7.127)

Variants are obtained either by changing the subgrid viscosity model used or by replacing the tensor L with the tensor Lm (7.105) or the tensor L (7.101). One-Parameter Mixed Dynamic Model. A mixed dynamic model was proposed by Zang, Street, and Koseﬀ [799]. This is based initially on the Bardina model coupled with the Smagorinsky model, but the latter can be replaced by any other subgrid viscosity model. The subgrid viscosity model constant is computed by a dynamic procedure. The subgrid tensors corresponding to the two ﬁltering levels are modeled by a mixed model: 1 1 m τij − τkk δij = −2νsgs S ij + Lm ij − Lkk δij 3 3

,

(7.128)

1 * + Q − 1Q δ Tij − Tkk δij = −2νsgs S ij ij kk ij 3 3

,

(7.129)

in which ** Qij = u/ i uj − ui uj

,

(7.130)

and νsgs = Cd ∆|S|

.

(7.131)

7.4 Mixed Modeling

The residual Eij is now of the form: 2 − H − −2C ∆ m + δ P Eij = Lm ij d ij ij kk ij in which

** , Hij = u/ i uj − ui uj m *i u *j , u/ L i uj − u ij = * 2 ∆ * * / mij = |S|S ij − |S|S ij ∆

,

241

(7.132)

(7.133) (7.134) ,

(7.135)

and where Pkk represents the trace of the subgrid tensor. The Germano–Lilly dynamic procedure leads to: Cd =

(Lm ij − Hij )mij mij mij

.

(7.136)

In simulations performed with this model, the authors observed a reduction in the value of the dynamic constant with respect to that predicted by the usual dynamic model (i.e. based on the Smagorinsky model alone). This can be explained by the fact that the diﬀerence between the Lm and H terms appears in the numerator of the fraction (7.136) and that this diﬀerence is small because these terms are very similar. This shows that the subgrid viscosity model serves only to model a residual part of the full subgrid tensor and not its entirety, as in the usual dynamic model. Vreman et al. [747] propose a variant of this model. For the sake of mathematical consistency, by making the model for the tensor Tij dependent only * on the velocity ﬁeld that corresponds to the same level of ﬁltering, i.e. u, these authors propose the following alternate form for the tensor Qij : ** */ *j − u * *j Qij = u iu iu

.

(7.137)

N-Parameter Dynamic Mixed Model. General Formulation and Formal Resolution. A general form of multiparameter dynamic model was derived by Sagaut et al. [630]. Considering a formal N -part parametrization of the subgrid tensor, each term being associated to a real constant Cl , k = 1, .., N Cl fijl (u, ∆) , (7.138) τij = l=1,N

where the functions fijl are the kernels of the diﬀerent parts of the complete model. The equivalent formulation obtained at the test ﬁlter level is * * ∆) Cl fijl (u, . (7.139) Tij = l=1,N

242

7. Structural Modeling

Inserting (7.138) and (7.139) into the Germano identity (5.138), we get the following deﬁnition of the residual Eij : Eij = Lij −

* − f2l (u, ∆) . * ∆) Cl mlij , mlij = fijl (u, ij

(7.140)

l=1,N

In order to obtain N linearly independent relations to compute the constants Cl , a ﬁrst solution is to operate the contraction of the residual (7.140) with N independent tensors Alij . The constants will then appear as the solutions of the following linear algebraic problem of rank N : Cl mlij Akji = Lij Akji , k = 1, N . (7.141) l=1,N

It is worth noting that the N constants are coupled, resulting in a global self-adaption of each constant. The particular case of the least-square minimization is recovered by taking Akij = mkij , k = 1, N . In the case where some constants are not computed dynamically but are arbitrarily set, the linear system (7.141) corresponds to a ill-posed problem containing more constraints than degrees of freedom. Assuming that the N ﬁrst constants are arbitrarily chosen, we recover a well-posed problem of rank N − N by replacing Lij with Lij , where Lij = Lij −

Cl mlij

.

(7.142)

l=1,N

Two-Parameter Dynamic Models. Mixed models have also been proposed by Salvetti [640, 643] and Horiuti [324] with two dynamic constants (one for the subgrid-viscosity part and one for the scale-similarity), corresponding to the N = 2 case in the previous section. These models have the advantage of avoiding any a priori preference for the contribution of one or the other model component. An extensive study of dynamic mixed model has been carried out by Sarghini et al. [646] for the plane channel case. Numerical simulations show that two-paramater mixed models may yield disappointing results, because of a too low dissipation level. This is due to the fact that the coupled dynamic procedure described in the previous section gives a heavy weigth to the scale-similarity part of the model, because its correlation coeﬃcient with the exact subgrid tensor is much higher than the one of the subgrid-viscosity model. This conclusion was conﬁrmed by Anderson and Meneveau [12] in isotropic turbulence. These authors also observed a serious lack of robustness in regard to the dynamic Smagorinsky model when the test ﬁlter cutoﬀ exceeds the integral scale of turbulence: negative values of the dynamic constant are returned in this case. To relieve this problem, Morinishi [542, 543] proposes to uncouple the computation of the dynamic constants. The modiﬁed algorithm for the dynamic procedure is:

7.5 Diﬀerential Subgrid Stress Models

243

1. Compute the constant associated to the subgrid-viscosity part of the model using a classical dynamic procedure, without taking the scale similarity part into account. This corresponds to the N = 1 case in the previous section. The resulting constant will ensure a correct level of dissipation. 2. Compute the constant associated to the scale-similarity part using a twoparameter dynamic procedure, but considering that the constant of the subgrid-viscosity part is ﬁxed. This corresponds to N = 2 and N = 1 in the previous section.

7.5 Diﬀerential Subgrid Stress Models A natural way to ﬁnd an expression for the subgrid stress tensor is to solve a prognostic equation for each component. This approach leads to the deﬁnition of six additional equations, and thus a very signiﬁcant increase in both the model complexity and the computational cost. But the expected advantage is that such a model would a priori be able to account for a large class of physical mechanisms, yielding the deﬁnition of a very robust model. Three proposal have been published by diﬀerent research groups: 1. The pioneering model of Deardorﬀ (p. 243). 2. The model by Fureby et al. (p. 244), which is an improved version of the Deardorﬀ model with better theoretical properties, such as realizability. 3. The models based of the use of transport equations for the velocity ﬁltered probability density function (p. 245), which do not rely on an explicit model for the triple correlations. 7.5.1 Deardorﬀ Model Another approach for obtaining a model for the subgrid tensor consists in solving an evolution equation for each of its components. This approach proposed by Deardorﬀ [173] is analogous in form to two-point statistical modeling. Here, we adopt the case where the ﬁlter is a Reynolds operator. The subgrid tensor τij is thus reduced to the subgrid Reynolds tensor Rij . We deduce the evolution equation of the subgrid tensor components from that of the subgrid modes (3.31)9 : ∂τij ∂t 9

=

−

∂ ∂uj ∂ui (uk τij ) − τik − τjk ∂xk ∂xk ∂xk

This is done by applying the ﬁlter to the relation obtained by multiplying (3.31) by uj and taking the half-sum with the relation obtained by inverting the subscripts i and j.

244

7. Structural Modeling

∂uj ∂ ∂ui − u u u +p + ∂xk i j k ∂xj ∂xi −

∂ ∂u ∂uj ∂ ui p − uj p − 2ν i ∂xj ∂xi ∂xk ∂xk

.

(7.143)

The various terms in this equation have to be modeled. The models Deardorﬀ proposes are: – For the pressure–strain correlation term: + 2 qsgs ∂u ∂u 2 2 2 2 j i + δij + qsgs S ij p = −Cm τij − qsgs ∂xj ∂xi 3 5 ∆

,

(7.144)

2 where Cm is a constant, qsgs the subgrid kinetic energy, and S ij the strain rate tensor of the resolved ﬁeld. – For the dissipation term:

ν

2 3/2 (qsgs ) ∂ui ∂uj = δij Ce ∂xk ∂xk ∆

,

(7.145)

where Ce is a constant. – For the triple correlations: + ∂ ∂ ∂ 2 ui uj uk = −C3m ∆ qsgs τjk + τik + τij ∂xi ∂xj ∂xk

.

(7.146)

The pressure–velocity correlation terms p ui are neglected. The values of the constants are determined in the case of isotropic hom*ogeneous turbulence: Cm = 4.13, Ce = 0.70, C3m = 0.2

.

(7.147)

Lastly, the subgrid kinetic energy is determined using evolution equation (5.119). 7.5.2 Fureby Diﬀerential Subgrid Stress Model An alternate form of the diﬀerential stress model is proposed by Fureby et al. [232], which has better symmetry preservation properties than the original Deardorﬀ model. The triple correlation term is approaximated as + ∂τij 2 ∆ −ui uj uk = cq qsgs ∂xk

,

(7.148)

νsgs

with cq = 0.07. Other terms are kept unchanged. Each term in the closed subgrid stress equations is now frame indiﬀerent. The set of closed equations

7.5 Diﬀerential Subgrid Stress Models

245

is realizable, and has the same transformation properties as the exact Navier– Stokes equations under a change of frame. An alternate form of the subgrid viscosity in (7.148) which allows the representation of backscatter is also deﬁned by Fureby: νsgs = −

τij S ij S ij S ij

.

(7.149)

7.5.3 Velocity-Filtered-Density-Function-Based Subgrid Stress Models A methodology referred to as the Velocity Filtered Density Function to close the ﬁltered Navier–Stokes equations is proposed by Gicquel et al. [264] on the grounds of the previous works by Pope and Givi. The subgrid scale stresses are reconstructed by considering the joint probability function of all of the components of the velocity vector. To this end, an exact evolution equation is derived for the velocity ﬁltered density function in which unclosed terms are modeled. Two implementations have been proposed: a ﬁrst one consists in discretizing transport equations for the subgrid stresses associated to the velocity ﬁltered density function (this is the one emphasized below); the second one consists in solving it via a Lagrangian Monte Carlo scheme, which leads to the deﬁnition of an equivalent stochastic system. This last form should be classiﬁed as a structural model based on a stochastic reconstruction of subgrid scales, and will be mentioned in Sect. 7.7. Deﬁnitions. The ﬁrst step consists in deﬁning the velocity ﬁltered density function PL +∞ PL (v; x, t) = ρ[v, u(x , t)]G(x − x)dx , (7.150) −∞

where G is the convolution ﬁlter kernel and ρ[v, u(x , t)] is the ﬁne-grained density ρ[v, u(x , t)] = δ(v − u(x, t)) .

(7.151)

It is observed that PL has all the properties of the probability density function when the ﬁlter kernel is positive. The conditional ﬁltered value of a dummy variable φ(x, t) is therefore deﬁned as ! +∞ φ(x, t)|u(x, t) = vL = φ|vL

−∞

φ(x , t)ρ[v, u(x , t)]G(x − x)dx PL (v; x, t)

.

(7.152) where α|βL denotes the ﬁltered value of α conditioned on β.

246

7. Structural Modeling

First Model. The starting point of the procedure is to write an expression for the time-derivative of PL . Such an expression is found by applying the time-derivative operator to Eq. (7.150), yielding ∂PL (v; x, t) ∂t

∂ui (x , t) ∂ρ[v, u(x , t)] G(x − x)dx ∂t ∂vi −∞ A @ ∂ui ∂ v = − P (v; x, t) (7.153) L ∂vi ∂t L +∞

=

The next step consists in eliminating the velocity time derivative using the momentum equation, leading to ∂PL ∂t

+ − −

∂PL ∂ ∂p ∂PL =− [(vk − uk ) PL ] + ∂xk ∂xk ∂xi ∂vi A @ ∂p ∂S ik ∂PL ∂ ∂p v PL 2ν + − ∂xk ∂vi ∂vi ∂xi ∂xi L A @ ∂Sik ∂ ∂S ik v 2ν − PL , ∂vi ∂xi ∂xi

uk

(7.154)

L

where the relation φ≡G φ=

+∞

φ|vL PL (v; x, t)dv

−∞

,

(7.155)

was used. The last two terms in the right hand side of (7.154) are unclosed terms which require the deﬁnition of ad hoc subgrid models. The ﬁrst term in the right hand side is related to the subgrid advection and is closed. The sum of the two unknown terms is modeled as follows ∂ ∂vi

@

A A @ ∂p ∂Sik ∂p ∂S ik ∂ − − P − 2ν PL v v L ∂xi ∂xi ∂vi ∂xi ∂xi L L −

1 ∂ 2 PL ∂ [Gij (vj − ui )PL ] C0 ε (7.156) , ∂vi 2 ∂vi ∂vi

with

1 + 2 with C0 = 2.1 and where the subgrid mixing frequency ω are evaluated as Gij = −ω

ε=

2 3/2 qsgs ∆

,

3 C0 δij , (7.157) 4 dissipation rate ε and the subgrid

ω=

ε 2 qsgs

.

(7.158)

7.5 Diﬀerential Subgrid Stress Models

247

2 The subgrid kinetic energy qsgs is taken equal to half the trace of the subgrid stress tensor. This closure is equivalent the the one proposed by Rotta for the Reynolds-averaged Navier–Stokes equations. The ﬁnal stage consists in writing the corresponding equations for the subgrid stresses, which are deﬁned here using the generalized central moment framework of Germano (Sect. 3.3.2), i.e. τij = τG (ui , uj ). Evolution equations consistant with the previous closed form of the velocity ﬁltered density function are

∂τij ∂t

∂ ∂τijk (uk τij ) = − + Gik τjk + Gjk τik ∂xk ∂xk ∂uj ∂ui −τik − τjk + C0 εδij . ∂xk ∂xk

+

(7.159)

The last unknown term is the third-order generalized central moment τijk = τG (ui , uj , uk ). Its value are taken from the Lagrangian Monte Carlo solver used to generate a prognostic stochastic velocity ﬁeld whose pdf PL satisﬁes equation (7.154). Details of the numerical implementation can be found in Ref. [264] and will not be reproduced here. Second Model. An alternative form for relation (7.154) is ∂PL ∂t

∂ ∂p ∂PL ∂PL =− [(vk − uk ) PL ] + ∂xk ∂xk ∂xi ∂vi A @ ∂p ∂ ∂p v PL − ∂vi ∂xi ∂xi L A @ ∂2 ∂ 2 PL ∂ui ∂uj v − PL ν ν ∂xk ∂xk ∂vi ∂vi ∂xk ∂xk L

uk

+ + +

.

(7.160)

Using the same procedure as in the previous case, the following closed form is obtained ∂PL ∂t

+

∂PL ∂ ∂p ∂PL =− [(vk − uk ) PL ] + ∂xk ∂xk ∂xi ∂vi ∂ui ∂uj ∂ 2 PL ∂ui ∂ 2 PL +ν + 2ν ∂xk ∂xk ∂vi ∂vj ∂xk ∂vi ∂vk ∂ 1 ∂ 2 PL − (Gij (vj − uj )PL ) + C0 ε . ∂vi 2 ∂vi ∂vj uk

(7.161)

The consistent subgrid stress equations are now ∂τij ∂t

+

∂ ∂τijk (uk τij ) = − + Gik τjk + Gjk τik ∂xk ∂xk ∂ 2 τij ∂uj ∂ui +ν − τik − τjk + C0 εδij . ∂xk ∂xk ∂xk ∂xk

(7.162)

248

7. Structural Modeling

The diﬀerence with the previous model is that viscous diﬀusion is taken into account in the subgrid stress transport equations, leading to a better accuracy in ﬂows where viscous eﬀects are inﬂuencial. The numerical implementation relies on the same Monte Carlo approach as the previous model. 7.5.4 Link with the Subgrid Viscosity Models We reach the functional subgrid viscosity models again starting with a model with transport equations for the subgrid stresses, at the cost of additional assumptions. For example, Yoshizawa et al. [794] proposed neglecting all the terms of equation (7.143), except those of production. The evolution equation thus reduced comes to: ∂τij ∂uj ∂ui = −τik − τjk ∂t ∂xk ∂xk

.

(7.163)

Assuming that the subgrid modes are isotropic or quasi-isotropic, i.e. that the extra-diagonal elements of the subgrid tensor are very small compared with the diagonal elements, and that the latter are almost mutually equal, the right-hand side of the reduced equation (7.163) comes down to the simpliﬁed form: 2 S ij , (7.164) −qsgs 2 in which qsgs = uk uk /2 is the subgrid kinetic energy. Let t0 be the characteristic time of the subgrid modes. Considering the relations (7.163) and (7.164), and assuming that the relaxation time of the subgrid modes is much shorter than that of the resolved scales10 , we get

1 2 S ij τij − τkk δij ≈ −t0 qsgs 3

.

(7.165)

The time t0 can be evaluated by dimensional argument using the cutoﬀ length ∆ and the subgrid kinetic energy: ∆ t0 ≈ + 2 qsgs

.

(7.166)

By entering this estimate into equation (7.165), we get an expression analogous to the one used in the functional modeling framework: + 1 2 S . (7.167) τij − τkk δij ≈ −∆ qsgs ij 3

10

We again ﬁnd here the total scale-separation hypothesis 5.4.

7.6 Stretched-Vortex Subgrid Stress Models

249

7.6 Stretched-Vortex Subgrid Stress Models 7.6.1 General Misra and Pullin [517, 608, 735], following on the works of Pullin and Saﬀman [609], proposed subgrid models using the assumption that the subgrid modes can be represented by stretched vortices whose orientation is governed by the resolved scales. Supposing that the subgrid modes can be linked to a random superimposition of ﬁelds generated by axisymmetrical vortices, the subgrid tensor can be written in the form: ∞ E(k)dkEpi Zpq Eqj , (7.168) τij = 2 kc

in which E(k) is the energy spectrum, Elm the rotation matrix used to switch from the vortex coordinate system to the reference system, Zij the diagonal tensor whose main elements are (1/2, 1/2, 0) and Epi Zpq Eqj , the moment of the probability density function P (α, β) of the Euler angles α and β giving the orientation of the vortex axis with respect to the frame of reference. The statistical average performed on the Euler angles of a function f is deﬁned as: π 2π 1 f (Eij ) = f (Eij )P (α, β) sin(α)dαdβ . (7.169) 4π 0 0 Two pieces of information are therefore needed to compute the subgrid term: the shape of the energy spectrum for the subgrid modes and the subgrid structure orientation distribution function. As the use of an evolution equation for the probability density function yielded no satisfactory results, Misra and Pullin propose modeling this function as a product of Dirac functions or a linear combination of such products. These are of the general form: P (α, β) =

4π δ(α − θ)δ(β − φ) sin(α)

,

(7.170)

where θ(x, t) and φ(x, t) determine the speciﬁc orientation considered. Deﬁning the two unit vectors e and ev : e1 = sin(α) cos(β), e2 = sin(α) sin(β), e3 = cos(α),

(7.171)

ev1 = sin(θ) cos(φ), ev2 = sin(θ) sin(φ), ev3 = cos(θ),

(7.172)

the subgrid tensor can be re-written in the form: ∞ 2 E(k)dk = δij − evi evj qsgs τij = δij − evi evj kc

.

(7.173)

250

7. Structural Modeling

The various models must thus specify the speciﬁc orientation directions of the subgrid structures. Three models are presented in the following. The sub2 grid kinetic energy qsgs can be computed in diﬀerent ways (see Sect. 9.2.3), for example by solving an additional evolution equation, or by using a double ﬁltering technique. A local evaluation procedure in the physical space based on the second-order velocity structure function is proposed by Voekl et al. [735]. 7.6.2 S3/S2 Alignment Model A ﬁrst hypothesis is to assume that the subgrid structures are oriented along the eigenvectors of the resolved strain rate tensor S ij that corresponds to its two largest eigenvalues. This is equivalent to assuming that they respond instantaneously to the forcing of the large scales. Using es2 and es3 to denote these two vectors, and λ2 and λ3 ≥ λ2 the associated eigenvalues, we get 3 4 2 s3 s2 λ δij − es3 + (1 − λ) δij − es2 , (7.174) τij = qsgs i ej i ej where the weighting coeﬃcient is taken proportional to the norms of the eigenvalues: λ3 . (7.175) λ= λ3 + |λ2 | 7.6.3 S3/ω Alignment Model The second model is derived on the assumption that the subgrid structures are oriented along the third eigenvector of the tensor S ij , denoted es3 as before, and the vorticity vector of the resolved ﬁeld. The unit vector it carries is denoted eω and is computed as: eω =

∇×u |∇ × u|

.

The subgrid tensor is evaluated as: 3 4 2 s3 ω τij = qsgs λ δij − es3 + (1 − λ) δij − eω i ej i ej

(7.176)

.

(7.177)

The weighting parameter λ is chosen arbitrarily. The authors performed tests considering the three values 0, 0.5, and 1. 7.6.4 Kinematic Model Starting with the kinematics of a vortex ﬁlament entrained by a ﬁxed velocity ﬁeld, Misra and Pullin propose a third model, for which the vector ev is obtained by solving an evolution equation. The equation for the ith component of this vector is: ∂ui ∂uk ∂evi = evj − evi evk evj . (7.178) ∂t ∂xj ∂xj

7.7 Explicit Evaluation of Subgrid Scales

251

The subgrid tensor is then evaluated by inserting the vector ev thus computed into the expression (7.173).

7.7 Explicit Evaluation of Subgrid Scales The models described in the present section are all based on an explicit evaluation of the subgrid scales u ≡ (Id − G) u. Because the subgrid modes correspond to scales of motion that can not be represented at the considered ﬁltering level (i.e. in practice on the computational grid), a new higher-resolution ﬁltering level is introduced. Numerically, this is done by introducing an auxiliary computational grid (or a set of embedded auxiliary grids), whose mesh size is smaller than the original one. The subgrid ﬁeld u is evaluated on that grid using one of the model presented below, and then the non-linear G ((u + u ) ⊗ (u + u )) is computed. The corresponding general algorithmic frame is 1. u is known from a previous calculation, on the computational grid, i.e. at the G ﬁltering level, whose characteristic length is ∆. 2. Deﬁne an auxiliary grid, associated to a new ﬁltering level F with char* < ∆, and interpolate u on the auxiliary grid. acteristic length ∆ 3. Compute the approximate subgrid ﬁeld ua = (F − G) u using a model on the auxiliary grid. 4. Compute the approximate non-ﬁltered non-linear term at the F level on the auxiliary grid: (u + ua ) ⊗ (u + ua ) . 5. Compute the approximate ﬁltered non-linear term at the G level on the computational grid: G ((u + ua ) ⊗ (u + ua ))

,

and use it to compute the evolution of u. It is worth noting that this class of models can be interpreted as a generalization of deconvolution-based models (see Sect. 7.2). Classical deconvolution models require the use of a second regularization to take into account the interactions with modes that cannot be reconstructed on the mesh. In the present case, these unresolved scales are explicitly reconstructed on a ﬁner grid, rendering the approach more general. Looking at the algorithmic framework presented above, we can see that (i) the second step (interpolation) is equivalent to the soft deconvolution, the third step is an extension of the hard deconvolution and, (iii) the ﬁfth step is associated to the primary regularization. Several ways to compute the subgrid motion on the auxiliary grid have been proposed by diﬀerent authors. They are classiﬁed by increasing order of complexity (computational cost):

252

7. Structural Modeling

1. Fractal Interpolation Procedure of the ﬂuctuations, as proposed by Scotti and Meneveau (p. 253). The subgrid ﬂuctuations are reconstructed in a deterministic way on the ﬁne grid using an iterative fractal interpolation technique (several similar fractal reconstruction techniques can be found in [364]). This model is based on geometrical considerations only, and does not take into account any information dealing with the ﬂow dynamics such as disequilibrium, anisotropy, ... But it provides an estimate of the subgrid motion at a very low cost. 2. Chaotic Map Model of McDonough et al. (p. 254). The subgrid ﬂuctuations are approximated in a deterministic way using a very simple chaotic dynamical system, which is chosen in order to mimic some properties of the real turbulent ﬂuctuations (amplitude, autocorrelation, distribution of velocity ﬂuctuations, ...). This model is the easiest to implement, and induces a very small overhead. A problem is that it requires the deﬁnition of a realistic dynamical system, and then a complete knowledge of the turbulent motion characteristics at each point of the numerical simulation. 3. One-Dimensional Turbulence model for the ﬂuctuations, as proposed by Kerstein and his co-workers (p. 257). This approach relies on the deﬁnition of a simpliﬁed model for the three subgrid velocity component along lines located inside the computational cells. This model can be seen as an improvement of the Chaotic Map Model, since the non linear cascade eﬀect are taken into account via the use of chaotic map, but diﬀusion eﬀects and resolved pressure coupling are incorporated by solving a differential equation. 4. Reconstruction of the subgrid velocity ﬁeld using kinematic simulations (p. 259). This approach, proposed by Flohr and Vassilicos, provides an incompressible, random, statistically steady, isotropic turbulent velocity ﬁeld with prescribed energy spectrum. This model does not require that we solve any diﬀerential equation, and thus has a very low algorithmic cost. 5. The Velocity Filtered Density Function proposed by Gicquel et al. (p. 260) is another model belonging to this family, based on the use of a stochastic model for the subgrid ﬂuctuations. Its equivalent diﬀerential formulation being presented in Sect. 7.5.3, the emphasis is put here on the equivalent stochastic system. In this form, it can be interepreted as the most advanced stochastic reconstruction technique for the subgrid scales, since it retains all the complexity of the Navier–Stokes dynamics. 6. Subgrid Scale Estimation Procedure proposed by Domaradzki and his coworkers (p. 261). The subgrid ﬂuctuation are now deduced from a simpliﬁed advection equation, deduced from the ﬁltered Navier–Stokes operator. An evaluation of the subgrid motion production term is derived, and integrated over a time interval associated to characteristic relaxation time of the subgrid scale. This model makes it possible to evaluate the

7.7 Explicit Evaluation of Subgrid Scales

253

subgrid motion at a very low computational cost, but requires the computation of an approximate inverse ﬁlter. 7. Multilevel Simulations (p. 263), which are based on the use of the exact Navier–Stokes equations on a set of embedded computational grids. The reduction of the computational eﬀort with respect to the Direct Numerical Simulation is obtained by freezing (quasi-static approximation) the high-frequencies represented on ﬁne grids for some time interval, leading to the deﬁnition of a cyclic strategy. These methods can be interpreted as a time-consistent extension of the classical multigrid procedures for steady computations. They correspond to the maximal computational eﬀort, but also to the most realistic approach. 7.7.1 Fractal Interpolation Procedure Scotti and Meneveau [661, 662] propose to reconstruct the subgrid velocity ﬁeld using two informations: (i) the resolved velocity ﬁeld, which is known on the coarsest grid, and (ii) the fractality of the velocity ﬁeld. The ﬂuctuations are evaluated by interpolating the resolved coarse-grid velocity ﬁeld on the ﬁne grid using a fractal interpolation technique. We ﬁrst describe this interpolation technique in the monodimensional case. It is based on an iterative mapping procedure. The ﬂuctuating ﬁeld ua is reconstructed within each interval of the coarse grid by introducing a local coordinate ξ ∈ [0, 1]. Let us consider the interval [xi−1 , xi+1 ], where i − 1 and i + 1 are related to the grid index on the coarse grid. We have ξ = (x − xi−1 )/2∆. The proposed map kernel W for a function φ to interpolated on the considered interval is:

W [φ](ξ) =

di,1 φ(2ξ) + qi,1 (2ξ) di,2 φ(2ξ) + qi,2 (2ξ)

if ξ ∈ [0, 1/2] if ξ ∈]1/2, 1]

,

(7.179)

where qi,j are polynomials and di,j are stretching parameters. The authors propose to use the following linear polynomials: qi,1 (ξ) =

(φ(xi ) − φ(xi−1 ) − di,1 (φ(xi+1 ) − φ(xi−1 ))ξ +φ(xi−1 )(1 − di,1 )

qi,2 (ξ) =

,

(7.180)

(φ(xi+1 ) − φ(xi ) − di,2 (φ(xi+1 ) − φ(xi−1 ))ξ −φ(xi−1 )di,2

.

(7.181)

The ﬂuctuation is deﬁned as ua = lim W n [u] = W ◦ W ◦ ... ◦ W [u] n→∞

n times

.

(7.182)

254

7. Structural Modeling

The stretching parameters are such that the Hausdorﬀ dimension D of the synthetic signal is equal to > log(|di,1 |+|di,2 |) if 1 < |di,1 | + |di,2 | < 2 1+ log(2) . (7.183) D= 1 if |di,1 | + |di,2 | ≤ 1 In order to conserve then mean value of the signal over the considered interval, we have di,1 = −di,2 = d. For three-dimensional isotropic turbulence, we have D = 5/3, yielding d = ∓21/3 . This procedure theoretically requires an inﬁnite number of iterations to build the ﬂuctuating ﬁeld. In practice, a ﬁnite number of iterations is used. The statistical convergence rate of process being exponential, it still remains a good approximation. A limited number of iterations can also be seen as a way to account for viscous eﬀects. The extension to the multidimensional case is straightforward, each direction of space being treated sequentially. This procedure also makes it possible to compute analytically the subgrid tensor. The resulting model will not be presented here (see [662] for a complete description). 7.7.2 Chaotic Map Model McDonough and his coworkers [552, 338, 469] propose an estimation procedure based on the deﬁnition of a chaotic dynamical system. The resulting model generates a contravariant subgrid-scale velocity ﬁeld, represented at discrete time intervals on the computational grid: ua = Au ζ V

,

(7.184)

where A is an amplitude coeﬃcient evaluated from canonical analysis, ζ an anisotropy correction vector consisting mainly of ﬁrst-order structure function of high-pass ﬁltered resolved scales, and V is a vector of chaotic algebraic maps. It is important noting that the two vectors are multiplied using a vector Hadamard product, deﬁned for two vectors and a unit vector i according to: (ζ V ) · i ≡ (ζ · i)(V · i) . (7.185) The amplitude factor is chosen such that the kinetic energy of the synthetic subgrid motion is equal to the energy contained in all the scales not resolved by the simulation. It is given by the expression: 1/6

Au = Cu u∗ Re∆ with

,

(7.186)

2

1/2

u∗ = (ν|∇u|)

, Re∆ =

∆ |∇u| ν

,

7.7 Explicit Evaluation of Subgrid Scales

255

where ν is the molecular viscosity. The scalar coeﬃcient Cu is evaluated from classical inertial range arguments. The suggested value is Cu = 0.62. The anisotropy vector ζ is computed making the assumption that the ﬂow anisotropy is smoothly varying in wave-number. In a way similar to the one proposed by Horiuti (see Sect. 6.3.3), the ﬁrst step consists in evaluating the anisotropy vector from the highest resolved frequency. In order to account for the anisotropy of the ﬁlter, the resolved contravariant velocity ﬁeld uc is considered. The resulting expression for ζ is: ζ=

√ 3

s |J −1

,

· s|

(7.187)

where J −1 is the inverse of the coordinate transformation matrix associated to the computational grid (and to the ﬁlter). The vector s is deﬁned according to √ |∇(2 uc · i)| , (7.188) s·i = 3 |∇2 uc |

2c is related to the test ﬁeld computed thanks to the use of the where the u * > ∆. test ﬁlter of characteristic length ∆ We now describe the estimation procedure for the stochastic vector V . In order to recover the desired cross-correlation between the subgrid velocity component, the vector V is deﬁned as: V = AM

,

(7.189)

where A is a tensor such that R = A · AT , where R is the correlation tensor of the subgrid scale velocity. In practice, McDonough proposes to use the evaluation: (∇* ui )j . (7.190) Aij = |∇* ui | Each component Mi , i = 1, 2, 3 of the vector M is of the form: al Mlm , Mi = σ l=0,N

(7.191)

m=1,Nl

where Nl is the binomial coeﬃcient N Nl ≡ l

,

and σ = 1.67 is the standard deviation for the variable, and the weights al are given by 1/2 √ al = 3 pl (1 − p)(N −l) , p = 0.7 . (7.192)

256

7. Structural Modeling

The maps Mlm are all independent instances of one of the three following normalized maps:

– The tent map:

m

(n+1)

⎧ ⎨ R(−2 − 3m(n) ) = R(3m(n) ) ⎩ R(−2 − 3m(n) )

if m(n) < −1/3 if − 1/3 ≤ m(n) ≤ 1/3 if m(n) > 1/3

,

(7.193)

where m(n) is the nth instance of the discrete dynamical system, and R ∈ [−1, 1]. – The logistic map: m(n+1) = RAR m(n) (1 − |m(n) |Am ) with √ AR = 2 + 2 2,

Am =

1 3 1+ AR 2

,

(7.194)

.

– The sawtooth map:

m

(n+1)

⎧ ⎨ R(2 + 3m(n) ) = R(3m(n) ) ⎩ R(−2 + 3m(n) )

if m(n) < −1/3 if − 1/3 ≤ m(n) ≤ 1/3 if m(n) > 1/3

.

(7.195)

The map parameter R is related to some physical ﬂow parameter, since the bifurcation and autocorrelation behaviors of the map are governed by R. An ad hoc choice for R will make it possible to model some of the local history eﬀects in a turbulent ﬂow in a way that is quantitatively and qualitatively correct. It is chosen here to set the bifurcation parameter R on the basis of local ﬂow values, rather than on global values such as the Reynolds number. That choice allows us to account for large-scale intermittency eﬀects. Selecting the ratio of the Taylor λ and Kolmogorov η scales, a possible choice is: 7 r

(λ/η) −1 tanh (Rc ) , (7.196) R = tanh (λ/η)c where r is a scaling exponent empirically assumed to lie in the range [4, 6], and (λ/η)c is a critical value of the microscale ratio that is mapped onto Rc , the critical value of R. Suggested values are given in Table 7.2. The last point is related to the time scale of the subgrid scales. Let te be the characteristic relaxation time of the subgrid scales, to be evaluated using inertial range considerations. If this time scale is smaller than the time step ∆t of the simulation (the characteristic ﬁlter time), then the stochastic

7.7 Explicit Evaluation of Subgrid Scales

257

Table 7.2. Parameters of the Chaotic Map Model. Map

Rc

(λ/η)c

r

26

5

Logistic

√ −(2 + 2 2)1/2

Tent

-1/3

28.6

5

Sawtooth

-1/3

28.6

5

variables Mi must be updated nu times per time step, with ∆t|∇u| ∆t −1/3 nu ≈ = , Re∆ te fM

(7.197)

where fM is a fundamental frequency associated with the chaotic maps used to generate the variables. It is deﬁned as: fM =

C θ

,

(7.198)

where C is some positive constant and θ the integral iteration scale θ=

m(n) m(n+l) 1 ρ(0) + ρ(l), ρ(l) = 2 m(n) m(n) l=1,∞

,

(7.199)

which completes the description of the model. This model is Galilean- and frame-invariant, and automatically generates realizable Reynolds stresses. It reproduces the desired root-mean-square amplitude of subgrid ﬂuctuations, along with the probability density function for this amplitude. Finally, the proper temporal auto-correlation function can be enforced. 7.7.3 Kerstein’s ODT-Based Method A more complex chaotic map model based on Kerstein’s One-Dimensional Turbulence (ODT) approach11 was also proposed [650, 387, 389, 390, 388, 305, 306, 208, 198, 386, 771]. This is a method for simulating turbulent ﬂuctuations along one-dimensional lines of sight through a three-dimensional turbulent ﬂows. The velocity ﬂuctuations evolve by two mechanisms, namely the molecular diﬀusion and turbulent stirring. The latter mechanisms is taken into account by a sequence of fractal transformations denoted eddy events. An eddy event may be interpreted as a model of an individual eddy, whose location, length scale and frequency are determined using a non-linear probabilistic model. 11

It is worth noting that ODT originates in the Linear Eddy Model [384, 385, 117, 118, 177, 696, 414, 413, 473].

258

7. Structural Modeling

The diﬀusive step consists in solving the following one-dimensional advection-diﬀusion equation for each subgrid velocity component along the line ∂ ∂ui ∂p ∂ 2 ui + (Vj ui ) = +ν ∂t ∂xj ∂xi ∂x2

,

(7.200)

where ν is the molecular viscosity and V is the local advective ﬁeld such that ui = Vi (ξ)dξ , (7.201) Ω

where Ω is the volume based on the cutoﬀ length ∆. The second step, which accounts for non-linear eﬀects, is more complex and consists in two mathematical operations. The ﬁrst one is a measurepreserving map representing the turbulent stirring, while the second one is a modiﬁcation of the velocity proﬁles in order to implement energy transfers among velocity components. These two steps can be expressed as ui (x) ←− ui (f (x)) + ci K(x) , where the stirring-related mapping ⎧ 3(x − x0 ) ⎪ ⎪ ⎨ 2l − 3(x − x0 ) f (x) = x0 + 3(x − x0 ) − 2l ⎪ ⎪ ⎩ x − x0

(7.202)

f (x) is deﬁned as if x0 ≤ x ≤ x0 + l/3 if x0 + l/3 ≤ x ≤ x0 + 2l/3 if x0 + 2l/3 ≤ x ≤ x0 + l otherwise

(7.203)

where l is the length of the segment aﬀected by the eddy event. The second term in the right hand side of (7.202) is implemented to capture pressureinduced energy redistribution between velocity components and therefore makes it possible to account for the return to isotropy of subgrid ﬂuctuations. The kernel K is deﬁned as K(x) = x − f (x)

(7.204)

The amplitude coeﬃcients ci are determined for each eddy to enforce = the two following constraints: (i) the total subgrid kinetic energy E = i Ei = = 1! u (x)u (x)dx remains constant, and (ii) the subgrid scale spectrum i i i 2 must be realizable, i.e. the energy extracted from a velocity component cannot exceed the available energy in this component. The resulting deﬁnition of the coeﬃcients is ⎛ ⎞ ' 27 ⎝ α −wi + sign(wi ) (1 − α)wi2 + (7.205) wj2 ⎠ , ci = 4l 2 j=i

7.7 Explicit Evaluation of Subgrid Scales

259

where 1 wi = 2 l

ui (f (x))K(x)dx

4 = 2 9l

x0 +l

ui (x)(l − 2(x − x0 ))dx

. (7.206)

x0

The degree of energy redistribution is governed by the parameter α, which is taken equal to 2/3 in Ref. [650] (corresponding to equipartition of the available energy among the velocity components). The last element of the method is the eddy selection step, which give access to the time sequence of eddy events. All events are implemented instantaneously, but occur with frequencies comparable to turnover frequencies of associated turbulent structures. At each time step, the event-rate distribution is obtained by ﬁrst associating a time scale τ (x0 , l) with every eddy event. Using l/τ and l3 /τ 2 as an eddy velocity scale and a measure of the energy of the eddy motion, respetively, the time scale τ is computed using the following relation 2 l α ν2 ∼ (1 − α)w12 + (w22 + w32 ) − Z 2 , (7.207) τ 2 l where Z is the amplitude of the viscous penalty term that governs the size of the smallest eddies for given local strain conditions. A probabilistic model can be derived deﬁning an event-rate distribution λ λ(x0 , l, t) =

C l2 τ (x0 , l, t)

,

(7.208)

where C is an arbitrary parameter which determines the relative strength of turbulent stirring. 7.7.4 Kinematic-Simulation-Based Reconstruction Following Flohr and Vassilicos [220], the incompressible, turbulent-like subgrid velocity ﬁeld is generated by summing diﬀerent Fourier modes u (x, t) = (an cos(kn · x + ωn t) + bn sin(kn · x + ωn t)) , (7.209) n=1,N

where N is the number of Fourier modes, an and bn are the amplitudes corresponding to wave vector kn , and ωn is a time frequency. The wave vectors are randomly distributed in spherical shells: kn = kn (sin θ cos φ, sin θ sin φ, cos θ) ,

(7.210)

where θ and φ are uniformly distributed random angles within [0, 2π[ and [0, π], respectively. The random uncorrelated amplitude vectors an and bn

260

7. Structural Modeling

are chosen such that an · kn = bn · kn = 0

,

(7.211)

to ensure incompressibility, and |an |2 = |bn |n = 2E(kn )∆kn

,

(7.212)

where E(k) is the prescribed energy spectrum, and ∆kn is the wave number increment between the shells. Recommended shell distributions in the spectral space are: – Linear distribution kn = k1 +

kN − k1 (n − 1) ; N −1

(7.213)

– Geometric distribution kn = k1

kN k1

(n−1)/(N −1) ;

(7.214)

– Algebraic distribution kn = k1 nlog(kN /k1 )/ log N

.

(7.215)

The time frequency ωn is arbitrary. Possible choices are ωn = Uc kn if all the modes are advected with a constant velocity Uc , and ωn = kn3 E(kn ) if it is proportional to the eddy-turnover time of mode n. In practice, Flohr and Vassilicos use this model to evaluate the dynamics of a passive tracer, but do not couple it with the momentum equations. Nevertheless, it could be used to close the momentum equation too. 7.7.5 Velocity Filtered Density Function Approach The reconstruction of the subgrid motion via a stochastic system which obeys the required probability density function is proposed by Gicquel et al. [264]. This method is also equivalent (up to the second order) to solving the diﬀerential equations for the subgrid stresses presented in Sect. 7.5.3. The bases of the method are presented in this section, and will not be repeated here. The key of the present method is the deﬁnition of a Lagrangian Monte Carlo method, which is used to evaluate both the position Xi (in space) and the value of a surrogate of the subgrid velocity, Ui , associated to a set of virtual particules. The value of the subgrid velocity in each cell of the LargeEddy Simulation grid is deﬁned as the statistical average over all the virtual particules that cross the cell during a ﬁxed time interval.

7.7 Explicit Evaluation of Subgrid Scales

261

The stochastic diﬀerential equations equivalent to the ﬁrst model presented in Sect. 7.5.3 is dXi (t) = Ui (t)dt

dUi (t) =

,

∂p ∂sik − + 2ν + Gij (Uj (t) − uj (t)) dt ∂xi ∂xk + C0 ε dWiv (t) ,

(7.216)

(7.217)

where Gij , C0 and ε are deﬁned in Sect. 7.5.3, and Wiv denotes and independent Wiener–Levy process. The second model proposed by Gicquel accounts for viscous diﬀusion and is expressed as √ dXi (t) = Ui (t)dt + 2ν dWix (t) , (7.218)

dUi (t) =

∂p ∂sik − + 2ν + Gij (Uj (t) − uj (t)) dt ∂xi ∂xk √ ∂ui dWjx (t) , + C0 ε dWiv (t) + 2ν ∂xj

(7.219)

where ν is the molecular viscosity and Wix is another independent Wiener– Levy process. In practice, convergence of the statistical average over the particules within each cell must be carefully checked to recover reliable results. 7.7.6 Subgrid Scale Estimation Procedure A two-step subgrid scale estimation procedure in the physical space12 is proposed by Domaradzki and his coworkers [458, 187, 394, 188]. In the ﬁrst (kinematic) step, an approximate inversion of the ﬁltering operator is performed, providing the value of the deﬁltered velocity ﬁeld on the auxiliary grid. In the second (non-linear dynamic) step, scales smaller than the ﬁlter length associated to the primary grid are generated, resulting in an approximation of the full solution. Let u be the ﬁltered ﬁeld obtained on the primary computational grid, and u• the deﬁltered ﬁeld on the secondary grid. That secondary grid is chosen such that the associated mesh size is twice as ﬁne as the mesh size of the primary grid. We introduce the discrete ﬁltering operator Gd , deﬁned such that Gd u• = u . 12

(7.220)

A corresponding procedure in the spectral space is described in reference [190].

262

7. Structural Modeling

It is important to note that in this two-grid implementation, the righthand side of equation (7.220) must ﬁrst be interpolated on the auxiliary grid to recover a well-posed linear algebra problem. To avoid this interpolation step, Domaradzki proposes to solve directly the ﬁltered Navier–Stokes equations on the ﬁnest grid, and to deﬁne formally the G ﬁltering level by taking * The deﬁltered ﬁeld u• is obtained by solving the inverse problem ∆ = 2∆. u• = (Gd )−1 u .

(7.221)

This is done in practice by solving the corresponding linear system. In practice, the authors use an three-point discrete approximation of the box ﬁlter for Gd (see Sect. 13.2 for a description of discrete test ﬁlters). This step corresponds to an implicit deconvolution procedure (the previous ones were explicit procedures, based on the construction of the inverse operator via Taylor expansions or iterative procedures), and can be interpreted as an interpolation step of the ﬁltered ﬁeld on the auxiliary grid. The ua subgrid velocity ﬁeld is then evaluated using an approximation of its associated non-linear production term: ua = θa N

,

(7.222)

where θa and N are a characteristic time scale and N the production rate. These terms are evaluated as follows. The full convection term on the auxiliary grid is ∂u• −u•j i , j = 1, 2, 3 . (7.223) ∂xj This term accounts for the production of all the frequencies resolved on the auxiliary grid. Since we are interested in the production of the small scales only, we must remove the advection by the large scales, and restrict the resulting term tho the desired frequency range. The resulting term Ni is ∂u• Ni = (Id − G) −(u•j − uj ) i ∂xj

.

(7.224)

In practice, the convolution ﬁlter G is replaced by the discrete operator Gd . The production time θa is evaluated making the assumption that the subgrid kinetic energy is equal to the kinetic energy contained in the smallest resolved scales: |ua |2 = θa2 |N |2 = α2 |u• − u|2 =⇒ θa = α

2|u• − u| |N |

,

(7.225)

where α is a proportionality constant, nearly equal to 0.5 for the box ﬁlter. This completes the description of the model.

7.7 Explicit Evaluation of Subgrid Scales

263

7.7.7 Multi-level Simulations This class of simulation relies on the resolution of an evolution equation for ua on the auxiliary grid. These simulations can be analyzed within the framework of the multiresolution representation of the data [293, 295, 17, 294], or similar theories such as the Additive Turbulent Decomposition [471, 338, 80]. Let us consider N ﬁlters G1 , ..., GN , with associated cutoﬀ lengths ∆1 ≤ ... ≤ ∆N . We deﬁne the two following sets of velocity ﬁelds: un = Gn ... G1 u = G1n u ,

(7.226)

v n = un − un+1 = (G1n − G1n+1 ) u = Fn u .

(7.227)

n

n

The ﬁelds u and v are, respectively, the resolved ﬁeld at the nth level of ﬁltering and the nth level details. We have the decomposition un = un−k + v n−l , (7.228) l=1,k

yielding the following multiresolution representation of the data: u ≡ {uN , v 1 , ..., v N −1 }

.

(7.229)

The multilevel simulations are based on the use of embedded computational grids or a hierarchical polynomial basis to solve the evolution equations associated with each ﬁltering level/details level. The evolution equations are expressed as ∂un + N S(un ) = −τ n = −[G1n , N S](u), ∂t

n ∈ [1, N ] ,

(7.230)

where N S is the symbolic Navier–Stokes operator and [., .] the commutator operator. The equations for the details are ∂vn + N S(v n ) = −τ n = −[Fn , N S](u), ∂t

n ∈ [1, N − 1] ,

(7.231)

or, equivalently, ∂v n + N S(un ) − N S(un+1 ) = −τ n + τ n+1 , ∂t

n ∈ [1, N − 1] .

(7.232)

There are three possibilities for reducing the complexity of the simulation with respect to Direct Numerical Simulation: – The use of a cycling strategy between the diﬀerent grid levels. Freezing the high-frequency details over some time while integrating the equations for the low-frequency part of the solution results in a reduction of the simulation complexity. This is referred to as the quasistatic approximation for the

264

7. Structural Modeling

high frequencies. The main problem associated with the cycling strategy is the determination of the time over which the high frequencies can be frozen without destroying the quality of the solution. Some examples of such a cycling strategy can be found in the Multimesh method of Voke [736], the Non-Linear Galerkin Method [174, 579, 705, 222, 223, 224, 91], the Incremental Unknowns technique [73, 201, 128, 200], Tziperman’s MTS algorithm [722], Liu’s multigrid method [452, 453] and the Multilevel algorithm proposed by Terracol et al. [711, 710, 633]. – The use of simpliﬁed evolution equations for the details instead of (7.231). A linear model equation is often used, which can be solved more easily than the full nonlinear mathematical model. Some examples among others are the Non-Linear Galerkin method, early versions of the Variational Multiscale approach proposed by Hughes et al. [332, 331, 336, 335, 333, 334], and the dynamic model of Dubrulle et al. [203, 426]. Another possibility is to assume that the nth-level details are periodic within the ﬁltering cell associated with the (n − 1)th ﬁltering level. Each cell can then be treated separately from the others. An example is the Local Galerkin method of McDonough [468, 466, 467]. It is interesting to note that this last assumption is shared by the hom*ogenization approach developed by Perrier and Pironneau (see Sect. 7.2.3). Menon and Kemenov [382] use a simpliﬁed set of one-dimensional equations along lines. A reminiscent approach was developed by Kerstein on the grounds of the stochastic One-Dimensional Turbulence (ODT) model (presented in Sect. 7.7.3). – The use of a limited number of ﬁltering levels. In this case, even at the ﬁnest description level, subgrid scales exist and have to be parametrized. The gain is eﬀective because it is assumed that simple subgrid models can be used at the ﬁnest ﬁltering level, the associated subgrid motion being closer to isotropy and containing much less energy than at the coarser ﬁltering levels. Examples, among others, are the Multilevel algorithm of Terracol [710, 633], the Modiﬁed Estimation Procedure of Domaradzki [193, 183], and the Resolvable Subﬁlter Scales (RSFS) model [806]. Some strategies combining these three possibilities can of course be deﬁned. The eﬃciency of the method can be further improved by using a local grid reﬁnement [699, 62, 378]. Non-overlapping multidomain techniques can also be used to get a local enrichment of the solution [611, 633]. These methods are presented in Chap. 11. We now present a few multilevel models for large-eddy simulation. The emphasis is put here on methods based on two grid levels, and which can be interpreted as models, in the sense that they rely on some simpliﬁcations and cannot be considered just as multilevel algorithms applied to classical largeeddy simulations. In these methods, the secondary grid level is introduced to compute the ﬂuctuations u , and not for the purpose of reducing the cost of the primary grid computation. This latter class of methods, which escapes the simple closure problem, is presented in Chap. 11. It is worth noting that

7.7 Explicit Evaluation of Subgrid Scales

265

Fig. 7.3. Schematic of the three-level models. Left: spectral decomposition; Middle: computational grid; Right: time cycling.

these two-grid methods correspond to a three-level decomposition of the exact solution: two ﬁlters are applied in order to split the exact solution into three spectral bands (see Fig. 7.3). Using Harten’s representation (7.229), this decomposition is expressed as u = {u2 , v 1 , v 0 }

,

(7.233)

where u2 is the resolved ﬁltered ﬁeld at the coarsest level, u1 = u2 + v 1 is the resolved ﬁltered ﬁeld at the ﬁnest level, and v 0 is the unresolved ﬁeld at the ﬁnest level, i.e. the true subgrid velocity ﬁeld. The detail v 1 corresponds to the part of the unresolved ﬁeld at the coarsest level which is resolved at the ﬁnest ﬁltering level. The coupling between these three spectral bands and the associated closure problem can be understood by looking at the nonlinear term. For the sake of simplicity, but without restricting the generality of the results, we will assume here that the ﬁltering operators perfectly commute with diﬀerential operators, and that the domain is unbounded. The remaining coupling comes from the nonlinear convective term. For the exact solution, it is expressed as B(u, u) = B(u2 + v 1 + v 0 , u2 + v 1 + v 0 ) ,

(7.234)

where B is the bilinear form deﬁned by relation (3.27). At the coarsest resolution level, the nonlinear term can be split as follows: 2

B(u, u)

2

= B(u2 , u2 ) I

266

7. Structural Modeling 2

2

2

+ B(u2 , v 1 ) + B(v1 , u2 ) + B(v 1 , v 1 )

II 2

2

+ B(u2 + v 1 , v 0 ) + B(v0 , u2 + v 1 + v 0 )

.

(7.235)

III

Term I can be computed directly at the coarsest grid level. Term II represents the direct coupling between the two levels of resolution, and can be computed exactly during the simulation. Term III represents the direct coupling with the true subgrid modes. It is worth noting that, at least theoretically, the non-local interaction between u2 and v 0 is not zero and requires the use of an ad hoc subgrid model. At the ﬁnest resolution level, the analogous decomposition yields 1

B(u, u)

=

1

B(u2 , u2 ) IV 1

1

+ B(u2 , v 1 ) + B(v1 , u2 )

V 1

+ B(v 1 , v 1 )

(7.236)

VI 1

1

1

+ B(u2 + v 1 , v 0 ) + B(v 0 , u2 + v 1 ) + B(v0 , v 0 ) .

V II

Terms IV and V represent the coupling with the coarsest resolution level, and can be computed explicitly. Term V I is the nonlinear self-interaction of the detail v 1 , while term V II is associated with the interaction with subgrid scales v 0 and must be modeled. By looking at relations (7.235) and (7.236), we can see that speciﬁc subgrid models are required at each level of resolution. Several questions arise dealing with this closure problem: 1. Is a subgrid model necessary in practice for terms III and V II? 2. What kind of model should be used? 3. Is it possible to use the same model for both terms? Many researchers have worked on these problems, leading to the deﬁnition of diﬀerent three-level strategies. A few of them are presented below: 1. The Variational Multiscale Method (VMS) proposed by Hughes et al. (p. 267). 2. The Resolvable Subﬁlter-scale Model (RSFR) of Zhou et al. (p. 269). 3. The Dynamic Subﬁlter-scale Model (DSF) developed by Dubrulle et al. (p. 270).

7.7 Explicit Evaluation of Subgrid Scales

267

4. The Local Galerkin Approximation (LGA), as deﬁned by McDonough et al. (p. 270). 5. The Two-Level-Simulation (TLS) method, proposed by Menon et al. (p. 271). 6. The Modiﬁed Subgrid-scale Estimation Procedure (MSEP) of Domaradzki et al. (p. 269). 7. Terracol’s multilevel algorithm (TMA) with explicit modeling of term III (p. 271). The underlying coupling strategies are summarized in Table 7.3. Table 7.3. Characteristics of multilevel subgrid models. + means that the term is taken into account, and − that it is neglected. Model

I

II

III

IV

V

VI

VII

VMS RSFR MSEP DSF LGA TLS TMA

+ + + + + + +

+ + + + + + +

− − − − − − +

+ + + + − − +

+ + + + + + +

+ + + − + + +

+ + − − − − +

Variational Multiscale Method. Hughes et al. [332, 331, 336, 335, 333, 334] ﬁrst introduced the Variational Multiscale Method within the framework of ﬁnite element methods, and then generalized it considering a fully general framework. The coupling term III is neglected (see Table 7.3), at least in the original formulation of the method. The need for a full coupling including term III was advocated by Scott Collis [660]. 2 In practical applications, the coarse resolution cutoﬀ length scale ∆ is 1 taken equal to twice the ﬁne resolution cutoﬀ length scale ∆ , and the solution is integrated with the same time step at the two levels. The subgrid term V II is parametrized using a Smagorinsky-like functional model ∂vj1 ∂vi1 + , (7.237) (V II)ij = −2νsgs ∂xi ∂xj with two possible variants: – The small–small model13 : 13

This form of the dissipation is very close to the variational embedded stabilization previously proposed by Hughes (see Sect. 5.3.4). Similar expressions for the dissipation term have been proposed by Layton [429, 428] and Guermond [280].

268

7. Structural Modeling

νsgs

= (CS ∆1 ) 2|S 1 |, 2

1 Sij

=

∂vj1 ∂v 1 + i ∂xi ∂xj

.

(7.238)

.

(7.239)

– The large–small model: νsgs

+ 1 = (CS ∆1 ) 2|S |, 2

1 S ij

=

∂u1j ∂u1 + i ∂xi ∂xj

The recommended value of the constant CS is 0.1 for isotropic turbulence and plane channel ﬂow. This value is arbitrary, and numerical experiments show that it could be optimized. Numerical results show that both variants lead to satisfactory results on academic test cases, including non-equilibrium ﬂow. This can be explained by 2 the fact that the subgrid tensor is evaluated using the S 1 tensor, instead of S in classical subgrid-viscosity methods. Thus, the subgrid model dependency is more local in Fourier space, the emphasis being put on the highest resolved frequency, yielding more accurate results (see Sect. 5.3.3). This increased localness in terms of wavenumber with respect to the usual Smagorinsky model is emphasized rewriting the VMS method as a special class of hyperviscosity models. The link between these two approaches is enlightened assuming that the following diﬀerential approximation for secondary ﬁlter utilized to operate the splitting u1 = u2 + v 1 holds14 : 2

u2 = G2 u1 (Id + α∆2 ∇2 )u1

(7.240)

where α is a ﬁlter-dependent parameter, yielding 2

v 1 = −α∆2 ∇2 u1

(7.241)

Inserting that deﬁnition for v 1 into previous expressions for both the Small-Small (7.238) and the Large-Small (7.239) models shows that these models are equivalent to fourth-order hyperviscosity models (see p. 121). Higher-order hyperviscosities are recovered using higher-order elliptic ﬁlters to operate the splitting. The splitting of the resolved ﬁeld into two parts can be interpreted as the deﬁnition of a more complex accentuation technique (see Sect. 5.3.3, p. 156). In the parlance of Hughes, the ﬁltered Smagorinsky model corresponds the the Small-Large model (while the classical Smagorinsky model is the Large-Large model). The accuracy of the results is observed to depend on the spectral properties of the ﬁlter used to extract v 1 from u1 . It is observed that kinetic energy pile-up can occur if a spectral sharp cutoﬀ is utilized. The reason why is that in this case the subgrid viscosity acts only on scales encompassed within v 1 14

It is known from results of Sect. 2.1.6 that this approximation is valid for smooth symmetric ﬁlters and for most discrete ﬁlters.

7.7 Explicit Evaluation of Subgrid Scales

269

and non-local energy transfers between largest scales and v 0 are neglected. The use of a smooth ﬁlter which allows a frequency overlap between u1 and v 1 alleviates this problem. The use of the classical Smagorinsky model may also lead to an excessive damping of scales contained in v 1 . To cure this problem, Holmen et al. [316] proposed to use the Germano-Lilly procedure to evaluate the constant of the Large-Small model. Another formulation for the Variational Multiscale Method is proposed by Vreman [745] who build some analogous models on the grounds of the test ﬁlter G2 . Keeping in mind that one essential feature of the VMS approach is that the action of the subgrid scale is restricted to the small scale ﬁeld v 1 , the following possibilities arise – Model 1: restriction of the usual Smagorinsky model + 1 1 1 (V II)ij = (Id − G2 ) −2(CS ∆ )2 2|S |S ij

.

(7.242)

– Model 2 : Smagorinsky model based on the small scales 1

(V II)ij = −2(CS ∆ )2

1 2|S 1 |Sij

.

(7.243)

– Model 3 : restriction of the Smagorinsky model based on the small scales 1 1 (V II)ij = (Id − G2 ) −2(CS ∆ )2 2|S 1 |Sij

.

(7.244)

Models 2 and 3 are equivalent if the ﬁlter G2 is a sharp cutoﬀ ﬁlter, but are diﬀerent in the general case. If a diﬀerential second-order elliptic ﬁlter is used, model 3 will be equivalent to a sixth-order hyperviscosity, while model 2 is associated to a fourth-order hyperviscosity. Resolvable Subﬁlter-scale Model. Zhou, Brasseur and Juneja [806] developed independently a three-level model which is almost theoretically equivalent to the Variational Multiscale Method of Hughes. The terms taken into account at the two resolution levels are the same (term III is ignored in both cases), and term V II is modeled using the Smagorinsky model. The model used is the small–small model (7.238) following Hughes’ terminology. As in the VMS implementation described above, simulations of hom*ogeneous 2 1 anisotropic turbulence are carried out with ∆ = 2∆ . Modiﬁed Subgrid-scale Estimation Procedure. Domaradzki et al. [782, 193, 183] proposed a modiﬁcation of the original subgrid-scale estimation procedure (see Sect. 7.7.6) in order to improve its robustness. The key point

270

7. Structural Modeling

of this method is to account directly for the production of small scales v 1 by the forward energy cascade rather than using the production estimate (7.222). The governing equations of this method are the same as those of VMS and RSFR, with the exception that the subgrid term V II in the detail equation is neglected. The main diﬀerence with VMS and RSFR is the time integration procedure: in the two previous approaches the coarsely and ﬁnely resolved ﬁelds are advanced at each time step, while Domaradzki and Yee proposed advancing the ﬁne grid solution u1 over an evolution time T between 1% and 3% of the large-eddy turnover time. Dynamic Subﬁlter-scale Model. The dynamic subﬁlter-scale model of Dubrulle et al. [203, 426] appears as a linearized version of the three-level approach: the nonlinear term V I in the evolution equation of the details is neglected, and the subgrid term V II is not taken into account. This linearization process renders the detail equation similar to those of the Rapid Distortion Theory. Only a priori tests have been carried out on parallel wallbounded ﬂows. Local Galerkin Approximation. The Local Galerkin Approach proposed by McDonough et al. [468, 466, 467, 470, 471] can be seen as a simpliﬁcation of a typical three-level model. All subgrid terms are neglected, and, due to the fact that the original presentation of the method is not based on a ﬁltering operator, term IV is not taken into account. The key idea of the method is to make the assumption that the ﬂuctuating ﬁeld v 1 is periodic in space within each cell associated with the coarse level resolution (see Fig. 7.4). Consequently, a spectral simulation is performed 2 within each cell of size ∆ , but the ﬁeld v 1 is not continuous at the interface of 1 each cell. The number of Fourier modes determine the cutoﬀ length scale ∆ .

Fig. 7.4. Schematic of the Local Galerkin Approach.

7.7 Explicit Evaluation of Subgrid Scales

271

This method can be seen as a dynamic extension of the Kinematic Simulation approach (see Sect. 7.7.4) and is very close, from a practical point of view, to the Perrier–Pironneau hom*ogenization technique (see Sect. 7.2.3). Menon’s Two-Level-Simulation Method. Menon and Kemenov [382] developed a two-level method based on a simpliﬁed model for the subgrid scales. Instead of deﬁning a three-dimensional grid to compute the subgrid modes inside each cell of the large-eddy simulation grid, the authors chose to solve simpliﬁed one-dimensional equations along lines (one in each direction in practice) inside each cell, leading to a large cost reduction. This feature makes it reminiscent of the Kerstein subgrid closure based on the stochastic ODT model (see Sect. 7.7.3). In Menon’s approach, terms III and V II are neglected. In each cell, the three-dimensional subgrid velocity ﬁeld is modeled as a family of one-dimensional velocity vector ﬁelds deﬁned on the underlying family of lines {l1 , l2 , l3 } (which are in practice aligned with the axes of the reference Cartesian frame). Assuming that the derivatives of the modeled subgrid velocity ﬁeld are such that ∂v 1 ∂v 1 ∂vi1 ∼ i ∼ i, ∂l1 ∂l2 ∂l3

i = 1, 2, 3 ,

(7.245)

and that the incompressibility constraint can be expressed as ∂ 1 1 1 + v3,j v + v2,j =0 ∂lj 1,j

,

(7.246)

1 refers to the jth component of the subgrid ﬁeld computed along where vk,j the line of index k, the following mometum-like equations are found: 1 1 ∂vi,j ∂ 2 vi,j ∂p1 1 + N L(vi,j , u, lj ) = − + 3ν 2 ∂t ∂lj ∂lj

,

(7.247)

1 , u, lj ) contains the surrogates for terms where the non-linear term N L(vi,j IV , V and V I.

Terracol’s Multilevel Algorithm. The last three-level model presented in this section is the multilevel closure proposed by Terracol et al. [710, 633, 711]. It is the only one which considers the full closure problem by taking into account the non-local interaction term III, and can then be considered as the most general one. The original method presented in [711] is able to handle an arbitrary number of ﬁltering levels, but the present presentation will be restricted to the three-level case.

272

7. Structural Modeling

The key points of the method are: – The use of a speciﬁc closure at each level u2 and u1 . The proposed model at the coarse level is an extension of the one-parameter dynamic mixed model (see Sect. 7.4.2, p. 240), where the scale-similarity part is replaced by the explicitly computable term II. Term III is modeled using the Smagorinsky part of the mixed dynamic model, with a dynamically computed constant. At the ﬁne resolution level, term V II is parametrized using a oneparameter dynamic mixed model. Numerical results demonstrate that the use of a speciﬁc model for term III is mandatory, since the use of a classical dynamic Smagorinsky model yields poor results. – The deﬁnition of a cycling strategy between the diﬀerent resolution levels, in order to decrease the computational cost while maintaining the accuracy of the results. The idea is here to freeze the details v 1 and to carry out the computation at the coarse level only during a time T .15 The problem is to ﬁnd the optimal T in order to maximize the cost reduction while limiting the loss of coherence between u2 and v 1 . A simple solution is to advance the solution for one time step at each level, alternatively, with the same value of the Courant number at each level. Numerical experiments show that this solution leads to good results with a gain of about a factor two. Results obtained using this method on a plane mixing layer conﬁguration are illustrated in Fig. 7.5.

7.8 Direct Identiﬁcation of Subgrid Terms Introduction. This section is dedicated to the presentation of approaches which aim at reconstructing the subgrid terms using direct identiﬁcation mathematical tools. Like all other subgrid models, either of functional or structural types, they answer the following question: given a ﬁltered velocity ﬁeld u, what is the subgrid acceleration? Subgrid models described previously were all based on some a priori knowledge of the nature of the interactions between resolved and subgrid scales, on a description of the ﬁlter, or on a structure of the subgrid scales hypothesized a priori. The models presented in this section do not require any of this information, and do not rely on any assumptions about the internal structure of the subgrid modes. They are based on mathematical tools which are commonly used within the framework of pattern recognition, and do not really correspond to what is usually called a “model”. Using Moser’s words, they represent a radical approach to large-eddy simulation. 15

This part is close to the quasistatic approximation for small scales introduced within the context of the nonlinear Galerkin method [174].

7.8 Direct Identiﬁcation of Subgrid Terms

273

Fig. 7.5. Terracol’s three-level method. Plane mixing layer. Streamwise energy spectrum during the self-similar phase. Crosses: large-eddy simulation on the coarse grid only. Other symbols and lines: direct numerical simulation and large-eddy simulation using the multilevel closure. The vertical dashed lines denote the cutoﬀ wave numbers of the two grids. Courtesy of M. Terracol, ONERA.

Let us consider a scalar-, vector- or tensor-valued variable φ, which is to be estimated, and a set of solutions (u, φ)n , 1 ≤ n ≤ N . What we are looking for is an estimation of the value of φ for any new arbitrary velocity ﬁeld. This problem is equivalent to estimating the following functional φ −→ Mφ (u, K(u))

,

(7.248)

where K(u) can be any arbitrary function based on the solution (gradient, correlations, ...). The two classes of approaches presented below are: 1. The approach developed by Moser et al., which relies on linear stochastic estimation (Sect. 7.8.1). 2. The proposal of Sarghini et al. dealing with the use of neural networks (Sect. 7.8.2). Both approaches share the same very diﬃcult practical problem: they require the existence of a set of realizations to achieve the identiﬁcation process (computation of correlation tensors in the ﬁrst case, and training phase of the neural network in the second case). In other approaches, this systematic identiﬁcation process is replaced by the subgrid modeling phase, in which the modeler plays the role of the identiﬁcation algorithm. As a consequence, the potential success of these identiﬁcation methods depends on trade-oﬀ between the increase of computing power and the capability of researchers to improve typical subgrid models.

274

7. Structural Modeling

7.8.1 Linear-Stochastic-Estimation-Based Model Moser et al. [419, 549, 548, 165, 296, 741, 797] proposed an identiﬁcation procedure based on Linear Stochastic Estimation. This approach can be interpreted in several ways. An important point is that it is closely tied to the deﬁnition of large-eddy simulation as an optimal control problem, where the subgrid model plays the role of a controller. This interpretation is discussed in Sect. 9.1.4 and will not be repeated here. The linear stochastic estimation approach can also be seen as the best linear approximation for the subgrid term in the least-squares sense. Starting from the general formulation (7.248), the linear estimation is written as Mφ (E) = φ + L · E T , with

E = (u, K(u))

,

(7.249)

where the tensorial dimension of L depends on those of φ, u and K(u). The linear stochastic estimation procedure leads to the best values of the coefﬁcient of L in the least-squares sense. Considering vectorial unknowns, we obtain the following spatially non-local estimation φi (x) = φi + Lij (x, x )Ej (x )dx ; (7.250) the best coeﬃcients Lij are computed by solving the following linear problem: (7.251) Ei (x )φj (x) = Ljk (x, x )Ej (x)Ek (x )dx . Local estimates can also be deﬁned using one-point correlations instead of two-point correlations to deﬁne L. Practial applications of this approach have been carried out in hom*ogeneous turbulence and plane channel ﬂow [549, 548]. Numerical experiments have shown that both the subgrid stresses and the subgrid energy transfer must be taken into account to obtain stable and accurate numerical simulations. This means that both ui τij and S ij τij must be recovered. This is achieved by estimating the subgrid acceleration φi =

∂ τij ∂xj

,

(7.252)

as a function of the velocity ﬁeld and its gradients E = (u, ∇u) .

(7.253)

The mean value φ is computed from the original data set and stored. In practice, some simpliﬁcations can be assumed in deﬁnitions (7.252) and (7.253) in parallel shear ﬂows.

7.9 Implicit Structural Models

275

7.8.2 Neural-Network-Based Model Sarghini et al. [645] proposed estimating the subgrid terms using a multilayer, feed-forward neural network. Rather than estimating directly the subgrid acceleration in the momentum equation, the authors decided to decrease the complexity of the problem by identifying the value of a subgrid-viscosity coeﬃcient, yielding a new dynamic Smagorinsky model. To this end, a threelayer network is employed. The number of neurons in each layer is 15, 12 and 6, respectively, with a single output (the subgrid viscosity). The input vector is (7.254) E = (∇u, u ⊗ u ) ∈ IR15 . The output is the subgrid model constant: φ = CS ∈ IR. The training of the neural network is achieved using data ﬁelds originating from a classical large-eddy simulation of the same plane channel ﬂow conﬁguration. The learning rule used to adjust the weights and the biases of the network is chosen so as to minimize the summed-squared-error between the output of the network and the original set of data. Six thousand samples were found necessary for the training and validation steps. The training was performed through a backpropagation with weight decay technique in less than 500 iterations. A priori and a posteriori tests show that the resulting model leads to stable numerical simulations, whose results are very close to those obtained using typical subgrid viscosity models. An interesting feature of the model is that the predicted subgrid viscosity exhibits the correct asymptotic behavior in the near-wall region (see p. 159).

7.9 Implicit Structural Models The last class of structural subgrid models discussed in this chapter is the implicit structural model family. These models are structural ones, i.e. they do not rely on any foreknowledge about the nature of the interactions between the resolved scales and the subgrid scales. They can be classiﬁed as implicit, because they can be interpreted as improvements of basic numerical methods for solving the ﬁltered Navier–Stokes equations, leading to the deﬁnition of higher-order accurate numerical ﬂuxes. We note that, because the modiﬁcation of the numerical method can be isolated as a new source term in the momentum equation, these models could also be classiﬁed as exotic formal expansion models. A major speciﬁcity of these models is that they all aim at reproducing directly the subgrid force appearing in the momentum equation, and not the subgrid tensor τ . They diﬀer from the stabilized numerical methods presented in Sect. 5.3.4 within the MILES framework because they are not designed to induce numerical dissipation.

276

7. Structural Modeling

The two models presented in the following are: 1. The Local Average Method of Denaro (p. 276), which consists in a particular reconstruction of the discretized non-linear ﬂuxes associated to the convection term. This approach incorporates a strategy to ﬁlter the subgrid-scale by means of an integration over a control volume and to recover the contribution of the subgrid scales with an integral formulation. It can be interpreted as a high-order space-time reconstruction procedure for the convective numerical ﬂuxes based on a deﬁltering process. 2. The Scale Residual Model of Maurer and Fey (p. 278). As for the Approximate Deconvolution Procedure, the purpose is to evaluate the commutation error which deﬁnes the subgrid term. This evaluation is carried out using the residual between the time evolution of the solutions of the Navier–Stokes equations on two diﬀerent grids (i.e. at two diﬀerent ﬁltering levels) and assuming some self-similarity properties of this residual. This model can be considered as: (i) a generalization of the previous one, which does not involve the deconvolution process anymore, but requires the use of the second computational grid and (ii) a generalization of the scale-similarity models, the use of a test ﬁlter for deﬁning the test ﬁeld being replaced by the explicit computation (by solving the Navier–Stokes equations) of the ﬁeld at the test ﬁlter level. Other implicit approaches for large-eddy simulation exist, which make it possible to obtain reliable results without subgrid scale model (in the common sense given to that term), and without explicit addition of numerical diﬀusion16 . An example is the Spectro-Consistent Discretization proposed by Verstappen and Veldman [730, 729]. Because these approaches rely on numerical considerations only, they escape the modeling concept and will not be presented here. 7.9.1 Local Average Method An other approach to the traditional large-eddy simulation technique was proposed by Denaro and his co-workers in a serie of papers [175, 169, 170]. It is based on a space-time high-order accurate reconstruction/deconvolution of the convective ﬂuxes, which account for the subgrid-scale contribution. As a consequence, it can be seen as a particular numerical scheme based on a diﬀerential approximation of the ﬁltering process. For sake of simplicity, we will present the method in the case of a dummy variable φ advected by a velocity ﬁeld u, whose evolution equation is (only convective terms are retained): ∂φ = −∇ · (uφ) = A(u, φ) . (7.255) ∂t 16

Dissipative numerical methods should be classiﬁed as Implicit Functional Modeling.

7.9 Implicit Structural Models

277

The local average of φ in a ﬁltering cell Ω is deﬁned as the mean value of φ in this cell17 : 1 φ(x, t) ≡ φ(ξ, t)dξ = φ(t), ∀x ∈ Ω , (7.256) V Ω where V is the measure of Ω. We now consider an arbitrary ﬁltering cell. Applying this operator to equation (7.255), and integrating the resulting evolution equation over the time interval [t, t + ∆t], we get: t+∆t n · uφ(ξ, t )dξdt , (7.257) (φ(t + ∆t) − φ(t))V = t

∂Ω

where ∂Ω is the boundary of Ω, and n the vector normal to it. The righthand side of this equation, which appears as the application of a time-box ﬁlter to the boundary ﬂuxes, can be approximated by means of a diﬀerential operator, exactly in the same way as for the space-box ﬁlter (see Sect. 7.2.1), yielding: ⎞ ⎛ t+∆t ∆tl−1 ∂ l ⎠ φ(ξ, t)dξ. n · uφ(ξ, t )dξdt ∆t n · u ⎝Id + l! ∂tl t ∂Ω ∂Ω l=1,∞

(7.258) The time expansion is then writen as a space diﬀerential operator using the balance equation (7.255): ⎛ ⎞ ⎞ ⎛ ∆tl−1 ∆tl−1 ∂ l ⎠ φ(ξ, t) = ⎝Id + ⎝Id + Al−1 (u, ·)⎠ φ(ξ, t) , l! ∂tl l! l=1,∞

l=1,∞

(7.259) with Al (u, φ) ≡ A(u, ·) ◦ A(u, ·) ◦ ... ◦ A(u, φ)

.

l times

The second step of this method consists in the reconstruction step. At each point x located inside the ﬁltering cell Ω, we have φ(x, t) = φ(x, t + ∆t) =

17

φ(t) + φ (x, t) , φ(t + ∆t) + φ (x, t + ∆t)

=

φ(t) + (φ(t + ∆t) − φ(t)) + φ (x, t + ∆t) +(φ (x, t + ∆t) − φ (x, t))

=

φ(x, t) + (φ(t + ∆t) − φ(t)) +(φ (x, t + ∆t) − φ (x, t)) .

(7.260)

(7.261)

This ﬁltering operator corresponds to a modiﬁcation of the box ﬁlter deﬁned in Sect. 2.1.5: the original box ﬁlter is deﬁned as a IR → IR operator, while the local average is a IR → IN operator. It is worth noting that the local average operator is a projector.

278

7. Structural Modeling

The ﬁrst term in the left-hand side of relation (7.261) is known. The second one, which corresponds to the contribution of the low frequency part of the solution (i.e. the local averaged part), is computed using equation (7.257). The third term remains to be evaluated. This is done using the diﬀerential operator (2.52), leading to the ﬁnal expression: φ(x, t + ∆t) = φ(x, t) + (Id − Pd ) (φ(t + ∆t) − φ(t)) with

⎞l ⎞ 1⎜1 ∂ ⎠⎟ ⎝ Pd = (ξi − xci ) ⎠ dξ ⎝ l! V Ω ∂xi ⎛

,

(7.262)

⎛

l=1,∞

,

(7.263)

i=1,d

where d is the dimension of space and xci the ith coordinate of the center of the ﬁltering cell. In practice, the serie expansions are truncated to a ﬁnite order. The repeated use of equation (7.262) makes it possible to compute the value of the new pointwise value at each time step. 7.9.2 Scale Residual Model Maurer and Fey [500] propose to evaluate the full subgrid term, still deﬁned as the commutation error between the Navier–Stokes operator and the ﬁlter (see Chap. 3 or equation (7.2)), by means of a two-grid level procedure. A deconvolution procedure is no longer needed, but some self-invariance properties of the subgrid term have to be assumed. First we note that a subgrid model, referred to as m(u), is deﬁned in order to minimize the residual E, with E = [N S, G ](u) − m(G u) .

(7.264)

Assuming that the ﬁlter G has the two following properties: – G is a projector, – G commutes with the Navier–Stokes operator in the sense that N S ◦ (G )u = N S ◦ (G ) ◦ (G )u = (G ) ◦ N S ◦ (G )u

,

the residual can be rewritten as E = (G ) ◦ (N S ◦ Id − N S ◦ (G ))u − m ◦ (G )u

.

(7.265)

We now introduce a set of ﬁlter Gk , k = 0, N , whose characteristic lengths ∆k are such that 0 = ∆N < ∆N −1 < .... < ∆0 . The residual Ek obtained for the kth level of ﬁltering is easily deduced from relation (7.265): Ek

= (Gk ) ◦ (N S ◦ (GN ) − N S ◦ (Gk ))u −m ◦ (Gk )u

(7.266)

7.9 Implicit Structural Models

= (Gk ) ◦

279

(N S ◦ (Gj ) − N S ◦ (Gj+1 ))u

j=0,k−1

−m ◦ (Gk )u

.

(7.267)

To construct the model m, we now make the two following assumptions: – The interactions between spectral bands are local, in that sense that the inﬂuence of each spectral band gets smaller with decreasing values of j < k. – The residuals between two ﬁltering levels have the following self-invariance property: (N S ◦(Gj+1 )−N S ◦(Gj ))u = α(N S ◦(Gj )−N S ◦(Gj−1 ))u, (7.268) where α < 1 is a constant parameter. It is important noting that this can only be true if the cutoﬀs occur in the inertial range of the spectrum (see the discussion about the validity of the dynamic procedure in Sect. 5.3.3). Using these hypotheses, the following model is derived: m ◦ (Gk )u = Gk (

αj )(N S ◦ (Gk ) − N S ◦ (Gk+1 ))u , (7.269)

j=1,k

where the operator (Gk ) ◦ N S ◦ (Gk+1 ) corresponds to a local reconstruction of the evolution of the coarse solution Gk+1 u according to the ﬂuctuations of the ﬁne solution Gk u. The implementation of the model is carried out as follows: a short history of both the coarse and the ﬁne solutions are computed on two diﬀerent computational grids, and the model (7.269) is computed and added as a source term into the momentum equations solved on the ﬁne grid. This algorithm can be written in the following symbolic form: = N S 2k,∆t + ω(N S 2k,∆t − N S 1k+1,2∆t ) un+1 un+1 k k

,

(7.270)

where un+1 designates the solution on the ﬁne grid (kth ﬁltering level) at k the (n + 1)th time step, (N S nk,∆t refers to n applications of the discretized Navier–Stokes operator on the grid associated to the ﬁltering level k with a time step ∆t (i.e. the computation of n time steps on that grid without any subgrid model), and ω is a parameter deduced from relation (7.269). The weight α is evaluated analytically through some inertial range consideration, and is assumed to be equal to the ratio of the kinetic energy contained in the two spectral bands (see equation (5.132)). An additional correction factor (lower than 1) can also be introduced to account for the numerical errors.

8. Numerical Solution: Interpretation and Problems

This chapter is devoted to analyzing certain practical aspects of large-eddy simulation. The ﬁrst point concerns the diﬀerences between the ﬁltering such as it is deﬁned by a convolution product and such as it is imposed on the solution during the computation by the subgrid model. We distinguish here between static and dynamic interpretations of the ﬁltering process. The analysis is developed only for subgrid viscosity models because their mathematical form makes this possible. However, the general ideas resulting from this analysis can in theory be extended to other types of models. The second point has to do with the link between the ﬁlter cutoﬀ length and the mesh cell size used in the numerical solution. It is important to note that all of the previous developments proceed in a continuous, non-discrete framework and make no mention of the spatial discretization used for solving the equations of the problem numerically. The third point addressed is the comparative analysis of the numerical error and the subgrid terms. We propose here to compare the amplitude of the subgrid terms and numerical discretization errors to try to establish criteria for the required numerical scheme accuracy so that the errors committed will not overly mar the computed solution.

8.1 Dynamic Interpretation of the Large-Eddy Simulation 8.1.1 Static and Dynamic Interpretations: Eﬀective Filter The approach that has been followed so far in explaining large-eddy simulation consists in ﬁltering the momentum equations explicitly, decomposing the non-linear terms that appear, and then modeling the unknown terms. If the subgrid model is well designed (in a sense deﬁned in the following chapter), then the energy spectrum of the computed solution, for an exact solution verifying the Kolmogorov spectrum, is of the form 2 (k) E(k) = K0 ε2/3 k −5/3 G

,

(8.1)

282

8. Numerical Solution: Interpretation and Problems

where G(k) is the transfer function associated with the ﬁlter. This is the classical approach corresponding to a static and explicit view of the ﬁltering process. An alternate approach is proposed by Mason et al. [495, 496, 497], who ﬁrst point out that the subgrid viscosity models use an intrinsic length scale denoted ∆f , which can be interpreted as the mixing length associated with the subgrid scales. A subgrid viscosity model based on the large scales is written thus (see Sect. 5.3.2): νsgs = ∆2f |S| .

(8.2)

The ratio between this mixing length and the ﬁlter cutoﬀ length ∆ is: ∆f = Cs ∆

.

(8.3)

Referring to the results explained in the section on subgrid viscosity models, Cs can be recognized as the subgrid model constant. Varying this constant is therefore equivalent to modifying the ratio between the ﬁlter cutoﬀ length and the length scale included in the model. These two scales can consequently be considered as independent. Also, during the simulation, the subgrid scales are represented only by the subgrid models which, by their eﬀects, impose the ﬁlter on the computed solution1 . But since the subgrid models are not perfect, going from the exact solution to the computed one does not correspond to the application of the desired theoretical ﬁlter. This switch is ensured by applying an implicit ﬁlter, which is intrinsically contained in each subgrid model. Here we have a dynamic, implicit concept of the ﬁltering process that takes the modeling errors into account. The question then arises of the qualiﬁcation of the ﬁlters associated with the diﬀerent subgrid models, both for their form and for their cutoﬀ length. The discrete dynamical system represented by the numerical simulation is therefore subjected to two ﬁltering operations: – The ﬁrst is imposed by the choice of a level of representation of the physical system and is represented by application of a ﬁlter using the Navier–Stokes equations in the form of a convolution product. – The second is induced by the existence of an intrinsic cutoﬀ length in the subgrid model to be used. In order to represent the sum of these two ﬁltering processes, we deﬁne the eﬀective ﬁlter, which is the ﬁlter actually seen by the dynamical system. To qualify this ﬁlter, we therefore raise the problem of knowing what is the share of each of the two ﬁltering operations mentioned above. 1

They do so by a dissipation of the resolved kinetic energy −τij S ij equal to the ﬂux ε* through the cutoﬀ located at the desired wave number.

8.1 Dynamic Interpretation of the Large-Eddy Simulation

283

8.1.2 Theoretical Analysis of the Turbulence Generated by Large-Eddy Simulation We ﬁrst go into the analysis of the ﬁlter associated with a subgrid viscosity model. This section resumes Muschinsky’s [551] analysis of the properties of a hom*ogeneous turbulence simulated by a Smagorinsky model. The analysis proceeds by establishing an analogy between the large-eddy simulation equations incorporating a subgrid viscosity model and those that describe the motions of a non-Newtonian ﬂuid. The properties of the latter are studied in the framework of isotropic hom*ogeneous turbulence, so as to bring out the role of the diﬀerent subgrid model parameters. Analogy with Generalized Newtonian Fluids. Smagorinsky Fluid. The constitutive equations of large-eddy simulation for a Newtonian ﬂuid, at least in the case where a subgrid viscosity model is used, can be interpreted diﬀerently as being those that describe the dynamics of a non-Newtonian ﬂuid of the generalized Newtonian type, in the framework of direct numerical simulation, for which the constitutive equation is expressed σij = −pδij + νsgs Sij

,

(8.4)

where σij is the stress tensor, S the strain rate tensor deﬁned as above, and νsgs will be a function of the invariants of S. Eﬀects stemming from the molecular viscosity are ignored because this is a canonical analysis using the idea of an inertial range. It should be noted that the ﬁltering bar symbol no longer appears because we now interpret the simulation as a direct one of a ﬂuid having a non-linear constitutive equation. If the Smagorinsky model is used, i.e. 2 (8.5) νsgs = ∆2f |S| = Cs ∆ |S| , such a ﬂuid will be called a Smagorinsky ﬂuid. Laws of Similarity of the Smagorinsky Fluid. The ﬁrst step consists in extending the Kolmogorov similarity hypotheses (recalled in Appendix A): 1. First similarity hypothesis. E(k) depends only on ε, ∆f and ∆. 2. Second similarity hypothesis. E(k) depends only on ε and ∆ for wave numbers k much greater than 1/∆f . 3. Third similarity hypothesis. E(k) depends only on ε and ∆f if ∆ ∆f . The spectrum can then be put in the form: E(k) = ε2/3 k −5/3 Gs (Π1 , Π2 ) ,

(8.6)

where Gs is a dimensionless function whose two arguments are deﬁned as: Π1 = k∆f ,

Π2 =

∆ 1 = ∆f Cs

.

(8.7)

284

8. Numerical Solution: Interpretation and Problems

By analogy, the limit in the inertial range of Gs is a quantity equivalent to the Kolmogorov constant for large-eddy simulation, denoted Kles (Cs ): Kles (Cs ) = Gs (0, Π2 ) .

(8.8)

By introducing the shape function fles (k∆f , Cs ) =

Gs (Π1 , Π2 ) Gs (0, Π2 )

,

(8.9)

the spectrum is expressed: E(k) = Kles (Cs )ε2/3 k −5/3 fles (k∆f , Cs ) .

(8.10)

By analogy with Kolmogorov’s work, we deﬁne the dissipation scale of the non-Newtonian ﬂuid ηles as: 1/4 3 νsgs ηles = . (8.11) ε For the Smagorinsky model, by replacing ε with its value, we get: ηles = ∆f = Cs ∆ .

(8.12)

Using this deﬁnition and postulating that Kolmogorov’s similarity theory for the usual turbulence remains valid, the third similarity hypothesis stated implies, for large values of the constant Cs : 2/3 −5/3 E(k) = lim Kles (Cs ) ε k , (8.13) lim fles (kηles , Cs ) Cs →∞

Cs →∞

which allows us to presume that the two following relations are valid: lim Kles (Cs ) = K0

,

Cs →∞

lim fles (x, Cs ) = f (x)

Cs →∞

(8.14) ,

(8.15)

where f (x) is the damping function including the small scale viscous eﬀects, for which the Heisenberg–Chandresekhar, Kovazsnay, and Pao models have already been discussed in Sect. 5.3.2. The corresponding normalized spectrum of the dissipation2 is of the form: gles (x, Cs ) = x1/3 fles (x, Cs ) ,

(8.16)

where x is the reduced variable x = kηles . 2

The dissipation spectrum, denoted D(k), associated with the energy spectrum E(k) is deﬁned by the relation: D(k) = k2 E(k)

.

8.1 Dynamic Interpretation of the Large-Eddy Simulation

285

By comparing the dissipation computed by integrating this spectrum with the one evaluated from the energy spectrum (8.10), the dependency of the Kolmogorov constant as a function of the Smagorinsky constant is formulated as: 1 1 . (8.17) ≈ ! Cs π Kles (Cs ) = ! ∞ 2 0 gles (x, Cs ) 2 0 gles (x, Cs ) When this expression is computed using the formulas of Heisenberg– Chandrasekhar and Pao, it shows that the function Kles does tend asymptotically to the value K0 = 1.5 for large values values of Cs , as the error is negligible beyond Cs = 0.5. The variation of the parameter Kles as a function of Cs for the spectra of Heisenberg–Chandrasekhar and Pao is presented in Fig. 8.1. When Cs is less than 0.5, the Kolmogorov constant is over-evaluated, as has actually been observed in the course of numerical experiments [478, 479]. These numerical simulations, carried out by Magnient et al. [479], have shown that: – The damping function depends on Cs . A clear bifurcation is observed in the behavior of the models. The theoretical value of Cs , referred to as Cs0 , obtained by the canonical analysis corresponds to the case where the resolved kinetic energy transfer is equal to the energy transfer across the wave number π/∆. In this case, we obtain fles = 1 for all subgrid viscosity models. For larger values, the resulting damping function is not equal to a Heaviside function, and depends on the subgrid model. An interesting feature is that scales larger than ∆ are progressively damped. This damping

Fig. 8.1. Variation of the Kolmogorov constant as a function of the Smagorinsky constant for the Heisenberg–Chandrasekhar spectrum and the Pao spectrum.

286

8. Numerical Solution: Interpretation and Problems

originates from two diﬀerent phenomena: (i) the energy drain induced by the subgrid-viscosity model, and (ii) the forward energy cascade, which is responsible for a net drain of kinetic energy by the modes located within the spectral band [(Cs /Cs0 )π/∆, π/∆]. – The damping function is subgrid-model dependent: each subgrid model leads to a diﬀerent equilibrium between the two sources of resolved energy drain, yielding diﬀerent spectra and associated damping functions. Interpretation of Simulation Parameters. Eﬀective Filter. The above results allow us to reﬁne the analysis concerning the eﬀective ﬁlter. For large values of the Smagorinsky constant (Cs ≥ 0.5), the characteristic cutoﬀ length is the mixing length produced by the model. The model then dissipates more energy than if it were actually located at the scale ∆ because it ensures the energy ﬂux balance through the cutoﬀ associated with a longer characteristic length. The eﬀective ﬁlter is therefore fully determined by the subgrid model. This solution criterion should be compared with the one deﬁned for hot-wire measurements, which recommends that the wire length be less than twice the Kolmogorov scale in developed turbulent ﬂows. For small values of the constant, it is the cutoﬀ length ∆ that plays the role of characteristic length and the eﬀective ﬁlter corresponds to the usual analytical ﬁlter. It should be noted in this case that the energy drainage induced by the model is less than the transfer of kinetic energy through the cutoﬀ, so the energy balance is no longer maintained. This is reﬂected in an accumulation of energy in the resolved scales, and the pertinence of the simulation results should be taken with caution. For intermediate values of the constant, i.e. values close to the theoretical one predicted in Sect. 5.3.2 (i.e. Cs ≈ 0.2), the eﬀective ﬁlter is a combination of the analytical ﬁlter and model’s implicit ﬁlter, which makes it diﬃcult to interpret the dynamics of the smallest resolved scales. The dissipation induced by the model in this case correctly insures the equilibrium of the energy ﬂuxes through the cutoﬀ. Microstructure Knudsen Number. It has already been seen (relation (8.12)) that the mixing length can be interpreted as playing a role analogous to that of the Kolmogorov scale for the direct numerical simulation. The cutoﬀ length ∆, for its part, can be linked to the mean free path for Newtonian ﬂuids. We can use the ratio of these two quantities to deﬁne an equivalent of the microstucture Knudsen number Knm for the large-eddy simulation: Knm =

∆ 1 = ∆f Cs

.

(8.18)

Eﬀective Reynolds Number. Let us also note that the eﬀective Reynolds number of the simulation, denoted Reles , which measures the ratio of the inertia

8.1 Dynamic Interpretation of the Large-Eddy Simulation

287

eﬀects to the dissipation eﬀects, is taken in ratio to the Reynolds number Re corresponding to the exact solution by the relation: Reles =

η ηles

4/3 Re ,

(8.19)

where η is the dissipative scale of the full solution. This decrease in the eﬀective Reynolds number in the simulation may pose some problems, if the physical mechanism determining the dynamics of the resolved scales depends explicitly on it. This will, for example, be the case for all ﬂows where critical Reynolds numbers can be deﬁned for which bifurcations in the solution are associated3 . Subﬁlter Scale Concept. By analysis of the decoupling between the cutoﬀ length of the analytical ﬁlter ∆ and the mixing length ∆f , we can deﬁne three families of scales [495, 551] instead of the usual two families of resolved and subgrid scales. These three categories, illustrated in Fig. 8.2, are the: 1. Subgrid scales, which are those that are excluded from the solution by the analytical ﬁlter. 2. Subﬁlter scales, which are those of a size less than the eﬀective ﬁlter cutoﬀ length, denoted ∆eﬀ , which are scales resolved in the usual sense but whose dynamics is strongly aﬀected by the subgrid model. Such scales exist only if the eﬀective ﬁlter is determined by the subgrid viscosity model. There is still the problem of evaluating ∆eﬀ , and depends both on the presumed shape of the spectrum and on the point beyond which we consider to be “strongly aﬀected”. For example, by using Pao’s spectrum and deﬁning the non-physically resolved modes as those for which the energy level is reduced by a factor e = 2.7181..., we get: ∆eﬀ =

Cs ∆ , Ctheo

(8.20)

where Ctheo is the theoretical value of the constant that corresponds to the cutoﬀ length ∆. 3. Physically resolved scales, which are those of a size greater than the eﬀective ﬁlter cutoﬀ length, whose dynamics is perfectly captured by the simulation, as in the case of direct numerical simulations. Characterization of the Filter Associated with the Subgrid Model. The above discussion is based on a similarity hypothesis between the properties of isotropic hom*ogeneous turbulence and those of the ﬂow simulated 3

Numerical experiments show that too strong a dissipation induced by the subgrid model in such ﬂows may inhibit the ﬂow driving mechanisms and consequently lead to unreliabable simulations. One known example is the use of a Smagorinsky model to simulate a plane channel ﬂow: the dissipation is strong enough to prevent the transition to turbulence.

288

8. Numerical Solution: Interpretation and Problems

Fig. 8.2. Representation of diﬀerent scale families in the cases of ∆eﬀ < ∆ (Right) and ∆eﬀ > ∆ (Left).

using a subgrid viscosity model. This is mainly true of the dissipative eﬀects, which are described using the Pao spectrum or that of Heisenberg–Kovazsnay. So here, we adopt the hypothesis that the subgrid dissipation acts like an ordinary dissipation (which was already partly assumed by using a subgrid viscosity model). The spectrum E(k) of the solution from the simulation can therefore be interpreted as the product of the spectrum of the exact solution Etot (k) by the square of the transfer function associated with the eﬀective eﬀ (k): ﬁlter G 2eﬀ (k) . (8.21) E(k) = Etot (k)G Considering that the exact solution corresponds to the Kolmogorov spectrum, and using the form (8.10), we get: ' eﬀ (k) = Kles (Cs ) fles (k∆f , Cs ) . G (8.22) K0 The ﬁlter associated with the Smagorinsky model is therefore a “smooth” ﬁlter in the spectral space, which corresponds to a gradual damping, very diﬀerent from the sharp cutoﬀ ﬁlter.

8.2 Ties Between the Filter and Computational Grid. Pre-ﬁltering The above developments completely ignore the computational grid used for solving the constitutive equations of the large-eddy simulation numerically. If we consider this new element, it introduces another space scale: the spatial discretization step ∆x for simulations in the physical space, and the maximum wave number kmax for simulations based on spectral methods.

8.2 Ties Between the Filter and Computational Grid. Pre-ﬁltering

289

The discretization step has to be small enough to be able to correctly integrate the convolution product that deﬁnes the analytical ﬁltering. For ﬁlters with fastly-decaying kernel, we have the relation: ∆x ≤ ∆ .

(8.23)

The case where ∆x = ∆ is the optimal case as concerns the number of degrees of freedom needed in the discrete system for performing the simulation. This case is illustrated in Fig. 8.3.

Fig. 8.3. Representation of spectral decompositions associated with pre-ﬁltering (Left) and in the optimal case (Right).

The numerical errors stemming from the resolution of the discretized system still have to be evaluated. To ensure the quality of the results, the numerical error committed on the physically resolved modes has to be negligible, and therefore committed only on the subﬁlter scales. The theoretical analysis of this error by Ghosal in the simple case of isotropic hom*ogeneous turbulence is presented in the following. As the numerical schemes used are consistent, the discretization error cancels out as the space and time steps tend toward zero. One way of minimizing the eﬀect of numerical error is to reduce the grid spacing while maintaining the ﬁlter cutoﬀ length, which comes down to increasing the ratio ∆/∆x (see Fig. 8.3). This technique, based on the decoupling of the two space scales, is called pre-ﬁltering [14], and aims to ensure the convergence of the solution regardless of the grid4 . It minimizes the numerical error but induces more computations because it increases the number of degree of freedoms in the numerical solution without increasing the number of degrees of freedom in the physically resolved solution, and requires that the analytical ﬁltering be 4

A simpliﬁed analysis shows that, for an nth-order accurate numerical method, the weight of the numerical error theoretically decreases as (∆/∆x)−n . A ﬁner estimate is given in the remainder of this chapter.

290

8. Numerical Solution: Interpretation and Problems

performed explicitly [14] [59]. Because of its cost5 , this solution is rarely used in practice. Another approach is to link the analytical ﬁlter to the computed grid. The analytical cutoﬀ length is associated with the space step using the optimal ratio of these quantities and the form of the convolution kernel is associated with the numerical method. Let us point out a problem here that is analogous to that of the eﬀective ﬁlter already mentioned: the eﬀective numerical ﬁlter and therefore the eﬀective numerical cutoﬀ length, are generally unknown. This method has the advantage of reducing the size of the system as best possible and not requiring the use of an analytical ﬁlter, but it allows no explicit control of the eﬀective numerical ﬁlter, which makes it diﬃcult to calibrate the subgrid models. This method, because of its simplicity, is used by nearly all the authors.

8.3 Numerical Errors and Subgrid Terms 8.3.1 Ghosal’s General Analysis Ghosal [259] proposes a non-linear analysis of the numerical error in the solution of the Navier–Stokes equations for an isotropic hom*ogeneous turbulent ﬂow whose energy spectrum is approximated by the Von Karman model. Classiﬁcation of Diﬀerent Sources of Error. In order to analyze and estimate the discretization error, we ﬁrst need a precise deﬁnition of it. In all of the following, we consider a uniform Cartesian grid of N 3 points, which are the degrees of freedom of the numerical solution. Periodicity conditions are used on the domain boundaries. A ﬁrst source of error stems from the approximation we make of a continuous solution u by a making a discrete solution ud with a set of N 3 values. This is evaluated as: |ud − P(u)|

,

(8.24)

where P is a deﬁnite projection operator of the space of continuous solutions to that of the discrete solutions. This error is minimum (in the L2 sense) if P is associated with the decomposition of the continuous solution on a ﬁnite base of trigonometric polynomials, with the components of ud being the associated Fourier coeﬃcients. This error is intrinsic and cannot be canceled. Consequently, it will not enter into the deﬁnition of the numerical 5

For a ﬁxed value of ∆, increasing the ratio ∆/∆x by a factor n leads to an increase in the number of points of the simulation by a factor of n3 and increases the number of time steps by a factor n in order to maintain the same ratio between the time and space steps. In all, this makes an overall increase in the cost of the simulation by a factor n4 .

8.3 Numerical Errors and Subgrid Terms

291

error discussed in this present section. The best possible discrete solution is uopt ≡ P(u). The equations of the continuous problem are written in the symbolic form: ∂u = N S(u) , ∂t

(8.25)

where N S is the Navier–Stokes operator. The optimal discrete solution uopt is a solution of the problem: ∂Pu = P ◦ N S(u) , ∂t

(8.26)

where P ◦ N S is the optimal discrete Navier–Stokes operator which, in the ﬁxed framework, corresponds to the discrete operators obtained by a spectral method. Also, we note the discrete problem associated with a ﬁxed discrete scheme as: ∂ud = N S d (ud ) . (8.27) ∂t By taking the diﬀerence between (8.26) and (8.27), it appears that the best possible numerical method, denoted N S opt , is the one that veriﬁes the relation: (8.28) N S opt ◦ P = P ◦ N S . The numerical error Enum associated with the N S d scheme, and which is analyzed in the following, is deﬁned as: Enum ≡ (P ◦ N S − N S d ◦ P) (u) .

(8.29)

This represents the discrepancy between the numerical solution and the optimal discrete one. To simplify the analysis, we consider in the following that the subgrid models are perfect, i.e. that they induce no error with respect to the exact solution of the ﬁltered problem. By assuming this, we can clearly separate the numerical errors from the modeling errors. The numerical error Enum (k) associated with the wave number k is decomposed as the sum of two terms of distinct origins: – The diﬀerentiation error Edf (k), which measures the error the discrete operators make in evaluating the derivatives of the wave associated with k. Let us note that this error is null for a spectral method if the cutoﬀ frequency of the numerical approximation is high enough. – The spectrum aliasing error Ers (k), which is due to the fact that we are computing non-linear terms in the physical space in a discrete space of ﬁnite dimension. For example, a quadratic term will bring in higher frequencies than those of each of the arguments in the product. While some of these frequencies are too high to be represented directly on the discrete base,

292

8. Numerical Solution: Interpretation and Problems

they do combine with the low frequencies and introduce an error in the representation of them6 . Estimations of the Error Terms. For a solution whose spectrum is of the form proposed by Von Karman: E(k) =

a k4

,

17/6

(b + k 2 )

(8.30)

with a = 2.682 and b = 0.417, and using a quasi-normality hypothesis for evaluating certain non-linear terms, Ghosal proposes a quantitative evaluation of the diﬀerent error terms, the subgrid terms, and the convection term, for various Finite Diﬀerence schemes as well as for a spectral scheme. The convection term is written in conservative form and all the schemes in space are centered. The time integration is assumed to be exact. The exact forms of these terms, available in the original reference work, are not reproduced here. For a cutoﬀ wave number kc and a sharp cutoﬀ ﬁlter, simpliﬁed approximate estimates of the average amplitude can be derived for some of these terms. The amplitude of the subgrid term σsgs (kc ), deﬁned as (

kc

σsgs (kc ) =

)1/2 |τ (k)|dk

,

(8.31)

where τ (k) is the subgrid term for the wave number k, is bounded by:

0.36 kc0.39 upper limit σsgs (kc ) = , (8.32) 0.62 kc0.48 lower limit that of the sum of the convection term and subgrid term by: σtot (kc ) = 1.04 kc0.97 6

,

(8.33)

Let us take the Fourier expansions of two discrete functions u and v represented by N degrees of freedom. At the point of subscript j, the expansions are expressed:

N/2−1

uj =

N/2−1

u n e(i(2π/N)jn) , vj =

n=−N/2

vm e(i(2π/N)jm) j = 1, N

.

m=−N/2

The Fourier coeﬃcient of the product wj = uj vj (without summing on j) splits into the form: wk = u n vm + u n vm . n+m=k

n+m=k±N

The last term in the right-hand side represents the spectrum aliasing error. These are terms of frequencies higher than the Nyquist frequency, associated the sampling, which will generate spurious resolved frequencies.

8.3 Numerical Errors and Subgrid Terms

293

in which

σsgs (kc ) ≈ kc−0.5 . σtot (kc ) The amplitude of the diﬀerentiation error σdf (kc ), deﬁned by: ( )1/2

(8.34)

kc

σdf (kc ) =

Edf (k)dk

,

(8.35)

is evaluated as:

⎧ 1.03 (second order) ⎪ ⎪ ⎪ ⎪ ⎨ 0.82 (fourth order) 0.70 (sixth order) σdf (kc ) = kc0.75 × ⎪ ⎪ ⎪ 0.5 (heigth order) ⎪ ⎩ 0 (spectral)

,

(8.36)

and the spectrum aliasing error σrs (kc ), which is equal to: ( )1/2 kc

σrs (kc ) =

Ers (k)dk

(8.37)

is estimated as: ⎧ 0.90 ⎪ ⎪ ⎨ 2.20 σrs = 0.46 ⎪ ⎪ ⎩ 1.29

kc0.46 kc0.66 kc0.41 kc0.65

(minimum estimation, spectral, no de-aliasing) (maximum estimation, spectral, no de-aliasing) . (minimum estimation, second order) (maximum estimation, second order) (8.38) The spectrum aliasing error for the spectral method can be reduced to zero by using the 2/3 rule, which consists of not considering the last third of the wave numbers represented by the discrete solution. It should be noted that, in this case, only the ﬁrst two-thirds of the modes of the solution are correctly represented numerically. The error of the ﬁnite diﬀerence schemes of higher order is intermediate between that of the second-order accurate scheme and that of the spectral scheme. From these estimations, we can see that the discretization error dominates the subgrid terms for all the ﬁnite diﬀerence schemes considered. The same is true for the spectrum aliasing error, including for the ﬁnite diﬀerence schemes. Finer analysis on the basis of the spectra of the various terms shows that the discretization error is dominant for all wave numbers for the second-order accurate scheme, whereas the subgrid terms are dominant at the low frequencies for the heigth-order accurate scheme. In the former case, the eﬀective numerical ﬁlter is dominant and governs the solution dynamics. Its cutoﬀ length can be considered as being of the order of the size of the computational domain. In the latter, its cutoﬀ length, deﬁned as the wavelength of the mode beyond which it becomes dominant with respect to the subgrid terms, is smaller and there exist numerically well-resolved scales.

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8.3.2 Pre-ﬁltering Eﬀect The pre-ﬁltering eﬀect is clearly visible from relations (8.32) to (8.38). By decoupling the analytical from the numerical ﬁlter, two diﬀerent cutoﬀ scales are introduced and thereby two diﬀerent wave numbers for evaluating the numerical error terms and the subgrid terms: while the cutoﬀ scale ∆ associated with the ﬁlter remains constant, the scale associated with the numerical error (i.e. ∆x) is now variable. By designating the ratio of the two cutoﬀ lengths by Crap = ∆x/∆ < 1, we see that the diﬀerentiation error σdf (kc ) of the ﬁnite diﬀerence scheme is −3/4 reduced by a factor Crap with respect to the previous case, since it varies 3/4 as kc . This reduction is much greater than the one obtained by increasing the order of accuracy of the the schemes. Thus, more detailed analysis shows that, for the second-order accurate scheme, the dominance of the subgrid term on the whole of the solution spectrum is ensured for Crap = 1/8. For a ratio of 1/2, this dominance is once again found for schemes of order of accuracy of 4 or more. These theoretical evaluations do not take into account the nonlinear feedback of the computed solution on the numerical error. Numerical experiments were conducted by Chow and Moin [138] in isotropic turbulence to assess Ghosal’s results. Their results show that to ensure that the subgrid terms will dominate the numerical error, a ﬁlter-grid ratio ∆/∆x of at least four is desired for a second-order centered ﬁnite diﬀerence scheme. This minimum is a decreasing function of the scheme order of accuracy, and a ratio of two is found to be suﬃcient for a sixth-order centered Pad´e scheme. The eﬃciency of the preﬁltering technique was exhaustively checked by Geurts and Fr¨ ohlich [256, 257] on the plane mixing layer conﬁguration. The main conclusion of this study is that the best solution for improving the results of a large-eddy simulation in practice is to reﬁne the computational grid, i.e. to lower ∆x, while keeping a low value of the ratio ∆/∆x. The numerical tests conducted by these authors show that ∆/∆x = 1 is optimal for highresolution large-eddy simulations, i.e. for simulations which are close to direct numerical simulation. For coarser grid simulation, the optimum was found to be ∆/∆x ≈ 2 − 3, combined with a fourth-order accurate non-dissipative numerical method. Nevertheless, the recommended strategy to improve the results at a ﬁxed computational cost is to reﬁne the grid using ∆/∆x = 1−2 rather than augmenting the ratio ∆/∆x for a given value of ∆. This conclusion is conﬁrmed by a large set of results published by Gullbrand and Chow [283], who carried out several simulations of turbulent plane channel ﬂow with second-order and fourth-order accurate ﬁnite diﬀerence methods. In these simulations, the use of a preﬁlter with size 2∆x didn’t lead to a clear improvement of the results. The ﬁnest study of the eﬀect of preﬁltering was conducted by Meyers, Geurts and Baelmans [515] in large-eddy simulation of isotropic turbulence using the Smagorinsky model. Deﬁning the error etotal (∆, ∆x) for a dummy

8.3 Numerical Errors and Subgrid Terms

295

variable φ as the diﬀerence between the ﬁltered exact solution (obtained by Direct Numerical Simulation) and the solution found using large-eddy simulation with cutoﬀ length ∆ on a mesh of size ∆x: etotal (∆, ∆x) = φDNS − φLES (∆, ∆x)

,

(8.39)

the authors investigate the relative inﬂuence of diﬀerent sources of error. Subgrid modeling and discretization errors being respectively deﬁned as emodel = φDNS − φLES (∆, 0) , ediscr = φLES (∆, 0) − φLES (∆, ∆x)

(8.40) ,

(8.41)

exhaustive analyses carried out using a direct numerical simulation database yields the following conclusions: – The global error behavior is parametrized by the subgrid activity parameter s deﬁned like ε s= , (8.42) ε + εν where ε = −τij S ij is the subgrid dissipation and εν = νSij Sij is the molecular dissipation. Direct numerical simulation corresponds to s = 0, while s = 1 characterizes large-eddy simulation in the limit of inﬁnite Reynolds number. Tests prove there there exists a threshold value sc 7 : for s ≤ sc the total error is dominated by discretization error eﬀects, while for s > sc the modeling error is the most important source of uncertainty. – For s ≤ sc , the relative error δerr : ! 2 dt e , (8.43) δerr = ! total2 φDNS dt can be either a decreasing or increasing function of s, depending on s and ∆x when it is based on the kinetic energy (see Fig. 8.4). No general trend is observed in this regime, in which strong interactions can occur between discretization errors and modeling errors. Partial cancellation sometimes occurs leading to a signiﬁcant reduction of the total error. For s > sc , the relative error is a monotone exponentially increasing function of s. – The relative error based on the Taylor length scale has a diﬀerent behavior in the s ≤ sc regime (see Fig. 8.5): strong interactions between modeling and discretization errors are observed, but the relative error exhibits a monotonic increasing behavior as a function of sc . Partial error cancellation is less intense than in the case of the turbulent kinetic energy. – Because of the strong non-linear interactions between modeling and discretization errors, which leads to a non-monotone behavior of the total 7

Typical values are sc = 0.4 for isotropic turbulence at Reλ = 50, sc = 0.8 for isotropic turbulence at Reλ = 100 and sc = 0.5 for a plane mixing layer.

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8. Numerical Solution: Interpretation and Problems

Fig. 8.4. Evolution of the relative error on kinetic energy as a function of the subgrid activity, in isotropic turbulence (Courtesy of J. Meyers and B. Geurts, Univ. Twente). Top: Reλ = 50, Bottom: Reλ = 100.

8.3 Numerical Errors and Subgrid Terms

297

Fig. 8.5. Evolution of the relative error on the Taylor scale as a function of the subgrid activity, in isotropic turbulence (Courtesy of J. Meyers and B. Geurts, Univ. Twente). Left: Reλ = 50, Right: Reλ = 100.

error. As a consequence, grid reﬁnement (decrease of ∆x at constant ∆) must be coupled to a change in the Smagorinsky constant CS to obtain an optimal error minimization. The ratio CS ∆/∆x which yields the minimum error follows non-trivial trajectories in both the (∆x, CS ∆) and the (∆x, s/sc ) planes. This behavior is illustrated in Fig. 8.6.

8.3.3 Conclusions This analysis can be used only for reference, because it is based on very restrictive hypotheses. It nonetheless indicates that the numerical error is not negligible and that it can even be dominant in certain cases over the subgrid terms. The eﬀective numerical ﬁlter is then dominant over the scale separation ﬁlter. This error can be reduced either by increasing the order of accuracy of the numerical scheme or by using a pre-ﬁltering technique that decouples the cutoﬀ length of the analytical ﬁlter of the discretization step. Ghosal’s ﬁndings seem to indicate that a combination of these two techniques would be the most eﬀective solution. These theoretical ﬁndings are conﬁrmed by the numerical experiments of Najjar and Tafti [563] and Kravenchko and Moin [409], who observed that the eﬀect of the subgrid models is completely or partially masked by the numerical error when second-order accurate methods are employed. It should be noted here that practical experience leads us to less pessimistic conclusions than the theoretical analyses: large-eddy simulations performed with a scheme accurate to the second order show a dependency with respect to the subgrid model used. The eﬀects of these models are not entirely masked, which justiﬁes using them. However, no precise qualiﬁcation exists today of the information loss due to the use of a given scheme. These observations are made empirically, case by case [76, 480, 94].

298

8. Numerical Solution: Interpretation and Problems

Fig. 8.6. Top: Map of the relative error in isotropic turbulence (Courtesy of J. Meyers and B. Geurts, Univ. Twente). Left: Reλ = 50, Right: Reλ = 100. The dashed-dotted line is related to the optimal trajectory corresponding to the minimal error. The dashed line corresponds to the trajectory associated to a ﬁxed value of the subgrid activity parameter. Bottom: Optimal reﬁnement strategy for diﬀerent Reynolds numbers and error deﬁnitions, shown in diﬀerent planes. The shadded areas are related to minimal value of the error.

Another eﬀect was emphasized by Geurts et al. [255, 515], who found that partial cancellation of modeling and numerical errors may occur, leading to a signiﬁcant improvement of the accuracy of the simulation. This cancellation was observed in both plane mixing layer conﬁguration and isotropic turbulence for fourth-order and spectral schemes. An important and counterintuitive conclusion is that using higher-order accurate schemes or improved subgrid models may lead to worse results with regards to a simulation in which this cancellation occurs. This behavior is observed to be strongly dependent on the value of the subgrid activity parameter s and the considered deﬁnition of the error. An important ﬁnding dealing with the preﬁltering technique is that reﬁning the the grid at constant ﬁlter cutoﬀ (i.e. ∆/∆x −→ ∞ at ﬁxed ∆) must be coupled to a change in subgrid viscosity model constant to obtain the optimal error reduction.

8.3 Numerical Errors and Subgrid Terms

299

8.3.4 Remarks on the Use of Artiﬁcial Dissipations Many comments have been made over recent decades on the sensitivity of large-eddy simulation results, for example concerning the formulation of the convection term [319, 409], the discrete form of the test ﬁlter [597, 563, 81, 641, 605], and the formulation of the subgrid term [634], but there are far too many, too dispersed, and too far from general to be detailed here. Moreover, countless analyses have been made of the numerical error associated with various schemes, especially as concerns the treatment of the non-linear terms, which will not be resumed here, but we will still take more special note of the ﬁndings of Fabignon et al. [211] concerning the characterization of the eﬀective numerical ﬁlter of several schemes. Special attention should still be paid to the discretization of the convective terms. To capture strong gradients without having the numerical solution polluted with spurious high-frequency wiggles, the scheme is very often stabilized by introducing artiﬁcial dissipation. This dissipation is added explicitly or implicitly using an upwind scheme for the convection term. Introducing an additional dissipation term for the large-eddy simulation is still controversial [563, 521, 239, 228, 665, 501, 94] because the eﬀective ﬁlter is then very similar in form to that which would be imposed by subgrid viscosity model, making for two competing resolved kinetic energy spectrum mechanisms. The similarity between the numerical dissipation and that associated with the energy cascade model is still being investigated, but a few conclusions have already been drawn. It seems that the total numerical dissipation induced by most upwind schemes is still greater than that of the subgrid viscosity models, if no preﬁltering method is used. This is true even for seventh-order accurate upwind schemes [52]. Garnier et al. [239] developed the generalized Smagorinsky constant as a tool to compare numerical and physical subgrid dissipations. The generalized Smagorinsky constant is the value that should take the constant of the Smagorinsky model to obtain a total dissipation equal to the numerical dissipation. Numerical tests carried out on decaying isotropic turbulence have shown that all the numerical upwind schemes, up to the ﬁfth order of accuracy, are more dissipative than the usual Smagorinsky model. Typical results are displayed in Fig. 8.7. It has also been demonstrated [501] that the use of a stabilized sixth-order accurate scheme may lead to the same quality of results as a second-order accurate scheme, because of the very high dissipation applied to the highest resolved frequency, which is responsible for the largest part of the interactions with subgrid modes. These two dissipations are correlated in space (especially in the case of the Smagorinsky model), but have diﬀerent spectral distributions: a subgrid viscosity model corresponds to a second-order dissipation associated with a spectrum of the form (k/kc )2 E(k), while an nth-order numerical dissipation is associated with a spectrum of the form (k/kc )n E(k).

300

8. Numerical Solution: Interpretation and Problems

Fig. 8.7. Time history of the generalized Smagorinsky constant for various dissipative schemes (second- to ﬁfth-order of accuracy) in freely decaying isotropic turbulence. Courtesy of E. Garnier, ONERA.

For n > 2 (resp. n < 2), the numerical (resp. subgrid) dissipation may be dominant for the highest resolved frequencies and the subgrid (resp. numerical) dissipation will govern the dynamics of the low frequencies. This point is illustrated in Figs. 8.8 and 8.9.

Fig. 8.8. Numerical and subgrid dissipations for a Von Karman spectrum. The peak of the Von Karman spectrum is at kc /5. The dissipation spectra have been normalized so that the total dissipation is the same in all cases. It is worth noting that typical numerical schemes lead to a total dissipation higher than the subgrid dissipation.

8.3 Numerical Errors and Subgrid Terms

301

Fig. 8.9. Turbulent kinetic energy spectra (isotropic turbulence) computed by large-eddy simulation. Solid line, ﬁfth-order accurate dissipative scheme without subgrid model. Dashed lines, second-order accurate non-dissipative scheme with (i) Smagorinsky model and (ii) dynamic Smagorinsky model. Courtesy of E. Garnier, ONERA.

The studies that have been made show a sensitivity of the results to the subgrid model used, which proves that the eﬀects of the model are not entirely masked by the numerical dissipation. The theoretical analysis presented above should therefore be taken relative to this. But consistent with it, Beaudan et al. [52] have observed a reduction in the numerical cutoﬀ length as the order of accuracy of the scheme increases. This type of ﬁnding should nonetheless be treated with caution, because the conclusions may be reversed if we bring in an additional parameter, which is the grid reﬁnement. Certain studies have shown that, for coarse grids, i.e. high values of the numerical cutoﬀ length, increasing the order of accuracy of the upwind scheme can lead to a degradation of the results [701]. But some speciﬁc numerical stabilization procedures can be deﬁned, which tune the numerical dissipation in such a way that the results remain sensitive to subgrid modeling [94, 15, 95]. These numerical methods allow the use of relatively coarse grids, but no general theory for them exist at present time. This relative similarity between artiﬁcial dissipation and the direct energy cascade model has induced certain authors to perform “no-model” large-eddy simulations, with the ﬁltering based entirely on the numerical method (leading to the Implicit Large-Eddy Simulation technique, see Sect. 5.3.4 for a detailed discussion). Thus many ﬂow simulations have been seen in complex geometries, based on the use of an third-order accurate upwind scheme proposed by Kawamura and Kuwahara [381], yielding interesting results. In the compressible case, this approach has been called the Monotone Integrated

302

8. Numerical Solution: Interpretation and Problems

Large-Eddy Simulation (MILES) method (see Sect. 5.3.4 for a description of ILES numerical methods). The “experimental” analysis of some particular numerical methods reveals that there exists some numerical dissipation procedures which mimic very well the theoretical subgrid viscosity [192]. This is illustrated in Fig. 8.10, which compares the computes spectral viscosity associated to the MPDATA scheme with the theoretical spectral viscosity proﬁle. The observed agreement prove that the Implict Large-Eddy Simulation approach can yield very good results if the numerical scheme is carefully chosen. The use of artiﬁcial dissipation therefore raises many questions, but is very common in simulations that are physically very strongly under-resolved in complex conﬁgurations, because experience shows that adding subgrid models does not ensure a monotonic solution. To ensure that certain variables remain positive, such as concentrations of pollutants or the temperature, it seems to be necessary to resort to such numerical methods. Alternatives based on local reﬁnement of the solution, i.e. decreasing the eﬀective cutoﬀ length by enriching or adapting the grid, have been studied by certain authors but no ﬁnal conclusion has been drawn. A few studies dealing with some particular stabilization techniques such as the Galerkin Least-Square method, reveal that the numerical dissipation may happen to be be too low to prevent a pile-up in the resolved kinetic energy, leading to the growth of bounded wiggles. In such a case, the use of a subgrid model seems to be required to recoved physical results. The main problem arising in this case is that the use of a basic subgrid model will lead to a global overdamping of the computed solution. The use of self-adaptive subgrid models able to tune their induced dissipation so that the sum of

Fig. 8.10. Analysis of the numerical viscosity associated to the MPDATA scheme. Dashed line: theoretical spectral subgrid viscosity. Solid line: MPDATA equivalent spectralk viscosity. Solid line with symbol: spectral viscosity recomputed from kinetic energy transfer of the unﬁltered velocity ﬁeld. From [192].

8.3 Numerical Errors and Subgrid Terms

303

the numerical and subgrid dissipation will reach the required level to obtain physical results. The identiﬁcation of subgrid models and/or procedures to deﬁne such subgrid models is still a open issue. An a priori requirement is that the subgrid model must be local in terms of wave number, i.e. it must emphasized the highest resolved frequencies. Numerical experiments show that self-adaptive models and some multilevel approaches have this property. While the Navier–Stokes equations contain energy information, they also contain information concerning the signal phase. Using centered schemes for the convection term therefore raises problems too, because of the dispersive errors they induce in the highest resolved frequencies. Generally, estimates of the wave number beyond which the modes are considered to be well resolved numerically vary from 2∆x to 20∆x, depending on the schemes and authors [604]. 8.3.5 Remarks Concerning the Time Integration Method Large-eddy simulation is ordinarily addressed using a spatial ﬁltering, but without explicitly stating the associated time ﬁltering. This is due to the fact that most computations are made for moderate time steps (CFL ≡ u∆t/ ∆x < 1) and it is felt that the time ﬁltering eﬀects are masked by those of the space ﬁltering. Choi and Moin [131], however, have shown by direct simulations of a plane channel ﬂow that the time ﬁltering eﬀects can be very large, even for CFLs of the order of 0.5, since the turbulence cannot be maintained numerically if the time step is greater than the characteristic time associated with the Kolmogorov scale. Most authors use second-order accurate integration methods, but no complete study has been published to date to determine what timescales are well resolved numerically and physically. We should also note the results of Beaudan and Moin [52] and Mittal and Moin [521], who showed that the use of artiﬁcial viscosity aﬀects the solution of a very large share of the simulated time frequencies (about 75% for the particular case studied).

9. Analysis and Validation of Large-Eddy Simulation Data

9.1 Statement of the Problem 9.1.1 Type of Information Contained in a Large-Eddy Simulation The solution to the equations that deﬁne the large-eddy simulation furnishes explicit information only on the scales that are resolved, i.e. those that are left after reduction of the number of degrees of freedom in the exact solution. We are therefore dealing with information that is truncated in space and time. The time ﬁltering is induced implicitly by the spatial ﬁltering because, as the ﬁltering eliminates certain space scales, it eliminates the corresponding time scales with them (see p. 19). The information of use for analysis or validation is what is contained in those scales that are both physically and numerically well-resolved. It should be remembered that, since the eﬀective numerical and physical ﬁlters are unknown, the usable scales are most often identiﬁed empirically. Adopting the assumption that all the scales represented by the simulation are physically and numerically well-resolved, the statistical average of the usable resolved ﬁeld is expressed u. The statistical ﬂuctuation of the resolved ﬁeld, denoted u , is deﬁned by: ui = ui − ui

.

(9.1)

The diﬀerence between the statistical average of the resolved scales and that of the exact solution is deﬁned as: ui − ui = ui ,

(9.2)

which corresponds to the statistical average of the unresolved scales. The Reynolds stresses computed from the resolved scales are equal to ui uj . e The diﬀerence from the exact stresses ue i uj , where the exact ﬂuctuation is e deﬁned as u = u − u, is: e ue i uj = (ui − ui )(uj − uj ) = ui uj − ui uj

= ui uj + τij − ui + ui uj + uj

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9. Analysis and Validation

= ui uj + τij − ui uj − ui uj − ui uj − ui uj = ui uj + τij − ui uj − ui uj − ui uj

,

where τij = ui uj + uj ui + ui uj is the deﬁltered subgrid tensor. Since the subgrid scales are not known, the terms containing the contribution u , cannot be computed from the simulation. When the statistical average of the subgrid modes is very small compared with the other terms, we get: e (9.3) ue i uj ui uj + τij . The two terms on the right-hand side can be evaluated from the numerical simulation, but the quality of the model’s representation of the subgrid tensor partly conditions that of the result. We can easily see that subgrid-viscosity models, which only account for the deviatoric part of the subgrid-stress tensor, make it possible only to recover the deviatoric part of the Reynolds stresses [760, 397], at least theoretically. 9.1.2 Validation Methods The subgrid models and their various underlying hypotheses can be validated in two ways [218]: – A priori validation. The exact solution, which is known in this case, is ﬁltered analytically, leading to the deﬁnition of a fully determined resolved ﬁeld and subgrid ﬁeld. The various hypotheses or models can then be tested. The exact solutions are usually generated by direct numerical simulations at moderate or low Reynolds numbers, which limits the ﬁeld of investigation. A priori tests like this have also been performed using experimental data, making it possible to reach higher Reynolds numbers. This type of validation raises a fundamental problem, though. By comparing the exact subgrid stresses with those predicted by a subgrid model evaluated on the basis of the ﬁltered exact solution, the eﬀects of the modeling errors are neglected and the implicit ﬁlter associated with the model is not considered1 . This means that the results of a priori validations are only relative in value. – A posteriori validation. Here, we perform a large-eddy simulation computation and validate by comparing its results with a reference solution. This is a dynamic validation that takes all the simulation factors into consideration, while the previous method is static. Experience shows that models yielding poor a priori results can be satisfactory a posteriori, and vice 1

This ﬁeld could not have been obtained by a large-eddy simulation since it is a solution of the ﬁltered momentum equations in which the exact subgrid tensor appears. In the course of a simulation, the subgrid model is applied to a velocity ﬁeld that is a solution of the momentum equation where the modeled subgrid tensor appears. These two ﬁelds are therefore diﬀerent in theory. Consequently, in order to be fully representative, an a priori test has to be performed on the basis of a velocity ﬁeld that can be obtained from the subgrid model studied.

9.1 Statement of the Problem

307

versa [597]. It is more advantageous to validate models a posteriori because it corresponds to their use in the simulation; but it is sometimes diﬃcult to draw any conclusions on a precise point because of the multitude of often imperfectly controlled factors at play in a numerical simulation. 9.1.3 Statistical Equivalency Classes of Realizations The subgrid models are statistical models and it seems pointless to expect them to produce deterministic simulations in which the resolved scales coincide exactly with those of other realizations, for example of the experimental sort. On the other hand, large-eddy simulation should correctly reproduce the statistical behavior of the scales making up the resolved ﬁeld. Equivalency classes can thus be deﬁned among the realizations [503] by considering that one of the classes consists of realizations that lead to the same values of certain statistical quantities computed from the resolved scales. Belonging to the same class of equivalency as a reference solution is a validation criterion for the other realizations. If we set aside the numerical errors, we can deﬁne the necessary conditions on the subgrid models such that two realizations will be equivalent, by verifying these validity criteria. These conditions will be discussed in the following sections. A subgrid model can thus be considered validated if it can generate realizations that are equivalent to a reference solution, in a sense deﬁned below. Theoretically, while we overlook the eﬀect of the discretization on the modeling, it can be justiﬁably thought that a model reproducing the interscale interactions exactly will produce good results, whereas the opposite proposition is not true. That is, the idea of suﬃcient complexity of a model has to be introduced in order to obtain a type of result on a given conﬁguration with a tolerated margin of error in order to say what a good model is. The idea of a universal or best model might not be rejected outright, but should be taken relatively. The question is thus raised of knowing what statistical properties two subgrid models should share in order for the two resulting solutions to have common properties. Let u and u∗ be the ﬁltered exact solution and the solution computed with a subgrid model, respectively, for the same ﬁlter. The exact (unmodeled) subgrid tensor corresponding to u is denoted τij , and the modeled subgrid tensor computed from the u∗ ﬁeld is denoted τij∗ (u∗ ). The two velocity ﬁelds are solutions of the following momentum equations: ∂u + ∇ · (u ⊗ u) = −∇ · p + ν∇2 u − ∇ · τ ∂t

,

(9.4)

∂u∗ + ∇ · (u∗ ⊗ u∗ ) = −∇ · p∗ + ν∇2 u∗ − ∇ · τ ∗ (u∗ ) . (9.5) ∂t A simple analysis shows that, if all the statistical moments (at all points of space and time) of τij conditioned by the u ﬁeld are equal to those of

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τij∗ (u∗ ) conditioned by u∗ , then all the statistical moments of u and u∗ will be equal. This is a full statistical equivalency, which implies that the subgrid models fulﬁll an inﬁnity of conditions. To relax this constraint, we deﬁne less restrictive equivalency classes of solutions which are described in the following sections. They are deﬁned in such a way as to bring out the necessary conditions applying to the subgrid models, in order to qualify them [503]. We try to deﬁne conditions such that the statistical moments of moderate order2 (1 and 2) of the ﬁeld resulting from the large-eddy simulation u∗ are equal to those of a reference solution u. Equivalency of First-Order Moments. The equivalency relation is built on the equality of the ﬁrst-order statistical moments of the realizations. A velocity and a pressure ﬁeld are associated with each realization. Let (u, p) and (u∗ , p∗ ) be the doublets associated with the ﬁrst and second realizations, respectively. The evolution equations of the ﬁrst-order statistical moments of the velocity ﬁeld of these two realizations are expressed: ∂u + ∇ · (u ⊗ u) ∂t

−∇ · p + ν∇2 u − ∇ · τ

=

−∇ · (u ⊗ u − u ⊗ u) , ∂u∗ + ∇ · (u∗ ⊗ u∗ ) = ∂t

(9.6)

−∇ · p∗ + ν∇2 u∗ − ∇ · τ ∗ (u∗ ) −∇ · (u∗ ⊗ u∗ − u∗ ⊗ u∗ ) , (9.7)

where designates an ensemble average performed using independent realizations. The two realizations will be called equivalent if their ﬁrst- and second-order moments are equivalent, i.e. ui = p = ui uj =

u∗i ,

(9.8)

∗

p , u∗i u∗j .

(9.9) (9.10)

Analysis of evolution equations (9.6) and (9.7) shows that one necessary condition is that the resolved and subgrid stresses be statistically equivalent. The last condition is expressed: τij = τij∗ + Cij

,

(9.11)

where Cij is a null-divergence tensor. This condition is not suﬃcient because a model that leads to a good prediction of the mean stresses can generate an error on the mean ﬁeld if the mean resolved stresses are not correct. To obtain a suﬃcient condition, the equivalency of the stresses ui uj and u∗i u∗j must be ensured by another relation. 2

Because these are the quantities sought in practice.

9.1 Statement of the Problem

309

Equivalency of Second-Order Moments. We now base the equivalency relation on the equality of the second-order moments of the resolved scales. Two realizations will be called equivalent if the following conditions are satisﬁed: ui = ui uj = ui uj uk = pui = pS ij = ; ∂u ∂u < i i = ∂xk ∂xk

u∗i

,

u∗i u∗j , u∗i u∗j u∗k , p∗ u∗i , ∗ p∗ S ij , ; ∂u∗ ∂u∗ < i i ∂xk ∂xk

(9.12) (9.13) (9.14) (9.15) (9.16) .

(9.17)

Analysis of the equation for the second-order moments ui uj shows that, in order for two realizations to be equivalent, the following necessary condition must be satisﬁed: ∂ (ui τjk + uj τik ) = ∂xk ∂ ∗ ∗ ∗ ∗ ∗ ∗ ∗ S kj + τjk S ki − . τik ui τjk + u∗j τik ∂xk τik S kj + τjk S ki −

This condition is not suﬃcient. To obtain such an realization, the equality of the third-order moments also has to be ensured. It is noted that the nonlinear coupling prohibits the deﬁnition of suﬃcient conditions on the subgrid model to ensure the equality of the nth-order moments of the resolved ﬁeld without adding necessary conditions on the equality of the (n + 1)th-order moments. Equivalency of the Probability Density Functions. We now base the deﬁnition of the equivalency classes on the probability density function fprob (V , x, t) of the resolved scales. The ﬁeld V is the test velocity ﬁeld from which the conditional average is taken. The function fprob is deﬁned as the statistical average of the one-point probabilities: fprob (V , x, t) ≡ δ(u(x, t) − V )

,

and is a solution of the following transport equation:

; ∂p <7 ∂fprob ∂ ∂τij ∂fprob 2 +Vj = + − ν∇ uj |u = V fprob ∂t ∂xj ∂Vj ∂xj ∂xj

(9.18)

. (9.19)

Two realizations can be called equivalent if: fprob (V , x, t) = ui (y)|u = V = ui (y)uj (y)|u = V =

∗ fprob (V , x, t) , ∗ ui (y)|u = V ,

(9.20) (9.21)

u∗i u∗j (y)|u = V .

(9.22)

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9. Analysis and Validation

Once the pressure gradient is expressed as a function of the velocity (by an integral formulation using a Green function) and the conditional average of the strain rate tensor is expressed using gradients of the two-point conditional averages, equation (9.19) can be used to obtain the following necessary condition: ∂2 ∂ xj − yj 3 1 d y + lim τik |u(x) = V τij |u(x) = V − y→x ∂yi 4π ∂yi ∂yk |x − y| 1 xj − yj 3 ∂2 ∗ d y =− τik |u∗ (x) = V 4π ∂yi ∂yk |x − y| ∂ τij∗ |u∗ (x) = V + Cj y→x ∂yi

+ lim

,

in which the divergence of vector Cj is null. It is noted that the condition deﬁned from the one-point probability density uses two-point probabilities. We again ﬁnd here the problem of non-localness already encountered when the equivalency class is based on statistical moments. A more restrictive condition is: ∗ |u∗ (x) = V . τik |u(x) = V = τik

(9.23)

9.1.4 Ideal LES and Optimal LES An abstract subgrid model can be deﬁned, which is in all senses ideal [419]. An LES using this model will exactly reproduce all single-time, multipoint statistics, and at the same time will exhibit minimum possible error in instantaneous dynamics. Such a LES will be referred to as ideal LES. Using the same notations as in Sect. 9.1.3, ideal LES is governed by the conditional average @ A du∗ du ∗ = u = u , (9.24) dt dt where u and u∗ are the solution of the exact LES equation and the LES equation with a subgrid model, respectively. It can be shown that such ideal LES is associated to the minimum mean-square error between the evolution of the LES ﬁeld u∗ (t) and the exact solution u(t), deﬁned as an instantaneous pointwise measurement on ∂u/∂t: ei (x) =

∂ui ∂u∗i − ∂t ∂t

.

(9.25)

Equivalently, this error can be evaluated using the exact and the modeled subgrid forces, referred to as M = ∇ · τ and m = ∇ · τ ∗ : e(x) = M (x) − m(x) .

(9.26)

9.1 Statement of the Problem

311

The ideal subgrid model τ ∗ is then such that ∇ · τ ∗ = m = M |u = u∗

.

(9.27)

This model is written as an average over the real turbulent ﬁelds whose resolved scales match the current LES ﬁeld, making it impossible to compute in practical applications. In order to approximate in an optimal sense this ideal model, several authors [419, 54, 6, 137] propose to formally approximate the conditional average by a stochastic estimation. These new models can be referred to as optimal or nearly-optimal models, leading to optimal LES. The estimation of the subgrid force is based on the convolution of an estimation kernel Kij with velocity event data at N points (ξ1 , ..., ξN ): (9.28) mi (x) = Kij (x, ξ1 , ..., ξN )Ej (u∗ ; ξ1 , ..., ξN )dξ1 ...dξN , where Ej is an event vector. Chosing E(ξ1 , ..., ξN ) = (1, u∗i (ξ1 ), u∗j (ξ1 )u∗k (ξ2 ), ...)

,

(9.29)

we recover the expansion mi (x) = Ai (x) + Bij (x, ξ1 )u∗j (ξ1 )dξ1 + Cijk (x, ξ1 , ξ2 )u∗j (ξ1 )u∗k (ξ2 )dξ1 dξ2

.

(9.30)

The random mean square error between Mi and mi is minimal when ei (x)Ek (η1 , ..., ηN ) = 0

,

(9.31)

yielding the following deﬁnition of the optimal kernel Kij Mi (x)Ek (η1 , ..., ηN ) = Kij (x, ξ1 , ..., ξN ) . × Ej (ξ1 , ..., ξN )Ek (η1 , ..., ηN ) dξ1 ...dξN (9.32) The resulting optimal subgrid models have the property that the correlation of the parametrized subgrid force with any event data is the same as the correlation of the exact subgrid force with the same event data: mi (x)Ej (ξ1 , ..., ξN ) = Mi (x)Ej (ξ1 , ..., ξN )

.

(9.33)

9.1.5 Mathematical Analysis of Sensitivities and Uncertainties in Large-Eddy Simulation The developments presented above deal with the nature of the information retrieved from large eddy simulation and some statistical equivalency constraints. But a key problem in practical cases is to evaluate the sensitivity of

312

9. Analysis and Validation

the computed results with respect to the subgrid model (or its inputs), i.e. to obtain an estimate for the uncertainties associated to the fact that all scales are not resolved and that small ones are parameterized. The mathematical framework for such an analysis was proposed by Anitescu and Layton [13]. Starting from the governing equations for large eddy simulation in the case where a subgrid viscosity model is utilized ∂u + ∇ · (u ⊗ u) = −∇p + ν∇2 u + ∇ · (νsgs (∆, u)S(u)) ∂t

,

∇·u = 0 ,

(9.34) (9.35)

where the notation S(u) was used instead of the usual S for the sake of convenience. Let us now introduce the sensitivities of the velocity and the pressure deﬁned as ∂p ∂u , q≡ . (9.36) v≡ ∂∆ ∂∆ These sensitivities are solutions of the following equations, which are obtained diﬀerentiating (9.34) and (9.35) with respect to ∆ ∂v ∂t

+

∇ · (v ⊗ u + u ⊗ v) = −∇q + ν∇2 v ∂ ∂ νsgs (∆, u) · v S(u) νsgs (∆, u) + +∇ · ∂u ∂∆ +∇ · νsgs (∆, u)S(v) , ∇·v =0

.

(9.37) (9.38)

Thus, once the large-eddy simulation ﬁelds u and p are computed, their sensitivities can be evaluated solving the linear problem given above. Let us illustrate this approach considering the Smagorinsky model (5.90), which corresponds to (9.39) νsgs (∆, u) = (CS ∆)2 |S(u)| . Applying the diﬀerentiation rules, one obtains the following Jacobian of the subgrid viscosity ∂ S(u) ∂ 2 2 νsgs · v = 2CS ∆|S(u)| + (CS ∆) νsgs + : S(v) . (9.40) ∂u |S(u)| ∂∆ That expression is also linear with respect to the sensitivity vector v and can easily be computed. Once the sensitivies of the solution are known, one can derive an estimate for the uncertainties on the quantities calculated from it. Let J (∆, u) be any smooth functional estimated using the large-eddy simulation results (e.g. drag and lift of an immersed body). The best value one can sought is the one associated to the exact, unﬁltered velocity ﬁeld u, i.e. J (0, u).

9.2 Correction Techniques

313

The error commited on J can be expressed as a function of the cutoﬀ length ∆ writing the following ﬁrst-order Taylor series expansion: J (0, u) = J (∆, u) − ∆J (∆, u) · v

,

(9.41)

where J is the Jacobian of J . The error is given by the second term in the right hand side. An interesting point is that this relation can also be used to estimate the exact value of the functional by correcting the value computed from the large-eddy simulation data.

9.2 Correction Techniques As relations (9.2) and (9.3) show, the statistical moments computed from the resolved ﬁeld cannot be equal to those computed from the exact solution. In order to be able to compare these moments for validation purposes, or analyze the large-eddy simulation data, the error term has to be evaluated or eliminated. Several possible techniques are described in the following for doing this. 9.2.1 Filtering the Reference Data The ﬁrst solution is to apply the same ﬁltering as was used for the scale separation to the reference solution [533, 7]. Strict comparisons can be made with this technique, but it does not provide access to theoretically usable values, which makes it diﬃcult to use the data generated by large-eddy simulation for predicting physical phenomena, because only ﬁltered data are available. In order for physical analyses to be fully satisfactory, they should be made on complete data. However, analysis is possible when the quantities considered are independent or weakly dependent on the subgrid scales3 . Moreover, this approach is diﬃcult to apply when the eﬀective ﬁlter is not known analytically, because the reference data cannot be ﬁltered consistently. It may also be diﬃcult to apply an analytical to experimental data, because in order to do so, access is needed to the data spectra that are to serve for validation or analysis. We see another source of problems cropping up here [562]: experimentally measured spectra are time spectra in the vast majority of cases, while the large-eddy simulation is based on space ﬁltering. This may introduce essential diﬀerences, especially when the ﬂow is highly anisotropic in space, as it is in the regions near a solid wall. Similar remarks can be made concerning the spatial ﬁltering of data from a direct numerical simulation for a priori test purposes: applying a one- or two-dimensional ﬁlter can produce observations that are diﬀerent from those that would be obtained with a three-dimensional ﬁlter. 3

As is generally the case for the mean velocity ﬁeld. See the examples given in Chap. 14.

314

9. Analysis and Validation

9.2.2 Evaluation of Subgrid-Scale Contribution A second solution is to evaluate the error term and reconstruct from the ﬁltered solution moments that are equal to those obtained from the full ﬁeld. Use of a De-ﬁltering Technique. One way is to try to reconstruct the full ﬁeld from the resolved one, and compute the statistical moments from the reconstructed ﬁeld. In theory, this makes it possible to obtain exact results if the reconstruction itself is exact. This reconstruction operation can be interpreted as de-ﬁltering, i.e. as an inversion of the scale separation operation. As was seen in Chap. 2, this operation is possible if the ﬁlter is an analytical one not belonging to the class of Reynolds operators. In other cases, i.e. when the eﬀective ﬁlter is unknown or possesses projector properties, this technique is not strictly applicable and we have to do with an approximate recontruction. We then use a technique based on the diﬀerential interpretation of the ﬁlter analogous to the one described in Sect. 7.2.1. With this interpretation, we can express the ﬁltered ﬁeld u as: ∞ 2n 2n ∂ u = Id + Cn ∆ u . (9.42) ∂x2n n=1 This relation can be formally inverted writing: u=

Id +

∞ n=1

2n

Cn ∆

∂ 2n ∂x2n

−1 u

,

(9.43)

and, by interpreting the diﬀerential operator as an expansion function of the small parameter ∆, we get: ∞ 2n 2n ∂ u = Id + Cn ∆ u . (9.44) ∂x2n n=1 By truncating the series at some arbitrary order, we thus get a recontruction method that is local in space and easy to use. The diﬃculty resides in the choice of the coeﬃcients Cn , which describe the eﬀective ﬁlter and can only be determined empirically. Use of a Subgrid Model. Another means that is easier to use is to compute the contribution of the subgrid terms by means of the subgrid stresses representation generated by the model used in the simulation. This technique cannot evaluate all the error terms present in (9.2) and (9.3) and can only reduce the error committed in computing the second-order moments. It does, however, oﬀer the advantage of not requiring additional computations as in the recontruction technique. It should be noted here that this technique seems to be appropriate when the models used are structural, representing the subgrid tensor, but that it

9.2 Correction Techniques

315

is no longer justiﬁed when functional models are used because these ensure only an energy balance. It is worth noting that diﬀerent subgrid models can be used for diﬀerent purposes: a model can be used during the computation to close the ﬁltered momentum equation, while another model can be used to recover a better reconstruction of subgrid contributions. This is more particulary true when the data extraction is related to multiphysics purposes. As an example, speciﬁc subgrid models have been derived for predicition of subgrid acoustic sources by Seror et al. [666, 667], while an usual functional model was used in the momentum equations. 9.2.3 Evaluation of Subgrid-Scale Kinetic Energy 2 = ui ui /2 The speciﬁc case of the evaluation of the subgrid kinetic energy qsgs (not to be assimilated to the generalized subgrid kinetic energy, see p. 54 received a lot of attention, since it is required in many applications. Several proposals have been made, which are presented below:

1. Yoshizawa’s method (p. 315), which relies on a simple dimensional analysis and requires the same inputs as the Smagorinsky functional model. Its dynamic variants are also discussed. 2. Knaepen’s dynamic model (p. 316), which is based on scale similarity concepts. 3. Models based on the integration of a spectrum shape (p. 317). These methods require the foreknowledge of the analytical spectrum shape, whose parameters (dissipation rate, Kolmogorov scale) are computed using simple subgrid models. This section presents the original version of the models. But it is observed that many elements of each model can be transposed in the other ones, leading to the deﬁnition of new models. Yoshizawa’s Model. A simple dimensional analysis yields 2

2 = 2CI ∆ |S|2 qsgs

.

(9.45)

The value of the constant is taken equal to CI = 1/π 2 in [231] and to 0.01 in [538]. A dynamic evaluation of the parameter CI based on the Germano identity was proposed by Moin et al. [538]. Denoting the test ﬁlter level with a tilde symbol, one obtains CI =

*k uk u 1 u/ k uk − * 2 / 2 2* *2 ∆ |S| − ∆ |S|2

.

(9.46)

This dynamic model may requires some numerical stabilization (like averaging over hom*ogeneous directions) in practical applications. Another

316

9. Analysis and Validation

dynamic evaluation procedure is proposed by Wong and Lilly [766], which is based on the so-called Kolmogorov scaling: 4/3

2 = CI ∆ qsgs

|S| ,

(9.47)

where the dimensional constant is evaluated as CI =

*k 1 u/ uk u k uk − * 4/3 2 * 4/3 * ∆ −∆ |S|

.

(9.48)

2 This new formulation ensures that the realizability constraint qsgs is fulﬁlled.

Knaepen’s Model. Rewriting the multiplicative Germano identity (3.80) and taking the trace of it, one obtains (the test ﬁlter level being noted with the tilde symbol) *i * *i = ui ui − u Lii ≡ u/ ui u ui i ui − *

,

(9.49)

showing that the trace of the Leonard tensor L is equal to twice the diﬀerence of the resolved kinetic energy at the two considered ﬁltering level. Assuming that the two ﬁlters are sharp cutoﬀ ﬁlters with cutoﬀ wavenumbers kc and kc , the following expression is recovered 1 Lii = 2

kc

E(k)dk

.

(9.50)

kc

If the case the spectrum shape is of the form E(k) = A k −5/3 , (9.50) yields Lii 1 . (9.51) A= 3 kc−2/3 − kc −2/3 Using this last relationship and assuming that the inertial range spectrum shape is valid at all subgrid wave numbers, one obtains the following estimate for the subgrid kinetic energy ∞ Lii 1 2 qsgs = E(k)dk = 2/3 . (9.52) * 2 kc ∆ −1 ∆ This model can be modiﬁed to account for the existence of the dissipative range of the spectrum, leading to ∞ Lii 1 2 qsgs = E(k)dk = 2/3 (9.53) (1 − γ 2/3 ) , * 2 ∆ kc − 1 ∆

9.2 Correction Techniques

317

where γ = ηK /∆ is the ratio of the ﬁlter cutoﬀ length and the Kolmogorov scale. The Kolmogorov scale can also be evaluated in the same way. Assuming that the local equilibrium hypothesis applies and that most of the viscous dissipation occurs in the inertial range, one obtains kη 2νk 2 E(k) = −τij S ij , (9.54) kc

where kη = 2π/ηK and τ is the subgrid tensor (to be evaluated with any subgrid model). As a consequence, one obtains kη =

3/4 2τij S ij kc4/3 − 3νA

,

(9.55)

leading to a closed model. Dissipation-based models. Another way to compute the subgrid kinetic energy is to assume a spectrum shape and to integrate it for all wave numbers larger than the cutoﬀ kc [517, 735, 608]. A simple spectrum shape is obtained assuming that the Kolmogorov spectrum shape is valid from kc to Jkη (with kη the Kolmogorov wave number and J a cutoﬀ parameter) and that no scale exist at higher wave number, leading to:

K0 ε2/3 k −5/3 kc ≤ k ≤ Jkη . (9.56) E(k) = 0 otherwise Integrating this expression, one obtains ⎧ 2/3 ⎨ 3K0 ε2/3 kc 1 − kc ≤ Jkη 2 2/3 Jkη 2kc qsgs = ⎩ 0 otherwise

.

(9.57)

The subgrid dissipation rate ε is not known a priori and must be evaluated. Assuming that the subgrid scale are in statistical equilibrium, the relation ε = −τij S ij is valid an can be used to compute ε, the subgrid stresses τij being parameterized using an arbitrary subgrid model. A simple subgrid viscosity model is usually suﬃcient to this end, but more complex models can be used to account for anisotropy, as proposed by Pullin [608] who uses a stretchedvortex structural model. Equation (9.57) is explicit if J = ∞. For ﬁnite values of J, the factor Jkη remains to be evaluated. A simple method to do that [517, 735] is to assume a local balance between the total dissipation on one side and the sum of the resolved-scale dissipation and the subgrid dissipation on the other side ε = ν|S|2 − τij S ij

.

(9.58)

Expressing ε and/or τij as a function of the kinetic energy spectrum, one obtains a non-linear equation for Jkη which can be solved thanks to Newton algorithm.

318

9. Analysis and Validation

9.3 Practical Experience Practice shows that nearly all authors make comparisons with reference data or analyze large-eddy simulation data with no processing of the data. The agreements observed with the reference data can then be explained by the fact that the quantities being compared are essentially related to scale ranges contained in the resolved ﬁeld. This is generally true of the ﬁrst-order moments (i.e. the mean velocity ﬁeld) and, in certain cases, of the second-order moments (the Reynolds stresses). This lack of processing prior to data analysis seems to be due mainly to the uncertainties in the techniques for evaluating the contributions of the subgrid scales and to the diﬃculty of ad hoc ﬁltering of the reference data. Large-eddy simulation also allows a satisfactory prediction of the time frequency associated with large-scale periodic or quasi-periodic phenomena (such as vortex shedding) and the ﬁrst harmonics of this frequency for ﬁne mesh. One has to be careful when trying to recover high-order statistical data from large-eddy simulation. The two main reasons are: – The properties of the exact ﬁltered solution, i.e. the solution of an ideal large-eddy simulation without numerical and modeling errors referred to as uΠ in Chap. 1, may be diﬀerent from those of the exact unﬁltered solution. – Numerical and modeling errors can corrupt the ﬁeld, yielding new errors.

Fig. 9.1. Comparison of time correlation versus a normalized time lag in isotropic turbulence. Upper curves are from Large-Eddy Simulation and lower curves from Direct Numerical Simulation. Courtesy of S. Rubinstein, NASA.

9.3 Practical Experience

319

Fig. 9.2. Time microscale versus wave number in isotropic turbulence. Solid line: Large-Eddy Simulation; Dashed line: Direct Numerical Simulation; Dotted line: theoretical −1 slope in the inertial range. Courtesy of S. Rubinstein, NASA.

Fig. 9.3. Ratio of LES time microscale with the DNS value (Symbols). Solid line shows the best linear ﬁt. Courtesy of S. Rubinstein, NASA.

320

9. Analysis and Validation

Fig. 9.4. Probability density function of ﬁltered velocity increments (experimental data), for diﬀerent ﬁlter size. Tails of the PDF is observed to be damped with increasing ﬁlter size. Courtesy of C. Meneveau, Johns Hopkins Univ.

Fig. 9.5. Comparison of probability density functions of ﬁltered velocity increments. Dashed line: ﬁltered experimental data. Others: LES computations based on diﬀerent subgrid models (Smagorinsky, dynamic Smagorinsky, dynamic functionalstrucrtural models). Courtesy of C. Meneveau, Johns Hopkins Univ.

9.3 Practical Experience

321

Fig. 9.6. Longitudinal velocity structure of order 6 and 10 plotted versus displacement. Squares: unﬁltered data; Circles: ﬁltered data. From [116].

322

9. Analysis and Validation

Very little information is available on these two sources of error. He et al. [300, 301] have shown that spectral viscosity models generate ﬁelds which are more correlated than the corresponding unﬁltered or perfect large-eddy simulation solutions (see Fig. 9.1). This increased correlation is shown to lead to a modiﬁcation of Eulerian time correlations. The subgrid closure is observed to have a signiﬁcant impact on the time microscale at all wave numbers (see Figs. 9.2 – 9.3): the value found in Large-Eddy Simulation is 1.8 times greater than in Direct Numerical Simulation, while theoretical analyses show that the value in ﬁltered Direct Numerical Simulation (or ideal LargeEddy Simulation) should lie in the range 1–1.15. From experimental grid and wake turbulence data, Cerutti and Meneveau [116] and Kang et al. [375] examined fundamental diﬀerences between ﬁltered and unﬁltered velocity ﬁelds through probability density functions and the scaling behavior of high-order structure functions (see Figs. 9.4 – 9.6). This comparative study dealing with probability density functions of velocity increments yields the conclusion that the tails of the distributions are aﬀected by the ﬁltering even at scales much larger than the ﬁlter scale. Large discrepancies are also observed with respect to the scaling of structure functions. But it is worth noting that using simple shell models of turbulence, Benzi et al. [53] have shown that the use of subgrid viscosity-like models does not preclude internal intermittency. Inertial intermittent exponents were observed to be fairly independent of the way energy is dissipated at small scales for this very simpliﬁed dynamical model of turbulence. Other eﬀects, such as the limited size of the computational domain, can also generate some discrepancies with experimental data [7]. Nevertheles, Large-Eddy Simulation can be used to recover some useful informations if the numerical error is controlled and the grid ﬁne enough: Alvelius and Johansson [11] observed a good prediction of two-point pressure-velocity correlations in hom*ogeneous turbulence.

10. Boundary Conditions

Like all the other approaches mentioned in the introduction, large-eddy simulation requires the setting of boundary conditions in order to fully determine the system and obtain a mathematically well-posed problem. This chapter is devoted to questions of determining suitable boundary conditions for largeeddy simulation computations. The ﬁrst section is a discussion of general order, the second is devoted to the representation of solid walls, and the third discusses methods used for representing an unsteady upstream ﬂow.

10.1 General Problem 10.1.1 Mathematical Aspects Many theoretical problems dealing with boundary conditions for large-eddy simulation can be identiﬁed. Two of the most important ones are the possible interaction between the ﬁlter and the exact boundary conditions, and the speciﬁcation of boundary conditions leading to the deﬁnition of a well-posed problem. From the results of Sects. 2.2 and 3.4, we deduce that the general form of the ﬁltered Navier–Stokes equations on a bounded domain Ω is ∂ui ∂ui ∂ ∂p ∂ ∂uj + (ui uj ) + −ν + = ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂ui (ξ) ∂uj (ξ) − G(x−ξ) ui (ξ)uj (ξ)+p(ξ)−ν + nj (ξ)dξ , (10.1) ∂xj ∂xi ∂Ω ∂ui =− G(x − ξ)ui (ξ)ni (ξ)dξ , (10.2) ∂xi ∂Ω where n is the outward unit normal vector to the boundary of Ω, ∂Ω. It must be noted that only terms induced by the interaction of the ﬁlter and the boundaries have been retained, i.e. other commutation errors are neglected. Right-hand side terms in (10.1) and (10.2) are additional source terms, that must be modeled because they involve the non-ﬁltered velocity and pressure ﬁelds.

324

10. Boundary Conditions

It is seen from these equations that two possibilities arise when ﬁltering the Navier–Stokes equations on bounded domains: – The ﬁrst one, which is the most commonly adopted (classical approach), consists of considering that the ﬁlter width decreases when approaching the boundaries such that the interaction term cancels out (see Fig. 10.1). Thus, the source term can be neglected and the basic ﬁltered equations are left unchanged. The remaining problem is to deﬁne classical boundary conditions for the ﬁltered ﬁeld. – The second solution (embedded boundary conditions), ﬁrst advocated by Layton et al. [206, 234, 429, 357, 207], consists of ﬁltering through the boundary (see Fig. 10.1). As a consequence, there exists a layer along the boundary, whose width is of the order of the ﬁlter cutoﬀ lengthscale, in which the source term cannot be neglected. This source term must then be explicitly computed, i.e. modeled. The discussions so far clearly show that the constitutive equations of large-eddy simulation can be of a degree diﬀerent from that of the original Navier–Stokes equations. This is trivially veriﬁed by considering the diﬀerential interpretation of the ﬁlters: the resolved equations are obtained by applying a diﬀerential operator of arbitrarily high order to the basic equations. Moreover, it has been seen that certain subgrid models generate high-order derivatives of the velocity ﬁeld. This change of degree in the resolved equations raises the problem of determining the associated boundary conditions, because those associated with the equations governing the evolution of the exact solution can no longer be used in theory for obtaining a mathematically well-posed problem [728, 234]. This problem is generally not considered, arguing the fact that the p higher-order terms appear only in the form of O(∆ ), p ≥ 1 perturbations of the Navier–Stokes equations and the same boundary conditions are used both for the large-eddy simulation and for direct numerical simulation of the Navier–Stokes equations. Moreover, when the eﬀective ﬁlter is unknown, it is no longer possible to derive suitable boundary conditions strictly, which also leads to the use of the boundary conditions of the basic problem. 10.1.2 Physical Aspects The boundary conditions, along with the similarity parameters of the equations, specify the ﬂow, i.e. determine the solution. These conditions represent the whole ﬂuid domain beyond the computational domain. To specify the solution completely, these conditions must apply to all of its scales, i.e. to all the space-time modes it comprises. So in order to characterize a particular ﬂow, the amount of information in the boundary conditions is a function of the number of degrees of freedom of the boundary condition system. This poses the problem of representing a particular solution, in order to be able to reproduce it numerically. We

10.1 General Problem

325

Fig. 10.1. Schematic of the classical approach (Top) and the embedded boundary condition approach (Bottom), in a solid wall conﬁguration. In the classical approach the cutoﬀ length ∆ is reduced in the vicinity of the wall so that the interaction term cancels out. In the second approach, the cutoﬀ length is constant. Courtesy of E. Garnier, ONERA.

326

10. Boundary Conditions

have a new modeling problem here, which is that of modeling the physical test conﬁguration. This diﬃculty is increased for the large-eddy simulation and direct numerical simulation, because these simulations contain a large number of degrees of freedom and require a precise space-time deterministic representation of the solution at the computational domain boundaries. Two special cases will be discussed in the following sections: that of representing solid walls and that of representing a turbulent inﬂow. The problem of the outﬂow conditions, which is not speciﬁc to the large-eddy simulation technique, will not be addressed1 .

10.2 Solid Walls 10.2.1 Statement of the Problem Speciﬁc Features of the Near-Wall Region. The structure of the boundary layer ﬂow has certain characteristics that call for special treatment in the framework of large-eddy simulation. In this section, we describe the elements characteristic of the boundary layer dynamics and kinematics, which shows up the diﬀerence with an isotropic hom*ogeneous turbulence. For a detailed description, the reader may refer to [648, 150]. Deﬁnitions. Here we adopt the ideal framework of a ﬂat-plate, turbulent boundary layer, without pressure gradient. The external ﬂow is in the (Ox) direction and the (Oz) direction is normal to the wall. The external velocity is denoted Ue . In the following, the Cartesian coordinate system will be denoted either (x, y, z) or (x1 , x2 , x3 ), for convenience. Similarly, the velocity vector is denoted (u, v, w) or (u1 , u2 , u3 ). We ﬁrst recall a few deﬁnitions. The boundary layer thickness δ is deﬁned as the distance from the plate beyond which the ﬂuid becomes irrotational, and thus where the ﬂuid velocity is equal to the external velocity. The wall shear stress τp is deﬁned as: + 2 2 + τp,23 , (10.3) τp = τp,13 in which τp,ij = νS ij (x, y, 0). The friction velocity uτ is deﬁned as: √ uτ = τp . In the case of the canonical boundary layer, we get: ∂u1 uτ = ν (x, y, 0) . ∂z 1

(10.4)

(10.5)

See [156] for a speciﬁc study of exit boundary conditions for the plane channel ﬂow case.

10.2 Solid Walls

327

We deﬁne the Reynolds number Reτ by: Reτ =

δuτ ν

.

(10.6)

The reduced velocity u+ , expressed in wall units, is deﬁned as: u+ = u/uτ

.

(10.7)

The wall coordinates (x+ , y + , z + ) are obtained by the transformation: (x+ , y + , z + ) = (x/lτ , y/lτ , z/lτ )

,

(10.8)

where the viscous length lτ is deﬁned as lτ = ν/uτ . Statistical Description of the Canonical Boundary Layer. The boundary layer is divided into two parts: the inner region (0 ≤ z < 0.2δ) and the outer region (0.2δ ≤ z). This decomposition is illustrated in Fig. 10.2. In the inner region, the dynamics is dominated by the viscous eﬀects. In the outer region, it is controlled by the turbulence. Each of these regions is split into several layers, corresponding to diﬀerent types of dynamics. In the case of the canonical boundary layer, we have three layers in the inner region in which the mean longitudinal velocity proﬁle follows special laws. The positions of these layers are referenced in the reduced coordinate system, because the dynamics of the inner region is dominated by the wall eﬀects and lτ is the pertinent length scale for describing the dynamics. The characteristic velocity scale is the friction velocity. These three layers are the: – Viscous sublayer: z + ≤ 5, in which + + u+ 1 (z ) = z

.

(10.9)

Fig. 10.2. Mean streamwise velocity proﬁle for the canonical turbulent boundary layer, and its decomposition into inner and outer regions. Left: mean velocity proﬁle (external units). Right: mean velocity proﬁle (wall units).

328

10. Boundary Conditions

– Buﬀer layer: 5 < z + ≤ 30, where + + u+ 1 (z ) 5 ln z − 3.05 .

(10.10)

– Prandtl or logarithmic inertial layer: 30 < z + ; z/δ 1, for which + u+ 1 (z )

1 ln z + + 5, 5 ± 0, 1, κ = 0, 4 . κ

(10.11)

The outer region includes the end of the logarithmic inertial region and the wake region. In this zone, the characteristic length is no longer lτ but rather the thickness δ. The characteristic velocity scale remains unchanged, though. The average velocity proﬁles are described by: – For the logarithmic inertial region: zuτ u1 (z) +B = A ln uτ ν

,

(10.12)

where A and B are constants; – For the wake region: Π z zuτ u1 (z) +B+ W = A ln uτ ν κ δ

,

(10.13)

where A, B and Π are constants and W the wake function deﬁned by Clauser as: (10.14) W (x) = 2 sin2 (πx/2) . Concerning the Dynamics of the Canonical Boundary Layer. Experimental and numerical studies have identiﬁed dynamic processes within the boundary layer. We will summarize here the main elements of the boundary layer dynamics that originate the turbulence in the near-wall region. Observations show that the ﬂow is highly agitated very close to the wall, consisting of pockets of fast and slow ﬂuid that organize in ribbons parallel to the outer velocity (streaks, see Fig. 10.3). The low-velocity pockets migrate slowly outward in the boundary layer (ejection) and are subject to an instability that makes them explode near the outer edge of the inner region. This burst is followed by an arrival of fast ﬂuid toward the wall, sweeping the near-wall region almost parallel to it. These highly intermittent events in time and space induce strong variation in the unsteady Reynolds stresses and originate a very large part of the production and dissipation of the turbulent kinetic energy. These variations produce ﬂuctuations in the subgrid dissipation that can reach 300% of the average value and can make it change sign. Analyses of direct numerical simulations [291, 598, 449, 363] indicate that a very intense small scale dissipation in the buﬀer region is correlated with the presence of sheared layers that form the interfaces between the ﬂuid pockets of diﬀerent velocities. These mechanisms are highly anisotropic. Their

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329

Fig. 10.3. Visualization of the streaks in a plane channel ﬂow (Reτ = 590). Dark and light gray denote opposite sign of streamwise vorticity. Courtesy of M. Terracol, ONERA.

characteristic scales in the longitudinal and transverse directions λx and λy , + respectively, are such that λ+ x ≈ 200 – 1000 and λy ≈ 100. The maximum + turbulent energy production is observed at z ≈ 15. This energy production at small scales gives rise to a high backward energy cascade and associated with the sweeping type events. The forward cascade, for its part, is associated with the ejections. In the outer regions of the boundary layer where the viscous eﬀects no longer dominate the dynamics, the energy cascade mechanism is predominant. Both cascade mechanisms are associated preferentially with the ejections. Numerical and theoretical results [353, 354, 355, 750] show that wallbounded turbulence below z + ≈ 80 is a relatively autonomous system, which is responsible for the generation of a signiﬁcant part of the turbulent kinetic energy dissipated in the outer part of the boundary layer. This turbulent cycle is often referred to as an autonomous cycle because it is observed to remain active in the absence of the outer ﬂow. This cycle involves the formation of velocity streaks from the advection of the mean velocity proﬁle by streamwise vortices, and the generation of the vortices from the instability of the streaks.

330

10. Boundary Conditions

The presence of the wall seems to be only necessary to maintain the mean shear. The way that this inner turbulent cycle and the outer ﬂow interact are still under investigation. H¨ artel and his coworkers [291, 290, 289] give a more precise analysis of the subgrid transfer in the boundary layer by introducing a new splitting2 of the subgrid dissipation ε ε = −τij S ij = εMS + εFS

,

(10.15)

with εMS = −τij S ij ε

FS

,

= −(τij − τij )(S ij − S ij )

(10.16) .

(10.17)

The εMS is related to the mean strain, and accounts for an enhancement of subgrid kinetic energy in the presence of mean-ﬂow gradients. The second term, which is linked to the strain ﬂuctuations, represents the redistribution of energy without aﬀecting the mean ﬂow directly. A priori tests [291, 290, 289] perfomed using plane channel ﬂow and circular pipe data reveal that the net eﬀect of the coupling is a forward energy transfer, and: – The mean strain part is always associated to a net forward kinetic energy cascade. – The ﬂuctuating strain part results in a net backward kinetic cascade in a zone located in the buﬀer layer, with a maximun near z + = 15. This net backward cascade is correlated to the presence of coherent events associated to turbulence production. Typical distributions of εMS and εFS are shown in Fig. 10.4. Kinematics of the Turbulent Boundary Layer. The processes described above are associated with existence of coherent structures [617]. The buﬀer layer is dominated by isolated quasi-longitudinal structures that form an average angle with the wall of 5◦ at z + = 15 and 15◦ at z + = 30. Their mean diameter increases with their distance from the wall3 . The logarithmic inertial region belongs both to the inner and outer regions, and thus contains characteristic space scales, which is compatible with the existence of two diﬀerent types of structures. The dynamics is governed by quasi-longitudinal and arch structures. The quasi-longitudinal structures can be connected to transverse structures and form an angle with the surface that varies from 15◦ to 30◦ . The span of the arch structures is of the order of the width of the slow-ﬂuid pockets at the bottom of the layer, and 2 3

It diﬀers from the splitting proposed by Shao (see Sect. 7.4.1). It should be noted that contradictory observations can be found. Lamballais [421] observes that the most probable angle of the vorticity (projected on a plane perpendicular to the wall) is close to 90◦ for 5 < z + < 25, which goes against the model of longitudinal vortices at the wall.

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331

Fig. 10.4. Distribution of mean strain (εMS ) and ﬂuctuating strain (εFS ) dissipations in a plane channel ﬂow. Fluctuating strain is observed to yield dominant backscatter near z + ∼ 15, but the total dissipation remains positive.

increases linearly with the distance from the wall. The relative number of quasi-longitudinal structures decreases with the distance from the wall, until it cancels out at the beginning of the wake region. The wake region is populated with arch structures forming an angle of 45◦ with the wall. Their x and y spacing is of the order of δ. Resolving or Modeling. The description we have just made of the boundary layer ﬂow structure clearly shows the problem of applying the large-eddy simulation technique in this case. Firstly, the mechanisms originating the turbulence, i.e. the ﬂow driving mechanisms, are associated with ﬁxed characteristic length scales on the average. Also, this turbulence production is associated with a backward energy cascade, which is largely dominant over the cascade mechanism in certain regions of the boundary layer. These two factors make it so that the subgrid models presented in the previous chapters become inoperative because they no longer permit a reduction of the number of degrees of freedom while ensuring at the same time a ﬁne representation of the ﬂow driving mechanisms. There are then two possible approaches [536]: – Resolving the near-wall dynamics directly. Since the production mechanisms escape the usual subgrid modeling, if we want to take them into account, we have to use a suﬃciently ﬁne resolution to capture them. The solid wall is then represented by a no-slip condition: the ﬂuid velocity is

332

10. Boundary Conditions

set equal to that of the solid wall. This equality implicitly relies on the hypothesis that the mean free path of the molecules is small compared with the characteristic scales of the motion, and that these scales are large compared with the distance of the ﬁrst grid point from the wall. In practice, this is done by placing the ﬁrst point in the zone (0 ≤ z + ≤ 1). To represent the turbulence production mechanisms completely, Schumann [655] + + + recommends a spatial resolution such that ∆1 < 10, ∆2 < 5 and ∆3 < 2. Also, Zang [798] indicates that the minimum resolution for capturing the + + existence of these mechanisms is ∆1 < 80, ∆2 < 30 and that three grid + points should be located in the z ≤ 10 zone. Zahrai et al. [795] indicate + + that ∆1 100, ∆2 = 12 should be used as an upper limit if a secondorder accurate numerical method is used. These values are given here only for reference, since larger values can also be found in the literature. For + example, Piomelli [591] uses ∆1 = 244 for a plane channel ﬂow. Chapman [119] estimates that representing the dynamics of the inner region, which contributes about one percent to the thickness of the full boundary layer, requires O(Re1.8 ) degrees of freedom, while only O(Re0.4 ) are needed + + to represent the outer zone. This corresponds to ∆1 100, ∆2 20 and + ∆3 < 2. Considering that non-isotropic modes must be directly resolved, Bagget et al. [32] show that the number of degrees of freedom of the solution (in space) scales as Re2τ . – Modeling the near-wall dynamics. To reduce the number degrees of freedom and especially avoid having to represent the inner region, we use a model for representing the dynamics of that zone. This is a special subgrid model called the wall model. Since the distance from the ﬁrst grid point to the wall is greater than the characteristic scales of the modes existing in the modeled region, the no-slip condition can no longer be used. The boundary condition will apply to the values of the velocity components and/or their gradients, which will be provided by the wall model. This approach makes it possible to place the ﬁrst point in the logarithmic layer (in practice, 20 ≤ z + ≤ 200). The main advantage of this approach is that the number of degrees of freedom in the simulation can be reduced greatly; but since a part of the dynamics is modeled, it constitutes an additional source of error. 10.2.2 A Few Wall Models In the following, we present the most popular wall models for large-eddy simulation. These models all represent an impermeable wall, and most of them have been implemented using a staggered grid (see Fig. 10.5). The discussion will be restricted to wall models deﬁned on Cartesian body-ﬁtted computational grids. Details dealing with implementation on curvilinear body-ﬁtted grids or on Cartesian grids using the immersed boundary technique [35, 713, 271, 270] will not be discussed, since they would require an

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333

Fig. 10.5. Illustration of a staggered grid system in the streamwise/wall-normal plane.

extensive discussion about numerical methods which is far beyond the scope of this book. Existing strategies for the deﬁnition of wall models for large-eddy simulation following the classical approach can be grouped in several classes: – Higher-order boundary conditions: velocity gradients at the wall are controlled by enforcing boundary conditions on second-order derivatives. This class is represented by Deardorﬀ’s model (p. 337). – Wall stress models: the ﬁrst mesh at the wall is chosen to be very large, so that it is not able to respresent correctly the dynamics of the inner layer. Typical dimensions are 100 wall units in the wall-normal and spanwise directions and 500 wall units in the streamwise direction. This is illustrated in Fig. 10.7, where it is clearly observed that the near-wall events are averaged over the ﬁrst grid cell. The wall model should provide the value of wall stresses, which cannot be accurately directly computed on the grid because of its coarseness (see Fig. 10.6), and the value of the wall-normal velocity component. The basic form of these stress models is τi,3 = F (u1 , z2 ) ,

i = 1, 2 ,

(10.18)

where z2 is the height of the ﬁrst cell. If the ﬂow is bidimensional and laminar, a trivial relation is τ1,3 = −ν

u1 z2 /2

.

(10.19)

A ﬁrst group of wall-stress models is based on the extrapolation of this linear, laminar law, the molecular viscosity being replaced by an eﬀective turbulent viscosity, νeﬀ . Recalling that the total mean shear stress is almost

334

10. Boundary Conditions

Fig. 10.6. Computed value of the mean wall shear stress (with reference to its exact value), expressed as a function of the distance of the ﬁrst oﬀ-wall grid point.

constant across the inner part of the boundary layer, we have: τ13 (z) = −νtot (z)

∂u1 (z) τp,13 , ∂z

(10.20)

where νtot (z) is the sum of the molecular and the turbulent viscosity. By integrating (10.20) in the wall-normal direction, we obtain: z z τ13 (z) ∂u1 (z) dz = − dz , (10.21) ∂z 0 νtot (z) 0 leading to

τp,13 0

z

1 dz = −u1 (z) , νtot (z)

(10.22)

and, ﬁnally, τp,13 = −

−1 z 1 1 u1 (z) dz z 0 νtot (z) z

.

(10.23)

νeﬀ

The wall-stress models discussed below are: 1. The Schumann model (p. 339), which relies on a linear relation between the wall stress and the velocity component at the ﬁrst oﬀ-wall grid point. The skin friction is an entry parameter for the model.

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335

Fig. 10.7. Representation of the wall-stress approach. The contours of a typical cell are superposed onto a boundary-layer instantaneous ﬂow obtained with a wallresolving mesh (instantaneous velocity vectors and isolevels of streamwise velocity ﬂuctuation). Length scales are expressed in wall units. The dimensions of the cell are 500 wall units in the streamwise direction and 100 wall units in the spanwise and wall-normal directions. Top: view in an (x, z) plane. Bottom: view in an (y, z) plane. Courtesy of E. Tromeur and E. Garnier, ONERA.

336

10. Boundary Conditions

2. The Gr¨ otzbach model (p. 339), which is an extension of the Schumann model. The skin friction is now computed assuming that the ﬂow corresponds to the canonical ﬂat-plate boundary layer. The skin friction is computed by inverting the logarithmic law proﬁle for the streamwise velocity. 3. The shifted correlations model (p. 340), which extends the Gr¨ otzbach model by taking into account explicitly the fact that ﬂuctuations are governed by coherent structures with given time- and lengthscales. 4. The ejection model (p. 341), which is another extension of the model of Gr¨ otzbach. It takes into account the eﬀects of sweep and ejection events on the wall shear stress. 5. The optimized ejection model (p. 341), which is based on experimental correlation data and yields better correlation coeﬃcient in a priori tests. 6. The model of Werner and Wengle (p. 344), which can be seen as a variant of the Gr¨ otzbach model based on power-law proﬁles for the streamwise velocity instead of the logarithmic law. The main advantage is that the power law can be inverted explicitly. 7. The modiﬁed Werner–Wengle model (p. 345), which accounts for the existence of ejection, as the model proposed by Piomelli. 8. The model of Murakami et al. (p. 343), which can be interpreted as a simpliﬁed version of the preceding model. 9. The model of Mason and Callen for rough walls (p. 340). 10. The suboptimal-control-based wall models (p. 345). Numerical experiments show that the previous wall stress models are not robust, i.e. they lead to disappointing results on very coarse mesh for high Reynolds numbers. This is mainly due to the fact that they can not account for large numerical and physical errors occuring on such coarse grids. These new models, developed within the framework of suboptimal control, aim at producing the best possible results. The second group of wall-stress models relies on the use of an internal layer near the wall, leading to the deﬁnition of two-layer simulations. Another set of governing equations is solved on an auxiliary grid located inside the ﬁrst cell of the large-eddy simulation grid. An eﬀective gain is obtained if the auxiliary simulation can be run on a much coarser grid than largeeddy simulation while guaranteeing the accuracy of the results, or if the auxiliary equations are much simpler than the Navier–Stokes equations. Models belonging to this group are: 1. The thin-boundary-layer equation model developed by Balaras et al. (p. 342). The governing equations solved in the inner layer are the boundary layer equations derived from the Navier–Stokes equations. The gain comes from the fact that the pressure is assumed to be constant across the boundary layer, and the pressure is not computed. 2. The wall model based on Kerstein’s ODT approach (p. 350). This model relies on the reconstruction of the solution in the inner layer via

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337

a simpliﬁed one-dimensional stochastic system. It can be interpreted as a surrogate of the model based on the full boundary layer equations. 3. Hybrid RANS/LES approaches, in which a RANS-like simulation is performed in the near-wall region, while the core of the ﬂow is treated by large-eddy simulation. All these models and techniques are discussed in Chap. 12, with emphasis in Sect. 12.2. – Oﬀ-wall boundary conditions: the mesh is built so that the ﬁrst point is located in the ﬂuid region, and not on the wall. The ﬁrst grid line parallel to the wall must be located in the inner region of the boundary layer. This approach is illustrated in Fig. 10.8. An important point is that the grid must still be able to represent details of the ﬂow, and thus the mesh should be the same as those used for wall-resolving simulation away from the wall. Several types of boundary conditions can be used following this approach, dealing either with the velocity components or their derivatives [30, 568, 356]. Poor results have been obtained using that approach, which requires very accurate structural information on the ﬂuctuations to yield accurate results. Jimenez and Vasco [356] showed that the ﬂow is very sensitive to the prescribed wall-normal velocity component (transpiration velocity), which must satisfy the continuity constraint.4 The need for an instantaneous non-zero transpiration velocity can easily be understood by looking at the velocity ﬁeld displayed in Fig. 10.8. The usual failure of these models leads to the appearance of a strong, spurious boundary layer above the artiﬁcial boundary. – Deterministic minimal boundary-layer unit simulation. Pascarelli et al. [580] proposed performing a wall-resolved temporal large-eddy simulation on the smallest domain allowing the existence of the near-wall autonomous cycle, and duplicating it. This approach leads to the deﬁnition of a crystal of elementary chaotic dynamical systems. The associated numerical technique corresponds to a multiblock approach. Nonlinear interactions are expected to scramble the data in the outer part of the boundary layer and to break possible periodicity. The wall-model developed by Das and Moser within the framework of embedded boundary conditions is presented on p. 349. Deardorﬀ ’s Model. In the framework of a plane channel simulation with inﬁnite Reynolds number, Deardorﬀ [172] proposes using the following conditions for representing the solid walls: 1 ∂ 2 u1 ∂ 2 u1 =− + 2 2 ∂z κ(z2 /2) ∂y 2 4

,

(10.24)

This conclusion must be considered together with the fact that the best results obtained with wall-stress models are with non-zero transpiration velocity (see suboptimal models, p. 345). A general conclusion is that even for impermeable walls a non-zero wall-normal velocity must be prescribed to accurately describe the near-wall dynamics.

338

10. Boundary Conditions

Fig. 10.8. Representation of the oﬀ-wall approach. The grid lines are superposed to a boundary-layer instantaneous ﬂow obtained with a wall-resolving mesh (instantaneous velocity vectors and isolevels of streamwise velocity ﬂuctuation). Length scales are expressed in wall units. The ﬁrst grid point is located at 100 wall units. Top: view in an (x, z) plane. Bottom: view in an (y, z) plane.

10.2 Solid Walls

u3 = 0 2

,

339

(10.25)

2

∂ u2 ∂ u2 = 2 ∂z ∂x2

,

(10.26)

where z2 is the distance from the ﬁrst point to the wall and κ = 0.4 the Von Karman constant. The ﬁrst condition assumes that the average velocity proﬁle veriﬁes the logarithmic law and that the second derivatives of the ﬂuctuation u = u − u in the y and z directions are equal. The impermeability condition (10.25) implies that the resolved stresses u1 u3 , u3 u3 and u2 u3 are zero at the wall. This model suﬀers from a number of defects. Namely, it shows no dependency as a function of the Reynolds number, and assumes that the shear-stress near the wall is entirely due to the subgrid scales. Schumann Model. Schumann [653] has developed a wall model for performing a plane channel ﬂow simulation at a ﬁnite Reynolds number. It is based on the extended turbulent relation (10.23). Using dimensional analysis, the eﬀective viscosity can be evaluated using νeﬀ =

τp z2 2 u1 (x, y, z2 )

.

The resulting boundary conditions are: u1 (x, y, z2 ) τp,13 (x, y) = τp u1 (x, y, z2 ) u3 = 0 , u3 (x, y, z2 ) 2 τp,23 (x, y) = Reτ z2

(10.27)

,

(10.28) (10.29)

,

(10.30)

where designates a statistical average (associated here with a time average), and z2 the distance of the ﬁrst point to the wall. The condition (10.28) is equivalent to adopting the hypothesis that the longitudinal velocity component at position z2 is in phase with the instantaneous wall shear stress. The mean velocity proﬁle can be obtained by the logarithmic law, and the mean wall shear stress τp is, for a plane channel ﬂow, equal to the driving pressure gradient. This wall model therefore implies that the mean velocity ﬁeld veriﬁes the logarithmic law and can be applied only to plane channel ﬂows for which the value of the driving pressure gradient is known a priori. The second condition is the impermeability condition, and the third corresponds to a no-slip condition for the transverse velocity component u2 . Gr¨ otzbach Model. Gr¨ otzbach [278] proposes extending the Schumann model to avoid having to know the mean wall shear stress a priori. To do this, the statistical average is now associated with a mean on the plane parallel to the solid wall located at z = z2 . Knowing u1 (z2 ), the mean wall

340

10. Boundary Conditions

shear stress τp is computed from the logarithmic law. The friction velocity is computed from (10.11), i.e.: u+ 1 (z2 ) = u1 (z2 )/uτ =

1 log(z2 uτ /ν) + 5.5 ± 0.1 , κ

(10.31)

then τp , by relation (10.4). This model is more general than Schumann’s, but it still requires that the mean velocity proﬁle verify the logarithmic law. Another advantage of Gr¨ otzbach’s modiﬁcation is that it allows variations of the total mass ﬂux through the channel. Shifted correlations Model. Another modiﬁcation of Schumann’s model can be made on the basis of the experimental works of Rajagopalan and Antonia [614]. These two authors observed that the correlation between the wall shear stress and the velocity increases when we consider a relaxation time between these two evaluations. This phenomenon can be explained by the existence of coherent inclined structures that are responsible for the velocity ﬂuctuations and the wall shear stress. The modiﬁed model is expressed [595]: u1 (x + ∆s , y, z2 ) τp,13 (x, y) = (10.32) τp , u1 (x, y, z2 ) τp,23 (x, y) =

u3 = 0

,

u2 (x + ∆s , y, z2 ) u1 (x, y, z2 )

(10.33)

τp ,

(10.34)

where the value of the length ∆s is given by the approximate relation: ⎧ for 30 ≤ z2+ ≤ 50–60 ⎨ (1 − z2 ) cot(8◦ ) ∆s = . (10.35) ⎩ (1 − z2 ) cot(13◦ ) for z2+ ≥ 60 Rough Wall Model. Mason and Callen [497] propose a wall model including the roughness eﬀects. The three velocity components are speciﬁed at the ﬁrst computation point by the relations: uτ (x, y) u1 (x, y, z2 ) = cos θ (10.36) ln(1 + z2 /z0 ) , κ uτ (x, y) u2 (x, y, z2 ) = sin θ (10.37) ln(1 + z2 /z0 ) , κ u3 (x, y, z2 ) = 0 ,

(10.38)

where z0 is the roughness thickness of the wall and angle θ is given by the relation θ = arctan(u2 (z2 )/u1 (z2 )). These equations can be used to compute the friction velocity uτ as a function of the instantaneous velocity components

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341

u1 and u2 . The instantaneous surface friction vector u2τ is then evaluated as: u2τ =

1 |u |u

M

,

(10.39)

where u is the vector (u1 (x, y, z2 ), u2 (x, y, z2 ), 0) and 1 1 = 2 ln2 (1 + z2 /z0 ) . M κ The instantaneous wall shear stresses in the x and y directions are then evaluated respectively as |u2τ | cos θ and |u2τ | sin θ. This model is based on the hypothesis that the logarithmic distribution is veriﬁed locally and instantaneously by the velocity ﬁeld. This becomes even truer as the grid is coarsened, and the large scale velocity approaches the mean velocity. Ejection Model. Another wall model is proposed by Piomelli, Ferziger, Moin, and Kim [595] in consideration of the fact that the fast ﬂuid motions toward or away from the wall greatly modify the wall shear stress. The impact of fast ﬂuid pockets on the wall causes the longitudinal and lateral vortex lines to stretch out, increasing the velocity ﬂuctuations near the wall. The ejection of fast ﬂuid masses induces the inverse eﬀect, i.e. reduces the wall shear stress. To represent the correlation between the wall shear stress and the velocity ﬂuctuations, the authors propose the following conditions: τp,13 (x, y) = τp − Cuτ u3 (x + ∆s , y, z2 ) , τp u2 (x + ∆s , y, z2 ) , τp,23 (x, y) = u1 (z2 ) u3 (x, y) = 0

(10.40) (10.41)

,

(10.42)

where C is a constant of the order of unity, τp is computed from the logarithmic law as it is for the Gr¨ otzbach model, and ∆s is computed by the relation (10.35). Marusic’s Optimized Ejection Model. Piomelli’s ejection wall model was futher improved by Marusic et al. [492] on the grounds of very accurate wind tunnel experiments. The proposed generalization for relation (10.40) based on the experimental correlations is τp,13 (x, y) = τp − αuτ (u1 (x + ∆s , y, z2 ) − u1 (x, y, z2 ))

,

(10.43)

where α is a parameter taken equal to 0.1 for zero pressure gradient ﬂows. This new ejection model, which is based on the streamwise velocity component instead of the wall-normal velocity in Piomelli’s original model, is found to yield better results on priori tests carried out using experimental data: the computed peak correlation coeﬃcient is in the range 0.34–0.53 for the new model while it is between 0.19 and 0.24 for the original model.

342

10. Boundary Conditions

Thin Boundary Layer Models. Balaras et al. [37] and Cabot [86, 87] propose more sophisticated models based on a system of simpliﬁed equations derived from the boundary layer equations. A secondary grid is embedded within the ﬁrst cell at the wall (see Fig. 10.9), on which the following system is resolved: ∂ ∂ui ∂ ∂p ∂ ∂ui + (u1 ui ) + (u3 ui ) = − + (ν + νsgs ) , i = 1, 2 , ∂t ∂x ∂z ∂xi ∂z ∂z (10.44) where z is the direction normal to the wall. Equation (10.44) can be recast as an equation for the shear stresses τ˜i3 = ∂ui /∂z, i = 1, 2: ∂ ∂ ∂ui ∂ ∂p ((ν + νsgs )˜ + (u1 ui ) + (u3 ui ) + τi3 ) = i = 1, 2 . (10.45) ∂z ∂t ∂x ∂z ∂xi Simpliﬁed models can be derived by neglecting some source terms in the right-hand side of (10.45) or by approximating them using values from the outer ﬂow [90]. This approach is equivalent to assuming that the inner zone of the boundary layer behaves like a Stokes layer forced by the outer ﬂow. Balaras et al. propose computing the viscosity νsgs by the simpliﬁed mixing length model: νsgs = (κz)2 Db (z)|S| ,

(10.46)

where z is the distance to the wall, κ the Von Karman constant, and Db (z) the damping function: , (10.47) Db (z) = 1 − exp(−(z + /A+ )3 )

Fig. 10.9. Representation of the primary and secondary grids.

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343

with A+ = 25. Cabot proposes the alternate deﬁnition: 2 νsgs = κus zDC (z) ,

(10.48)

in which DC (z) = (1 − exp(−zud/Aν))

,

(10.49)

where us and ud are velocity scales to be determined, and A = 19. The simplest choice is us = ud = uτ . Cabot and Moin [90] observed that the constant of the mixing length model must be lowered in regard to its usual value in RANS computations in order to account for resolved stresses. A dynamic evaluation of this constant was achieved by Wang [751] to deal with ﬂows with strong favorable/adverse pressure gradient and incipient separation. The dynamic adjustment is performed by imposing that the mixing-length viscosity and the subgrid viscosity are equal at the interface of the two simulations. When this system is solved, it generates longitudinal and transverse velocity component distributions at each time step, so that the value of the wall shear stress can be calculated for solving the ﬁltered Navier–Stokes equations on the main grid. The pressure gradient appears as a source term, because this is obtained using the relation ∂p/∂xn = 0. The vertical velocity component is obtained from the continuity equation: z2 ∂u1 ∂u2 (x, y, ξ) + (x, y, ξ) dξ . (10.50) u3 (x, y, z2 ) = − ∂x ∂y 0 The boundary conditions applied to the secondary system are: – On the solid wall: no-slip condition; – On the upper boundary: Dirichlet condition obtained from the value of the velocity ﬁeld computed on the ﬁrst cell of the main grid. Model of Murakami et al. Murakami, Mochida, and Hibi [556] developed a wall model for dealing with the case of the separated ﬂow around a cube mounted on a ﬂat plate. This model is based on power-law solutions for the mean longitudinal velocity proﬁle of the form: u1 (z) z n Ue δ

.

(10.51)

The authors recommend using n = 1/4 on the ﬂat plate and n = 1/2 on the cube surface. When the ﬁrst grid point is located close enough to the wall, the following boundary conditions are used: n z2 ui (x, y) = ui (x, y, z2 + ∆z), i = 1, 2 , (10.52) z2 + ∆z u3 (x, y) = 0

,

(10.53)

344

10. Boundary Conditions

where ∆z is the size of the ﬁrst cell. The ﬁrst equation is obtained by assuming that the instantaneous proﬁle also veriﬁes the law (10.51). When the distance of the ﬁrst point from the wall is too large for the convection eﬀects to be neglected, the relation (10.53) is replaced by: ∂u3 =0 ∂z

.

(10.54)

Werner–Wengle Model. In order to be able to compute the same ﬂow as Murakami et al., Werner and Wengle [757] propose a wall model based on the following hypotheses: – The instantaneous tangential velocity components at the wall u2 (x, y, z2 ) and u3 (x, y, z2 ) are in phase with the associated instantaneous wall shear stresses. – The instantaneous velocity proﬁle follows the law:

z+ if z + ≤ 11.81 u+ (z) = , (10.55) + B otherwise A(z ) in which A = 8.3 and B = 1/7. The values of the tangential velocity components can be related to the corresponding values of the wall shear stress components by integrating the velocity proﬁle (10.55) over the distance separating the ﬁrst cell from the wall. This allows a direct analytical evaluation of the wall shear stress components from the velocity ﬁeld: – If |ui (x, y, z2 )| ≤

ν 2/(1−B) , 2zm A

then:

τp,i3 (x, y) =

2νui (x, y, z2 ) z2

,

(10.56)

– and otherwise: τp,i3 (x, y) =

( 1+B 1+B ν ui (x, y, z2 ) 1 − B 1−B A |ui (x, y, z2 )| 2 z2 2 ) 1+B B 1+B ν + |ui (x, y, z2 )| , A z2

(10.57)

where zm is the distance to the wall that corresponds to z + = 11.81. This model has the advantage of not using average statistical values of the velocity and/or wall shear stresses, which makes it easier to use for inhom*ogeneous conﬁgurations. An impermeability condition is used to specify the value of the velocity component normal to the wall: u3 = 0

.

(10.58)

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345

Werner–Wengle-Type Ejection Model. A version of the Werner–Wengle model which accounts for the shift that exists in the correlation between the wall friction and the instantaneous velocity is proposed by Hassan and Barsamian [298]. The authors recommend to account for this shift as in Piomelli’s shifted model, leading to the following modiﬁcations for relation (10.57) ( 1+B 1+B ν ui (x + ∆s , y, z2 ) 1 − B 1−B A τp,i3 (x, y) = |ui (x + ∆s , y, z2 )| 2 z2 2 ) 1+B B 1+B ν + |ui (x + ∆s , y, z2 )| A z2 −Cuτ u3 (x + ∆s , y, z2 ) ,

(10.59)

where C is a constant of the order of the unity and ∆s is computed using the relation (10.35). Equation (10.56) is kept unchanged, since the shift is assumed to be negligible in the viscous sublayer. Suboptimal-Control-Based Wall Models. The goal of this approach is to provide numerical boundary conditions so that the overall error (to be deﬁned) is minimum in a given norm. The boundary conditions (wall stresses and wall-normal velocity component) are used as a control to minimize a cost function at each time step. Many variants of this approach can be deﬁned [566, 33], considering diﬀerent degrees of freedom at the boundary (i.e. diﬀerent controllers), diﬀerent cost functions and diﬀerent ways to evaluate the gradient of the cost function with respect to the controller. Nicoud et al. [566] and Bagget et al. [33] considered a control vector φ on the boundary whose components are the usual output of a wall-stress model: φ = (τp,13 , u3 , τp,23 ) .

(10.60)

The control can then be exerted by modifying both the stress at the wall and the wall-normal transpiration velocity. The general form of the cost function J (u; φ) is deﬁned as J (u; φ) =

3 i=1

Jmean,i (u; φ) +

3 i=1

Jrms,i (u; φ) +

3

Jpenalty (φ)

, (10.61)

i=1

where the terms appearing on the right-hand side are, from left to right: the part of the cost based on the mean ﬂow, the part of the cost based on the rms velocity ﬂuctuations, and a penalty term representing the cost of the control. The mean-ﬂow part of the cost function is typically a measure of the diﬀerence between the computed mean ﬂow and a target mean ﬂow uref . For the plane channel ﬂow, it can be expressed as h 1 eu (z)2 dz , (10.62) Jmean,i (u; φ) = αi 2h −h i

346

10. Boundary Conditions

1 (10.63) eui (z) = (ui (x, y, z) − Uref,i (z))dxdy , A where A is the surface of computational planes parallel to the walls and αi is an arbitrary weighting factor. The target mean ﬂow can be prescribed using experimental data, RANS simulations or theory. In a similar way, the rms-based cost function is deﬁned as h 1 eu (z)2 dz , (10.64) Jrms,i (u; φ) = βi 2h −h i with

with 1 eui (z) = A

(ui (x, y, z) − ui (z))2 − u2 ref,i (z) dxdy

,

(10.65)

where ui is the average of the computed ﬂow over hom*ogeneous directions, βi is an arbitrary parameter, and u2 ref,i (z) are prescribed rms velocity proﬁles. Bagget et al. proposed the following form of the penalty term: γi λ φ2ui dxdy + δi3 φ4u3 dxdy , (10.66) Jpenalty (φ) = A z=±h A z=±h where γi and λ are arbitrary constants. The last term of the penalty term prevents the transpiration velocity from becoming too high. Several tests have been carried out, dealing with diﬀerent weights of the three parts of the cost function and the possibility of having a non-zero transpiration velocity. The main results are: – Suboptimal-control-based models lead to better results than usual wallstress models (see Fig. 10.10). An interesting feature of these models is that they are able to break the spurious linear dependence of the predicted wall stresses with respect to the instantaneous velocity at the ﬁrst oﬀ-wall grid point. This is observed by looking at Figs. 10.11 and 10.12. – The use of a non-zero transpiration velocity makes it possible to improve signiﬁcantly the mean ﬂow proﬁle compared to usual wall-stress models, if the rms part of the cost function is not considered (βi = 0). But, in that case, rms velocity proﬁles are not improved and prediction can even be worse. – Rms velocity proﬁles can be improved if both Jmean,i and Jrms,i are taken into account, but the improvement of the prediction of the mean velocity proﬁle is less important than in the previous case. The use of the non-zero transpiration velocity is also observed to be beneﬁcial. This class of wall model based on the suboptimal control theory can be considered as the best achievable wall-stress model, and it seems diﬃcult to get much better results controlling the same parameters (wall stresses and transpiration velocity). Then, an interesting conclusion is that getting the

10.2 Solid Walls

347

Fig. 10.10. Large-eddy simulations of plane channel ﬂow (Reτ = 4000) on a uniform 32 × 32 × 32 grid. Mean velocity proﬁle: Symbols: 2.41 log(z + ) + 5.2; Solid line: suboptimal wall stress model, without transpiration velocity; Upper dashed line: suboptimal estimation of τ13 , with shifted wall-stress model for τ23 ; Lower dashed line: shifted wall-stress model. Courtesy of F. Nicoud, University of Montpellier.

correct mean velocity proﬁle and the correct rms velocities may be competing objectives. The preceding models are based on suboptimal control theory, and necessitate computing the gradient of the cost function. This involve a large computational eﬀort, whatever solution is adopted to compute the gradient (ﬁnite diﬀerences or solving the adjoint problem).5 The use of the incomplete gradient approach of Mohammadi and Pironneau [535] was observed to yield poor results for this problem by Templeton et al. [707]. Following the pioneering works of Bagwell et al. [34], Nicoud et al. [566] proposed a more practical wall model based on linear stochastic estimation. This model is the best possible least-square estimate of the suboptimal wall stresses as explicit functions of the local velocity ﬁeld. It can be expressed as the conditional average of the wall stress given the local velocity ﬁeld: τi,3 |E ,

(10.67)

where E is a vector of events containing the local instantaneous velocity. This formal expression does not lead to a tractable wall model, and it is approximated via a polynomial expansion. Restricting this expansion to the 5

The reported cost of a large-eddy simulation based on these models is 20 times greater than that of a simulation on the same grid with explicit wall-stress models.

348

10. Boundary Conditions

Fig. 10.11. Large-eddy simulations of plane channel ﬂow (Reτ = 4000) on a uniform 32 × 32 × 32 grid. Shifted correlation model. Instantaneous isocontours of predicted stresses and velocity components at the ﬁrst oﬀ-wall grid point. Courtesy of F. Nicoud, University of Montpellier.

ﬁrst term, one obtains the following linear stochastic estimate for the wall stresses: (10.68) τi,3 |E ≈ Lij Ej , where the estimation coeﬃcients Lij are governed by τi,3 Ek = Lij Ej Ek .

(10.69)

In practice, these coeﬃcients are computed using reference data (direct numerical simulation data or suboptimal-control-based prediction of the wall stresses). The resulting model is an explicit model, whose cost is of the same order as those of the other explicit wall-stress models, which are able to reproduce accurately the results of the suboptimally controlled simulations. Unfortunately, numerical experiments have shown this model to be very sensitive to numerical and modeling errors, indicating that it might be impossible to ﬁnd an accurate and robust6 linear wall stress model. 6

A model whose accuracy will be the same whatever numerical method and subgrid model are employed.

10.2 Solid Walls

349

Fig. 10.12. Large-eddy simulations of plane channel ﬂow (Reτ = 4000) on a uniform 32 × 32 × 32 grid. Suboptimal prediction of the wall stresses, without transpiration velocity. Instantaneous isocontours of predicted stresses and velocity components at the ﬁrst oﬀ-wall grid point. Courtesy of F. Nicoud, University of Montpellier.

Das–Moser Embedded Wall Model. In the case of an impermeable wall, the general equations (10.1) and (10.2) simplify to ∂ui ∂p ∂ ∂uj ∂ui ∂ = (ui uj ) + −ν + + ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂ui ∂uj − G(x − ξ) p(ξ) − ν (ξ) + (ξ) nj (ξ)dξ , (10.70) ∂xj ∂xi ∂Ω ∂ui =0 . ∂xi

(10.71)

The source term in the continuity equation cancels out, while the momentum source term is associated with the total force exerted on the wall. When ﬁltering through the wall, it is necessary to prolong the velocity ﬁeld inside the solid wall. This is done by assuming a zero velocity ﬁeld inside the wall in a buﬀer zone Ω∗ .

350

10. Boundary Conditions

In the present formulation, the problem is to compute the source term in (10.70). Because the unﬁltered velocity is set to zero in the wall, the wall stress is the surface forcing required to ensure that momentum is not transfered to a buﬀer domain located inside the wall, i.e. that the velocity will remain zero. Using this remark, Das and Moser [164] suggested computing the source term at each time step in order to minimize the transport of momentum from the ﬂuid domain toward the buﬀer zone. The embedded wall boundary condition is then a control minimizing the following cost function: 2 ∂u dx , J = |u|2 + α (10.72) ∂t Ω∗ where the ﬁrst term forces the energy in the buﬀer zone to be small, and the second one ensures that the momentum transfer of energy across the wall surface is small. The constant α scales like ∆t2 in plane channel ﬂow computation, with ∆t being the time step. ODT-Based Wall Model. Structural models presented in Sect. 7.7 can be used to reconstruct subgrid ﬂuctuations inside the ﬁrst cell near the solid walls. This strategy was adopted by Schmidt et al. [650], who used the ODTbased model (see p. 257) to predict the near wall dynamics, the outer part of the boundary layer being computed using a classical large-eddy simulation method. It is worth noting that this approach is reminiscent of the one proposed by Balaras based on thin boundary layer equations within the ﬁrst grid cell. The proposed coupling strategy, based on the existence of an overlap region between ODT and large-eddy simulation (see p. 257 for a detailed description of the ODT-based model), is the following. This zonal decomposition is illustrated in Fig. 10.13. An ODT-type simulation is performed within the ﬁrst grid cell of the large-eddy simulation grid. The large-eddy simulation provides the ODT with boundary conditions at the top of the ODT domain, and also appears in the evaluation of the advection speed in the ODT equation. The ODT simulation provides the large-eddy simulation with subgrid ﬂuxes in two diﬀerent regions: (i) in the inner region (i.e. the ﬁrst grid cell oﬀ the wall), the full ﬂuxes appearing in the large-eddy simulation equations is computed using the ODT ﬂuctuations and (ii) in the outer region, which has a thickness L (L being the maximum size for eddy events allowed in the ODT simulation within the inner region). In the outer region, ODT equations are not solved, but the inﬂuence of large events belonging to the ODT solution computed in the inner region is taken into account. This is achieved by adding the contribution of ODT eddy events of suﬃcient size to reach the considered position in the large-eddy simulation grid to the conventional ﬂuxes in that region.

10.2 Solid Walls

351

Fig. 10.13. Schematic of the ODT-based wall model.

10.2.3 Wall Models: Achievements and Problems Most of the wall models presented above have been developed on the grounds of the dynamics of the zero-pressure gradient, equilibrium ﬂate plate boundary layer. In practical cases, most of them exhibit the same behavior: – In attached, equilibrium ﬂows, satisfactory results (skin friction predicted within a 20% error) on the mean ﬂow are recovered on medium grids, i.e. on grids such that the ﬁrst cell oﬀ the wall has the following dimensions: ∆x+ ≤ 500,

∆y + ≤ 200 − 300,

∆z + = 50 − 150 .

On such a grid, spurious bumps are observed in the turbulence intensities just near the ﬁrst grid point (see Fig. 10.14). These unphysical overshoots are associated to the existence of large spurious streaky structures, whose size can be governed by either the mesh size or the numerical and subgrid model dissipation. The growth of these structures might be explained by the same mechanisms as the physical streaky vortices: they might be parts of an autonomous self-sustaining cycle feeded by the mean shear involving low-/high-speed streaks and streamwise vortices. Another possible cause is that they could arise from the splatting of turbulent eddies coming from the outer part of the boundary layer on the boundary condition (almost all models account for the impermeability constraint enforcing a vanishing wall normal velocity). No general cure for this problem is known. These bumps can be damped by scrambling the spurious streamwise vortices by adding a random noise in the region where they are detected [592]. Partial error cancellation when using a very dissipative subgrid model has also been observed: the spurious

352

10. Boundary Conditions

Fig. 10.14. Resolved Reynolds stresses in a plane channel ﬂow with Gr¨ otzbach wall stress model. Courtesy of Y. Benarafa and F. Ducros, CEA.

streaks are then damped. It was also shown by Nicoud that some control on the turbulence intensities can be achieved playing of the wall stresses and the wall normal velocity, but with a higher error level on the mean ﬂow proﬁle. It is worth noticing that the problem of the reduction of this error is close to the one of the active control of the boundary layer dynamics. – On very coarse grids, the models are no longer ables to yield an accurate prediction of the mean ﬂow proﬁle: the correct logarithmic slope is not recovered in the logarithmic layer, the skin friction is poorly predicted (see Fig. 10.15), ... It must be observed that on very coarse grids the problem is much more complicated: (i) the near-wall layer dynamics is not resolved, (ii) large subgrid modeling errors occur in the core of the ﬂow since a large part of the turbulent kinetic energy is contained in subgrid scales (and most subgrid scale models are not good at taking into account a large part of the full turbulent kinetic energy), and (iii) since the mesh is coarse, numerical errors may become dominant. – Most models are not very eﬃcient at predicting separation, since they are based on very stringent assumptions. This was observed, among other studies, by Temmerman et al. [706] in a wavy channel conﬁguration where the location of the separation point (and therefore of the reattachment point) is seen to be sensitive to the wall model. A noticeable exception is the Thin Boundary Layer Model, which has been proved to yield a satisfac-

10.2 Solid Walls

353

Fig. 10.15. Error on the computed skin friction versus the size of the mesh in the streamwise direction (in wall units) in plane channel ﬂow computation, with diﬀerent wall models. Steady RANS solution is shown for comparison. Courtesy of Y. Benarafa and F. Ducros, CEA.

tory prediction of an incipient separated region on smooth airfoil trailing edge [754] and on a circular cylinder at high Reynolds number [113]. In conﬁgurations where the separation point is imposed like in the backward facing step case investigated by Cabot [87], main features of the ﬂow are relatively insensitive to the way the solid wall is treated inside the separated region. But skin friction in the seprated region is of course subject to consequent errors. First trials for the deﬁnition of wall models devoted to separated regions have been done [706, 370], but satisfactory results have not yet been obtained. A main diﬃculty is the lack of universal scaling law for the mean velocity proﬁle in separated ﬂows. – Some wall-stress models require values related to the mean ﬂow as inputs. This need lead to a severe limitation of the models, since the mean ﬂow must be known, or some hom*ogeneous directions must exists to enable the evaluation of some statistiscal moments of the instantaneous ﬁelds at each time step. Possible solutions to overcome this problem are: (i) to run several statistically equivalent simulations in parallel and to perform true statistical average or (ii) to use a local spatial averaging instead of a statistical average. The latter solution is used by Hassan and Barsamian [298] to obtain a localized version of the Gr¨ otzbach model.

354

10. Boundary Conditions

10.3 Case of the Inﬂow Conditions 10.3.1 Required Conditions Representing the ﬂow upstream of the computational domain also raises difﬁculties when this ﬂow is not fully known deterministically, because the lack of information introduces sources of error. This situation is encountered for transitional or turbulent ﬂows that generally contain a very large number of space–time modes [631]. Several boundary condition generation techniques are used for furnishing information about all the modes contained to the large-eddy simulation computation. Apart the purely mathematical problem of deﬁning well-posed inﬂow boundary conditions, the Large-Eddy Simulation and the Direct Numerical Simulation techniques raise the problem of reconstructing the turbulent ﬂuctuations at the inlet plane in an accurate way. The exact deﬁnition of accurate turbulent inﬂow conditions is still an open question, but the accumlulated experience proves that both kinetic energy and coherence of the inlet ﬂuctuations must be taken into account to minimize the size of the buﬀer region that exists downstream the inlet plane, in which turbulent ﬂuctuations consistent with the Navier–Stokes dynamics are reconstructed by the non-linear eﬀects. This need is illustrated in Fig. 10.16, which displays results obtained in Direct Numerical Simulation of a two-dimensional mixing layer. 10.3.2 Inﬂow Condition Generation Techniques Stochastic Recontruction from a Statistical One-Point Description. When the freestream ﬂow is described statistically (usually the mean velocity ﬁeld and the one-point second-order moments), the deterministic information is deﬁnitively lost. The solution is then to generate instantaneous realizations that are statistically equivalent to the freestream ﬂow, i.e. that have the same statistical moments. In practice, this is done by superimposing random noises having the same statistical moments as the velocity ﬂuctuations, on the mean statistical proﬁle. This is expressed as u(x0 , t) = U (x0 ) + u (x0 , t) ,

(10.73)

where the mean ﬁeld U is given by experiment, theory or steady computations, and where the ﬂuctuation u is generated from random numbers. This technique makes it possible to remain in keeping with the energy level of the ﬂuctuations as well as the one-point correlations (Reynolds stresses) in the directions of statistical hom*ogeneity of the solution, but does not reproduce the two-point (and two-time) space–time correlations [432, 540, 493]. The information concerning the phase is lost, which can have very harmful consequences when the consistency of the ﬂuctuations is important, as is the case

10.3 Case of the Inﬂow Conditions

355

Fig. 10.16. Illustration of the inﬂuence of the turbulent inlet boundary condition (DNS of a 2D mixing layer). Iso-contours of instantaneous vorticity are shown. Top: reference 2D simulation. Below: Truncated simulation using as inﬂow conditions: a) exact instantaneous velocity ﬁeld stored at the x0 section; b) random velocity ﬂuctuations spatially and temporally uncorrelated (white noise) having the same Reynolds stress tensor components as in case (a); c) instantaneous velocity ﬁeld preserving temporal two point correlation tensor of case (a); d) instantaneous velocity ﬁeld preserving spatial two point correlation tensor of case (a); e) reconstructed velocity ﬁeld with the aid of Linear Stochastic Estimation procedure from the knowledge of exact instantaneous velocity ﬁeld at 3 reference locations (center of the mixing layer and ±δω /2 where δω is the local vorticity thickness). Courtesy of Ph. Druault and J.P. Bonnet, LEA.

356

10. Boundary Conditions

for shear ﬂows (mixing layer, jet, boundary layer, and so forth). That is, the computations performed show the existence of a region in the computational domain in which the solution regenerates the space–time consistency speciﬁc to the Navier–Stokes equations [139]. The solution is not usable in this region, which can cover a large part7 of the computational domain, and this entails an excess cost for the simulation. Also, it appears that this technique prevents the precise control of the dynamics of the solution, in the sense that it is very diﬃcult to reproduce a particular solution for a given geometry. A few ways to generate the random part of the inlet ﬂow are now presented: 1. The Lee–Lele–Moin procedure (p. 356). 2. The Smirnov–Shi–Celik procedure and Batten’s simpliﬁed version (p. 356). 3. The Li–Wang procedure (p. 358). 4. The Weighted Amplitude Wave Superposition procedure (p. 358). 5. The digital ﬁlter based method (p. 359). 6. The Arad procedure (p. 360). 7. The Yao–Sandham model (p. 361). The LLM Procedure. The ﬁrst method was proposed by Lee, Lele and Moin [432] for a ﬂow evolving in the direction x and hom*ogeneous in the two other directions, and statistically stationary in time. Assuming that the energy spectrum of a ﬂow variable φ, Eφφ , is prescribed in terms of frequency and two transverse wave numbers, the Fourier coeﬃcients of the ﬂuctuating part of φ are prescribed as + ˆ y , kz , ω, t) = Eφφ (ky , kz , ω) exp [ıψr (ky , kz , ω, t)] , (10.74) φ(k √ where ψr is the phase factor and ı = −1. The dependence of this phase factor on time and transverse wave numbers is necessary so that the signal generated is not periodic. The authors propose changing ψr only once in a given time interval Tr at a random instance by a random bounded amount ∆ψr . The resulting signal is not continuous, and the frequency spectrum of the generated turbulence is not equal to Eφφ . The authors get satisfactory results for decaying isotropic turbulence by applying this procedure to each ﬂuctuating velocity component ui , but its application to advanced transitional ﬂows or turbulent ﬂows yields the occurrence of large non-physical transition regions. The SSC Procedure. Another procedure was proposed by Smirnov, Shi and Celik [677] with application to wall-bounded ﬂows. It involves scaling and orthogonal transformation operations applied to a continuous ﬁeld generated as a superposition of harmonic functions. 7

Numerical experiments show that this region can cover more than 50% of the total number of simulation points.

10.3 Case of the Inﬂow Conditions

357

Let Rij = ui uj be the (anisotropic) velocity correlation tensor at the inlet plane (see Appendix A for a precise deﬁnition). The ﬁrst step of the SSC procedure consists of ﬁnding an orthogonal transformation tensor Aij that would diagonalize Rij (without summation over Greek indices) Aαi Aβj Rij = δαβ λ2β Aik Akj = δij

,

(10.75)

,

(10.76)

where the coeﬃcients λ1 , λ2 and λ3 play the role of turbulent ﬂuctuating velocities u1 , u2 and u3 in the new coordinate system. It is worth noting that both the transformation matrix and new coeﬃcients are functions of space. The second step consists of generating a transient ﬂow-ﬁeld V in a threedimensional domain and rescaling it. This ﬁeld is computed using the modiﬁed Kraichnan’s method: N # 2 " n ai cos(k˜jn x ˜j + ωn t˜) + bni sin(k˜jn x ˜j + ωn t˜) , (10.77) Vi (x, t) = N n=1 with x ˜j =

xj t L0 c , t˜ = , c= , k˜jn = kjn L0 T0 T0 λj n n ani = ijm ζjn km , bni = ijk ξjn km ,

ζjn , ξjn , ωn

∈ N (0, 1),

kin

∈ N (0, 1/2) ,

,

(10.78) (10.79) (10.80)

where L0 and T0 are the length- and timescales of turbulence, ijm is the permutation tensor, and N (M, σ) is a normal distribution with mean M and standard deviation σ. Quantities ωn and kjn represent a sample of n frequencies and wave number vectors of the turbulent spectrum. In practice, the authors use the following model spectrum: 2 4 k exp(−2k 2 ) . (10.81) E(k) = 16 π The last step consists of applying scaling and orthogonal transformations to V to recover the synthetic ﬂuctuating ﬁeld in physical space: Wα = λα Vα ,

ui = Aik Wk

.

(10.82)

The resulting ﬂuctuating ﬁeld is nearly divergence-free, and has correlation scales L0 and T0 with the correlation tensor Rij . This method was successfully applied to boundary-layer ﬂows. A simpliﬁed formulation is proposed by Batten et al. [49], which does not require the use of the orthogonal transformation. This simpliﬁcation is achieved redeﬁning the parameters λj as ' j 3Rlm klj km λj = , (10.83) j j 2kn kn and using the relation (10.77) to prescribe directly the velocity ﬂuctuations u instead of V .

358

10. Boundary Conditions

The Li–Wang Procedure. Another random generation technique for ﬂuctuations in a boundary-layer ﬂow was proposed by Li and Wang [446]. Fluctuations are reconstructed using the following equation: u (x, y, z, 0) =

N1 N2 N3 + √ 2 Eu (ωxl , ωy m , ωz n )∆ωx ∆ωy ∆ωz l=1 m=1 n=1

× cos(ωx l x + ωy m y + ωz n z + φlmn ) ,

(10.84)

where Eu is the target spectrum, φlmn a random phase with uniform distribution, and ωx l = (l − 1)∆ωx the angular frequency in the x direction. The periodicity is eliminated by deﬁning ωx l = ωx l + δωx

,

(10.85)

where δωx is a small random frequency. The time-evolving ﬂuctuating ﬁeld at the inlet plane is then reconstructed using Taylor’s frozen turbulence hypothesis: u (0, y, z, t) = u (x , y, z, 0),

x = Uc t ,

(10.86)

where Uc is an advecting velocity. Weighted Amplitude Wave Superposition (WAWS) spectral representation method. Another procedure relying on modiﬁed random time series to generate velocity ﬂuctuations was proposed by Glaze and Frankel [265]. This method, referred to as the WAWS method, is capable of simulating both spatial and temporal correlation. It is based on the regeneration of the ﬂuctuating signal from its cross-spectral density at the inlet plane. As a result, both spatial correlation across the inlet plane and the power spectrum of each velocity component can be enforced. Let M and N be the number of grid points at the inlet plane and the number of frequencies to be prescribed, respectively. Each velocity component is synthetized at the ith grid point using the following relation i N √ |Him (ωn )| 2∆ω cos(ωn t + θim (ωn ) + φmn ) , (10.87) ui (t) = 2 m=1 n=1

with ωn =

1 n− ∆ω , 2

ωn = ωn + δωn

,

(10.88)

where ∆ω = ωu /N , ωu and δωn ∈ [−∆ω /20, +∆ω /20] are the frequency resolution, the maximum frequency and a small random perturbation, respectively. The parameters φmn are random phases perturbations uniformly distributed between 0 and 2π. The key parameters are the components of the transfer function matrix, Him (ωn ), which are linked to the cross spectral

10.3 Case of the Inﬂow Conditions

359

density matrix components Sij (ωn ) through the relation ∗ Sij (ωn ) = Hik (ωn )Hkj (ωn )

(10.89)

where H ∗ is the Hermitian transpose of H. Since the H matrix is not unique for a given S, a method should be chosen to compute it. Glaze and Frankel used a Cholesky decomposition to obtain a lower-triangular matrix, leading to simple calculations. The θim (ωn ) parameter in (10.87) are deﬁned writing the transfer function matrix components in polar form: Him (ωn ) = |Him (ωn )|eıθim (ωn )

.

(10.90)

Since the foreknowledge of the cross-spectral density matrix S for each velocity component is not a realistic requirement in practice, the next step consists in modeling it from available data. Noticing that it can be expressed as a function of one-point spectral density and the complex coherence function γim : Sim (ωn ) = γim Sii (ωn )Smm (ωn ) , (10.91) the problem is equivalent to prescribing the power spectrum at each grid point and the coherence function. This is achieved using informations available on each ﬂow. The case of the near-ﬁeld of a turbulent jet studied in [265] is given below as an illustration. The coherence function is estimated like follows 2 A(rim + Brim ) , A = 1, B = 4 , (10.92) γim = exp − Uim where rim is the distance between points i and m and Uim is the average mean velocity between these points. The power spectrum is obtained using the von Karman model: Sii (f ) =

4f˜(u )2 f (1 + 70.8f˜2)5/6

,

(10.93)

where u is the rms turbulence intensity, f the frequency (in hertz), f˜ = Lu f /U the associated Strouhal number, with U the mean velocity and Lu the turbulent integral scale. Digital Filter Based Method. Klein, Sadiki and Janicka [396] introduced a new approach based on signal modeling through the use of linear nonrecursive ﬁlters. The general form of the discrete time series for the u velocity component at any grid point of the inﬂow plane is um =

N n=−N

bn rm+n

,

(10.94)

360

10. Boundary Conditions

where rm is a series of random data with zero mean and such that rm rn = δmn and bn are the digital ﬁlter coeﬃcients. The autocorrelation of the synthetized signal is um um+k = um um

=N

j=−N +k bj bj+k =N 2 j=−N bj

.

(10.95)

The problem consists in inverting this relation to compute the coeﬃcients of the digital ﬁlter associated to a given autocorrelation tensor. The authors propose to use a model autocorrelation function to obtain a simple form of the ﬁlter coeﬃcients. They use the following form, which is valid for fully developed hom*ogeneous turbulence (autocorrelation of the u component in the direction associated to it): 2 πr Ru u (r, 0, 0) = exp , (10.96) 4L2 where L is a prescribed integral length scale. Combining this model autocorrelation function with relation (10.95) and setting L = n∆x, one obtains =N

j=−N +k bj bj+k =N 2 j=−N bj

π(k∆x)2 πk 2 = exp − = exp − 4(n∆x)2 4n2

whose accurate approximate explicit solution is ak πk 2 bk = += , ak = exp − 2 N 4n 2 a j=−N k

.

,

(10.97)

(10.98)

This method yields the generation of a set of values with the targeted autocorrelation. Cross-correlations between velocity component can be enforced using the same change of variable as in the SSC procedure. Extension to the three-dimensional case straightforward, applying the procedure in sequentially in the three directions. This method was shown to give satifactory results in plane jet simulations. Arad’s Procedure. Arad [16] proposed a reconstruction technique for an initial condition based on physical observations related to the turbulence production process in a boundary layer. Assuming a general form of the perturbation corresponding to linearly unstable modes of the mean proﬁle, ˆi (z) exp(ı(αx + βy − ωt)), ui (x, y, z, t) = u

u ˆi (z) = Ai φ exp(−γz 2 ) , (10.99) where Ai is the amplitude of the mode and φ a random number. This idea of Arad is to design the ﬂuctuations so that their growth rate will be maximized, resulting in a short unphysical transient region near the inlet plane.

10.3 Case of the Inﬂow Conditions

361

Remarking that in a boundary layer the turbulence production term has the following form u w dU + P = , (10.100) u2τ dz + and taking into account the fact that quadrant Q2 (ejection: u < 0, w > 0) and Q4 (sweep: u > 0, w < 0) events have a dominant contribution to the shear stress u w , Arad proposed introducing a phase shift between u and w : u (x, y, z, t) = w (x, y, z, t) =

u ˆ(z) cos(αx + βy − ωt) , u ˆ(z) cos(αx + βy − ωt + π)

(10.101) (10.102)

.

The spanwise component ﬂuctuation, v , is assumed to be in phase with u in Arad’s work. The resulting ﬂuctuations correspond to sweep and ejection events. The Yao–Sandham Procedure. A more sophisticated procedure was proposed by Yao and Sandham [781, 644], which relies on the observations that ﬂuctuations in the inner and outer parts of the boundary layer have diﬀerent characteristic scales. As a consequence, speciﬁc disturbances are introduced in each part of the boundary layer. The inner-part ﬂuctuations, uinner are + : designed to represent lifted streaks with an energy maximum at zp,j + (y, z, t) = cij exp(−z + /zp,j ) sin(ωj t) cos(ky,j y + φj ) uinner i

.

(10.103)

The outer-part ﬂuctuation is assumed to be of the following form (with a peak at zp,j ) (y, z, t) = cij uouter i

z zp,j

exp(−z/zp,j ) sin(ωj t) cos(ky,j y + φj )

,

(10.104)

where subscripts i = 1, 2 and j are related to the velocity component and to the mode indices, respectively, and cij are constants. The + superscript refers to inner coordinates (wall units). The φj are phase shifts, ωj are forcing frequencies, and ky,j are spanwise wave numbers. These parameters are to be adjusted using information on the boundary-layer dynamics. The spanwise velocity component is deduced from the continuity constraint. In the inner region of the boundary layer, it is assumed that the disturbances travel downstream for a distance of 1000 wall units at a convective velocity Uc ≈ 10 uτ , where uτ is the friction velocity within a time period. The wave numbers ky,j are chosen such that there will be four streaks with a typical characteristic length of 100 wall units. In the outer region, the downstream travelling distance is taken equal to 16 and the convection velocity is Uc ≈ 0.75 U∞, where U∞ is the external velocity. The spanwise wave number is chosen to be of the order of the spanwise extent of the computational domain.

362

10. Boundary Conditions

Yao and Sandham applied this procedure to a turbulent boundary layer, taking one mode in the inner region and three in the outer region. Corresponding parameters are given in Table 10.1. They also add a random noise with a maximum amplitude of 4% of the external velocity to prevent possible spurious symmetries. Table 10.1. Coeﬃcients of the four-mode Yao–Sandham model of ﬂuctuations for boundary-layers.

inner region outer region outer region outer region

j

c1j

c2j

ωj

ky,j

φj

+ zp,j

zp,j

0 1 2 3

0.1 0.3 0.3 0.3

−0.0016 −0.06 −0.06 −0.06

0.1 0.25 0.125 0.0625

π 0.75π 0.5π 0.25π

0. 0. 0.1 0.15

12 − − −

− 1. 1.5 2.0

Deterministic Computation. Precursor Simulation. One way of minimizing the errors is to perform a simulation of the upstream ﬂow [740, 226, 649], called a precursor simulation, with a degree of resolution equivalent to that desired for the ﬁnal simulation (see Fig. 10.17). This technique almost completely eliminates the errors encountered before, and oﬀers very good results. On the other hand, it is hardly practical in the general case because it requires reproducing the entire history

Fig. 10.17. Schematic of the precursor simulation technique. A precursor simulation of an attached boundary layer ﬂow is performed. An extraction plane is deﬁned, whose data are used as an inlet boundary condition for a simulation of the ﬂow past a trailing edge.

10.3 Case of the Inﬂow Conditions

363

of the ﬂow which, for complex conﬁgurations, implies very high computation costs. Another problem stemming from this approach is that of causality: since the precursor is computed separately, no feedback of information from the second simulation is possible. This is a one-way coupling between two simulations that can become problematic when a signal (acoustic wave, for example) is emitted by the second. Li et al. [445] reduced the cost of the precursor technique by storing the results of the precursor simulation over a (relatively) short time, and cycling in time over these data to generate the ﬂow at the inlet plane of the main computation. In practice, the precursor results are stored over a time of the order of the integral timescale of the ﬂow, and windowed to get a periodic signal. This technique has been applied to a plane mixing layer ﬂow. Numerical results show that non-linear interactions quickly eliminate the spurious periodicity imposed at the inlet plane (about 25% of the total computational domain is contaminated). The eﬃciency of the method for ﬂows with lower scrambling eﬀects, such as wall-bounded ﬂows, remains to be investigated. Lund’s Extraction/Rescaling Technique. Lund et al. [463] developed a variant of the precursor approach for boundary layers, in which the information at the inlet plane is produced from that contained in the computation. There is no longer any need for a precursor. The heart of the method is a means of estimating the velocity at the inlet plane, based on the velocity ﬁeld extracted from the simulation on a plane downstream, as illustrated in Fig. 10.18. The main diﬃculty arises from the fact that the mean ﬂow is not parallel, i.e. the boundary layer thickness increases, and the ﬂow at the extraction plane must ﬁrst be rescaled before being used at the inlet plane. The ﬁrst step consists of decomposing the extracted ﬂow, ue (x, t), as the sum of a mean and a ﬂuctuating part: e uei (x, y, z, t) = ue i (x, y, z, t) + Ui (y, z) .

(10.105)

The second step consists of rescaling the mean ﬂow part using classical scalings related to the mean velocity proﬁle of the turbulent boundary layer (see Sect. 10.2.1). In practice, the rescaling is carried out according to the law of the wall in the inner region and the defect law in the outer region of the boundary layer, leading to the following relations for the streamwise component: U inner U∞ − U outer

= =

uτ (x)f1 (z + ) , uτ (x)f2 (η) ,

(10.106) (10.107)

where x is assumed to be the streamwise direction, z the wall-normal direction, U∞ the external velocity, uτ the friction velocity, η = z/δ the outer coordinate, δ the boundary-layer thickness, and f1 and f2 two universal functions to be determined. These two scaling laws dictate that the extracted mean

364

10. Boundary Conditions

Fig. 10.18. Schematic of Lund’s extraction/rescaling technique. Instantaneous isolevels of streamwise velocity in a boundary layer are shown. Courtesy of E. Tromeur and E. Garnier, ONERA.

velocity U e and the rescaled mean velocity at the inﬂow, U r , are related in the inner and outer regions via U r,inner

=

γU e (z +,r ) ,

U r,outer

=

γU e (η r ) + (1 − γ)U∞

(10.108) ,

(10.109)

with γ=

urτ ueτ

,

(10.110)

and where urτ and ueτ are skin friction at the inlet plane and extraction plane, respectively, z +,r is the inner coordinate computed at the inlet plane, and η r is the external coordinate computed at the inlet plane. A linear interpolation is used between the grid points of the planes. A similar technique is used to rescale the mean wall-normal component, yielding: W r,inner W

r,outer

= =

W e (z +,r ) , e

r

W (η ) .

(10.111) (10.112)

10.3 Case of the Inﬂow Conditions

365

The third step consists of rescaling the ﬂuctuating part of the instantaneous ﬁeld: ui,r,inner

= γu,e (y, z +,r , t) ,

(10.113)

ui,r,outer

= γu,e (y, η r , t) .

(10.114)

The last step consists of writing a composite proﬁle for the full instantaneous velocity at the inlet plane, ur , that is approximately valid over the entire boundary layer. It is deﬁned as a weighted average of the inner and outer proﬁles # " uri = Uir,inner + ui,r,inner (1 − β(η r )) 3 4 + Uir,outer + u,r,outer β(η r ) , i

(10.115)

with β(η) =

1 α(η − b) 1 + tanh /tanh(α) 2 (1 − 2b)η + b

,

(10.116)

where α = 4 and β = 0.2. This extraction/rescaling technique is observed to be eﬃcient in practice, but must be used with care. The ﬁrst point is that the extraction plane must be located far enough from the inlet to prevent spurious couplings in the computed solution. This constraint is satisﬁed by taking a distance between the two planes larger than the correlation length of the ﬂuctuations in the streamwise direction. The second point is that it is valid for fully turbulent self-similar boundary layers only, and that the scaling laws must hold to obtain a relevant procedure. Spille-Kohoﬀ–Kaltenbach Method. The extraction/rescaling technique presented above suﬀers some lack of generality, because it relies on self-similarity assumptions and can introduce some spurious couplings inside the computational domain. A more general method was proposed by Spille-Kohoﬀ and Kaltenbach [686], with application to a boundary-layer. The core of the method is the deﬁnition of a buﬀer region, referred to as the control region, near the inlet plane, where a body force is adjusted in order to recover targeted proﬁles of turbulent ﬂuctuations at a position located downstream of this buﬀer region. The body force is deﬁned within the closed-loop control theory, and makes it possible, at least theoretically, to control both rms proﬁles and integral properties of the boundary layer. This procedure is illustrated in Fig. 10.19. A random ﬂuctuation is ﬁrst speciﬁed at the inlet plane, together with a mean velocity proﬁle. The body force is applied to the wall-normal velocity component only. The rationale for this is the observation that −w w dU/dz

366

10. Boundary Conditions

Fig. 10.19. Schematic of the Spille-Kohoﬀ–Kaltenbach method. Courtesy of H. Kaltenbach, University of Berlin.

is the dominant production term in the balance equation for the shear stress −u w . The amplitude of the body force at a streamwise position x0 is adjusted via a PI controller in order to achieve a prescribed shear–stress proﬁle −u w target . This target can be deﬁned using experimental data or RANS simulations. Another way to deﬁne the targeted proﬁle is to extract and rescale it from a position xR downstream of the control region.8 The instantaneous body force in the plane x = x0 is computed as follows f (x0 , y, z, t) = A(z, t) [u(x0 , y, z, t) − uy,t (x0 , z)] 8

,

(10.117)

It is important to remark that only mean proﬁles are extracted, and not instantaneous ﬁelds as in Lund’s approach. This prevents the occurrence of spurious feedback.

10.3 Case of the Inﬂow Conditions

367

where the amplitude A(z, t) is given by A(z, t) = αe(z, t) + β

t

e(z, t )dt

,

(10.118)

where α and β are two arbitrary parameters. The averaging operator y,t is associated with the average in the spanwise (hom*ogeneous) direction and in time over a sliding window of width equal to O(10)δ/U∞ . The error term e(z, t) is a measure of the diﬀerence between the computed and the prescribed shear stress: e(z, t) = u w target (x0 , z) − u w y,t (x0 , z, t) .

(10.119)

In order to prevent unphysically large values of the shear stress, the body force is applied only at grid points satisfying the following four instantaneous constraints: |u | < 0.6 U∞ ,

|w | < 0.4 U∞ ,

u w < 0,

2 |u w | > 0.0015 U∞ . (10.120) If the extraction technique is used to specify target values inside the control region, another closed-loop controller is used to control the boundarylayer thickness at the inlet plane until the target value is obtained at the extraction plane. Similarly, the wall-normal velocity component at the top of the computational domain is controlled in order to obtain the desired streamwise pressure gradient. In this case, the authors observed that the adjustment time for reaching statistically steady values is of the order of 100 δ/U∞ .

Semideterministic Recontruction. Bonnet et al. [64, 199] propose an intermediate approach between the two previous ones, to recover the twopoint correlations of the inﬂow with no preliminary computations. The signal at the inﬂow plane is decomposed in the form u(x0 , t) = U (x0 ) + Uc (x0 , t) + u (x0 , t) ,

(10.121)

where U (x0 ) is the mean ﬁeld, Uc (x0 , t) the coherent part of turbulent ﬂuctuations, and u (x0 , t) the random part of these ﬂuctuations. In practice, this last part is generated by means of random variables and the coherent part is provided by a dynamical system with a low number of degrees of freedom (like the POD, as seen in the Introduction), or by linear stochastic estimation, which gives access to the two-point correlations.

11. Coupling Large-Eddy Simulation with Multiresolution/Multidomain Techniques

11.1 Statement of the Problem This chapter is devoted to the presentation of the coupling of large-eddy simulation with multiresolution and/or multidomain approaches. The main purpose of these couplings is to decrease the computational cost of the largeeddy simulations by clustering the degrees of freedom in regions of interest. The key idea is to adapt locally the cutoﬀ length scale of the simulation, i.e. to reﬁne the computational grid.1 This grid reﬁnement is associated with the deﬁnition of diﬀerent subdomains with varying resolution.2 The methods proposed by various research groups can be classiﬁed as follows: – Methods relying on fully overlapping subdomains (Sect. 11.2), as shown in Fig. 11.1. The term fully overlapping means here that the subdomain with the ﬁnest resolution is totally embedded within the coarsest resolution subdomain.

Fig. 11.1. Multiresolution decomposition with full overlap.

1

2

This corresponds to h-adaptivity within the framework of h–p methods. We will focus on h-adaptivity only, because these are the most employed methods, even within the ﬁnite-element framework. The problem of mesh adaptation on unstructured grids for large-eddy simulation will not be discussed here, because it has not yet been treated.

370

11. Multiresolution/Multidomain LES Techniques

The full overlap feature makes it possible to deﬁne two diﬀerent strategies: – Cycling between the diﬀerent grid resolutions, which will be considered as a time-consistent extension of the multigrid acceleration technique for steady simulations. The emphasis is put here on the fact that the cycling strategies discussed below are not associated with convergence acceleration for implicit methods for unsteady simulations but are based on consistent time-integration at each grid level. – Global resolution methods, in which time integration on all the subdomains is carried out at each time step. – Methods with partial or no overlap between the subdomains (Sect. 11.3), as illustrated in Fig. 11.2.

Fig. 11.2. Partial-overlap conﬁguration.

From mathematical and physical points of view, the underlying problem can be interpreted as coupling two solutions obtained with diﬀerent ﬁlter kernels and diﬀerent cutoﬀ length scales. An idealized problem with two domains is discussed below, which shows the full complexity of the problem. Let us note G1 and G2 , the ﬁlter kernels associated with the ﬁne and coarse resolution levels, respectively. The associated cutoﬀ length scales are ∆1 and ∆2 . The domains with ﬁne and coarse resolutions are referred to as Ω1 and Ω2 , respectively. The solution is decomposed as u = u1 + u1 in Ω1 ,

u = u2 + u2 in Ω2

.

(11.1)

The key problem is the transfer of information between Ω1 and Ω2 . Let v 12 be the complementary ﬁeld deﬁned as v 12 = u1 − u2 = G1 u − G2 u = (G1 − G2 ) u = G12 u

,

(11.2)

11.2 Methods with Full Overlap

371

where G12 is the restriction operator governing the transfer from the ﬁne to the coarse resolution level. The problem of interfacing the two domains is thus the following: – Transfer from Ω1 to Ω2 : restrict u1 on Ω2 , or equivalently substract v 12 from u1 . This corresponds to the deﬁnition of the new ﬁltering operator G21 such that G2 u = G21 G1 u . (11.3) – Transfer from Ω2 to Ω1 : deﬁlter u2 at the ∆1 level, or equivalently add v 12 to u2 . It is worth noting that most methods presented in this chapter can be deﬁned as static methods, because the number of levels of resolution is arbitrarily ﬁxed before the computation. Results dealing with dynamic methods, in which the number of levels is not ﬁxed but automatically adjusted, are very rare. Most advanced results dealing with the coupling of large-eddy simulation with Adaptive Mesh Reﬁnement (AMR) are given in Sect. 11.4.

11.2 Methods with Full Overlap According to the notation used in the previous section, the full overlap corresponds to Ω1 ∩ Ω2 = Ω1 . The methods described in this section are: – Methods with separate time-integration at each level. These methods are true multidomain methods, in the sense that the solution is integrated separately on each subdomain. The solutions are coupled via information transfer from time to time. The methods presented below are: 1. The one-way coupling procedure of Khanna and Brasseur (Sect. 11.2.1). This method is the simplest one, and represents the minimal degree of coupling between the two domains, the coarse level solution being independent of the ﬁne level solution. 2. The two-way coupling procedure of Sullivan et al. (Sect. 11.2.2). This method can be seen as an extension of the previous one, because the information transfer is now taken into account in both ways: from ﬁne to coarse level, and from coarse to ﬁne level. 3. The multilevel of Terracol et al. (Sect. 11.2.3), which is the most complete one. It relies on a two-way coupling, and also incorporates a dynamic cycling strategy. It is also the only one to incorporate a speciﬁc subgrid modeling for multilevel computations.3 – Methods with a single time-integration step. These methods are not true multidomain techniques, but rather multiblock methods. Time-integration 3

This multilevel method is an extension of the multilevel closure presented in Sect. 7.7.7 (p. 271).

372

11. Multiresolution/Multidomain LES Techniques

is carried out at the same time at each level, without any distinction between the diﬀerent resolution levels. The example given below is the method proposed by Kravchenko et al. (Sect. 11.2.4), which is based on a Galerkin method with overlapping trial functions. 11.2.1 One-Way Coupling Algorithm Khanna and Brasseur [365] developed a one-way coupling embedded grid technique for large-eddy simulation of atmospheric boundary layers. The ﬁne resolution (i.e. ﬁne grid) domain Ω1 is located in the near-wall region, in order to permit a more accurate capture of details of the ﬂow in that zone. The coarse resolution domain is noted Ω2 . The solution is integrated in each domain independently for an arbitrary time. The coupling is enforced by imposing boundary conditions on Ω1 using data coming from the coarse solution Ω2 . The proposed boundary conditions are the following: – Dirichlet conditions on the velocity components: u1 ∂Ω = u2 ∂Ω + v 12 , 1

1

(11.4)

where the coarse resolution ﬁeld u2 is obtained on the boundary of Ω1 , ∂Ω1 by a simple linear interpolation procedure. The complementary ﬁeld v 12 deﬁned by relation (11.2) is here approximated by adding random noise perturbation following a k −5/3 law within the spectral band k ∈ [π/∆2 , π/∆1 ]. – Rescaling of the subgrid viscosity. The second boundary condition is applied to the subgrid viscosity, which is rescaled in order to take into account the grid reﬁnement. Assuming that both cutoﬀs are within the inertial subrange, (5.36) leads to the following rescaling law for the subgrid viscosity at the boundary 1 4/3 ∆ 1 2 νsgs , (11.5) νsgs ∂Ω = 2 ∂Ω1 1 ∆ 1 2 where νsgs and νsgs are the values of the subgrid viscosity in Ω1 and Ω2 , respectively.

In practice, the coupling is operated at the end of each time step. 11.2.2 Two-Way Coupling Algorithm A two-way coupling procedure was proposed by Sullivan, McWilliams and Moeng [698] (also used in [62]). These authors consider a set of nested grids with increasing resolution for simulating the planetary boundary layer. The presentation of the method is restricted to a two-grid case for the sake of simplicity. The extension to an arbitrary number of grids is straightforward.

11.2 Methods with Full Overlap

373

The coupling is achieved at each time step in the following way: – From the coarse resolution level to the ﬁne resolution level: boundary conditions on ∂Ω1 are obtained by interpolating the low-resolution ﬁeld u2 (see (11.4)). The diﬀerence between this and the one-way coupling algorithm presented above is that the complementary ﬁeld v 12 is now neglected. – From the ﬁne resolution level to the coarse resolution level: the numerical ﬂuxes at the coarse resolution level are computed by ﬁltering the ﬂuxes computed at the ﬁne resolution level on the overlap region. This can be written as follows: ∂u2 + G21 N S(u1 ) = 0, in Ω1 ∩ Ω2 ∂t

,

(11.6)

where N S denotes the Navier–Stokes operator. Another possibility, as proposed by Manhart and Friedrich [483, 482] for direct numerical simulation, is to replace the coarse velocity ﬁeld with the restricted ﬁnely resolved velocity ﬁeld: u2 = G21 u1 in Ω1 ∩ Ω2 . (11.7) The data transfer is done at each time step. 11.2.3 FAS-like Multilevel Method We now discuss the most general approaches, which can be seen as generalizations of the Full Approximation Scheme (FAS) classical multigrid acceleration technique. The method proposed by Terracol et al. [633, 711, 710, 712] appears as the most general one. It deals with the use of an arbitrary number of nested resolution levels, and the gain in computational time is optimized by considering a self-adaptive time-cycling strategy between the diﬀerent levels. Other FAS-type methods have been proposed by Voke [736], Tziperman et al. [722] and Liu et al. [452, 453]. Using the multilevel framework developed in Sect. 7.7.7 (p. 271), the governing equations at each resolution level can be expressed as ∂un + N S(un ) = −τ n = −[Gn , N S](u), ∂t

n = 1, 2 .

(11.8)

Equation (11.8) shows that the coupling between the diﬀerent levels is theoretically achieved through the generalized subgrid term which arises on the right-hand side. This is seen by decomposing the commutation error at the coarse resolution level as τ2

=

[G2 , N S](u)

= =

G21 G1 N S(u) − N S(G21 G1 u) G21 (N S(G1 u) + [G1 , N S](u)) − N S(G21 u1 )

=

G21 N S(u1 ) + G21 τ 1 − N S(u2 ) .

(11.9)

374

11. Multiresolution/Multidomain LES Techniques

The ﬁrst and last terms on the right-hand side of (11.9) correspond to the direct coupling term between the two levels of resolution. It is worth noting that it is equivalent to the so-called forcing function in FAS multigrid methods. The remaining term represents the coupling between unresolved scales (“true” subgrid scales) and the coarsest level of resolution. It is the only term which requires a physical modeling, all the other terms being directly computable. On the grounds of the preceding developments, Terracol proposed a multilevel algorithm whose main elements are: – The subgrid term at level ∆2 is computed according to relation (11.9). The subgrid term at the ﬁnest resolution level can be computed with any subgrid model. In Liu’s multigrid approach, the subgrid term is replaced by a numerical stabilization term, which is essentialy equivalent to low-pass ﬁltering. – Time cycling is optimized by freezing the velocity ﬁeld at the ﬁnest level during a time T . During this period, time integration is carried out at the coarse resolution level only. The coupling from the ﬁne to the coarse level is carried out when evaluating τ 2 . The coupling from the coarse to the ﬁne level is performed by refreshing the low-frequency part of u1 at the end of the period T using the new value of u2 : u1 (t + T ) = u2 (t + T ) + v 12 (t) .

(11.10)

The time scale T is evaluated in order to satisfy the following constraint: ∂|v 12 |2 ≤ max u1 · u1 T , (11.11) 2 ∂t 2 where max is a prescribed error tolerance. Typical values for this parameter range from 10−4 to 10−3 . The resulting self-adaptive cycling obtained in a plane mixing layer conﬁguration is illustrated in Fig. 11.3. These bounds are experimentally observed to allow a decrease of the cost by a factor N for an N -grid simulation for both wall-bounded and free shear ﬂows. Voke’s and Liu’s methods are based on static cycling strategies, with empirically determined integration times at each level. Typical results are illustrated in Fig. 11.4, which displays the computed time evolution of the momentum thickness of the mixing layer obtained with diﬀerent subgrid closures at the ﬁnest resolution level.

11.2.4 Kravchenko et al. Method A zonal embedded grid technique for wall bounded ﬂows was proposed by Kravchenko et al. [411]. Contrary to the other methods with full overlap presented in this section, it does not rely on a separated time integration

11.2 Methods with Full Overlap

375

Fig. 11.3. Terracol’s self-adaptive time-cycling algorithm. Two-grid simulation of a plane mixing layer ﬂow. Time history of the number of consecutive time-steps on the coarse grid. Courtesy of M. Terracol, ONERA.

Fig. 11.4. Large-eddy simulation of a plane mixing layer using Terracol’s selfadaptive time-cycling algorithm with two resolution levels. Time evolution of the momentum thickness for diﬀerent subgrid closures at the ﬁnest resolution level. Crosses correspond to classical large-eddy simulation on the coarse level. Courtesy of M. Terracol, ONERA.

376

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on each grid. The main element of the method is the use of a Galerkintype numerical method with wide-stencil trial and weighting functions (fully non-local, such as Fourier basis, or B-splines). The transfer of information is intrinsically achieved thanks to the fact that the basis functions are not orthogonal. In fact, this method can be interpreted as a particular case of the implementation of Galerkin methods on unstructured grids.

11.3 Methods Without Full Overlap We now describe the methods without full overlap, i.e. methods designed for the case where the ﬁne-resolution domain Ω1 is not totally embedded into the coarse-resolution domain Ω2 : Ω1 ∩ Ω2 = Ω1 . The partial overlap precludes the use of multigrid-type algorithms like those presented in the previous section, and time integration must be carried synchronously in each domain. The data transfer is done at each time step (or substep for multistep schemes) in order to provide refreshed boundary conditions to the subdomains. Qu´em´er´e et al. [612] carried out an extensive analysis of the interfacing problem between subdomains with diﬀerent resolutions within the large-eddy simulation framework. The main results of their analysis are the following: – From ﬁne to coarse resolution subdomain: boundary conditions are obtained by ﬁltering the data (point values or ﬂuxes): u2 Γ = G21 u1 Γ , and/or G2 N S(u)|Γ = G21 G1 N S(u)|Γ , (11.12) with (11.13) Γ = ∂Ω1 ∩ (Ω1 ∩ Ω2 ) . – From coarse to ﬁne resolution subdomain: the deﬁnition of boundary conditions for u1 requires two successive steps: 1. Interpolation of u2 at the ﬁne resolution level ∆1 . This step is easy to implement, and does not induce speciﬁc theoretical problems. 2. Reconstruction of the complementary ﬁeld v 12 . If Γ corresponds to an exit boundary for the subdomain Ω1 , numerical experiments show that the enrichment step is not mandatory and v 12 = 0 can be used. On the contrary, if Γ is an inﬂow boundary condition for Ω1 , the reconstruction of v 12 is necessary, and the enrichment step is theoretically equivalent to the deﬁnition of turbulent inﬂow conditions (see Sect. 10.3). All the methods proposed for generating inﬂow turbulence can be used to predict v 12 . A simple extrapolation procedure from the interior of Ω1 can also be used, if the elementary convection length Uc ∆t (with Uc the characteristic advection velocity across Γ and ∆t the time step) is much smaller than the local correlation length scale of turbulent ﬂuctuations.

11.4 Coupling Large-Eddy Simulation with Adaptive Mesh Reﬁnement

377

Qu´em´er´e et al. emphasized the fact that a weighted extrapolation can be used to account for the spatial variation of the mean Reynolds stress proﬁles. 3. Rescaling of the subgrid models. The coupling can also be completed by enforcing some compatibility conditions on the subgrid terms at the interface. Inertial range arguments make it possible to derive proportionality factors. The following technique for rescaling subgrid viscosity models was proposed in [612]. Starting with the classical relations 1 2 νsgs ∝ ∆1 u1 · u1 , νsgs ∝ ∆2 u2 · u2 , (11.14) and assuming the orthogonality property u2 · u2 = (u1 + v 12 ) · (u1 + v 12 ) ≈ u1 · u1 + v 12 · v 12 we obtain the following relation ∆2 2 νsgs ∝ ∆1 u1 · u1 + v 12 · v 12 ∆1

.

,

(11.15)

(11.16)

From this last relation, we deduce a scaling law between the two viscosities 2 2 2 νsgs v 12 · v 12 ∆2 ∝ 1+ , (11.17) 1 νsgs u1 · u1 ∆1 which can be used at the interface.

11.4 Coupling Large-Eddy Simulation with Adaptive Mesh Reﬁnement 11.4.1 Statement of the Problem The theoretical advantage of coupling with Adaptive Mesh Reﬁnement (AMR) is twofold: – Decrease in the mesh size ∆x results in a reduction of the discretization error, yielding an improved numerical accuracy. – If the cutoﬀ length ∆ is tied to ∆x, mesh reﬁnement will also induce a decrease in ∆ and make the subgrid model less inﬂuential on the results (since a larger part of the exact solution is directly captured). The philosophy of the AMR approach is that the grid reﬁnement must be performed automatically during the computation, leading to the deﬁnition of several open problems:

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– Finding an error estimate (the global error or each error source individually) and deﬁning a criterion (usually a threshold value) that will trigger the reﬁnement. – Finding a bound for the algorithm: since projection error, discretization error and modeling error are present, an accurate error sensor should detect all of them, and a consistent unbounded AMR algorithm will converge toward a Direct Numerical Simulation, leading to very high computational cost. The method must then be bounded in the sense that the ﬁnal solution should correspond to a large-eddy simulation with potentially large projection error. This bound can be imposed in several ways: by ﬁxing a maximum number of degrees of freedom, by giving a minimum mesh size or playing on the threshold level in the evaluation of the error. 11.4.2 Error Estimation The criterion used to reﬁne the grid is based on the deﬁnition of an error estimate. Therefore, it relies on an a priori choice dealing with the quantities whose accuracy is the most important for the considered application. This is an arbitrary, user-dependent decision. Since it should account for both the numerical and modeling errors, the error estimate is closely tied to the numerical method and the subgrid model used in the simulation. The optimal choice is therefore case dependent, and reveals a certain degree of empiricism. The most striking achievements have been obtained by Hoﬀman [314, 310, 312, 313] using an a posteriori error estimate within a ﬁnite element framework. Less developed methods [149, 518, 268] will not be described here. The deﬁnition of an error estimate brings in many questions dealing with mathematical analysis that will not be mentioned here. The interested reader can refer to the original publications and the references given therein for more details. The presentation given below will be restricted to the main features of the method. The purpose is to improve the accuracy of the simulation by error control of the quantity (uΠ − ud ) · ψdtdx , (11.18) err(uΠ − ud ) = Q

where ψ is a vectorial test function and the integration domain Q is deﬁned as Q = Ω × I, where Ω is the volume of the computational domain and I a time interval. The two velocity ﬁelds uΠ and ud are related to the ideal solution in which only the projection error is present and the computed solution including modeling and discretization errors, respectively. The aim of an AMR procedure based on (11.18) is therefore to eliminate modeling and discretization errors. This deﬁnition of the error is the most general one, and allows the deﬁnition of optimal control methods.

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379

Hoﬀman introduces the following linearized dual problem to deﬁne an optimal error control method: −

∂uφ − uΠ · ∇uφ + ∇ud · uφ + ∇pφ − ν∇2 uφ = ψ ∂t ∇ · uφ = 0 ,

,

(11.19) (11.20)

where uφ and pφ are the dual variables. The dual system is supplemented with adequate boundary and initial conditions which account for the deﬁnition of the physical quantity of interest (drag/lift of a body, vortex shedding frequency, ...). The exact projected ﬁeld uΠ being unknown, it is replaced in practice by ud . Using these new variables, the error can be expanded as err(uΠ − ud ) = (τ (ud ) − ν∇ud ) : ∇uφ dxdt Q

discretization error ∂ud + + ud · ∇ud · uφ dxdt − ∂t Q

discretization error + (∇ · ud )pφ + pd (∇ · uπ )dxdt Q

discretization error + (Π(τ (u)) − τ (ud )) : ∇uφ dxdt , (11.21) Q

modeling error where τ (ud ) denotes the modeled subgrid tensor and Π(τ (u)) the projection of the exact subgrid tensor onto the considered basis of degrees of freedom. This expression enlights the deﬁnition of both the modeling and the numerical errors. The discretization error is a closed quantity which can be used in a straightforward way, to the contrary of the modeling error, which requires the knowledge of the projection of the exact subgrid tensor and therefore appears as an unclosed quantity. To close this equation, Hoﬀman proposes to use a structural subgrid model which exhibits a high degree of correlation with the exact subgrid tensor, like the Bardina model or approximate deconvolution models. More accurate formula including interpolation errors on the dual variables can be derived, which are not presented here for the sake of simplicity. It is worth noting that the error estimate is based on a volume integral, even in the case where the physical quantities can be expressed as surface integrals (e.g. drag, lift). This volumic formulation makes it possible to reﬁne the grid in regions where the error originates.

380

11. Multiresolution/Multidomain LES Techniques

Fig. 11.5. LES-AMR simulation of the ﬂow around a surface-mounted cube. View of the instantaneous ﬂow Top: (x–y) plane: Bottom: (x–z) plane. Courtesy of J. Hoﬀman, Courant Institute.

Fig. 11.6. LES-AMR simulation of the ﬂow around a surface-mounted cube. View of the grid after 9 reﬁnement steps. Courtesy of J. Hoﬀman, Courant Institute .

11.4 Coupling Large-Eddy Simulation with Adaptive Mesh Reﬁnement

381

Fig. 11.7. LES-AMR simulation of the ﬂow around a surface-mounted cube. Computed cube drag versus the number of grid points (in arbitrary units). Courtesy of J. Hoﬀman, Courant Institute .

Fig. 11.8. LES-AMR simulation of the ﬂow around a surface-mounted cube. Discretization (Circle) and modelling (Cross) errors (as deﬁned in 11.21) versus the number of grid points (in arbitrary units). Courtesy of J. Hoﬀman, Courant Institute.

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11. Multiresolution/Multidomain LES Techniques

The resulting algorithm is 1. Compute the solution (ud , pd ) solving the large-eddy simulation governing equation over a time interval I. 2. Solve the dual problem (11.19) – (11.20) to obtain (uφ , pφ ) over I and compute the discretization and modeling error according to (11.21). 3. Reﬁne the grid in zones where one or both error sources are higher than a ﬁxed threshold, taking into account possible bounds. 4. Loop until the targeted level of accuracy is reached, i.e. until err(uΠ −ud ) is lower than a required value. Good results were obtained dealing with the prediction of aerodynamic forces on bluﬀ bodies, using a ﬁnite-element method on unstructured meshes. Typical results obtained dealing with the prediction of the drag of a surfacemounted cube are displayed in Figs. 11.5 – 11.8.

12. Hybrid RANS/LES Approaches

12.1 Motivations and Presentation This chapter is devoted to the presentation of hybrid RANS/LES methods. The main motivation for hybridizing the two methods is to decrease the cost of the traditional large-eddy simulation method, which is large because of: – the requirement to directly capture all the scales of motion responsible for turbulence production; – the observed inability of most subgrid models to correctly account for anisotropy and disequilibrium. These two weaknesses lead to the use of very-ﬁne-resolution meshes, which can be a real problem if the characteristic lengthscale of turbulence-production events is a decreasing function of the Reynolds number. A famous problematic example is the inner region of boundary layers, whose intrinsic scale is the wall unit (see Sect. 10.2). In order to alleviate this problem, a possible solution is to blend large-eddy simulation with another technique which must be able to provide relevant lower-frequency solutions at a much lower cost. A natural proposition is to use the RANS approach, which relies on a statistical average of the exact solution and leads to a very large reduction of the number of degrees of freedom in comparison with large-eddy simulation. The presently existing techniques can be classiﬁed as follows: – Zonal decomposition (Sect. 12.2): the global computational domain is divided into subdomains, some of them being treated with the RANS method, the other ones being computed using large-eddy simulation. The gain comes from the fact that the grid resolution can be coarser in the RANS subdomains. Further cost reduction can be obtained when one or two spatial directions can be suppressed in the RANS subdomains thanks to statistical hom*ogeneity of the ﬂow in these directions. – Nonlinear Disturbance Equations (Sect. 12.3): the idea here is to compute the mean ﬂowﬁeld or the low-frequency part of the solution using a RANS or unsteady RANS simulation, and to reconstruct the missing part of the ﬂuctuating ﬁeld using a large-eddy-type simulation. This approach can be interpreted as a zonal multidomain approach in the frequency domain.

384

12. Hybrid RANS/LES Approaches

– Universal modeling (Sect. 12.4): the subgrid model is replaced by a new model, which appears as a generalized turbulence model deﬁned as a combination of a RANS model and a typical subgrid model. The hope here is to obtain robust subgrid models, which are able to deal with very coarse grids similar to those used in unsteady RANS computation. The merging with a RANS model is expected to introduce more physics into the subgrid model, rendering it eﬃcient if the cutoﬀ is located in the low-frequency part of the spectrum, outside or at the very beginning of the inertial range.

12.2 Zonal Decomposition 12.2.1 Statement of the Problem The zonal decomposition approach can be recast as a generalized multidomain/multiresolution problem. The main diﬀerence with the multidomain methods presented in Chap. 11 is that the coupling is now performed between subdomains where the scale separation is performed using operators of a diﬀerent nature: ﬁltering in large-eddy simulation and statistical averaging in RANS. Equations presented in Sect. 11.1 can be reused in the present framework, keeping in mind that now G2 is related to statistical averaging: 1 2 φ ≡ G2 (φ) = φi = φ , (12.1) N i=1,N

where N is the number of samples chosen to compute the statistical average, while G1 is still deﬁned as a convolution ﬁlter. All the theoretical analyses presented in Sect. 11.1 can be directly extended to the RANS/LES coupling method. The main practical diﬀerence is that now the problem of the reconstruction of the complementary ﬁeld v 12 is strictly equivalent to that of the deﬁnition of turbulent inﬂow conditions for large-eddy simulation described in Sect. 10.3. The mean ﬂow is now predicted using the RANS computation. Another pratical diﬀerence is the deﬁnition of the restriction operator G21 , which is deﬁned so that: G21 (G1 u) = G2 (u)

,

(12.2)

or, equivalently, and using the usual notation: G21 (u1 ) = u

.

(12.3) G21

From a purely theoretical point of view, we see that must be deﬁned as the sequential application of a deconvolution operator and statistical averaging. It can be simpliﬁed as a simple statistical average when the mean velocity proﬁles of the exact and ﬁltered solutions are equal.1 1

This condition is satisﬁed if the ﬁlter is applied in hom*ogeneous directions only.

12.2 Zonal Decomposition

385

Two main approaches are identiﬁed: – Sharp transition (Sect. 12.2.2): the reconstruction of the complementary ﬁeld is carried out explicitly at the interface between RANS and LES subdomains, yielding a sharp transition over one grid mesh from one solution to the other. – Smooth transition (Sect. 12.2.3): the reconstruction of v 12 is not performed at the interface, and the high-frequency part of the ﬂuctuating ﬁeld must be regenerated by instabilities of the mean ﬂow and the forward energy cascade. This approach leads to the existence of a transition region near the interface, whose width is case-dependent. 12.2.2 Sharp Transition A method based on the sharp transition approach was proposed by Qu´em´er´e et al. [610], and is an extension of the multidomain method proposed by the same authors (see Sect. 11.3). It relies on a strict analogy between the reconstruction of the complementary ﬁeld v 12 and the deﬁnition of the turbulent inﬂow condition. The G21 restriction operator is simply computed as a statistical average, without incorporating a deﬁltering operator. Several techniques for reconstructing the ﬂuctuations at the interface have been implemented (listed in decreasing order of eﬃciency): use of a predictor simulation, Lund’s extraction/rescaling technique, and random perturbation. Numerical experiments show that the smaller the transition region near the interface the more realistic the ﬂuctuating ﬁeld option becomes. When a predictor simulation is implemented, it is reduced to one grid cell. The method was applied with varying success to plane channel ﬂow and ﬂow around a blunt trailing edge. The RANS model was the Jones–Launder k–ε model. The boundary conditions for the turbulent quantities at the interface were derived from a simple extrapolation procedure. Typical results for a plane channel ﬂow conﬁguration are presented in Figs. 12.1 and 12.2. A simpliﬁed version of Qu´em´er´e’s approach was implemented by Georgiadis et al. [241], who operated the switch from RANS to LES while neglecting the reconstruction of the complementary ﬁeld. The purpose was the simulation of a mixing layer ﬂow, in which the primary instabilities have a very large growth rate, resulting in a small inﬂuence of the exact deﬁnition of the complementary ﬁeld. Applications of this approach to the deﬁnition of sophisticated two-layer wall models for large-eddy simulation have also been proposed by Davidson et al. [167] and Diurno et al. [180]. In both cases, the complementary ﬁeld is not reconstructed. The resolved ﬁeld is assumed to be continuous at the interface in Davidson’s method, which is based on a k −ω RANS model, while the coupling is achieved by transmiting wall stresses at the interface in Diurno’s simulations relying on the Spalart–Allmaras model. Additional boundary conditions must be provided for turbulent quantities at the RANS/LES

386

12. Hybrid RANS/LES Approaches

Fig. 12.1. Large-eddy simulation of plane channel ﬂow with zonal RANS/LES coupling. RANS subdomains are used in the near-wall region, while the core of the channel is simulated using LES. The computed mean velocity proﬁle is compared with reference data, for two position of the interface Γ . Solid and dashed lines: hybrid RANS/LES computations. Squares: classical RANS simulation. Crosses: classical LES simulation. Courtesy of P. Qu´em´er´e, ONERA.

Fig. 12.2. Large-eddy simulation of plane channel ﬂow with zonal RANS/LES coupling. RANS subdomains are used in the near-wall region, while the core of the channel is simulated using LES. The computed rms velocity proﬁles are compared with reference data obtained via typical LES simulation, for two positions of the interface Γ . Solid and dashed lines: hybrid RANS/LES computations. Symbols: classical LES simulation. Courtesy of P. Qu´em´er´e, ONERA.

12.2 Zonal Decomposition

387

interface. Davidson assumed no coupling at the interface for variables related to models and imposed arbitrary conditions, while explicit coupling is enforced in Diurno’s method. The latter assumes that the total stress is equal on both sides of the interface. 12.2.3 Smooth Transition Hybrid RANS/LES strategies presenting a smooth transition have all been derived to alleviate the classical grid resolution requirements of classical large-eddy simulation in the near-wall region. Consequently, they can be interpreted as hybrid wall-models for large-eddy simulation. Combinations between the two basic techniques are derived in two diﬀerent ways: – By performing an explicit blending of the two basic models. Bagget [31] classiﬁed the existing approaches in two groups. The ﬁrst one corresponds to the linear combination of the predicted turbulent/subgrid stresses: 4 3 1 τij − τkk δij = −νsgs S ij − (1 − β(z))S ij − β(z)νrans S ij , (12.4) 3 where νsgs and νrans are the subgrid and turbulent viscosities, respectively. The arbitrary weight β(z) is expressed as a function of the distance to the wall, and should be prescribed. This form of hybrid RANS/LES model is an extension of the splitting technique proposed by Schumann (see Sect. 6.3.3, p. 200) to account for anisotropy in the near-wall region. Methods belonging to the second group are based on the blending of the modeled viscosities: 1 τij − τkk δij = − [(1 − β(z))νsgs + β(z)νrans ] S ij 3

.

(12.5)

In both cases β = 0 corresponds to classical large-eddy simulations and β = 1 to classical RANS simulations. – Replacing the characteristic scales which explicitly appear in the RANS model equations by new scales corresponding to the LES ﬁlter. The important point is that the turbulent viscosity or the turbulent stresses are not directly rescaled, and consequently these quantities exhibit a continuous behaviour, yielding smooth transition. The most famous example belonging to this category is the Detached Eddy Simulation proposed by Spalart et al. [680, 569]. This model is based on a modiﬁcation of the Spalart– Allmaras RANS model, in which the distance to the wall d is replaced ˜ by d: (12.6) d˜ = min(d, Cdes ∆) , in the destruction term of the transport equation. The constant Cdes is calibrated in order to recover a Smagorinsky-like behaviour in isotropic turbulence at equilibrium, yielding Cdes = 0.65. The Detached Eddy Simulation approach has been extented by Strelets [16] to a large family of

388

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RANS two-equation models. For a k–ω model, the characteristic length √ lk−ω ∝ k/ω is replaced in the dissipative term of the k-transport equation by ˜l = min(lk−ω , Cdes ∆), Cdes = 0.78 . (12.7) A similar approach was developed by Tucker and Davidson [720] which relies on two-equation k–l models. In the RANS domain, a usual k–l model is used, while the Yoshizawa k–l subgrid model is used in the LES subdomain. The equations of these two models are formally equivalent, but with diﬀerent values of the constants appearing in them. The switch is achieved by changing the length scale in the transport equations: the usual RANS length scale is used in the RANS subdomain, while the subgrid lengthscale ∆ is considered in the LES subdomain. This switch results in a discontinuous value of the length scale at the interface. Tucker and Davidson obtain a regular distribution at the interface by applying a 3-point test ﬁlter across the interface to the lengthscale. Another smooth transition method was proposed by Hamba [284]. In this method, the RANS and LES subdomains are overlapping, and the length scale in the subgrid model is assumed to vary linearly from the RANS length scale to the usual subgrid length scale tied to the computational mesh. Hamba applied this technique to a channel ﬂow, hybridizing a k–ε model and a Smagorinsky model. This method yields some error in the mean velocity proﬁle, and was further improved in [285]. In the improved method, the dissipation ε is also linearly interpolated between the RANS value and the one predicted using the subgrid model. But the most important improvement factor is that the convective ﬂuxes on each side at the interface are estimated using an ad hoc interpolated velocity ﬁeld which is more compatible with the discrete mass preservation constraint. This blending of the two solutions at the interface can also be interpreted as a surrogate for the complementary ﬁeld v 12 . The improved method is observed to yield much satisfactory results on a plane channel conﬁguration. 12.2.4 Zonal RANS/LES Approach as Wall Model Both sharp- and smooth-transition hybrid methods have been used to derive new robust wall models for large-eddy simulation. Results are generally disappointing for methods which neglect the reconstruction of the complementary ﬁeld at the interface. An artiﬁcial turbulent boundary layer develops, which is composed of overly large streamwise streaks and vortices. This turbulent process is observed to be self-sustaining. This can be explained by theoretical and numerical results dealing with the existence of a near-wall autonomous cycle [353, 354, 355, 750], which is created by the mean shear. For sharp-transition methods, the problem is identical to that of wall models for large-eddy simulation, and it seems that zonal modeling cannot

12.2 Zonal Decomposition

389

perform better than the suboptimal-control-theory-based wall stress models (see Sect. 10.2). For smooth-transition methods, the interface is replaced by a transition region. An explanation [31] for the existence of a spurious near-wall cycle is that the no-slip boundary condition produces a near-wall viscous region in which the mean pressure gradient is balanced only by the mean viscous stress. The ﬂow is fully turbulent outside the transition region, so that the mean shear scales as 1/z, z being the wall-normal coordinate. In the transition region, the mean shear has to be reduced by wall-normal streamwise momentum transport, i.e. mean Reynolds stress, to couple the core ﬂow to the wall. If the subgrid model does not carry a large enough amount of Reynolds stress, the resolved motion will adjust in order to enforce the necessary balance. Some authors tried to cure this problem by artiﬁcially enhancing the dissipation in the transition region, but this method did not prove to be eﬃcient and general. For high-Reynolds-number boundary layers, the use of too coarse a grid also yields signiﬁcant errors in the mean velocity proﬁles [569]. This demonstrates that, even if the RANS region is able to provide the large-eddy simulation with correct boundary conditions, the grid resolution in the LES region must be the same as for classical wall-resolving large-eddy simulations. Another point is that Qu´em´er´e’s results seem to indicate that accounting for the complementary ﬁeld v 12 at the interface improves the results and can prevent the appearance of the spurious cycle.2 Applications to massively separated ﬂows have also been perfomed [180, 167, 146, 680, 695] which demonstrate that these hybrid approaches can yield much more satisfactory results than for attached ﬂows. The main reason is that massively separated ﬂows are mostly governed by inviscid, large-scale instabilities of the mean ﬂow, which occur in the LES region. The quality of the results is then equivalent to that of a typical large-eddy simulation, but with an eﬀective cost reduction. Strelets [695] remarked that the numerical scheme must be adjusted to the selected approach in each region to obtain reliable results: a low-order accurate dissipative method can be used in the RANS region, but not in the LES region. As a consequence, the zonal approach for physical modeling can also imply the deﬁnition of a zonal approach for the numerical method. The concept of hybridizing RANS and large-eddy simulation in the nearwall region is fully general and does not depends, from a purely formal standpoint, on the modeling approaches used in both parts. The RANS simulation is almost always carried out using models relying on zero to three additional equations, while subgrid models involving zero to two equations are very often met. A noticeable exception is provided by Tucker [719], who couples a oneequation RANS model with a MILES approach in the large-eddy simulation domain. 2

This is consistent with the ﬁnding that the use of a non-zero transpiration velocity for wall stress models yields improved results.

390

12. Hybrid RANS/LES Approaches

12.3 Nonlinear Disturbance Equations Another hybrid RANS/LES method was proposed by Morris et al. [545, 140, 288, 547], which is referred to as the Nonlinear Disturbance Equations (NLDE) method. The underlying idea is to split the ﬁeld into a low-frequency or steady part on the one hand, and a high-frequency ﬂuctuating part. The former can be computed using steady or unsteady RANS simulations, or theoretical laws, while the latter is computed via large-eddy simulation. The resulting set of governing equations appears as a generalized form of the Navier–Stokes equations in perturbation form: ∂u2 + N S(u2 ) = τ 2 ∂t

,

(12.8)

∂v 12 + N S(u1 ) − N S(u2 ) = τ 1 − τ 2 . (12.9) ∂t An interesting diﬀerence with other methods presented above is that the equation for the detail v 12 is solved, rather than the equation for the ﬁltered ﬁeld u1 . If the carrying ﬁeld u2 is steady, the time derivative in (12.8) cancels out, leading to ∂u1 ∂v 12 = . (12.10) ∂t ∂t If it is laminar, then τ 2 = 0. Using the bilinear form (3.27), the nonlinear convective term appearing in (12.9) can be recast as B(u1 , u1 ) − B(u2 , u2 ) = B(u2 , v 12 ) + B(v12 , u2 ) + B(v12 , v 12 )

I

. (12.11)

II

Terms I and II are related to the coupling between the two levels of resolution and to the nonlinear interactions between the ﬂuctuations, respectively. Another coupling is achieved through the source term in the right-hand side of (12.9), which is deﬁned as the diﬀerence between the subgrid force and the Reynolds forcing term. This approach can be interpreted as a generalized multilevel simulation (see Sects. 7.7.7 and 11.2), in which the coarsest level is deﬁned using a statistical-average operator and not a convolution ﬁlter. This technique was ﬁrst developed by Morris et al. to evaluate acoustic sources from a steady jet computation, on the basis of a nonlinear inviscid model equation for the ﬂuctuations. Their second main approximation is that mean-ﬂow source terms in the nonlinear disturbance equations can be neglected. More recently, these authors added a subgrid model to the disturbance equations, in order to take into account the eﬀect of unresolved modes, leading to a true hybrid RANS/LES approach. They also reintroduced a part of the mean ﬂow source term into the nonlinear disturbances equation. The

12.4 Universal Modeling

391

NLDE approach was used by Hansen et al. [288] to simulate the unsteady, two-dimensional laminar ﬂow around a circular cylinder, corresponding to the ﬁrst coupling between NLDE and an unsteady mean ﬂow. The ﬁrst attempt to use it for wall-bounded ﬂow is due to Chyczewski et al. [140], with fairly good results due to the use of very coarse grids. A fully general expression of the governing equations for nonlinear disturbance equations for compressible ﬂows was derived by Labourasse and Sagaut [416] who also proposed using this method to locally reconstruct the unsteady turbulent motion in subdomains embedded within the full ﬂow conﬁguration. This corresponds to a coupling between the nonlinear disturbance equations approach and the multidomain method presented above. The expected gain with respect to the classical large-eddy simulation approach is twofold: – Firstly, the mean ﬁeld being prescribed, the errors commited on turbulent ﬂuctuations will not pollute it, and we can expect this method to be more robust than classical large-eddy simulation. – Secondly, if the turbulence-producing events are localized in a small region, it will be possible to restrict the LES-type computation to a small subdomain included in the global domain, while classical large-eddy simulation would require us to consider the full domain. Numerical experiments show that this method is more robust than the usual large-eddy simulation if the mean ﬂow is correctly prescribed, i.e. coarser grids can be employed without loss of accuracy. Errors on the mean ﬂow ﬁeld u2 can be corrected using the hybrid approach on a ﬁne grid: the ﬂuctuating ﬁeld will adjust such that (u2 + G21 v 12 ) will be correct.

12.4 Universal Modeling The last class of hybrid RANS/LES methods presented in this chapter is that of the universal models. The underlying idea here is to design new models which can asymptotically recover typical RANS or typical LES capabilities. The existing models are based on the rescaling of the RANS models thanks to inertial range arguments. They all aim at decreasing the resolved kinetic energy dissipation induced by the model for the unresolved scales. This decrease in the induced dissipation leads to a weaker damping of high frequencies/high wavenumbers which can be sustained on the selected computational grid, yielding the recovery more irregular “LES-like” ﬂowﬁelds. Despite this strategy is clear and does not suﬀer any ﬂaw, it is worth noting that the diﬀerent methods used to achieve this reduction in the small scale damping3 are mostly empirical. The most common strategies found in the literature 3

The most common method is to reduce the amplitude of the turbulent viscosity of the RANS model.

392

12. Hybrid RANS/LES Approaches

deal with the modiﬁcation of eddy-viscosity-type RANS models, and can be grouped in two classes: 1. The methods based on a rescaling of the eddy viscosity provided by the RANS model. 2. The methods based on the modiﬁcation of some terms in the transport equations of the original RANS model, so as to reduce the amplitude of the eddy viscosity. In the case of most RANS models which include an evolution equation for the turbulent kinetic energy, k, this is achieved by increasing the destruction of k and/or modifying the second variable (if any). A few hybrid models are presented below: 1. 2. 3. 4.

Germano’s mixed modeling (Sect. 12.4.1). Speziale’s general rescaling method for turbulent stresses (Sect. 12.4.2). Arunajatesan’s modiﬁed two-equation model (Sect 12.4.4). Bush–Mani limiters (Sect. 12.4.5).

12.4.1 Germano’s Hybrid Model A general approach for the deﬁnition of hybrid RANS/LES models was proposed by Germano [249]. This approach relies on two basic hypotheses: – The statistical value of a ﬁltered quantity is equal to the statistical value of the unﬁltered quantity: (12.12) φ = φ . – The ﬁltered solution is characterized by the associated mean production of subgrid kinetic energy: τles,ij S ij = Cf τrans,ij Sij

,

(12.13)

where τles (u) and τrans (u) are the subgrid and Reynolds stress tensor, respectively, and Cf is a constant. The ﬁrst hypothesis makes it possible to decompose the modeled Reynolds stress tensor as follows: τrans (u) = τles (u) + τrans (u) ,

(12.14)

while the second hypothesis leads to τles,ij Sij =

Cf τrans,ij (u)Sij . 1 − Cf

(12.15)

This equation establishes a bridge between the modeled tensors at both ﬁltered and statistically averaged levels. For subgrid viscosity models, we

12.4 Universal Modeling

393

obtain the following deﬁnition: νles =

Cf |S| τrans,ij (u)Sij 2(1 − Cf ) |S|S ij Sij

.

(12.16)

The parameter Cf ∈ [0, 1] determines the respective weights of the ﬁltered and the Reynolds-averaged parts. The RANS solution is recovered as Cf −→ 1. The convergence of the method depends on the capability of the RANS model for τrans,ij , once applied to the ﬁltered ﬁeld u, to alleviate the singular behavior of (12.16). 12.4.2 Speziale’s Rescaling Method and Related Approaches Models belonging to this category can be expressed as τij = FR τijrans

,

(12.17)

where FR is the rescaling function to be deﬁned, and τijrans the modeled turbulent stress tensor predicted using any RANS model. Speziale [685, 684] proposed a general rescaling function 3 4n , (12.18) FR = 1 − exp(−β∆/ηK ) where β is a constant, n an arbitrary power, and ηK the Kolmogorov length scale computed from the variables of the RANS model. The direct numerical simulation regime is recovered as the limit of a very ﬁne resolution, ∆/ηK −→ 0. As the resolution becomes very coarse, ∆/ηK −→ ∞, the RANS model is recovered. As noted by Magnient [477], a consistency problem of the Speziale’s transition function is that large-eddy simulation is not recovered as a limit for very high Reynolds number.4 In practice, Speziale recommended applying this scaling law to complex RANS models able to deal with anisotropy and disequilibrium, such as Reynolds Stress Models or Explicit Algebraic Reynolds Stress Models. Proposed values in [685, 684] are β = 0.001 and n = 1, but the model is expected to be very sensitive to these parameters. Fasel [212] later used β = 0.004, ∆ = (∆1 ∆2 ∆3 )1/3 and ηK = Re−3/4 ε−1/4 together with an Explicit Algebraic Resynolds Stress Model to compute a ﬂat plate boundary layer (ε being evaluated from the outputs of the RANS model). The same authors used a slightly modiﬁed version to compute a wall jet ﬂow. Parameters for this conﬁguration are + 3 4 1 2 2 2 ∆1 ∆2 ∆3 , FR = 1 − exp(− max(0, 5∆ − 10ηK )/N ηK ) , ∆ = 3 (12.19) 4

A well known example is isotropic turbulence in the limit of an inﬁnite Reynolds number.

394

12. Hybrid RANS/LES Approaches

where N is an adjustable parameter that governs the cutoﬀ, which taken in the range 1000–2000 in the wall-jet application. A much simpler rescaling for computing subgrid viscosity was proposed by Peltier et al. [796, 584]. Using inertial range argument and numerical tests, these authors proposed computing the subgrid viscosity νsgs from the RANS eddy viscosity νrans as 2 ∆ νsgs = νrans , (12.20) l where l is the turbulent length computed using the RANS model outputs. Magnient [477] proposed a similar rescaling law for the RANS viscosity with the power 4/3 instead of 2. The rescaling approach was further developed by Batten and his coworkers [46, 47, 48, 49] as the Limited-Numerical-Scales method, who introduced the following rescaling factor: FR =

min [(Lsgs Usgs ), (Lrans Urans )] (Lrans Urans )

,

(12.21)

where Lsgs , Usgs , Lrans , Urans are the characteristic subgrid length scale, subgrid velocity scale, turbulent length scale and turbulent velocity scale, respectively. In the simple case where eddy viscosity type models are used in both cases, the products LU are equal to the eddy viscosities. The original implementation by Batten for several ﬂows rely on the nonlinear k–ε model of Goldberg and the Smagorinsky model. It is important noticing that the subgrid closure is used here only to evaluate the rescaling factor FR through (12.21). This method, coined as the Limited Numerical Scale approach, was mainly used through a zonal implementation. To improve the results, Batten strongly recommend to add explicit ﬂuctuations at the interface, i.e. to synthetize explicitly the ﬁeld v 12 . 12.4.3 Baurle’s Blending Strategy A more complex method, which can be interpreted as an evolution of Batten’s proposal using the same strategy as the one underlying Menter’s hybrid RANS model, is presented by Baurle and his co-workers [50]. The underlying idea is still to hybridize a classical RANS model with a standard subgrid scale model. A requirement is that both RANS and LES models include an equation on the unresolved turbulent kinetic energy (TKE). The hybrid model is therefore deﬁned by a linear combination of each model equation set: Hybrid RANS/LES

TKE equation = F × [RANS TKE equation] +(1 − F) × [LES TKE equation] , (12.22)

12.4 Universal Modeling

395

and νhybrid = F νrans + (1 − F)νsgs

.

(12.23)

where F is a weighting function to be discussed below. As remarked by Baurle, an algebraic subgrid viscosity model or even an implicit model can also be considered in this approach by constructing a production/destruction balance equation for the subgrid kinetic energy that recovers the desired subgrid viscosity. The key parameter in this approach is the weighting function F . In the case F depends only on geometric factors such as the distance to solid walls and on grid topology, the blending strategy results in a purely zonal approach reminiscent to those described in Sect. 12.2. The blending function proposed in Ref. [50] is more general and includes some information tied to the level of resolution of the ﬂow. It is deﬁned as F = max(tanh(ξ 4 ), F1 ) ,

whith ξ = max

Lrans Cd Lrans , 500ν √ d krans d2

(12.24) ,

(12.25)

where Lrans , d and krans are the turbulent length scale computed from the RANS model outputs, the distance to the nearest wall and the turbulent kinetic energy provided by the RANS model, respectively. The constant Cd is taken equal to 0.01. The function F1 is deﬁned on the grounds of Batten’s blending function (12.21):

1 if νsgs < νrans F1 = . (12.26) 0 otherwise The resulting weigthing function is such that the RANS treatment will be invoked near solid surfaces, while the LES treatment will govern the simulation in separated and free shear regions. The introduction of the F1 parameter allows to maintain the RANS treatment when sudden grid reﬁnements are encountered and that no resolved ﬂuctuations are present. A simpliﬁed version of the weighting function was developed in [772], which is expressed as 2 νsgs . (12.27) F = tanh νrans This approach is now illustrated considering the case of the two-equations k − ζ model (ζ being the turbulent enstrophy) treated in [772]. The RANS equation for the turbulent kinetic energy k is: µ µrans ∂k ∂ρk ∂ρ uj k ∂ui ∂ + + = + τij ∂t ∂xj ∂xj 3 σk ∂xj ∂xj ρk 1 µrans ∂ρ ∂p − − C1 − µζ , (12.28) 2 Ck ρ ∂xi ∂xi τp

396

12. Hybrid RANS/LES Approaches

where νrans =

k2 µrans = Cµ ρ νζ

,

(12.29)

with Cµ = 0.09, Ck , C1 and σk model-dependent parameters. The subgrid viscosity being evaluated as √ νsgs = Cs ∆ k, Cs = 0.01 , (12.30) the hybrid viscosity is deﬁned according to (12.23) while the original turbulent kinetic energy equation (12.28) is transformed into µ µhybrid ∂k ∂ρk ∂ρ uj k ∂ui ∂ + + = + τij ∂t ∂xj ∂xj 3 σk ∂xj ∂xj ρk 1 µhybrid ∂ρ ∂p −(1 − F) + C1 + µζ Ck ρ2 ∂xi ∂xi τp −FCd ρ

k 3/2 ∆

,

(12.31)

where Cd = 0.01 is a constant. The RANS turbulent length scale appearing in the computation of the blending function is Lrans =

k 3/2 νζ

.

(12.32)

12.4.4 Arunajatesan’s Modiﬁed Two-Equation Model A modiﬁed two-equation k–ε model was proposed by Arunajatesan et al. [20, 21, 22] as a basis for hybrid RANS/LES simulations. The key idea is similar to that of the Detached Eddy Simulation, i.e. some characteristic scales appearing in the classical RANS model are replaced by new ones associated with the ﬁltering operator. Two transport equations are solved along with the Navier–Stokes equations: one for the subgrid kinetic energy qsgs and one for the total dissipation rate ε. Based on the values of these quantities and the ﬁlter length ∆, a RANS and a subgrid viscosity referred to as νrans and νsgs , respectively, are computed. The subgrid viscosity is used in the equations for momentum and subgrid kinetic energy, while the RANS viscosity is used in the equation for the turbulent dissipation rate. The proposed transport equations for the subgrid kinetic energy and the dissipation rate are ∂ ∂qsgs ∂qsgs + ui qsgs − (ν + νsgs /σq ) (12.33) = Pq − ε , ∂t ∂xi ∂xi ∂ε ∂ ∂ε + ui ε − (ν + νrans /σε ) (12.34) = Pε − Dε , ∂t ∂xi ∂xi

12.4 Universal Modeling

397

where σk and σε are modeling constants equal to their usual RANS counterparts. Production and dissipation terms appearing on the right-hand sides of (12.33) and (12.34) are identical to those of the original RANS model and will not be detailed here (see [424]). Once these two quantities are known, it is assumed that the local turbulent energy spectrum can be represented everywhere using the following hybrid Von Karman/Pao form: ˘ = C1 k˘∗−5/3 ˘ k) E(

k˘ k˘∗

4 ⎛ ⎝1 +

k˘ k˘∗

2 ⎞−17/6 ⎠

exp

−9 ˘4/3 k 4

,

(12.35)

where k˘ = kηK is the normalized wave number, ηK the Kolmogorov length ˘ = E/(ν 5 ε)1/4 the normalized scale, k∗ the energy-containing wave number, E spectrum, and C1 a constant to be determined. The Kolmogorov scale is computed as ηK = (ν 3 /ε)1/4 . The constant C1 is chosen so that the integral of the dissipation range spectrum is equal to the local turbulence dissipation rate. The energy-containing wave number k∗ is computed in order to enforce the following relation: ∞

qsgs =

E(k)dk

.

(12.36)

π/∆

Once the local turbulence spectrum is completely determined, the authors propose computing the value of the subgrid viscosity for any cutoﬀ wave number using the constant spectral subgrid viscosity model (5.19), yielding ' E(π/∆) νsgs = 0.28 , (12.37) π∆ ' E(k∗ ) νrans = 0.28 . (12.38) k∗ 12.4.5 Bush–Mani Limiters Another possibility proposed by Bush and Mani [792] for deriving a hybrid model is to generalize the basic idea underlying Spalart’s Detached Eddy Simulation by applying it to all the turbulent variables. Considering the twoequation RANS model of the general form k − φ, the proposed solution is summarized in Table 12.1. In order to account for both space- and timeﬁlterings, the authors proposed redeﬁning the ﬁlter length in the dissipation term as √ * = max(∆, |u|∆t, k∆t) , ∆ (12.39)

398

12. Hybrid RANS/LES Approaches

Table 12.1. Bush–Mani limiters for hybrid RANS/LES models. Ci are constants, * is the cutoﬀ length. and ∆ φ

limiter

l ε ω

* min(l, C1 ∆) * max(ε, C2 k3/2 /∆) 1/2 * max(ω, C3 k /∆)

where ∆ is the usual cutoﬀ length used in large-eddy simulation, and ∆t the time step of the computation. The same approach was followed by Allen and Mendon¸ca [10], who tested extended versions of Spalart’s Detached Eddy Simulation based on several two-equations models. The values of the constants appearing in Table 12.1 given by these authors are C2 = 0.73 and C3 = 0.61. But numerical experiments reported by several authors indicate that the optimal value is casedependent. 12.4.6 Magagnato’s Two-Equation Model A modiﬁed two-equation model approach is proposed by Magnagato and Gabi [476], which includes an explicit random backscatter term. The original formulation of the model is based on either a non-linear k − ε or a non-linear k − τ RANS model, in which the characteristic length scale L is evaluated as L = max ∆, |u|∆t . (12.40) Here, quantities computed solving the equations of the RANS model with the modiﬁed length scale are assumed to be related to the unresolved turbulent ﬂuctuations. The dissipative part of the model is then written as 2 τij = −Cµ kτ S ij + kδij 3

,

(12.41)

,

(12.42)

for the k − τ model, and τij = −Cµ

k2 2 S ij + kδij ε 3

for the k − ε model. In both cases, the value Cµ = 0.09 is retained. The authors thus add a non-dissipative term to the expressions given above for the subgrid tensor to account for the backscatter phenomenon. The generic form of the backscatter model is 2 τij = vi vj − kδij 3

(12.43)

12.5 Toward a Theoretical Status for Hybrid RANS/LES Approaches

399

where the random velocities vi are calculated at each time step using a Langevin-type equation. Introducing ζi an independent random vector ranging from −1 to 1, the value of vi at the nth time step is deﬁned as ' ∆t ∆t ∆t 2 n n−1 n kζ , (12.44) vi = vi 1− + 2− τ τ τ 3 i where the subgrid time scale τ is directly provided by the k − τ model and is evaluated as L , (12.45) τ= √ k∞ for the k − ε model, where the length scale L is given by relation (12.40) and k∞ is the sum of the resolved and the unresolved kinetic energy. The full model formulation is obtained by adding the backscatter term to the selected dissipative part.

12.5 Toward a Theoretical Status for Hybrid RANS/LES Approaches Since they are based on the hybridization of the classical Large-Eddy Simulation and Reynolds-Averaged Numerical Simulation methods, most of the hybrid approaches presented above escape the usual theoretical framework developed to present them. As a matter of fact, the resulting ﬂow ﬁeld can neither be interpreted in terms of statistical average of the exact solution nor as the result of a conventional ﬁltering operation. Therefore, the question arises of ﬁnding a relevant paradigm to analyze and understand these hybrid approaches. A powerful framework to understand the properties of these methods is the unresolved-scale-model induced eﬀective ﬁlter paradigm developed by Muchinsky (see Sect. 8.1.1). This analysis is based on the observation that, during the computation, the only term which carries some information about the unresolved scales is the turbulence/subgrid model5 . Therefore, it is the model for the unresolved scales which determines the amount of damping (the numerical errors are assumed to be negligible here for the sake of clarity) of the resolved scales and governs the eﬀective ﬁltering of the exact solution. This is illustrated writing the resolved discrete problem as δud + Fd (ud , ud ) = Sd δt

,

(12.46)

where ud , δ/δt and Fd (·, ·) are the discrete approximations of the exact terms u, ∂/∂t and F (·, ·) on the computational grid, respectively. Here, F (·, ·) is 5

In the case of the Implicit Large-Eddy Simulation approach, this information is contained in the numerical errors.

400

12. Hybrid RANS/LES Approaches

related to the ﬂuxes in the exact Navier–Stokes equations. The source term Sd stands for the model for the unresolved scales. The discrete equation for the resolved kinetic energy is 1 δu2d + ud · Fd (ud , ud ) = ud · Sd 2 δt

.

(12.47)

The term in the right hand side of this new equation shows how the model for unresolved scales governs the dissipation process. A decrease in this dissipation will results in a larger amount of resolved kinetic energy, and therefore to the existence of smaller scales (the limit being ﬁxed by the grid resolution). All hybrid RANS/LES methods based on the modiﬁcation or the rescaling of a RANS model aim at decreasing this term. Following Muchinsky’s analysis, which relies on the idea that the RANS or LES computations can be seen as direct numerical simulations of a nonnewtonian ﬂuid, the observed eﬀects of the modiﬁcation of the original RANS models can be easily interpreted. The steady RANS solution corresponds to a steady laminar non-newtonian ﬂow. Reduction in the dissipation results in a higher Reynolds number. Unsteady RANS solutions are therefore similar to direct numerical simulations of unsteady non-newtonian ﬂow. The bifurcations sometimes observed in unsteady RANS simulations when the total dissipation is further reduced can be interpreted as analogous to bifurcations of a laminar newtonian ﬂow at low-Reynolds number (transtition to three-dimensional modes, growth of small scales, ...). The further dissipation reduction achieved by the deﬁnition of hybrid models leads to higher Reynolds numbers, and to the possiblity for smaller scales to be sustained.

13. Implementation

This chapter is devoted to the practical details of implementing the largeeddy simulation technique. The following are described: – Cutoﬀ length computation procedures for an arbitrary grid; – Discrete test ﬁlters used for computing the subgrid models or in a preﬁltering technique; – Computing the Structure Function model on an arbitrary grid. This part of the implementation of large-eddy simulation is more and more recognized as one of the keys of the success of a simulation. Most of the theoretical developments rely on an abstract ﬁlter, which is characterized by its cutoﬀ frequency. But practical experience show that computational results can be very sensitive to the eﬀective properties (transfer function, cutoﬀ length) of the ﬁtering operators used during the simulation. This is especially true of subgrid models relying on the use of a test ﬁlter, such as dynamic models and scale-similarity models. Thus, the problem of the consistency of the ﬁlter with the subgrid model must be taken into account [597, 563, 81, 641, 605].

13.1 Filter Identiﬁcation. Computing the Cutoﬀ Length The theoretical developments of the previous chapters have identiﬁed several ﬁlters of diﬀerent origins: 1. Analytical ﬁlter, represented by a convolution product. This is the ﬁlter used for expressing the ﬁltered Navier–Stokes equations. 2. Filter associated with a given computational grid. No frequency higher than the Nyquist frequency associated with this grid can be represented in the simulation. 3. Filter induced by the numerical scheme. The error committed by approximating the partial derivative operators by discrete operators modiﬁes the computed solution mainly the high-frequency modes. 4. Filter associated with the subgrid model, which acts like a control process on the computed solution.

402

13. Implementation

The computed solution is the result of these four ﬁltering processes constituting the simulation eﬀective ﬁlter. When performing a computation, then, the question arises as to what the eﬀective ﬁlter is, that governs the dynamics of the numerical solution, in order to determine the characteristic cutoﬀ length. This length is needed for several reasons. – In order to be able to determine the physically and numerically wellresolved scale beyond which we will be able to start using the results for analysis. – In order to be able to use the subgrid models like the subgrid viscosity models that use this cutoﬀ length explicitly. While the ﬁlters mentioned above are deﬁnable theoretically, they are almost never quantiﬁable in practice. This is particularly true of the ﬁlter associated with the numerical schemes used. In face of this uncertainty, practitioners have one of two positions they can adopt: 1. Arrange it so that one of the four ﬁlters becomes predominant over the others and is controllable. The eﬀective ﬁlter is then known. This is done in practice by using a pre-ﬁltering technique. Normally, this is done by ensuring the dominance of the analytical ﬁlter, which allows us strict control of the form of the ﬁlter and of its cutoﬀ length, so that we can get the most out of the theoretical analyses and thereby minimize the relative uncertainty concerning the nature of the computed solution. In the numerical solution, an analytical ﬁlter is then applied here to each computed term. In order for this ﬁlter to be dominant, its cutoﬀ length must be large compared with the other three. Theoretically, this analytical ﬁlter should be a convolution ﬁlter which, to keep the computation cost within acceptable limits, can only be applied for simulations performed in the spectral space1 . For the simulations performed in the physical space, discrete ﬁlters are used, based on weighted averages with compact support. These operators enter into the category of explicit discrete ﬁlters, which are discussed in the following section. We may point out here that the methods based on implicit diﬀusion with no physical subgrid model can be re-interpreted as a pre-ﬁltering method, in which case it is the numerical ﬁlter that is dominant. We can see the major problem of this approach looming here: the ﬁlter associated with a numerical method is often unknown and is highly dependent on the simulation parameters (grid, boundary conditions, regularity of the solution, and so forth). This approach is therefore an empirical one that oﬀers little in the way of an a priori guarantee of the quality of the results. It does, however, have the advantage of minimizing the computation costs because we are then limited to solving the Navier–Stokes equations without implanting any subgrid model or explicit discrete ﬁlter. 1

The convolution product is then reduced to a simple product of two arrays.

13.1 Filter Identiﬁcation. Computing the Cutoﬀ Length

403

2. Considering that the eﬀective ﬁlter is associated with the computational grid. This position, which can be qualiﬁed as minimalist on the theoretical level, is based on the intuitive idea that the frequency cutoﬀ associated with a ﬁxed computational grid is unavoidable and that this ﬁlter is therefore always present. The problem then consists in determining the cutoﬀ length associated with the grid at each point, in order to be able to use the subgrid models. In the case of a Cartesian grid, we take the ﬁltering cell itself as Cartesian. The cutoﬀ length ∆ is evaluated locally as follows: – For uniform grid, the characteristic ﬁltering length in each direction is taken equal to the mesh size in this same direction: ∆i = ∆xi

.

(13.1)

The cutoﬀ length is then evaluated by means of one of the formulas presented in Chap. 6. – For a variable mesh size grid, the cutoﬀ length in the ith direction of the grid point of index l is computed as: ∆i |l = (xi |l+1 − xi |l−1 )/2 .

(13.2)

The cutoﬀ length is then computed locally according to the results of Chap. 6. In the case of a curvilinear structured grid, two options are possible depending on the way the partial derivative operators are constructed: – If the method is of the ﬁnite volume type in the sense of Vinokur [733], i.e. if the control volumes are deﬁned directly on the grid in the physical space and their topologies are described by the volume of the control cells, by the area and the unit normal vector to each of their facets, the ﬁlter cutoﬀ length can be computed at each point either by taking it equal to the cube root of the control volume to which the point considered belongs, or by using what Bardina et al. propose (see Sect. 6.2.3). – If the method is of the ﬁnite diﬀerences type in the Vinokur sense [733], i.e. if the partial derivative operators are computed on a uniform Cartesian grid after a change of variables whose Jacobian is denoted J, then the cutoﬀ length can be evaluated at the point of index (l, m, n) either by Bardina’s method or by the relation: ∆l,m,n = (Jl,m,n ∆ξ∆η∆ζ)

1/3

,

(13.3)

where ∆ξ, ∆η and ∆ζ are the grid steps in the reference space. In the case of an unstructured grid, we use the same evaluations as for a structured curvilinear grid with a ﬁnite volume type method, in the sense given above.

404

13. Implementation

13.2 Explicit Discrete Filters Several techniques and subgrid models described in the previous chapters use a test ﬁlter. For reference, these are the: – – – – – –

Pre-ﬁltering technique; Soft deconvolution models and scale similarity models; Mixed Scale Model; Dynamic constant adjustment procedures; Models incorporating a structural sensor; Accentuation procedure.

The corresponding theoretical developments all assume that we are able to apply an analytical ﬁlter in the simulation. This operation comes down to a product of two arrays in the spectral space, which is a simple operation of little cost, and all the analytical ﬁlters whose transfer function is known explicitly can be used. The problem is very diﬀerent, though, when we consider the simulations performed in the physical space on bounded domains: applying a convolution ﬁlter becomes very costly and non-local ﬁlters cannot be employed. In order to be able to use the models and techniques mentioned above, we have to use discrete ﬁlters with compact support in the physical space. These are described in the rest of this section. These discrete ﬁlters are deﬁned as linear combinations of the values at points neighboring the one where the ﬁltered quantity is computed [632, 728, 563, 461]. The weighting coeﬃcients of these linear combinations can be computed in several ways, which are described in the following. We ﬁrst present the one-dimensional case and then that of the Cartesian grids of more than one dimension, and lastly extend this to arbitrary grids. The discrete approximation of the convolution ﬁlters is then discussed. 13.2.1 Uniform One-Dimensional Grid Case We restrict ourselves here to the case of a uniform one-dimensional grid of mesh size ∆x. The abscissa of the grid point of index i is denoted xi , such that we can say xi+1 − xi = ∆x. The ﬁltered value of the variable φ at the grid point of index i is deﬁned by the relation: φi ≡

N

al φi+l

,

(13.4)

l=−N

where N is the radius of the discrete ﬁlter stencil. The ﬁlter is said to be symmetrical if al = a−l ∀l and anti-symmetrical if a0 = 0 and al = −a−l ∀l = 0. The constant preservation property is represented by the following relation: N l=−N

al = 1 .

(13.5)

13.2 Explicit Discrete Filters

405

A discrete ﬁlter deﬁned by the relation (13.4) is associated with the continuous convolution kernel: N

G(x − y) =

al δ(x − y + l∆x)

,

(13.6)

l=−N

where δ is a Dirac function. Simple computations show that the associated transfer function G(k) is of the form: N

G(k) =

al eıkl∆x

.

(13.7)

l=−N

The real and imaginary parts of this transfer function are: (G(k)) =

a0 +

N

(al + a−l ) cos(kl∆x)

,

l=1

(G(k)) =

N

(al − a−l ) sin(kl∆x)

.

l=1

The continuous diﬀerential operator can be associated with the discrete ﬁlter (13.4). To do this, we introduce the Taylor expansion of the variable φ about the point i: φi±n

∞ (±n∆x)l ∂ l φ = l! ∂xl i

.

(13.8)

,

(13.9)

l=0

By substituting in relation (13.4), we get: ∞ l ∗ l ∂ φi = 1 + al ∆x φi ∂xl l=1

in which a∗l =

N 1 an n l l!

.

n=−N

We note that these ﬁlters belong to the class of elliptic ﬁlters as deﬁned in Sect. 2.1.3. In practice, the ﬁlters most used are the two following three-point symmetrical ﬁlters: a0 =

1 1 , a−1 = a1 = 2 4

,

(13.10)

a0 =

2 1 , a−1 = a1 = 3 6

.

(13.11)

406

13. Implementation

Table 13.1. Coeﬃcients of discrete nonsymmetrical ﬁlters. N is the number of vanishing moments N

a−2

1 2 2 3 3 3

a−1

a0

a1

a2

a3

1/4

1/2 7/8 5/8 15/16 3/4 5/8

1/4 3/8 3/8 1/4 3/8 1/4

−3/8 −1/8 −3/8 −1/4 −1/16

1/8

1/8 −1/16

1/16 1/4

1/4 1/16

a4

−1/16

Vasilyev et al. [728] have deﬁned nonsymmetric ﬁlters, which have a large number of vanishing moments2 . These ﬁlters are presented in Table 13.1. Linearly constrained ﬁlters can also be deﬁned, which satisfy additional constraints. Optimized ﬁlters, whose coeﬃcients are computed to minimize the functional π/∆x π/∆x t (k)})2 dk + t (k)})2 dk , (13.12) ({G(k) −G ({G(k) −G 0

t (k) is the targeted transfer function, have been proposed [632, 728]. where G These ﬁlters ensure a better spectral response of the ﬁlter, resulting in a better localization of the information in spectral space. For certain uses, such as in the Germano-Lilly dynamic procedure, the characteristic length of the discrete ﬁlter, denoted ∆d , has to be known. For a deﬁnite positive ﬁlter, one measure of this length is obtained by computing the standard deviation of the associated convolution ﬁlter [563, 461]: ' +∞

∆d =

ξ 2 G(ξ)dξ

12

.

(13.13)

−∞

The characteristic lengths of the two three-point ﬁlters mentioned above √ are 2∆x for the (1/6, 2/3, 1/6) ﬁlter and 6∆x for the (1/4, 1/2, 1/4) ﬁlter. This method of evaluating the characteristic lengths of the discrete ﬁlters is ineﬃcient for ﬁlters whose second-order moment is zero. One alternative is work directly with the associated transfer function and deﬁne the wave number associated with the discrete ﬁlter, as for the one for which the transfer function takes the value 1/2. Let kd be this wave number. The discrete ﬁlter cutoﬀ length is now evaluated as: ∆d = 2

π kd

.

(13.14)

These ﬁlters are necessary to obtain high-order commuting discrete ﬁlters (see Sect. 2.2.2).

13.2 Explicit Discrete Filters

407

Implementation of test ﬁlters for the dynamic procedure within the spectral element framework is discussed by Blackburn and Schmidt [60]. The general unstructured case is discussed in [297]. The case of the ﬁnite element method is addressed by Kollman et al. [400]. 13.2.2 Extension to the Multi-Dimensional Case For Cartesian grids, we extend to the multidimensional case by applying a one-dimensional ﬁlter in each direction of space. This application can be performed simultaneously or sequentially. When simultaneously, the multidimensional ﬁlter is written symbolically as a summation: 1 Gi n i=1 n

Gn =

,

(13.15)

where n is the dimension of the space and Gi the one-dimensional ﬁlter in the ith direction of space. If applied sequentially, the resulting ﬁlter takes the form of a product: n n G = Gi . (13.16) i=1

The multidimensional ﬁlters constructed by these two techniques from the same one-dimensional ﬁlter are not the same in the sense that their transfer functions and equivalent diﬀerential operators are not the same. In practice, it is the product construction that is most often used, for two reasons: – This approach makes it possible to call the easily implemented onedimensional ﬁltering routines sequentially. – Such ﬁlters are more sensitive to the cross modes than are the ﬁlters constructed by summation, and allow a better analysis of the threedimensional aspect of the ﬁeld. 13.2.3 Extension to the General Case. Convolution Filters For structured curvilinear grids (or Cartesian grids with variable mesh size), one method is to employ the ﬁlters deﬁned in the uniform Cartesian grid and take no account of the variations of the metric coeﬃcients. This method, which is equivalent to applying the ﬁlter in a reference space, is very easy to implement but allows no control of the discrete ﬁlter transfer function or its equivalent diﬀerential operator. So it should be used only for grids whose metric coeﬃcients vary slowly. Another method that is completely general and applicable to unstructured grids consists in deﬁning the discrete ﬁlter by discretizing a chosen diﬀerential operator. The weighting coeﬃcients of the neighboring nodes are then the coeﬃcients of the discrete scheme associated with this diﬀerential operator. In

408

13. Implementation

practice, this method is most often used by discretizing second-order elliptic operators: 2 φ = (Id + α∆ ∇2 )φ , (13.17) where α is a positive constant and ∆ the desired cutoﬀ length. Limiting the operator to the second order yields ﬁlters with compact support using only the immediate neighbors of each node. This has the advantages of: – Making it possible to deﬁne operators that cost little to implement; – Making a multiblock and/or multidomain technique easier to use, and the boundary conditions easier to process. The fast-decay convolution ﬁlters (box or Gaussian) can thus be approximated by discretizing the diﬀerential operators associated with them. These operators are described in Sect. 7.2.1. The sharp cutoﬀ ﬁlter, which is not of compact support, is used only when fast Fourier transforms are usable, which implies that the grid step is constant and the data periodic. Another possibility for deriving discrete ﬁlters on general meshes is to compute the weight of neighbouring points by solving a linear system based on Taylor series expansions [632, 490]. 13.2.4 High-Order Elliptic Filters Convolution ﬁlters are non-local, and may sometimes be diﬃcult to use together with complex numerical algorithms (multidomain topology, unstructured grid, ...). An alternative, that can be implemented with all numerical methods, consists in high-order elliptic ﬁlters [553]. The ﬁltered variable is computed as being the solution of the general elliptic equation: [−(∇2 )m + αId]φ = αφ,

m≥1 .

(13.18)

High values of m make it possible to obtain very sharp ﬁlters in the spectral space. Mullen and Fischer show that the solution of equation (13.18) can be approximated through numerical solution of a much simpler problem, namely the Poisson equation −∇2 ψ = φ .

(13.19)

13.3 Implementation of the Structure Function Models In order to use the subgrid viscosity model based on the second-order structure function or the third-order structure function of the velocity (see p. 124 and p. 126), we have to establish a discrete approximation of the operator: 2 [u(x, t) − u(x + x , t)] d3 x . (13.20) DLL (x, r, t) = |x |=r

13.3 Implementation of the Structure Function Models

409

In practice, this integration is approximated as a sum of the contributions of the neighboring points. In the case of uniform Cartesian grid with ∆x = r, the structure function is evaluated at the index point (i, j, k) by the relation: DLL (∆x, t)i,j,k

1 |ui,j,k − ui+1,j,k |2 + |ui,j,k − ui−1,j,k |2 6 |ui,j,k − ui,j+1,k |2 + |ui,j,k − ui,j−1,k |2 |ui,j,k − ui,j,k+1 |2 + |ui,j,k − ui,j,k−1 |2 . (13.21)

= + +

When the grid is non-uniform or when ∆x = r, an interpolation technique has to be used to compute the integral. Rather than use a linear interpolation, it is recommended that the interpolation method be based on physical knowledge. So in the isotropic hom*ogeneous turbulence case, when we see that we have: DLL (x, r, t) = 4.82K0(εr)2/3

,

2/3

DLL (x, r , t) = 4.82K0(εr )

,

we deduce the proportionality relation: DLL (x, r, t) = DLL (x, r , t)

r 2/3 r

.

(13.22)

Relation (13.21) is thus generalized to the form: 1 |u(x) − u(x + ∆i )|2 n i=1 n

DLL (x, r, t) =

r ∆i

2/3 ,

(13.23)

where n is the number of neighboring points retained for computing the structure function and ∆i the distance of the ith point to the point where this function is evaluated. It has already been said that the second-order Structure Function model in its original form exhibits defects similar to those of the Smagorinsky model because of the uncertainty relation that prevents any good frequency localization of the information. One way of at least partly remedying this problem is to look for the structure function evaluation information only in the directions of statistical hom*ogeneity of the solution. This is done by evaluating the structure function only from points located in the directions of periodicity of the solution. This way, the mean gradient of the solution is not taken into account in the evaluation of the subgrid viscosity. We again ﬁnd here an idea similar to the one on which the splitting technique is based, in Sect. 6.3.3.

14. Examples of Applications

This chapter gives a few examples of large-eddy simulation applications that are representative of their accomplishments in the sense that they correspond either to ﬂows that are very frequently treated or to conﬁgurations that stretch the technique of today to its limits.

14.1 hom*ogeneous Turbulence 14.1.1 Isotropic hom*ogeneous Turbulence Problem Description. Isotropic hom*ogeneous turbulence is the simplest turbulent ﬂow on which subgrid models can be validated. The physical description of this ﬂow is precisely the one on which the very great majority of these models are constructed. Moreover, the ﬂow’s statistical hom*ogeneity makes it possible to use periodicity conditions for the computation, and highaccuracy numerical methods: pseudo-spectral methods can be used, optimally reducing the eﬀect of the numerical error on the solution. Because of the great simplicity of this ﬂow, most subgrid models yield very satisfactory results in terms of the statistical moments of the velocity ﬁeld and the integral scales, which reduces the discriminatory range of this test case. It is nonetheless widely used for fundamental type studies of turbulence and modeling. Two types of such ﬂow are considered: – Freely decaying isotropic hom*ogeneous turbulence in which the energy is initially distributed in a narrow spectral band and then, as the energy cascade sets in, is directed toward the small scales and ﬁnally dissipated at the cutoﬀ by the subgrid model. During the time the cascade is setting in, the kinetic energy remains constant, and later declines. The computation can be validated by comparison with decay laws developed by analytical theories (see [439]) or by comparison with experimental data. – Sustained isotropic hom*ogeneous turbulence, in which total dissipation of the kinetic energy is prevented by injecting energy at each time step, for example by maintaining a constant energy level in the wave vectors of a given norm. After a transitory phase, an equilibrium solution is established including an inertial range. The computation is validated by comparison

412

14. Examples of Applications

with theoretical or experimental data concerning the inertial region, and quantities associated with the large scales. A few Realizations. The ﬁrst large-eddy simulations of the free-decaying type were performed at the end of the seventies and early eighties [136] with resolutions of the order of 163 and 323 . Self-similar solutions could not be obtained with these resolutions because the integral scale becomes larger than the computational domain. However, the comparison with ﬁltered experimental data turns out to be satisfactory [40]. More recent simulations (for example [441, 514]) performed with diﬀerent subgrid models on grids of 1283 points have yielded data in agreement analytical theories for the kinetic energy decay. Higher-resolution simulations have been performed. In the sustained case, Chasnov [120] is an example of achieving self-similar solutions in agreement with theory for resolutions of 643 and 1283, though with an over-evaluation of the Kolmogorov constant. More recently, Fureby et al. [231] have tested six subgrid models and a case of implicit numerical diﬀusion on a 323 grid. The conclusions of this work are that the diﬀerent realizations, including the one based on artiﬁcial dissipation, are nearly indiscernable in terms of the quantities linked to the resolved ﬁeld, and are in good agreement with data yielded by a direct numerical simulation. Though isotropic hom*ogeneous turbulence is statistically the simplest case of turbulent ﬂow, it possesses a complex dynamics resulting from the interactions of very many elongated vortex structures called “worms”. These structures are illustrated in Fig. 14.1, which comes from a large-eddy simulation of freely decaying isotropic hom*ogeneous turbulence on a 1283 grid. Obtaining good results therefore implies that the simulation is capable of reﬂecting the dynamics of these structures correctly. We clearly see here the diﬀerence with the RANS approach (see Chap. 1), for which isotropic hom*ogeneous turbulence is a zero-dimension problem: for the large-eddy simulation, this problem is fully three-dimensional and reveals all the aspects of this technique (modeling errors, ﬁlter competition, and so forth). 14.1.2 Anisotropic hom*ogeneous Turbulence Anisotropic hom*ogeneous turbulence allows a better analysis of the subgrid models because the dynamics is more complex, while optimal numerical methods are retained. So it can be expected that this type of ﬂow oﬀers more discriminatory test cases for the subgrid models than do isotropic ﬂows. Bardina et al. [40] performed a set of simulations corresponding to the following three cases in the early eighties: – hom*ogeneous turbulence subjected to a solid-body rotation. Good agreement is measured with experimental data using a de-ﬁltering technique, on a 323 grid with a Smagorinsky model (5.90). The eﬀects of rotation on the turbulence are conﬁrmed, i.e. a reduction in the dissipation of the kinetic energy.

14.1 hom*ogeneous Turbulence

413

Fig. 14.1. Isotropic hom*ogeneous turbulence. Instantaneous view of vortices (illustrated by an iso-value surface of the vorticity). Courtesy of E. Garnier, ONERA.

– hom*ogeneous turbulence subjected to pure strain: still a 323 grid, with results in good agreement with experimental data concerning the turbulent intensity, using the Smagorinsky model and mixed Smagorinsky–Bardina model (7.125). The best results are obtained with the latter. – hom*ogeneous turbulence subjected to a deformation and rotation: simulations are performed on a 323 grid with the two previously mentioned models. No validation is presented, for lack of reference data. Simulations of hom*ogeneous turbulence subjected to sequential shearing have also been performed by Dang [162] on a 163 grid with several eﬀective viscosity models, yielding good results concerning the prediction of the

414

14. Examples of Applications

anisotropy of the resolved scales. Similar computations have also been performed by Aupoix [23].

14.2 Flows Possessing a Direction of Inhom*ogeneity These ﬂows represent the next level of complexity. The presence of a direction of inhom*ogeneity prompts the use of lower-order numerical methods, at least for this inhom*ogeneity, and boundary conditions. Also, more complex physical mechanisms are at play that can exceed the possibilities of the subgrid models. 14.2.1 Time-Evolving Plane Channel Problem Description. The time-evolving plane channel ﬂow is a ﬂow between two inﬁnite parallel ﬂat plates having the same velocity. The time character is due to the fact that we consider the velocity ﬁeld as being periodic in both directions parallel to the plates. Since the pressure is not periodic, a forcing term corresponding to the mean pressure gradient is added in the form of a source term in the momentum equations. The ﬂow is characterized by the ﬂuid viscosity, the distance between the plates, and the ﬂuid velocity. This academic conﬁguration is used for investigating the properties of a turbulent ﬂow in the presence of solid walls, and is a widely used test case. Turbulence is generated within the boundary layers that develop along each solid wall (see Sect. 10.2.1). It is the driving mechanism here, which must imperatively be simulated with accuracy to obtain reliable results. To do so, the grid has to be reﬁned near the surfaces, which raises numerical problems with respect to the hom*ogeneous turbulence. Moreover, the subgrid models must be able to preserve these driving mechanisms. The ﬂow topology is illustrated in the iso-value surface plot of the streamwise velocity in Fig. 14.2. A Few Realizations. There are dozens of numerical realizations of plane channel ﬂows. The ﬁrst are from Deardorﬀ [172] in 1970. The ﬁrst landmark results obtained by solving the dynamics of the near-wall region are due to Moin and Kim [537] in 1982. The characteristics of the computations presented in the four reference works [537, 591, 653, 411] are reported in Table 14.1. These computations are representative of the various techniques employed by most authors. The Table summarizes the following information: – The Reynolds number Rec referenced to the channel mid-height and mean velocity at the center of the channel. – The dimensions of the computational domain expressed as a function of the channel mid-height. The domain dimensions must be greater than those of the driving mechanisms in the near-wall region (see Sect. 10.2.1).

14.2 Flows Possessing a Direction of Inhom*ogeneity

415

Fig. 14.2. Plane channel ﬂow. Iso-surface of instantaneous streamwise velocity ﬂuctuations. Courtesy of E. Montreuil, ONERA.

– The number of grid points. Simulations generally include few points because the solution is bi-periodical. The computations at high Reynolds number without wall model presented [411] use a hierarchic grid technique with nine grid levels (symbol “+H”). – The subgrid model used (“Sc” is the Schumann subgrid viscosity model (6.59) and “Dyn” the dynamic Smagorinsky model (5.149)). Only two Table 14.1. Characteristics of time-evolving plane channel ﬂow computations. Ref. Rec Lx × Ly × Lz Nx × Ny × Nz SGS Model Wall O(∆xα ) O(∆tβ )

[537] 13800 2π × π × 2 64 × 64 × 128 Sc – S/2 2

[591] 47100 5π/2 × π/2 × 2 64 × 81 × 80 Dyn – S/T 3

[653]

[411]

≈ 1, 5.10 4×2×1 64 × 32 × 32 Sc MSc 2 2 5

1, 09.105 2π × π/2 × 2 2.106 +H Dyn – S/Gsp 3

416

14. Examples of Applications

models are used in the computations presented, but most existing models have been applied to this conﬁguration. – The treatment of the solid walls (“–” is the no-slip condition, “MSc” the Schumann wall model (10.28) to (10.30)). A single computation based on a wall model is presented, knowing that nearly all the models mentioned in Chap. 10 have been used for dealing with this ﬂow. – The accuracy of the space discretization schemes. Since the directions of statistical hom*ogeneity are linked to directions of periodicity in the solution, pseudo-spectral methods are often used for processing them. This is true of all the computations presented, identiﬁed by an “S”, except for reference [653], which presents a second-order accurate ﬁnite volume method. In the normal direction, three cases are presented here: use of second-order accurate schemes (identiﬁed by a “2”), of a Chebyshev method (“T”), and a Galerkin method based on B-splines (“Gsp”). The eﬀect of the numerical error on the solution can be reduced by using higher-order methods, which are consequently recommended by many authors. – The accuracy of the time integration. The convection term is usually treated explicitly (Runge-Kutta or Adams-Bashforth scheme) and the diffusion terms implicitly (Crank-Nicolson or second-order backward Euler scheme). Nearly all the computations are performed with second- or thirdorder accuracy. The results obtained on this conﬁguration are usually in good agreement with experimental data, and especially as concerns the ﬁrst-order (mean ﬁeld) and second-order (Reynolds stresses) statistical moments. Examples of data for these quantities are shown in Figs. 14.3 and 14.4. The mean longitudinal

Fig. 14.3. Plane channel ﬂow. Mean longitudinal velocity proﬁle referenced to the friction velocity, compared with a theoretical turbulent boundary layer proﬁle. Small circle symbols: LES computation. Lines: theoretical proﬁle. Courtesy of E. Montreuil, ONERA.

14.2 Flows Possessing a Direction of Inhom*ogeneity

417

Fig. 14.4. Plane channel ﬂow. Proﬁles of solved Reynolds stresses with respect to the friction velocity, compared with data from a direct numerical simulation computation. Dot symbols: direct numerical simulation. Lines: LES computation. Courtesy of E. Montreuil, ONERA.

velocity proﬁle is compared here with a theoretical turbulent boundary layer solution, and very good agreement with it is observed. It should be noted that the logarithmic region is relatively small, which is due to the fact that the Reynolds number for the computation is low (Reτ = 180). The proﬁles of the three main Reynolds stresses are compared with those obtained by direct numerical simulation on a grid including about twenty times more degrees of freedom. Although these stresses are calculated only from the resolved ﬁeld, such that the contribution of the subgrid scales is not included, we observe that the agreement with the reference solution is very satisfactory. This illustrates the fact that data obtained by large-eddy simulation can be used directly in practice without recourse to a de-ﬁltering technique. In the present case, the very good quality of the results can be explained by the fact that a large part of the kinetic energy of the exact solution is contained in the resolved scales. The quality of the results is essentially due to the resolution of the dynamics in the near-wall region (z + < 100). This implies that, if a wall model is not used, the computational grid is ﬁne enough to represent the dynamics of the vortex structures present, and that the subgrid models employed do not alter this dynamics. Because of the necessary volumes of the grids, this resolution constraint limits the possible Reynolds number. The largest friction Reynolds number achieved to date, using a hierarchic grid generation method, is Reτ = 4000 [411]. More results dealing with high Reynolds number simulations can be found in [36]. The standard subgrid viscosity models (Smagorinsky, Structure Function, and so forth) are generally too dissipative and have to be used with caution (modiﬁ-

418

14. Examples of Applications

cation of the value of the model constant, wall damping function, and so forth) [635]. Results concerning the transition to turbulence in this conﬁguration are available in [600, 599]. Lastly, the results obtained for this ﬂow have been found to be very sensitive to numerical errors induced either by the discrete numerical scheme or by the continuous form of the convection term [409, 563]. 14.2.2 Other Flows Other examples of shear ﬂows treated in the framework of the time-evolution approximation can be found for: – – – – – – – –

plane mixing layer, in [674, 38]; rotating boundary layer, in [152]; free-surface ﬂow, in [672]; boundary layers, in [401, 496, 498, 533]; round jet, in [214]; plane wake, in [263]; rotating plane channel, in [523, 524, 702, 422, 596, 421]; plane jet, in [431].

As in the case of the plane channel ﬂows described above, periodicity conditions are used in the directions of statistical hom*ogeneity. The numerical methods are generally dedicated to the particular conﬁguration being treated (with spectral methods used in certain directions) and are therefore optimal. A forcing term is added in the momentum equations to take the driving pressure gradient into account or avoid diﬀusion of the base proﬁle. Transitional ﬂows are more sensitive to the subgrid model and to the numerical errors, as an inhibition of the transition or re-laminarization of the ﬂow are possible. This is more especially true of ﬂows (for example boundary layers) for which there exists a critical Reynolds number: the eﬀective Reynolds number of the simulation must remain above the threshold within which the ﬂow is laminar. It should be noted that the boundary conditions in the inhom*ogeneous direction raises little diﬃculty for the ﬂow conﬁgurations mentioned above. These are either solid walls that are easily included numerically (except for the procedure of including the dynamics), or outﬂow conditions in regions where the ﬂow is potential. In the latter case, the computation domain boundary is generally pushed back as far as possible from the region being studied, which reduces any spurious eﬀects. The types of results obtained, and their quality, are comparable to what has already been presented for the plane channel ﬂow.

14.3 Flows Having at Most One Direction of hom*ogeneity

419

14.3 Flows Having at Most One Direction of hom*ogeneity This type of ﬂow introduces several additional diﬃculties compared with the previous cases. The limited number or total absence of directions of hom*ogeneity makes it necessary in practice to use numerical methods of moderate order of accuracy (generally two, rarely more than four), and highly anisotropic grids. The eﬀect of the numerical error will therefore be high. Moreover, most of these ﬂows are in spatial expansion and the problems related to the deﬁnition of the inﬂow and outﬂow conditions then appear. Lastly, the ﬂow dynamics becomes very complex, which accentuates the modeling problems. 14.3.1 Round Jet Problem Description. The example of the round jet ﬂow in spatial expansion is representative of the category of free shear ﬂows in spatial expansion. The case is restricted here to an isothermal, isochoric round jet ﬂow piped into a uniform, steady outer ﬂow in a direction parallel to that of the jet. Two main regions can be identiﬁed: – First, we ﬁnd a region at the pipe exit where the ﬂow consists of a laminar core called a potential cone, which is surrounded by an annular mixing layer. The mixing layer is created by the inﬂectional instability associated

Fig. 14.5. Round jet. Iso-surface of instantaneous vorticity (LES computation). Exit plane in black. Courtesy of P. Comte, LEGI.

420

14. Examples of Applications

Fig. 14.6. Round jet. Iso-surface of instantaneous vorticity (LES computation). Courtesy of P. Comte, LEGI.

with the deﬁcit velocity proﬁle of the boundary layer on the wall of the circular pipe. As the mixing layer thickens while moving away from the pipe exit section, it reduces the diameter of the potential cone and also induces an increase in the jet diameter. – After the potential cone disappears, we have a “pure jet” region where the ﬂow gradually reaches a regime corresponding to a similarity solution. The ﬁrst region can be decomposed into two: a “transition” region where the mixing layer has not yet reached its self-similar regime, and the similarity region where it has. This organization is illustrated in Figs. 14.5 and 14.6, representing respectively the iso-surfaces of vorticity and pressure obtained from large-eddy simulation results. The vorticity ﬁeld very clearly shows the transition of the annular mixing layer. The topology of the pressure ﬁeld shows the existence of coherent structures. Experimental and numerical analyses have shown that this ﬂow is strongly dependent on many parameters, which makes it highly discriminatory. A Few Realizations. There are far fewer round jet simulations in the literature than there are plane channel ﬂows. This is mainly due to the increased diﬃculty. Four of these realizations are described in the following, with their characteristics listed in Table 14.2, which gives: – the Reynolds number ReD referenced to the initial jet diameter D and the maximum of the mean initial velocity proﬁle; – the computational domain dimensions referenced to the length D;

14.3 Flows Having at Most One Direction of hom*ogeneity

421

Table 14.2. Characteristics of the round jet computations. Ref. ReD Lx × Ly × Lz Nx × Ny × Nz SGS Model Inﬂow O(∆xα ) O(∆tβ )

[634] 21000 10 × 11 × 11 101 × 121 × 121 MSM U+b 2+up3 2

[573] 4

50.10 12 × 8 × 8 ≈ 270000+H Dyn U+b 3+up3 2

[83] 21000 10 × 11 × 11 101 × 288 × 288 FSF U+b S/6 3

– the number of grid points. All the grids used by the authors mentioned are Cartesian. The symbol H designates the use of embedded grids (four grid levels for [573]). – the subgrid model (“MSM” standing for the Mixed Scale Model (5.127); “Dyn” the dynamic model (5.149); “FSF” the ﬁltered Structure Function model (5.264)). It should be noted that, for all the known realizations of this ﬂow, only the subgrid viscosity models have been used. – the freestream condition generation mode. The symbol U + b indicates that the non-steady inﬂow condition was generated by superimposing an average steady proﬁle and a random noise, as indicated in Sect. 10.3.2. – the overall order of accuracy of accuracy in space of the numerical method. The symbol +up3 indicates that a third-order accurate upwind scheme is used for the convection term to ensure computation stability. The computations presented in [83] rely on spectral schemes in the directions normal to that of the jet. – the time accuracy of the method employed. Examples of results obtained on this conﬁguration are compared with experimental data in Figs. 14.7 to 14.11. The axial evolution of the location of the point where the mean velocity is equal to half the maximum velocity is represented in Fig. 14.7. This quantity, which gives some indication concerning the development of the annular mixing layer, remains constant during the ﬁrst phases of evolution of the jet, which conﬁrms the existence of a potential cone. After the cone disappears, this quantity increases, which indicates the beginning of the pure jet region. It is observed that the length of the potential cone predicted by the computation is less than is observed experimentally. Similar conclusions can be drawn from the axial evolution of the average longitudinal velocity, which is presented in Fig. 14.8. The toorapid expansion of the pure jet region is accompanied by a strong decay of the mean velocity1 . These symptoms are observed on all known large-eddy simulation computations on this conﬁguration and still have no precise explanation. Several hypotheses have been formulated concerning the dependency 1

This results from the conservation of the mass.

422

14. Examples of Applications

Fig. 14.7. Round jet. Axial evolution of the radial position of the point where the mean velocity is half the maximum velocity. Dots: experimental data. Dotdashed lines: extrapolation of this data. Solid line: LES computation. Courtesy of P. Comte, LEGI.

Fig. 14.8. Round jet. Axial evolution of the mean longitudinal velocity. Dots: experimental data. Line: LES computation. Courtesy of P. Comte, LEGI.

14.3 Flows Having at Most One Direction of hom*ogeneity

423

on the initial perturbation, on the boundary conditions, or on the computational grid. The axial proﬁles of two main Reynolds stresses are presented in Figs. 14.9 and 14.10. These results are qualitatively correct. The Reynolds stresses increase along the axis and exhibit a maximum in a region close to the tip of the potential cone, which is in agreement with the experimental observations. It is noted that the level of the longitudinal stress predicted by the computation is higher than the experimental level in the pure jet region. The peak observed on the downstream boundary of the computational domain is a spurious eﬀect that is no doubt related to the outﬂow condition. Generally, it is noticed that the quality of the results is not as good as in the case of the plane channel ﬂow, which illustrates the fact that this ﬂow is a more complicated case for large-eddy simulation. Lastly, the velocity spectra generated from the computation are presented in Fig. 14.11. Over a decade, the computations recover a slope close to the −5/3 predicted by theory, and which is the foundation of the theoretical analyses presented in the previous chapters. This indicates that the resolved turbulent scales have “physical” behavior. More generally, the conclusions given by the various authors are the following. – The dynamics of the numerical solution is consistent, i.e. the values produced are located within the bounds ﬁxed by the collected set of experimental measurements and the topology of the simulated ﬂow exhibits the expected characteristics (potential cone, annular mixing layer, and so forth). – While the dynamics is consistent, it is nonetheless very diﬃcult to reproduce a particular realization (for example with ﬁxed potential cone length and maximum turbulent intensity). – The numerical solution exhibits a strong dependency on many parameters, among which we ﬁnd: – the subgrid model, which allows a more or less rapid transition of the annular mixing layer are consequently inﬂuences the length of the potential cone and the turbulent intensity by modifying the eﬀective Reynolds number in the simulation. More dissipative models delay the development of the mixing layer, inducing the existence of a very long potential cone. – the inﬂow condition: the mixing layer transition is also strongly dependent on the amplitude and shape of the perturbations. – the numerical error, which can aﬀect the turbulent of the annular mixing layer and of the developed jet, especially during the transition phases. Here it is a matter of an error controlled by the computational grid and the numerical method. A dispersive error will have a tendency to accelerate the transition and thereby shorten the potential cone. A dissipative error will have the inverse eﬀect. With too coarse a grid, the annular mixing layer cannot be represented correctly, which can induce

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Fig. 14.9. Round jet. Axial evolution of the normalized longitudinal turbulent intensity. Dots: experimental data. Line: LES computation. Courtesy of P. Comte, LEGI.

Fig. 14.10. Round jet. Axial evolution of the normal turbulent intensity. Dots: experimental data. Line: LES computation. Courtesy of P. Comte, LEGI.

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Fig. 14.11. Round jet. Time (Left) and space (Right) spectra of solved turbulent kinetic energy at diﬀerent positions along the axis. Courtesy of P. Comte, LEGI.

it to thicken too quickly and thereby decrease the length of the potential cone. – the size of the computational domain. The computation is sensitive to the size of the computational domain, which modulates the eﬀect of the boundary conditions, especially the outﬂow condition.

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– All the computations predict the dominant time frequency of the jet correctly, which is therefore not a pertinent parameter for analyzing the models ﬁnely. – The quality of the simulation is not a global character. Certain parameters can be correctly predicted while others are not. This diversity in the robustness of the results with respect to the simulation parameters sometimes makes it diﬃcult to deﬁne discriminatory parameters. Other Examples of Free Shear Flows. Other examples of free shear ﬂows in spatial expansion have been simulated: – – – – – –

plane mixing layer (see [674, 38]); planar jet (see [159]); rectangular jet (see [272]); swirling round jet (see [472]); controlled round jet (see [157]); plane wake (see [269, 740, 237]).

The conclusions drawn from the analysis of these diﬀerent cases corroborate those explained previously for the round jet as concerns the quality of the results and their dependency as a function of the computation parameters (subgrid model, grid, inﬂow condition, computational domain, and so forth). These conclusions are therefore valid for all free shear ﬂows in spatial expansion. 14.3.2 Backward Facing Step Problem description. The ﬂow over a backward facing step of inﬁnite span is a generic example for understanding separated internal ﬂows. It involves most of the physical mechanisms encountered in this type of ﬂow and is doubtless the best documented, both experimentally and numerically, of the ﬂows in this category. Its dynamics can be decomposed as follows. The boundary layer that develops upstream of the step separates at the step corner, becoming a free shear layer. This layer expands in the recirculation region, thereby entraining turbulent ﬂuid volumes. This entrainment phenomenon may inﬂuence the development of the shear layer, which curves inward toward the wall in the reattachment region and impacts with it. After the reattachment, the boundary layer re-develops while relaxing toward a proﬁle in equilibrium. The topology of this ﬂow is illustrated in Fig. 14.12, which is developed from large-eddy simulation data. We observe ﬁrst the transition of the separated shear layer, the formation of vortex structures in the impact area, and then of hairpin structures in the boundary layer after the reattachment. This ﬂow brings out diﬃculties in addition to those of the round jet, because it adds the dynamics both of the free shear layers and of the near wall region.

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427

Fig. 14.12. Backward Facing Step. Iso-surface of instantaneous vorticity. Courtesy of F. Delcayre, LEGI.

A Few Realizations. The methods used for simulating this ﬂow are illustrated by four computations presented in Table 14.3. The parameters indicated are. – Reynolds number ReH , referenced to the step height H and the inﬂow velocity proﬁle; – the dimensions of the computational domain, referenced to the length H; – the number of grid points used; – subgrid model used (“Sc” means the Schumann model (6.59); “MSM” the Mixed Scale Model (5.127); “SF” the Structure Function model (5.102); “DynLoc” the constrained localized dynamic model (5.207)). As before for the round jet, only subgrid viscosity models have been used in the conﬁguration. – inﬂow condition generation mode: U + b means the same thing as before, while P designates the use of a precursor, which in this case is a large-eddy simulation of a plane channel ﬂow in [226]; Ca indicates the use of an inﬂow channel to allow the development of a “realistic” turbulence upstream of the separation. Depending on the author, the length of this channel is between four and ten H. – treatment of the solid walls: “–“ is the no-slip condition; MSc the Schumann wall model, (10.28) to (10.30); MGz the Gr¨ otzbach wall model (10.31). It

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Table 14.3. Backward facing step computation characteristics. Ref. ReH Lx × Ly × Lz Nx × Ny × Nz SGS Model Inﬂow Wall O(∆xα ) O(∆tβ )

[226]

[637] 5

1, 65.10 16 × 4 × 2 128 × 32 × 32 Sc P MSc 2 2

11200 20 × 4 × 2, 5 201 × 31 × 51 MSM U+b – 2+up3 2

[673] 38000 30 × 5 × 2, 5 200 × 30 × 30 FS U+b MLog 2+up3 2

[261] 28 000 30 × 3 × 4 244 × 96 × 96 DynLoc U+b,Ca – 2 3

can be seen that the use of wall models reduces the number of points considerably and makes it possible to simulate ﬂows with high Reynolds numbers. – spatial accuracy of the numerical method; – time accuracy of the numerical method. The results the various authors have obtained are generally in good qualitative agreement with experimental data: the ﬂow topology is recovered and the realizations show the existence of coherent structures similar to those observed in the laboratory. On the other hand, there is much more diﬃculty obtaining quantitative agreement whenever this is possible at all (only reference [261] produces results in satisfactory agreement on the mean velocity ﬁeld and turbulent intensity). This is due to the very high sensitivity of the result to the computation parameters. For example, variations of the order of 100% of the average length of the recirculation region have been recorded when the inﬂow boundary condition or subgrid model are manipulated. This sensitivity stems from the fact that the ﬂow dynamics is governed by that of the separated shear layer, so the problems mentioned before concerning free shear ﬂows crop up here. We also note a tendency to under-estimate the value of the mean velocity in the recirculation area. However, as in the case of the round jet, the simulated physics does correspond to that of a backward facing step ﬂow. This is illustrated by the mean velocity proﬁles and resolved Reynolds stresses in Fig. 14.13, and the pressure spectra in Fig. 14.14. The good agreement with experimental data in the prediction of the mean ﬁeld and Reynolds stresses proves the theoretical consistency of the computation. This agreement is even clearer if we analyze the spectra. Near the step, the mixing layer dynamics is dominated by frequencies associated with the Kelvin-Helmholtz instability. The predicted value of the dominant frequency is in very good agreement with experimental observations. The double peak at the second measurement point shows that the simulation is capable of reﬂecting the low-frequency ﬂapping mechanism of the separated region, and still remain in good agreement with experimental observations.

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LES Le & Moin Jovic & Driver

Fig. 14.13. Backward Facing Step. Mean velocity and Reynolds stresses proﬁles at the reattachment point. Triangles and solid line: LES computations. Squares: experimental data. Courtesy of F. Delcayre, LEGI.

Fig. 14.14. Backward Facing Step. Pressure spectra. Squares: in the free shear layer near the step corner. Triangles: in the separated region near the reattachment point. Courtesy of F. Delcayre, LEGI.

Also, it seems that the use of wall models does not aﬀect the dynamics of this shear layer greatly. It becomes possible to deal with higher Reynolds numbers, but at the price of losing some of the quality of the results as concerns the wall surface terms (friction, pressure coeﬃcient) in the recirculation zone [87]. Solutions for this conﬁguration turn out to be highly dependent on the subgrid model: a too dissipative model will delay the development of the separated shear layer, pushing away the position of the reattachment point.

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The results are also found to be dependent on the size of the domain and the ﬁneness of the mesh in the spanwise direction, because these parameters aﬀect the development of the mixing layer emanating from the step corner. A spanwise domain width of 4 to 6 H is considered to be a minimum in order to be able to capture the three-dimensional mechanisms at low frequencies. Lastly, the time frequencies associated with the separated zone dynamics are robust parameters in the sense that they are often predicted with precision. 14.3.3 Square-Section Cylinder Problem description. The square-section inﬁnite-span cylinder is a good example of separated external ﬂows around bluﬀ bodies. This type of conﬁguration involves phenomena as diversiﬁed as the impact of the ﬂow on a body, its separation (and possible reattachment) on the body surface, the formation of a near-wake region, and alternating vortex street, and the development of the wake up to a self-similar solution. Each of these phenomena poses its own particular numerical and modeling problems. Realizations. This ﬂow was chosen as an international test case for largeeddy simulation, and has consequently served as a basis for many computations, which are mostly summarized in [618] (see also [555, 729, 730, 678, 233, 94, 652] for a discussion of this test case). The test case parameters are: a Reynolds number ReD , referenced to the length D of the cylinder edge and the freestream velocity, equal to 22,000, and a computational domain of 20D × 4D × 14D. The span is assumed to be inﬁnite and a periodicity condition is used in this direction. None of the sixteen computations collected in [618] produces an overall good agreement with experimental data, i.e. is capable of predicting all of the following parameters with an error of less than 30%: average lift and drag; drag and lift variances; main vortex shedding frequency; and average length of the separated region behind the cylinder. Average lift and drag, as well as the vortex shedding frequency, are very often predicted very satisfactorily. This is due to the fact that these quantities do not depend on the small scale turbulence and are governed by Von Karman structures, which are very large in size. The length of the recirculation region behind the cylinder is very often under-estimated, as is the amplitude of the mean velocity in this region. Also, the mean velocity in the wake is only very rarely in agreement the experimental data. The numerical methods used are of moderate order of accuracy (at most second-order in space and third-order in time), and third-order upwind schemes are often used for the convection term. Only subgrid viscosity models have been used (Smagorinsky model, various dynamic models, Mixed Scale Model). Certain authors use wall models at the cylinder surface. The lack of agreement with experimental data is explained by the very high sensitivity of the diﬀerent driving mechanisms to the numerical errors

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and to the diﬀusion introduced by the models. So we again ﬁnd here the problems mentioned above for the case of the backward facing step, but now they are ampliﬁed by the fact that, in order to master the impact phenomenon numerically, the numerical diﬀusion introduced is much stronger than in the former case. Also, as most of the grids used are Cartesian and monodomain, the resolution near the cylinder is too weak to allow a satisfactory representation of the boundary layers. 14.3.4 Other Examples Many other ﬂows have been examined by large-eddy simulation. Among wall-bounded ﬂows without separation, we may mention: ﬂat plate boundary layer [237, 205, 463, 457, 585]; boundary layer on a curved surface in the presence of G¨ortler vortices [460]; ﬂow in a circular-section toric pipe [653]; three-dimensional boundary layer [767, 366]; juncture ﬂow [687]; ﬂow in a rotating pipe [777, 215]; ﬂow in a rotating square duct [576]; ﬂow in a square/rectangular annular duct [774, 65]. Examples of recirculating ﬂows are: conﬁned coaxial round jets [9]; ﬂow around a wing section of inﬁnite span at incidence [347, 371, 348, 349, 425, 78, 502, 494] (see Fig. 14.15); ﬂow in a planar asymmetric diﬀuser [367, 213, 372, 369]; ﬂow around a cube mounted on a ﬂat plate [618, 556, 757, 493, 652, 407]; ﬂow around a circular-section cylinder [52, 519, 520, 362, 75, 410, 715, 77, 360, 76]; ﬂow in tube bundles [620]; ﬂow in a lid-driven cavity [799, 179]; ﬂow in a ribbed channel/duct [779, 142, 778, 151, 361]; jet

Fig. 14.15. Flow around a wing at high incidence: isosurface of instantaneous vorticity. Courtesy of R. Lardat and L. Ta Phuoc, LIMSI.

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impacting a ﬂat plate [739, 616, 574, 51, 718]; boundary layer on a wavy surface or a bump [412, 197, 770, 302, 768, 769, 642, 522]; ﬂow over a swept wedge [330]; ﬂow past a blunt trailing edge [484, 754, 564, 571]; separated boundary layer [88, 758]; aircraft wake vortices [286]; axisymmetric pistoncylinder ﬂow [732]; ﬂow around an oscillating cylinder [721]; ﬂow around a road vehicle [731, 408]; ﬂow around a 3D wing [329]; ﬂow around a square cylinder [399, 393]; ﬂow around a forward-backward facing step [5]; ﬂow in reversing systems [61]; ﬂow in a blade passage [147, 516]; ﬂow pas a swept fence [178]; ﬂow over a sphere [146].

14.4 Industrial Applications 14.4.1 Large-Eddy Simulation for Nuclear Power Plants The increasing power of supercomputers and of numerical description of unsteady ﬂows allows the simulation of complex ﬂows relevant to industrial conﬁgurations in accident scenarios obviously not available through experiments. Simulation is used to capture large features of the ﬂow and to focus on major events, at least concerning qualitative behavior. The present example deals with induced rupture of the primary circuit in a pressurized water reactor. The simulation domain includes the reactor vessel, the steam generator and their connection through the hot leg (see Fig. 14.16): the dimensions of the real domain cover several meters. For the calculation, the steam generator tubes have been grouped into nine, thus decreasing the number of control volumes (a model has been developed to take into account this change in the geometry). Furthermore, the tube’s length has been reduced in order to limit the computer time consumption. In the case of the total loss of the heat sink (very hypothetical severe accident scenario) the ﬂow circulation driven by the primary pump is stopped, the coolant is pure vapour (T > 1000 ◦C) and natural convection develops in the circuit. The question here is to investigate the temperature distribution in the wall of the hot leg and in the tubes of the steam generator, in order to analyze their mechanical constraints. For that it is crucial to get the best discretization of the related zones. This is achieved by concentrating the grid points in this area. Figure 14.16 shows the mesh of each tube of a “simpliﬁed steam generator” composed of 400 tubes (3300 in reality). Figure 14.17 (top) provides a global view of the ﬂow in the hot leg, showing a clear stratiﬁcation and a return of ﬂow into the core. Figure 14.17 (bottom) displays a cut in the hot leg at diﬀerent locations, showing an increase of turbulence and mixing as the section is closer to the steam generator: this mixing leads to hom*ogenization of the temperature and reduced mechanical constraints. The present simulation was performed using the TRIO-U/PRICELES code developed at CEA Grenoble (in the DTP/SMTH/LDTA laboratory): it required around 106 grid points using a tetrahedral discretization. The underlying numerical

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Fig. 14.16. Top: global view of the unstructured mesh. A coarse discretization is chosen for the vessel, taking into account here a realistic boundary condition. The resolution is ﬁner in a region of interest, say the steam generator region (tubes and plenum). Bottom: local view of the mesh of the steam generator: 400 individual tubes are independently meshed. Courtesy of IRSN.

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Fig. 14.17. Top: a cut in the instantaneous temperature ﬁeld in the whole system, showing a stratiﬁcation in the “hot leg” that propagates in the direction of the steam generator tubes; a back ﬂow develops in the vessel. Bottom: cuts of the same ﬁeld near the steam generator, plus velocity vectors. Stratiﬁcation is perturbated by the turbulent ﬂow. Courtesy of IRSN.

method is a centered second-order accurate Finite Element based discretization, used together with a standard Smagorinsky model. All simulations were performed by U. Bieder (CEA) and H. Mutelle (IRSN) for the French Nuclear Safety Institute (IRSN).

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For other situations where experimental data are available, qualitative as well as quantitative features are looked at, as in more traditional disciplines. 14.4.2 Flow in a Mixed-Flow Pump This example of the use of large-eddy simulation for turbomachinery ﬂows is due to C. Kato et al. [380]. These authors computed the internal ﬂow in a mixed-ﬂow pump stage with a high designed speciﬁc speed. A view of the computational domain is shown in Fig. 14.18. The computational domain includes: the upstream inlet pipe, the two regulation plates, a four-blade impeller and the diﬀuser, leading to a very complex conﬁguration, both from the geometrical and physical points of view. In order to take the rotation into account, the authors make use of a Chimera-type technique: a moving boundary interface in the ﬂow ﬁeld is treated with overset grids from multiple frames of reference. A computational grid that rotates along with the impeller is used to compute the ﬂow within the impeller, and stationary grids are used for stationary parts of the pump. Each grid includes appropriate margins of overlap with its neighboring grids in order to allow appropriate interpolations, and coordinate transformation to take into account the diﬀerent frames of reference. Two mesh resolutions have been investigated: a coarse grid with a total of 1.7 × 106 points and a ﬁne grid with 5 × 106 points. A partial view of the surface mesh on the impeller is shown in Fig. 14.19. The boundary conditions are the following: no-slip boundary conditions are used on solid walls, and the turbulent pipe ﬂow at the inlet is obtained by performing an auxiliary large-eddy simulation of an inﬁnite pipe ﬂow.

Fig. 14.18. Flow in a mixed-ﬂow pump. View of the global computational domain. Courtesy of C. Kato, University of Tokyo.

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Fig. 14.19. Partial view of the surface mesh. Courtesy of C. Kato, University of Tokyo.

A classical Smagorinsky model is employed, with a Van Driest damping function enforcing a vanishing subgrid viscosity in the near-wall region. A streamwise-upwind ﬁnite-element method is used to discretize the Navier– Stokes equations. Time integration is based on the explicit Euler scheme, but shifts the spatial residuals of the governing equations in the upstream direction of the local ﬂow. The magnitude of this shift is one half of the time increment multiplied by the magnitude of the local ﬂow velocity, yielding an exact cancellation of the ﬁrst-order error terms. The resulting method is second-order accurate in space and time. Several operating conditions of the pump have been investigated using large-eddy simulations. All of them correspond to partial load conditions, i.e. oﬀ-design operating conditions. Comparison of computed and measured head-ﬂow characteristics is shown in Fig. 14.20, showing a very good agreement between ﬁne-grid simulations and experiments. Coarse grid simulation is seen to yield larger discrepancies when design operating conditions are approached, because the mesh is not ﬁne enough to capture accurately the attached boundary-layer dynamics. The normalized head-ﬂow characteristic is deﬁned here as the relation between the ﬂow-rate of the pump and its total-pressure rise. After stall onset (Q/Qd ≤ 0.55, where Q is the mass ﬂowrate), both grids yield very similar descriptions of the ﬂow. As in the previous case, Qd is the mass ﬂow-rate related to the design point.

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437

Fig. 14.20. Comparison of measured and predicted head-ﬂow characteristics as a function of the mass ﬂow rate Q with reference to the design ﬂow rate Qd . Courtesy of C. Kato, University of Tokyo.

An example of comparisons of computed and experimental data is shown in Fig. 14.21. 14.4.3 Flow Around a Landing Gear Conﬁguration The ﬂow around a realistic landing gear conﬁguration was studied by Souliez et al. [679]. The main purpose of this work was the prediction of the far-ﬁeld noise generated by the turbulent ﬂuctuations. The most complex geometry includes four wheels and two lateral struts. An unstructured mesh with 135 000 triangles on the surface of the landing gear and 1.2 million tetrahedral cells was used. The simulation is run without an explicit subgrid model, and then belongs to the MILES-like group of large-eddy simulations. The PUMA code used for this simulation is based on the ﬁnite volume approach. Explicit timeintegration is carried out using a Runge–Kutta scheme. Typical results dealing with the mean surface pressure and the topology of the instantaneous ﬂow are shown in Figs. 14.22 and 14.23. 14.4.4 Flow Around a Full-Scale Car Large-eddy simulations is also used in the automotive industry during the design of new cars. The main purposes are the prediction of traditional aerodynamic parameters (drag, lift, etc.) and aeroacoustics.

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Fig. 14.21. Predicted and measured phase-averaged velocity proﬁles at the impeller’s inlet cross-section for Q/Qd = 0.43 (post-stall operating condition). Courtesy of C. Kato, University of Tokyo.

The example presented here deals with the large-eddy simulations of the ﬂow around a full-scale car model. The wheels and the ﬂoor are ﬁxed, as in the wind tunnel experiments. The Reynolds number is 2 684 563 per meter. The numerical simulation is carried out using the Powerﬂow code, which is based on the Lattice Boltzmann approach (see [127] for an introduction). A Cartesian grid is used, which is composed of 20 × 106 cells. The size of the smallest mesh is 7 mm. Subgrid scales are taken into account using a twoequation k − τ subgrid model. A wall model is used at solid boundaries.

Fig. 14.22. Predicted mean surface pressure on the landing gear. Courtesy of F. Souliez and L. Long, Pennsylvania State University.

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439

Fig. 14.23. Instantaneous vorticity ﬁlaments. Courtesy of F. Souliez and L. Long, Pennsylvania State University.

Typical results are displayed in Figs. 14.24–14.27. The very complex topology of the ﬂow is recovered, exhibiting very intense vortical structures and pressure variations. The drag is predicted to be within 1% error in comparison with experimental data.

14.5 Lessons 14.5.1 General Lessons We can draw the following lessons concerning the large-eddy simulation technique from the computations mentioned above: – When the technique is used for dealing with the ideal case in which it was derived (hom*ogeneous turbulence, optimal numerical method), it yields very good results. The vast majority of subgrid models produce results that are indiscernable from reality, which removes any discriminatory character from this type of test case, which in fact can only be used to assess the consistency of the method. – Extending the technique to inhom*ogeneous cases brings up many other problems, concerning both the physical modeling (subgrid models) and the numerical method. The latter point becomes crucial because the use of numerical methods of moderate order of accuracy (generally two) greatly increases the eﬀect of the numerical error. This is accentuated by the use of artiﬁcial dissipations for stabilizing the simulation in “stiﬀ” cases (strong under-resolution, strong gradients). This error seems to be reducable by reﬁning the computational grid, which is done more and more by using adaptive grids (local adaptation or enrichment).

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Fig. 14.24. Isocontours of total pressure. Courtesy of Renault-Aerodynamic Department.

Fig. 14.25. Streamlines. Courtesy of Renault-Aerodynamic Department.

Fig. 14.26. Streamlines. Courtesy of Renault-Aerodynamic Department.

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Fig. 14.27. Streamlines. Courtesy of Renault-Aerodynamic Department.

– Shear ﬂows show themselves to be very strongly dependent on the inﬂow condition when this is unsteady. Generating these conditions is still an open problem. – The quality of the results is variable but, for each conﬁguration, robust, correctly predicted parameters exist. The physics simulated is often consistent in that it exhibits the generic features that are observed experimentally but does not necessarily correspond to a desired target realization. This is due to the dependency on the many simulation parameters. – The quality of the results is subordinate to the correct representation of the ﬂow driving mechanisms (transition, near-wall dynamics, and so forth). Low numerical error and consistent modeling are therefore mandatory in those regions where these mechanisms occur. The other regions of the ﬂow where the energy cascade is the dominant mechanism are of lesser importance. – When the ﬂow dynamics becomes complex, subgrid viscosity models are often used. This is because they provide a clear kinetic energy dissipation and therefore stabilize the simulation. This stabilizing character seems to become predominant compared with the physical quality of the modeling insofar as the numerical diﬃculties increase (with the presence of strong shear and highly heterogeneous grids, and so forth). – There is a consensus today that the numerical method used must be accurate to at least the second order in space and time. First-order accurate numerical dissipations are totally proscribed. Third-order accurate methods in time are rarely used. As concerns the spatial accuracy, satisfactory results are obtained by certain authors with second-order accurate methods, but higher-order accurate schemes are often used. Numerical stabilization methods (upwind scheme, artiﬁcial dissipation, smoothing, and so forth) should be used only when absolutely necessary. – Large-eddy simulation is presently a powerful tool for investigating massively separated industrial ﬂows, where the large scales and the turbulence production is not driven by ﬁne details of the dynamics of the boundary

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layers. For other ﬂows, such as fully attached ﬂows, large-eddy simulation is still too expensive to be used on full-scale conﬁgurations for realistic Reynolds numbers because of the huge number of grid points required to obtain an accurate resolution of the boundary layers. Existing wall models have not been validated in realistic cases up to now, and the question of their validity for such ﬂows remains an open question. 14.5.2 Subgrid Model Eﬃciency Here we will try to draw some conclusions concerning the eﬃciency of the subgrid models for processing a few generic ﬂows. These conclusions should be taken with caution. As we have seen all through this book, very many factors (numerical method, grid, subgrid model, and others) are involved and are almost indissociable, so it is very diﬃcult to try to isolate the eﬀect of a model in a simulation. The conclusions presented are statistical in the sense that they are the fruit of an analysis of simulations performed on similar (at least geometrically) ﬂow conﬁgurations with diﬀerent methods. A “deterministic” analysis could lead to contradictory conclusions, depending on the other. Also, there is no question of ranking the models, as the available information is too lacunary to draw up a reliable list. Lastly, very many factors like the discretization of the subgrid models still remain to be studied. We may, however, sketch out the following conclusions. 1. To simulate a hom*ogeneous ﬂow: a) All subgrid models including a subgrid viscosity yield similar results. The eﬃciency of functional models for the forward energy cascade in isotropic turbulence, despite their lack of accuracy in representing the subgrid tensor eigenvectors, was analyzed by Jimenez [351]. It is explained by the existence of a feedback between the resolved scales and the net energy drain provided by the subgrid model. Errors in subgrid models are localized at the highest resolved frequency, and do not contaminate the low frequency which is responsible for the turbulence production. The error accumulation at high frequencies leads to an adjustment of the subgrid model, which is expressed as a function of the resolved scales. This is easily seen by writing the induced subgrid dissipation: ε = −τij S ij = 2νsgs |S|2 . A classical example is the Smagorinsky model, which leads to ε = CS |S|3 . A local underestimation of CS will result in an energy accumulation at the cutoﬀ, leading to an increase of the resolved shear |S| and an increase of the net drain of kinetic energy. This drain mostly aﬀects the highest resolved frequencies, leading to a decrease of |S|. The global eﬀect on the ﬂow is small, because the highest resolved frequencies contain a small part of the total resolved kinetic energy. The eﬃciency of this adjustment depends of course on the way the subgrid viscosity is computed.

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For anisotropic ﬂows the loss of eﬃciency of the basic subgrid viscosity models is a well recognized fact. This is explained by the fact that the shear magnitude |S| is not governed by the highest resolved frequency any more, but by very large scales. Thus, the dynamic feedback loop described above is not suﬃcient any more to yield good results. Self-adaptive models (dynamic, ﬁltered, selective, etc.) must be used to ensure the quality of the results. The available experimental data suggest that the correlation obtained using the subgrid-viscosity model is of the order of 20%. This implies that it is impossible to get both the right energy spectrum and the right stresses from a subgridviscosity model if its eﬀect on the resolved ﬁeld is not negligible. Grid reﬁnement is known to yield improved results for two reasons: the error is commited on a smaller fraction of the total kinetic energy, and the Kolmogorov theory predicts that the small scales are more isotropic than the large ones, rendering the structural error less important. For a weak shear S, the Kolmogorov theory leads to the following scaling laws for the normal-stresses spectrum E11 and oﬀ-diagonal stress cospectrum E12 : E11 ∝ ε2/3 k −5/3 , and

τ12 ∝ (πLS /∆)−4/3 ,

E12 ∝ Sε1/3 k −7/3

,

τ12 /τ11 ∝ (πLS /∆)−2/3 , where the shear length LS = ε/S 3 is proportional to the integral scale Lε in equilibrium ﬂows. These scaling laws show that reﬁning the resolution, i.e. decreasing ∆, makes it possible to reduce quickly the anisotropy of the subgrid scales. Jimenez estimates that 1% error is obtained for Lε /∆ = 10–20. b) Scale similarity or soft-deconvolution-type models do not yield good results if used alone. This is also true for all the other types of ﬂows. A possible explanation for the improvement observed when using mixed models or explicit random models for the backward energy cascade is that these models weaken the spurious correlation between the resolved strain rate tensor and the modeled subgrid tensor. This improvement is also expected from theoretical results dealing with the full deconvolution problem or the rapid/slow decomposition introduced by Shao et al. It is important to note that on very coarse grids mixed models may not result in improved results. c) These results hold locally for all other ﬂows. 2. To simulate a free shear ﬂow (mixing layer, jet, wake): a) Subgrid viscosity models based on large scales can delay the transition. This problem can be remedied by using a dynamic procedure, a selection function, or an accentuation technique. The other subgrid viscosity models seem to allow the transition without any harmful effects.

444

3.

4.

5.

6.

14. Examples of Applications

b) Using a mixed structural/functional model improves the results obtained with a subgrid viscosity model based on the large scales. To simulate a boundary layer or plane channel ﬂow: a) Subgrid viscosity models based on the resolved scales may inhibit the driving mechanisms and relaminarize the ﬂow. As before, this problem is resolved by using a dynamic procedure, selection function, or accentuation technique. The other subgrid viscosity models do not seem to exhibit this defect. b) Using a mixed functional/structural model can improve the results by better taking the driving mechanisms into account. c) Using a model for the backward cascade can also improve the results. For separated ﬂows (backward facing step, for example), use a model that can yield good data on a free shear ﬂow (to capture the dynamics of the recirculating area) and on a boundary layer (to represent the dynamics after the reattachment point). For transitional ﬂows: a) Subgrid viscosity models based on the gradients of the resolved scales generally yield poor results because they are too dissipative and damp the phenomena. This problem can be remedied by using a dynamic procedure, a selection function, or the accentuation technique. b) Anisotropic tensorial models can inhibit the growth of certain threedimensional modes and lead to unexpected scenarios of transition to turbulence. For fully developed turbulent ﬂows, the problems with subgrid viscosity models based on the large scales are less pronounced than in the previous cases. Because these ﬂows have a marked dissipative character, they produce results that are sometimes better than the other models because they ensure numerical stability properties in the simulation.

14.5.3 Wall Model Eﬃciency Numerical experiments show that wall stress models based on a linear relationship between wall stresses and the instantaneous velocity component yield satisfactory results for well-resolved large-eddy simulation of attached ﬂows. For separated ﬂows, this class of wall models is inadequate, leading to a poor prediction of the skin friction inside the separation zone. For very coarse grids, i.e. when the cutoﬀ is not located inside the inertial range of the spectrum, large errors are observed and the mean velocity proﬁle is not recovered [363, 806, 89]. Several reasons for this can be identiﬁed: – On coarse grids, the numerical error is large and pollutes the solution, yielding erroneous input for the wall model. – On coarse grids, most subgrid models induce large errors on the resolved scales. This is especially true for all subgrid viscosity models, which are

14.5 Lessons

445

not able to account for the strong ﬂow anisotropy in the near-wall region. It has been shown [363, 806] that this error is mainly due to the fact that the subgrid acceleration term in the ﬁltered momentum equations is not properly predicted on very coarse grids.2 This bad prediction can lead to the occurrence of spurious coupling with linear wall models, yielding very bad results. The use of dynamic models may result in worse results if the test ﬁlter is applied outside the inertial range [89]. A very important conclusion of studies dealing with suboptimal-based wall models is that predicting the mean velocity proﬁle and recovering the best possible rms velocity proﬁle seem to be competing objectives. Wall models based on auxiliary simulations performed on secondary embedded grids (RANS, thin boundary layer equations) may be an alternative, which remains to be assessed in critical situations. The use of more complex subgrid models, at least in the resolved near-wall region, which are able to predict the subgrid acceleration is also expected to improve the response of wall stress models. Multilevel subgrid models have demonstrated a clear superiority in academic test cases. 14.5.4 Mesh Generation for Building Blocks Flows We discuss here the basic rules for mesh generation for two classical building blocks of complex ﬂows, namely the attached equilibrium boundary layer and the plane mixing layer. The two basic rules are: – The key idea is that the driving mechanisms, i.e. the events responsible for turbulence production and mean proﬁle instabilities, must be correctly captured by the simulation to recover reliable results. – The size of the computational domain must be larger than the correlation length of the ﬂuctuations, in each direction of space. Too small a domain size will yield spurious coupling between the dynamics of the ﬂow and the boundary conditions. Exact values of mesh size, number of grid points and their repartition, and domain size for ‘plug and play’ simulations are not available: these parameters depend on many parameters, including numerical methods. What are presented below are the commonly accepted ideas underlying the design of a large number of published works. Equilibrium Boundary Layer. As discussed in Sect. 10.2, the boundary layer exhibits two diﬀerent scalings. – In the inner layer the viscous length lτ is relevant to describe the dynamics. The typical correlation lenghts are 1000 wall units in the streamwise 2

It is recalled that subgrid viscosity models are designed to yield the correct amount of dissipation, not the proper subgrid acceleration.

446

14. Examples of Applications Z

Y X

-1000

x

z +

300 200 100 0 -200

+

-500

500

0 y + 200

1000

Fig. 14.28. Streaks in the near-wall region of a plane channel ﬂow (Reτ = 180). Left: well-resolved large-eddy simulation. Right: Coarse-grid large-eddy simulation. Courtesy of E. Montreuil, ONERA.

(x) direction, and 100 wall units in the spanwise direction (y). Based on a study of the anisotropy of the ﬂow, Bagget et al. [32] derived the following general criterion: ∆/L 0.1, where the local inertial length L = k 3/2 /ε is associated with the peak of the turbulent energy spectrum (with k the turbulent kinetic energy and ε the dissipation rate). Typical criteria dealing with the mesh size for a wall-resolving simulation are: ∆x+ 50,

∆y + ≤ 15 .

(14.1)

The ﬁrst grid point must be located at one wall unit from the wall, with three grid points in the viscous region 1 ≤ z + ≤ 10. Thirty to ﬁfty grid points across the boundary layer are generally enough to get acceptable results. Grids which are too coarse to allow a good resolution of the nearwall events usually yield to the occurrence of an overshoot in the streamwise rms velocity proﬁle. This peak is associated with fat streaky structures (see Fig. 14.28). In order to prevent spurious correlations the minimum size of the computational domain in the streamwise direction should be larger than 2000 wall units, with a spanwise extent of 400 wall units. – In the outer layer, the visual boundary layer thickness δ99 is the relevant scale. The large scales have a correlation length which scales with δ99 , and are advected in the streamwise direction at a speed roughly equal to 0.8 U∞ . Consequently, a minimum domain size of 3 δ99 to 5 δ99 is required in the streamwise direction, with a spanwise extent of 2 δ99 to 3 δ99 . Plane Mixing Layer. The choice of the extent of the computational domain and the mesh size for the plane mixing layer is often guided by results from the linear instability theory. For a hyperbolic tangent type mean velocity proﬁle, it is known that the two-dimensional linearly most unstable mode has a streamwise wavelength λx nearly equal to 7δω , where the vorticity

14.5 Lessons

447

thickness is deﬁned as δω = ∆U/max(dU /dz), with ∆U the velocity jump across the shear layer. The associated spanwise length scale is λy 2λx /3. Classical analysis of numerical errors shows that at least 5 to 20 grid points per wavelength are required to get a reliable description of physical phenomena. Thus, in the streamwise direction, large scales associated with Kelvin–Helmholtz instability should be captured with ∆x δω /2. Square-like meshes in the (x, y) plane are recommended, yielding ∆x = ∆y. Numerical experiments show that a minimum of 20 grid points is required across the shear layer, yielding ∆z δω /20.

Fig. 14.29. Large-eddy simulation of a time-developing plane mixing layer. Instantaneous view of the coherent structures at three diﬀerent times. Left: early transition stage. Right: advanced transition stage. Bottom: fully developed ﬂow. Courtesy of M. Terracol, ONERA.

448

14. Examples of Applications

The size of the domain is dictated by the purpose of the simulation. For unforced mixing layers, the turbulent self-similar state is reached after the second pairing. For temporal simulations, this indicates that the size of the domain in the streamwise direction must be greater than or equal to 8λx = 56 δω . The need for a third pairing originates from the fact that, after the last pairing permitted by the size of the computational domain, ﬂuctuations are correlated over the domain, leading to corrupted results. The self-similar state can be observed between the second and the last pairing. Typical results for a time-developing plane mixing layer are displayed in Fig. 14.29.

15. Coupling with Passive/Active Scalar

15.1 Scope of this Chapter This chapter is devoted to the extension of the previous results dealing with Large-Eddy Simulation of incompressible ﬂows to the case where a scalar quantity is added to the velocity and the pressure to describe the physical system. Depending on the application, the scalar can be related to the temperature (or temperature increment with respect to a given value), the density, the concentration of a pollutant, ... The chapter details two diﬀerent cases: 1. The case of the passive scalar (Sect. 15.2), where there is no feedback of the scalar equation in the momentum equation. In this regime, which represents a one-way coupling between the velocity ﬁeld and the scalar ﬁeld, the scalar dynamics is enslaved to the velocity ﬁeld, while the dynamics of the later is not aﬀected. The new closure problem is therefore restricted to the scalar equation, the treatment of the momentum equation being exactly the same as in the previous chapters. 2. The case of the active scalar (Sect. 15.3), which corresponds to physical systems in which a two-way coupling exist between the velocity and the scalar. The emphasis will be put on simple (destabilizing) buoyancy and stabilizing stratiﬁcation eﬀects. In the ﬁrst case the turbulent production is enhanced by the coupling with the scalar dynamics, while in the second case a new turbulence damping mechanism is involved. These two simple cases are used to illustrate the new problems dealing with subgrid modeling induced by the active scalar model: the ﬁltered scalar equation need to be closed, but the subgrid closure in the momentum equation must a priori be changed to account for the fact that the dynamics is now more complex than the simple kinetic energy cascade process. For the sake of simplicity, the chapter is not made exhaustive: in many cases, strategies developed to close the scalar equation are nothing but direct extrapolations of models/methods developed to close the mometum equations. In these cases, the models for the scalar equations will not be extensively described, and the way the original model is extended is indicated. A detailed description will be given only when the closure strategy or the model is not a straightforward extension of a former proposal. It is worth

450

15. Coupling with Passive/Active Scalar

noticing here that many models developed for the momentum equation have not yet been extended to the scalar problem, despite it may be done very easily. This chapter is restricted to presentation of the classical subgrid closure problem, and the issues dealing with the development of wall models and turbulent inlet conditions for the scalar (and the velocity ﬁeld in the active scalar case) will not be discussed.

15.2 The Passive Scalar Case 15.2.1 Physical Model Deﬁnitions and Filtered Equations in Physical Space. The following passive scalar equation is used as a starting point: ∂θ + ∇ · (uθ) = κ∇2 θ ∂t

,

(15.1)

where θ is the scalar quantity (temperature, pollutant concentration, ...) and κ the associated molecular diﬀusivity. Since there is no feedback in the momentum equation, the later will not be considered here. Applying a ﬁlter1 to (15.1), one obtains ∂θ + ∇ · (uθ) = κ∇2 θ ∂t

.

(15.2)

Splitting the ﬁltered non-linear term uθ into a resolved and a subgrid part, one obtains ∂θ + ∇ · (u θ) = κ∇2 θ − ∇ · τθ ∂t

,

(15.3)

where the subgrid scalar ﬂux τθ is deﬁned as τθ ≡ (uθ − u θ) .

(15.4)

The subgrid scalar ﬂux can be further decomposed in exactly the same way as the subgrid tensor in the momentum equation (see Sect. 3.3), yielding τθ = u − u θ + u θ + uθ + u θ θ Lθ

1

Cθ

.

(15.5)

Rθ

As for the momentum equation, the mathematical model used here to represent the true Large-Eddy Simulation problem is the convolution ﬁlter paradigm. But other mathematical models presented in Chap. 4 can also be used to this end.

15.2 The Passive Scalar Case

451

The subgrid ﬂuxes Lθ , Cθ and Rθ are analogous to the Leonard stress tensor, the Cross stress tensor and the subgrid Reynolds stress tensor, respectively. The subgrid scalar ﬂux can also be decomposed using Germano’s consistent decomposition approach (Sect. 3.3.3): τθ = τG (u, θ) = τG (u, θ) + τG (u, θ ) + τG (u , θ) + τG (u , θ ) ,

Lθ

Cθ

(15.6)

Rθ

where τG (φ, ψ) ≡ φψ − φ ψ is the generalized central moment of ψ and φ associated to the ﬁlter kernel G. Two quantities of interest to characterize the scalar dynamics are the subgrid scalar ﬂux τθ (which is equal to u θ if the ﬁlter belongs to the class 2 of the Reynolds operators) and the scalar subgrid variance θsgs ≡ θ θ . The former is related to the mixing/stirring transport process of the scalar ﬁeld at small scales, while the latter is tied to the existence of ﬂuctuations of the scalar ﬁeld at the subgrid level and therefore is a measure of its unmixedness. It is worth noting that these two quantities are generalized within the LargeEddy Simulation framework when arbitrary convolution ﬁlters are used in the same way that the subgrid kinetic energy ui ui /2 is extended considering the trace of the subgrid tensor. In the scalar case, u θ and the subgrid scalar variance τθ are extended as τ G(u, θ) and τG (θ, θ), respectively. Transport equations for these two quantities are easily derived using the scalar equation and the momentum equation. The most general expressions based on the generalized central moments are ∂τG (ui , θ) ∂τG (ui , θ) + uk ∂t ∂xk

∂θ ∂ui = − τG (uk , ui ) − τG (uk , θ) ∂xk ∂xk

I II ∂ui ∂θ ∂ + , θ + κτG , ui ντG ∂xk ∂xk ∂xk

III ∂ui ∂θ ∂p − (ν + κ)τG , − τG θ, ∂xk ∂xk ∂xi

IV

∂τG (uk , ui , θ) − , ∂xk

VI

for the generalized subgrid scalar ﬂux and ∂τG (θ, θ) ∂θ ∂τG (θ, θ) + uk = − 2τG (uk , θ) ∂t ∂xk ∂xk

V II

V

(15.7)

452

15. Coupling with Passive/Active Scalar

∂ 2 τG (θ, θ) +κ − 2κτG ∂xk ∂xk

V III

∂θ ∂θ , ∂xk ∂xk

IX

∂τG (uk , θ, θ) − , ∂xk

(15.8)

X

for the generalized subgrid scalar variance, where it is recalled that τG (a, b, c) = abc − aτG (b, c) − bτG (a, c) − cτG (a, b) − a b c

.

The physical meaning of the terms appearing in the preceding equations are – I: Production by interaction between the subgrid stresses and the resolved scalar gradient – II: Production by interaction between the subgrid scalar ﬂuxes and the resolved velocity gradient – III: Viscous diﬀusion – IV : Viscous dissipation – V : Scalar-pressure subgrid ﬂux – V I: Diﬀusion by subgrid motion – V II: Subgrid scalar variance production by interaction with the resolved scalar gradient – V III: Viscous diﬀusion – IX: Subgrid scalar variance diﬀusion, referred to as εθ – X: Diﬀusion by subgrid motion. Deﬁnitions and Filtered Equations in Spectral Space. Physical quantities and corresponding equations can be rewritten in the spectral space per forming a Fourier transform. Denoting θ(k) the Fourier transform of θ(x), one has the following relation dealing with the two-point correlation 6 ; < 1 )θ(k) = e−ı(k ·x+k·k ) θ(x)θ(x ) dxdx θ(k 2π Eθ (k) = δ(k + k ) , (15.9) 2πk 2 where Eθ (k) is the scalar spectrum (deﬁned here as an average over shell k = cste.). The scalar variance is equal to +∞ 1 θ(x)θ(x) = Eθ (k)dk . (15.10) 2 0 The spectral analogue of (15.1) is ∂ 2 + κk θ(k) = T θ (k) ∂t

,

(15.11)

15.2 The Passive Scalar Case

in which the non-linear term T θ (k) is equal to θ 3 T (k) = −ıkj p u j (k − p)θ(p)d

.

453

(15.12)

The corresponding evolution equation for the non-stationary spectrum of scalar variance is ∂ 2 + 2κk Eθ (k) = T θθ (k) , (15.13) ∂t where the scalar spectrum transfer is expressed as ; < − p) um (p)θ(−k) d3 p T θθ (k) = −8ık 2 km θ(k

.

(15.14)

The conservation property for the scalar variance takes the following form: T θθ (k)d3 k = 0 . (15.15) θ(k), The ﬁltered equation associated to the ﬁltered variable θ(k) ≡ G(k) where G(k) denotes the transfer function of the selected ﬁlter, is ∂ θ + κk 2 θ(k) (k) (15.16) = G(k)T ∂t =

θ Trθ (k) + Tsgs (k)

,

(15.17)

θ (k) are the resolved and subgrid spectral scalar ﬂuxes, where Trθ (k) and Tsgs only, while the respectively. The former involves resolved quantities θ and u latter contains all terms involving at least one subgrid component among (1 − G(k)) θ(k) and (1 − G(k)) u(k). The closure problem in the spectral θ space consists in ﬁnding an expression for Tsgs (k) which involves only known quantities.

15.2.2 Dynamics of the Passive Scalar This section is devoted to a brief survey of the dynamics of the passive scalar in isotropic turbulence. The main purpose here is to enlight the fact that the modeling task is far from being a trivial one, even within this simpliﬁed framework. The very reason for this is that several physical regimes exist, which are associated to diﬀerent values of the molecular Prandtl number (or Schmidt number, or Peclet number depending on the physical signiﬁcance of the scalar ﬁeld) deﬁned as ν . (15.18) Pr ≡ κ

454

15. Coupling with Passive/Active Scalar

The spectral properties of the three ideal regimes P r 1, P r ∼ 1 and P r 1 are surveyed in Sect. 15.2.2. The interested reader can refer to specialized books [714, 439, 464, 708] for a detailed discussion. Results obtained via Direct Numerical Simulation and EDQNM analysis dealing with the spectral dynamics of the passive scalar are displayed ﬁrst (see p. 456). The concept of subgrid Prandtl number is introduced and discussed in a second step (p. 459). Diﬀerent Regimes and Associated Dynamics. Three regimes for the passive scalar in isotropic turbulence are identiﬁed, each one being associated with a range of values for the Prandtl number. The existence of these three archetypal2 regimes originates in the diﬀerence between the viscous cutoﬀ scales for the scalar and the velocity ﬁeld. Each regime is associated to a speciﬁc scalar spectrum shape (the kinetic energy spectrum does not vary, since the scalar is strictly passive). To analyze the spectral characteristic features of these dynamical regimes, we introduce the scalar diﬀusion cutoﬀ length, referred to as the Obukhov– Corrsin scale, which is the analogue of the Kolmogorov scale ηK for the velocity: ε 1/4 1 3/4 = ηK . (15.19) ηθ = κ3 Pr The ratio of the wave numbers associated to these cutoﬀs is therefore estimated as kθ = P r3/4 . (15.20) kη The deﬁnition of Obukhov–Corrsin cutoﬀ scale is not valid if P r 1, since it is based on Kolmogorov-type hypotheses on the nature of the ﬂuctuations which are no longer adequate. In this case, Batchelor derived the following diﬀusive cutoﬀ wave number (referred to as the Batchelor wave number): ε 1/4 . (15.21) kB = νκ2 These three regimes are 1. P r 1 : the molecular diﬀusivity is much larger than the molecular viscosity. Looking at relations (15.19) and (15.20), it is seen that kθ is much smaller than kη , meaning that the the scalar diﬀusive cutoﬀ occurs within the inertial range of the Kolmogorov spectrum for the kinetic energy. One can therefore infer that two inertial ranges will be observed in the scalar spectrum (see Fig. 15.1): a) The inertial-convective range, which corresponds to wave numbers k kθ kη . These scales are not subject to viscous and diﬀusive eﬀects accordingly to Kolmogorov’s picture. Scalar ﬂuctuations are 2

It is important noticing that the results presented here deal with asymptotic cases, and that real-life ﬂows are more complex.

15.2 The Passive Scalar Case

455

Fig. 15.1. Schematic of the kinetic energy spectrum and the scalar variance spectrum in the very-low Prandtl number case.

driven by the stirring induced by velocity ﬂuctuations. The corresponding form of the scalar spectrum is Eθ (k) = βεθ ε−1/3 k −5/3

,

(15.22)

where Obukhov–Corrsin constant β is in the range 0.68 – 0.83. b) The inertial-diﬀusive range, which is observed for scales which belong to the Kolmogorov intertial range, meaning that the velocity ﬂuctuations are not directly sensitive to the molecular viscosity, but at which scalar ﬂuctuations experience a strong action of the molecular diﬀusivity. The corresponding wave number band is kθ k kη . The dynamics of the scalar at these scales is governed by a balance between the turbulent advection and the molecular diﬀusion. The scalar spectrum shape is Eθ (k) =

K0 εθ ε2/3 κ−3 k −17/3 3

.

(15.23)

Smaller scales (k ≥ kc ) are governed by molecular viscosity and diﬀusivity eﬀects, and exhibit an exponentially decaying behavior. 2. P r 1. The two cutoﬀ scales are very close, leading to the existence of a unique inertial-convective inertial range. The scalar spectrum shape is the same as in the previous case (15.22). 3. P r 1. This last case corresponds to conﬁgurations in which the diffusive cutoﬀ scale is much smaller than the viscous cutoﬀ scale for the velocity ﬂuctuations. Two inertial ranges exist (see Fig. 15.2):

456

15. Coupling with Passive/Active Scalar

Fig. 15.2. Schematic of the kinetic energy spectrum and the scalar variance spectrum in the very-high Prandtl number case.

a) An inertial-convective range, for wave numbers such that k kθ kB , which is similar to those already described above. b) A viscous-convective range, which is associated to scales (kθ k kB ) where the velocity ﬂuctuations are severely damped by viscous eﬀects but scalar ﬂuctuations are not aﬀected by molecular diﬀusion. The associated spectrum is 1 ν Eθ (k) = εθ k −1 . (15.24) 2 ε This brief survey obviously shows that the spectral scalar transfers across the large-eddy simulation cutoﬀ may strongly depend on the type of inertial range it is located in. As a consequence, the development of a closure based on the functional modeling approach may be much more diﬃcult than in the previous case of the momentum equations, since the latter exhibits only one type of inertial range dynamics. Interscale Scalar Transfers. We now present some results dealing with the analyis of passive scalar ﬂuxes in the spectral space. These results are the corner stone of the functional modeling approach. The emphasis is put on theoretical results coming from the EDQNM analysis of the passive scalar dynamics in isotropic turbulence (see B.3 for a short presentation). The main results of the EDQNM analysis deal with the non-local interactions in the scalar variance equation. Considering a non-local triad (k, p, q) satisfying the constraint k + p + q = 0, two cases are identiﬁed – The considered scale, associated with k, is interacting with one much larger scale : p k q. The EDQNM form of the associated non-local scalar

15.2 The Passive Scalar Case

457

variance ﬂux is T+θθ (k)

= + −

ak 2 θ 2 2 ∂Eθ (k) Θkkq q E(q)dq 2kEθ (k) − k 15 0 ∂k 1 ak θ 3 Θkkq q Eθ (q)dqE(k) 4 0 1 ak θ 5 E(k)Eθ (k) Θkkq q dq . (15.25) 4 0 k2

The parameter a deﬁnes the local/non-local triads (see p. 93). The ﬁrst term in the right hand side corresponds to interactions which are responsible for the existence of the viscous-convective range. – The scale k is involved in a non-local triadic interaction with much smaller scales: k p q. The asymptotic expression for the scalar variance ﬂux is +∞ 4 k 2 θ T−θθ (k) = − q Eθ (q)dq Θqpp E(p)dp 3 0 sup(k,q/a) +∞ 4 k 4 θ E(p)Eθ (p) + q dq Θqpp dp . (15.26) 3 0 p2 sup(k,q/a) A simpliﬁed expression, which is valid at low wave numbers, is +∞ 4 2 θθ θ T− (k) = − k Eθ (k) Θ0pp E(p)dp 3 ak 4 4 +∞ θ E(p)Eθ (p) k + Θ0pp dp . 3 p2 ak

(15.27)

The ﬁrst term in the right hand side is associated with a forward scalar variance cascade, whose intensity is governed by both the kinetic energy and scalar variance spectral distribution. The second term represents a backward scalar cascade. The analysis of Direct Numerical Simulation results [783] reveals that, in the inertial-convective range, the overall spectral transfer is forward cascading. It is dominated by the energy-containing scales in the velocity ﬁeld. More precisely, the dominant physical process at high wave numbers belonging to this inertial range is a local transfer associated with a non-local triadic interaction that links two high wave number scalar modes and one low wave number velocity mode. The other non-local interactions (one high wave number scalar mode, one low wave number scalar mode and one high wave number velocity mode) and local interactions are much weaker. Therefore, the suggested dominant physical process is the breaking of “blobs” of scalar ﬂuctuations into smaller-scale fragments by large energetic eddies.

458

15. Coupling with Passive/Active Scalar

The large-eddy simulation closure problem is now directly addressed by writing the equation associated to the resolved scalar variance density, de 2 (k)Eθ (k): noted E θ (k) = G ∂ 2 θθ + 2κk E θ (k) = Trθθ (k) + Tsgs (k) , (15.28) ∂t θθ where Trθθ (k) and Tsgs (k) are the resolved and the subgrid spectral scalar variance transfer, respectively. In a way similar to the one used to characterize subgrid kinetic energy transfers in Sect. 5.1.2, the intensity of these transfers across a cutoﬀ wave number kc can be parameterized deﬁning an eﬀective spectral diﬀusivity κe (k|kc ) such that θθ Tsgs (k) = −2κe (k|kc )Eθ (k) .

(15.29)

In the case the cutoﬀ is located within the inertial-conductive range, and assuming that all the basic hypotheses of the canonical analysis (inertial ranges extending to inﬁnity, sharp cutoﬀ ﬁlter, see Sect. 5.1.2), both the EDQNM analysis conducted by Chollet and Zhou’s RNG theory [811] show that the spectral eﬀective diﬀusivity shape is strictly similar to the one of the eﬀective spectral viscosity νe (k|kc ): – for k kc , it is independent of the wave number. The value of the plateau deduced from the non-local forward cascade term in (15.27) is 2 +∞ θ ∞ Θ0pp E(p)dp (15.30) κe (k|kc ) = κe ≡ 3 kc In practice, this plateau is observed for wave numbers in the range 0 ≤ k ≤ kc /3. Theoretical analysis carried out by Chollet and Lesieur using the EDQNM analysis in the case of a very wide inertial-convective range also shows that the value of the plateau depends on both the kinetic energy at the cutoﬀ and the slope of the spectrum. A general expression is [514]: ⎧ ' √ ⎪ ⎪ 4 3 − m E(kc ) ⎪ ⎨ m<3 3a m + 1 kc , (15.31) = κ∞ e ⎪ 1 1 1 E(kc ) ⎪ ⎪ √ m>3 ⎩ 3 m − 1 Dr kc where −m is the slope of the kinetic energy spectrum, a = 0.218K0 a struc!k tural EDQNM parameter and Dr = 0 c k 2 E(k)dk a norm of the resolved velocity gradient. Considering a Kolmogorov spectrum (m = 5/3, K0 = 1.4) one obtains κ∞ e = 0.3. – Near the cutoﬀ, i.e. at wave numbers kc /3 ≤ k ≤ kc , the eﬀective diﬀusivity exhibits a cusp, showing that the spectral transfers are more intense. In this region, local interactions are not negligible. It is important noting that this cusp is as sensitive as the one in the eﬀective viscosity: it is

15.2 The Passive Scalar Case

459

not observed with smooth ﬁlters, and can also disappear if the cutoﬀ is located at the very beginning of the inertial range. In the canonical case of an inﬁnite inertial-convective range, the maximum value of the subgrid diﬀusivity found using EDQNM analysis is κe (kc |kc ) = 0.6. Detailed investigations of the spectral transfers across a cutoﬀ located within the viscous-convective or the inertial-diﬀusive range are missing. This lack in the theory may be not very important for most practical applications, since putting the cutoﬀ within one of these inertial range requires the use of very ﬁne computational grids. Numerical experiments carried out by several authors [717] also show that the eﬀective diﬀusivity spectral shape is very sensitive to both the scalar variance spectrum shape and the kinetic energy spectrum: it has been observed to be either an increasing or a decreasing function of the wave number. Eﬀective viscosities computed from simulated isotropic turbulence by M´etais and Lesieur are displayed in Figs. 15.3 and 15.4. The Subgrid Prandtl Number Paradigm. The molecular Prandtl number (15.18) is a useful tool to compare turbulent scales which characterize dissipative/diﬀusive cutoﬀ scales of the velocity and scalar ﬁelds. This is why the idea of introducing a subgrid Prandtl number, P rsgs , is attractive. But preceding remarks dealing with the sensitivity of the eﬀective spectral diﬀu-

Fig. 15.3. Eﬀective subgrid viscosity (Solid line) and subgrid diﬀusivity (Dashed line) normalized by E(kc )/kc in the Lesieur–M´etais simulation (Case 1).

460

15. Coupling with Passive/Active Scalar

Fig. 15.4. Eﬀective subgrid viscosity (Solid line) and subgrid diﬀusivity (Dashed line) normalized by E(kc )/kc in the Lesieur–M´etais simulation (Case 2).

sivity indicate that a universal distribution for the spectral eﬀective subgrid Prandtl number νe (k|kc ) e , (15.32) (k|kc ) = P rsgs κe (k|kc ) cannot exist. As a consequence, subgrid closure strategies based on the concept of subgrid Prandtl number necessarily introduce some errors, since they do not account for subtle discrepancies that exist between the velocity and the scalar dynamics. In the very simple case of a quasi-inﬁnite inertial-convective range, the value found for wave numbers located within the plateau (k ≤ 0.3kc ) via the EDQNM analysis is [514] e P rsgs (k|kc )

5−m 20

.

(15.33)

But such a simple expression should not mask the fact that the subgrid Prandtl number is fully case dependent, since it characterizes the diﬀerences that may exist between kinetic energy and scalar spectral transfers. As an example, in the very simple case of passive scalar in isotropic turbulence, Lesieur and Rogallo [441] observed two very diﬀerent spectral subgrid Prandtl distributions (see Fig 15.5). In the ﬁrst case, the scalar ﬁeld exhibits a k −1

15.2 The Passive Scalar Case

461

Fig. 15.5. Eﬀective Prandtl number versus the wavenumber in Lesieur–Rogallo simulations of isotropic turbulence.

inertial range (this anomalous exponent is supposed to be due to shearing by large scales), and the subgrid Prandtl number approximately follows the law e P rsgs (k|kc ) = 0.35 + 0.2 log(k/kc ) + 9e−3.099kc /k

,

(15.34)

while in the second set of simulations the following distribution was observed e P rsgs (k|kc ) = 0.45 + 0.25 log(k/kc ) .

(15.35)

15.2.3 Extensions of Functional Models We now discuss the extension of functional models in the physical space. The construction of functional models in the Fourier space is a straightforward utilization of spectral subgrid difusivity laws given in the preceding section, and will not be further detailed. All explicit functional models for the passive scalar equations are based on the subgrid diﬀusivity paradigm, yielding the following general closure relation: (15.36) τθ = −κsgs ∇θ . As a consequence, the emphasis is put on the dominant mechanism observed in isotropic turbulence, namely the forward scalar cascade. This approach is strictly equivalent to the use of a subgrid viscosity to close the

462

15. Coupling with Passive/Active Scalar

ﬁltered momentum equations. Almost all functional subgrid models and/or closure strategies proposed for the momentum equations and presented in Chaps. 5 and 6 can be very easily extended to the scalar equation. An exhaustive description of all possibilities is not of interest, and it is worth noting that many straightforward extensions have not been published yet3 . The implicit Large-Eddy Simulation approach can also be used, if an adequate numerical scheme for the scalar equation is utilized. General Expression. Scalar Subgrid Diﬀusivity. As in the case of subgrid viscosity, a fully general expression of the subgrid diﬀusivity κsgs as a function of a set of selected basic quantities can be used to obtain a uniﬁed view of most published models. A possible expression is 2 2 κsgs = κsgs (P r, ∆, |S|, qsgs , ε, |∇θ|, θsgs , εθ ) ,

(15.37)

2 2 where quantities directly tied to subgrid quantities (i.e. qsgs , ε, θsgs , εθ ) can be evaluated solving corresponding prognostic equations, or using some local equilibrium and/or scale similarity assumptions to recover expressions involving resolved scales only. The Prandtl number is a priori included as an input of this general expression, since it can be used to distinguish the diﬀerent inertial range regimes. Weighting coeﬃcients in the constitutive equation (15.37) are found performing a simple dimensional analysis. An example is the model proposed by Schmidt and Schumann [651]: + 2 , (15.38) κsgs = Cκ ∆ qsgs

where interial-conductive range considerations lead to Cκ =

2 1 3K0 3βπ

.

(15.39)

Yoshizawa [789] proposes a more complex model: κsgs = C

2 2 ) ε (θsgs , 2 εθ

C = 0.446 .

(15.40)

A subgrid diﬀusivity model which accounts for molecular Prandtl number eﬀects was developed by Gr¨ otzbach [277, 279] to describe the temperature ﬁeld in liquid metals, which have very small Prandtl number (P r = 0.007 for liquid sodium under nuclear reactor conditions). We reproduce here the method developed by Gr¨ otzbach to account for small Prandtl number eﬀects rather than its model, since it can be applied to any subgrid viscosity model with one arbitrary constant. Let us consider a generic subgrid diﬀusivity 3

The interested reader will quickly ﬁnd material for a few dozens of papers dealing with “new improved models”!

15.2 The Passive Scalar Case

463

model under the form κsgs = CK

,

(15.41)

where C is the constant to be adjusted and K a constant-free parameter which has the dimension of a diﬀusivity. The method proposed by Gr¨ otzbach relies on the local equilibrium assumption (i.e. production = dissipation) for the scalar variance: τθ · ∇θ = εθ − κ∇θ · ∇θ , (15.42) where εθ is the full turbulent scalar variance dissipation. Now introducing the model in the left hand side of this relation, one obtains the following relation for the constant C: εθ − κ∇θ · ∇θ C= . (15.43) −K∇θ · ∇θ The last parameter to be evaluated is εθ . Assuming that the cutoﬀ occurs in a very wide inertial-convective range, an analytic expression for the scalar variance dissipation as a function of the kinetic energy dissipation can be found, yielding 1 −4/3 1 − βκε−1/3 ∆ , (15.44) C √ β K0 where the subgrid kinetic energy dissipation rate ε is retrieved from the work done to close the momentum equation, leading to a fully determined deﬁnition of the subgrid diﬀusivity. The Subgrid Prandtl Number Approach. A very common closure approach is to use a subgrid Prandtl number, leading to κsgs =

νsgs P rsgs

,

(15.45)

where the subgrid viscosity νsgs can be evaluated using any model described in the preceding chapters devoted to the momentum equation closure. Despite it is ﬂawed from a purely theoretical point of view, this approach is very often used in simulations in the physical space and the subgrid Prandtl number appears as an adjustable parameter which can be tuned in an ad hoc way to obtain the best ﬁt with the reference data. Values found in the literature range from 0.1 to 1, the most common one being 0.6. It is important noting that this approach is not valid in cases where the velocity ﬁeld if fully resolved, yielding νsgs = 0, while some subgrid scalar ﬂuctuations exist. A typical example would be to put the resolution cutoﬀ within the viscous-convective range. The subgrid Prandtl number can be made more accurate, but still not adequate to represent the cases mentioned above, rendering it self-adaptive, meaning that the value of the Prandtl number will be made space and time dependent. Such a modiﬁcation is expected to make the subgrid Prandtl number based models adequate to treat the case where all scalar ﬂuctuations are resolved while subgrid kinetic energy transfers exist. A common

464

15. Coupling with Passive/Active Scalar

example is to use the Germano identity to deﬁne a dynamic procedure, as proposed by Moin et al. [538]. The Germano identity for the scalar equation is 2 −u 2 −u ** ** uθ θ = uθ θ − uθ/ − uθ , (15.46) Lθ

τ*θ

Tθ

where the tilde symbol refers to the test ﬁlter level. Vectors Tθ and τθ are the subgrid scalar ﬂuxes at the grid and test ﬁlter levels, respectively. As in the analogous relation for the momentum equation, the left hand side of (15.46) can be directly computed, while replacing subgrid ﬂuxes in the right hand side by the corresponding subgrid models makes it possible the ﬁnd the best value of the model constant in the least-square sense. * the subgrid viscosity values computed at the Denoting νsgs (∆) and νsgs (∆) grid and test ﬁlter levels, respectively, the residual associated to the Germano identity (15.46) is equal to Eθ = L θ +

* (∆) νsgs (∆) νsgs/ ∇* θ− ∇θ P rsgs P rsgs

.

(15.47)

Assuming that the subgrid Prandtl number is the same at the two levels * the and that it does not vay signiﬁcantly over distances of the order of ∆, least-square minimization of Eθ leads to mθ · mθ P rsgs = − , (15.48) mθ · L θ where

* * mθ ≡ νsgs (∆)∇ θ − νsgs/ (∆)∇θ

.

(15.49)

Other expressions can be derived. As an example, Moin et al. [538] do not use the least-square minimization but chose to ﬁnd the value of P rsgs which corresponds to a zero of E · ∇* θ: θ

P rsgs = −

θ mθ · ∇* * ∇θ · L

.

(15.50)

θ

Wong and Lilly [766] deﬁne a dynamic procedure based on dimensional parameters, yielding P rsgs =

* (L · ∇* θ) (L : S) θ * * 2 |S| ∇θ · ∇* θ

.

(15.51)

All these dynamic subgrid Prandtl number deﬁnitions suﬀer some numerical stability problems. The methods used to cure this problem are the same as for the momentum equation (see Sect. 5.3.3 ): clipping, time/space averaging, ... All variants of the dynamic procedure described in preceding chapters can be applied to compute the dynamic subgrid Prandtl number.

15.2 The Passive Scalar Case

465

Anisotropic Subgrid Diﬀusivity. All models mentioned above are based on inertial range considerations, which are characteristic features of hom*ogeneous isotropic turbulence. Practical applications involve much more complex conﬁgurations, in which the scalar ﬁeld can be non hom*ogeneous while the velocity ﬁeld remains isotropic, or even more complex ﬂows were both the velocity and the scalar ﬁelds are not hom*ogeneous. Therefore, the question arises of accounting for the velocity ﬁeld anisotropy in the scalar diﬀusion. This task is expected to be much more complex than the equivalent one for the velocity ﬁeld, since the scalar variance spectrum exhibits much less universal features than the kinetic energy spectrum. Experiments conducted by Kang et al. [376] also show that the trend of decreasing anisotropy at small scales observed on velocity ﬂuctuations is not recovered on scalar ﬂuctuations. It is also known that, in the presence of a mean scalar gradient, structure functions and the derivative skewness of the scalar ﬁeld do not follow predictions from isotropy at inertial and even dissipative scales. This local isotropy breakdown is tied to the direct action of large velocity structures on small scalar scales (this is consistent with the EDQNM analyis results dealing with dominant triadic interactions). Taking anisotropy into account can be an important issue when the cutoﬀ is such that the subgrid scalar ﬂuctuations are governed by velocity ﬂuctuations with high local Reynolds number (the localness is understood here in terms of wave number). In such cases, anisotropy in the velocity ﬁeld will have an eﬀect on the scalar diﬀusion process, and the most simple idea consists in deﬁning a tensorial subgrid diﬀusivity. Yoshizawa [789] proposes the following anisotropic diﬀusivity, which is derived using a two-scale expansion: (κsgs )ij = −C

2 θsgs εθ

3 εS ij

,

(15.52)

where C = 0.366. The subgrid scalar dissipation rate and the subgrid scalar variance remain to be evaluated. Models for these quantities are presented in Sect. 15.2.6. The use of the resolved strain tensor S is coherent with the simple physical picture that scalar “blobs” are stretched by the velocity ﬂuctuations, and will therefore be elongated in the principal shear direction, leading to the development of smaller scalar scales in transverse directions. A similar model is proposed by Peng and Davidson [587]. Rotational eﬀects can also be included by taking into account the skewsymmetric part of the resolved velocity gradient, Ω ij : (κsgs )ij = −2ε with C1 = 0.29 and C2 = 0.08.

2 θsgs εθ

3

3 4 C1 S ij + C2 Ω ij

,

(15.53)

466

15. Coupling with Passive/Active Scalar

Another tensorial subgrid diﬀusivity model was proposed by Pullin [608], who extended the stretched-vortex approach (see Sect. 7.6) to the passive scalar case. In the extended model, it is assumed that the subgrid mixing is restricted to the plane normal to the vortex axis. A secondary property of this model is that the subgrid scalar gradient induced by the transport is orthogonal to the subgrid vorticity, the latter being represented by the stretched vortex. This is consistent with the observations that the probability density of the alignement between the vorticity and the scalar gradient is maximum when these vectors are orthogonal. The corresponding formulation is γπ + 2 , (15.54) qsgs δij − evi evj (κsgs )ij = 2kc where the vector ev is the same as in Sect. 7.6. Assuming that the cutoﬀ occurs within an inﬁnite inertial-convective range, the structural parameter γ is given by 2 2 1 . (15.55) γ= π 3K0 β 15.2.4 Extensions of Structural Models Many structural models have been applied to the subgrid scalar ﬂux vector. The most illustrative ones are presented below. Approximate Deconvolution and Scale Similarity Models. Scale similarity, or, in an equivalent way, soft approximate deconvolution models are deﬁned writing the following approximation: τθ ≡ uθ − uθ ≈ u• θ• − uθ

,

(15.56)

where the approximate deﬁltered ﬁelds are expressed as u• = G−1 l u,

θ• = G−1 l θ

,

(15.57)

an approximate inverse of the ﬁlter (see Sect. 7.2.1 for a detailed with G−1 l presentation). A large number of structural models can be generated using this simple form, as for the momentum equation: models based on iterative deconvolution, models based on Taylor series expansions and tensor diﬀusivity models. Because they are all approximations of the exact soft deconvolution solution, these models do not take into account scalar transfer towards subgrid scales and they must be supplemented with a dissipative term, which can be either a numerical regularization or a functional subgrid model. That leads to the deﬁnition of linear combination models for the subgrid scalar ﬂuxes. The Bardina-type model obtained using the lowest-order deconvolution is τθ = uθ − uθ

.

(15.58)

15.2 The Passive Scalar Case

467

The associated gradient-type model is (for a Gaussian or Box ﬁlter) 2

τθ =

∆ 2 ∇ (uθ) . 24

(15.59)

Diﬀerential Scalar Fluxes Model. Another solution consists in solving a diﬀerential equation for each component of the subgrid scalar ﬂux vector. This approach requires to close (15.7), in which terms V (pressure-scalar correlations) and V I (diﬀusion by subgrid ﬂuctuations) are not directly computable. Deardorﬀ [173] proposes to close term V as + 2 qsgs ∂p (15.60) = −C1 τG (ui , θ) , τG θ, ∂xi ∆ and to express the third-order term as + ∂ ∂ 2 τG (uk , ui , θ) = −C2 ∆ qsgs τG (uk , θ) + τG (ui , θ) ∂xi ∂xk

,

(15.61)

with C1 = 4.8 and C2 = 0.2. Other closures are derived by Sheikhi et al. [671], who use the velocityscalar ﬁltered density function approach. This method is strictly equivalent to the one developed by Gicquel et al. for the momentum equation (see Sect. 7.5.3): a stochastic ﬁeld whose probability density function is the solution of the evolution equation of the ﬁltered density probability function is computed via a Lagrangian-Monte-Carlo method. In the present case, the method is utilized to reconstruct both u and θ . These reconstructed stochastic variables can be used to close the ﬁltered momentum and scalar equations directly, or to close the equations for the subgrid scalar ﬂuxes and the subgrid scalar variance. The latter approach is considered here, since the former belongs to the family of the methods based on a direct evaluation of the subgrid scales (to be discussed below). This approach yields the following closure relation for the sum of the pressure-scalar correlations and the dissipation: ∂ui ∂θ 3 3 ∂p + C0 τG (ui , θ) , (15.62) , 2ντG + τG θ, =ω ∂xk ∂xk ∂xi 2 4 where C0 = 2.1 and the time scale ω is equal to + 2 qsgs ω= . ∆

(15.63)

The triple correlation term is directly computed using the stochastic subgrid ﬁeld.

468

15. Coupling with Passive/Active Scalar

Explicit Evaluation of Subgrid Scales, Multilevel Simulations and Others. The technique consisting in regenerating the subgrid scales of the scalar ﬁeld on a ﬁner mesh using a low-cost stochastic model, a simpliﬁed determinitic model or multilevel simulation has also been extended to the passive scalar case. These extensions being very reminiscent of their counterparts for the momentum equation, they will not be detailed here. 15.2.5 Generalized Subgrid Modeling for Arbitrary Non-linear Functions of an Advected Scalar The preceding sections addressed the problem of ﬁnding subgrid closures for the convection term, the other ones being assumed to be linear. But in many ﬂows, additional physical processes associated to scalar source/sink are present, which appear as non-linear functions of the scalar ﬁeld θ. Common examples are chemical reactions, ﬂuids with temperature-dependent molecular viscosity/diﬀusivity, or coupling of the scalar ﬁeld (density, temperature, concentration) with micro-physics such as radiative transfer. Let us consider a general form for this additional source term: ∂θ + ∇ · (uθ) = κ∇2 θ + ∂t

f (θ)

.

(15.64)

source term

The corresponding source term in the ﬁltered scalar equation is f (θ), which is decomposed as f (θ) = f (θ) + f (θ) − f (θ) . (15.65)

computable

τf :subgrid term

The new closure problem consists in ﬁnding a surrogate for τf . Since we are addressing a fully general problem which can cover a very wide range a physical mechanisms, the deﬁnition of a general functional model is hopeless. Speciﬁc functional models can be found for individual process, as it is done for the convection term. The most general approach being the structural modeling, Pantano and Sarkar [578] propose to use an approximate deconvolution approach to model general subgrid source terms (in practice, they applied this strategy to a chemical reaction term) . The key idea developed by these authors is to approximate θ by a synthetic ﬁeld θ• which is optimized so that it will minimize an error functional, leading to f (θ) − f (θ) = f (θ• ) − f (θ• ) .

(15.66)

Since physical processes may be very sensitive to small errors and are associated to high-order non-linearities in terms of θ (e.g. the radiative transfer

15.2 The Passive Scalar Case

469

ﬂux behaves as θ4 , where θ is the temperature), it is important to enforce some accuracy requirements when deﬁning θ• . The simple Taylor series expansion f (θ) − f (θ) =

1 2 f (θ)(θ2 − θ ) + ... , 2

(15.67)

illustrates the fact that a good control of the error committed on the subgrid scalar variance is necessary to ensure a satisfactory representation of the subgrid source/sink term. As a consequence, Pantano and Sarkar propose to modify the usual soft deconvolution procedure, leading to θ• = θ + C0 (θ − θ) + C1 (θ − 2θ + θ) + ... ,

(15.68)

were the coeﬃcients Ci are allowed to vary, instead of being ﬁxed to be unity as in the original approximate deconvolution procedure based on the Van Cittert iterative method (7.8). These coeﬃcients are chosen in order to make the statistical mean ﬁltered moments that appear in the Taylor series expansion of the modeled ﬁeld equal to their counterparts deﬁned using the exact ﬁeld. In practice, it is aimed to enforce the following global constraints k k θ•k − θ• dx = θk − θ dx, k = 2, ..., N , (15.69) Ω

Ω

where N is an arbitrary parameter and Ω is the ﬂuid domain under consideration. Practical subgrid models are derived truncating expansion (15.68) at an arbitrary order and inserting it in (15.69). To get a closed set of non-linear polynomial equations for the coeﬃcients Ci , it is also required to select an analytic expression of the scalar spectrum and the ﬁlter transfer function. 15.2.6 Models for Subgrid Scalar Variance and Scalar Subgrid Mixing Rate This section is devoted to models aiming at evaluating the two following subgrid quantities, which characterize the dynamics of the subgrid ﬂuctuations of the scalar ﬁeld θ: 2 – The subgrid scalar variance, θsgs , also referred to as the scalar unmixedness, since it measures the degree of local non hom*ogeneity of θ within the volume of characteristic diameter ∆: a uniform distribution is associated with a zero subgrid variance. – The scalar variance dissipation rate, εθ , also referred to as the scalar variance destruction rate, which is related to the stirring and mixing process at subgrid scales. A high destruction rate means that diﬀusion is quickly hom*ogenezing the scalar ﬁeld.

470

15. Coupling with Passive/Active Scalar

A large number of subgrid models for these two quantities have been proposed, since they are important inputs of subgrid models for combustion related terms. Since the speciﬁc problem of reactive ﬂows is beyond the scope of this chapter, this section illustrates the diﬀerent modeling ways and does not provide the reader with an exhaustive description. Models for the Subgrid Scalar Variance. It is worth noting that the 2 is reminiscent of the one dealing with the evaluation issue of predicting θsgs 2 . Therefore, all methods and models deﬁned of the subgrid kinetic energy, qsgs to compute the latter (see Sect. 9.2.3) can be modiﬁed to evaluate the former. This work is straightforward and is left to the interested reader. Only general approaches will be discussed in this section. The subgrid scalar variance being deﬁned as 2 ≡ θ2 − θ θsgs

2

,

(15.70)

a natural way to model it is to use a deconvolution-type approach (or its optimized version proposed by Pantano and Sarkar for arbitrary non-linear functions of θ, see Sect. 15.2.5). The usual deconvolution procedure yields 2 θ•2 − θ• θsgs

2

,

(15.71)

where θ• = G−1 l θ is the approximate deﬁltered ﬁeld. Using the zeroth-order expansion of the deconvolution operator, one obtains a scale-similarity type model: 2 2 2 C(θ − θ ) , (15.72) θsgs where the constant C is equal to one if the Van Cittert method is used. Other values can be found using a dynamic procedure based on a Germanotype identity. A gradient type model is recovered replacing the convolution ﬁlters by their equivalent diﬀerential expansion, and then truncating these expansion at an arbitrary order. For box and Gaussian ﬁlters, the ﬁrst order term is 2 2 C ∆ |∇θ|2 , (15.73) θsgs where the parameter C is ﬁlter dependent and can be adjusted using a dynamic procedure. The subgrid scalar variance can also be easily retrieved if a structural model based on an explicit reconstruction of the subgrid scales is used to close the ﬁltered scalar equation. In this case, it is directly computed from the synthetic subgrid scalar ﬁeld. It is also straightforwardly extracted if a diﬀerential model for the scalar ﬂuxes is used. 2 , in Another possiblity consists in solving a prognostic equation for θsgs a way similar to what is done for the subgrid kinetic energy. Using (15.8) as a starting point, a closed equation is obtained. The resulting model is

15.2 The Passive Scalar Case

471

presented in (15.135) for the sake of brevity. Prognostic equations with dynamic coeﬃcients can also be derived using techniques presented in Sect. 5.4 (p. 173) for the transport equation of the subgrid kinetic energy. A simple expression is obtained in the simpliﬁed case where the inertialconvective range spectrum (15.22) is assumed to be valid at all subgrid wave numbers: +∞ 3β 2/3 2 = Eθ (k)dk = εθ ε−1/3 ∆ . (15.74) θsgs 2π 2/3 π/∆ Models for the Subgrid Scalar Dissipation Rate. A ﬁrst simple evaluation of εθ is obtained assuming that the cutoﬀ is located within an inﬁnite inertial-convective range: + 2 2 θsgs qsgs εθ = C , (15.75) ∆ where the constant C can by either computed analytically or adjusted using a dynamic procedure. Considering an inﬁnite inertial-convective range, one obtains 2 2π . (15.76) C= 3β 3K0 The other way to evaluate the subgrid scalar variance dissipation rate is to assume that the local equilibrium hypothesis holds: in this case, it is equal to the local subgrid scalar variance production rate, leading to εθ = −τθ · ∇θ

,

(15.77)

where the subgrid scalar ﬂux τθ can be evaluated using any adequate subgrid model. If a subgrid diﬀusity model is utilized, one obtains εθ = 2κsgs |∇θ|2

.

(15.78)

This expression can be modiﬁed to account for the dissipation at the resolved scales, yielding εθ = 2(κ + κsgs )|∇θ|2

.

(15.79)

In order to obtain a more general expression which does not rely on the local equilibrium assumption, Jimenez et al. [352] assumed that the characteristic subgrid mixing time is proportional to the subgrid turbulent time: εθ ε ∝ 2 2 θsgs qsgs

,

(15.80)

,

(15.81)

yielding the following model εθ = C

2 ε θsgs 2 qsgs

472

15. Coupling with Passive/Active Scalar

where C is a parameter. Tests show that C = 1/P r leads to satisfactory results. All quantities which appear in the right hand side of (15.81) can be evaluated using ad hoc models. 15.2.7 A Few Applications – – – – – – – – – – – – –

Heat transfer at free surface [93] Heat transfer in plane channel ﬂow [92, 755, 196, 753] Isotropic turbulence [121, 441, 538, 122, 220, 608, 344, 345, 752] Complex cavities and ducts [279, 590, 639] hom*ogeneous turbulence [344, 345] Time-developing shear layer [344, 345] Flow past a backward facing step [27] Jet impinging on a plate [738, 155, 126, 236] Flow in a corrugated passage [141] Mixing Layer [168] Flow in rotating/steady duct with/without rib [557, 558, 559, 560] Flow in S-shaped duct [561] Wall-mounted cube matrix [565]

15.3 The Active Scalar Case: Stratiﬁcation and Buoyancy Eﬀects We now turn to the case of the coupling with an active scalar, i.e. with a ﬁeld which has a feedback eﬀect on the velocity ﬁeld, leading to a two-way coupling between the Navier–Stokes and the scalar equations. Since there are a very huge number of possible interactions, it is chosen to put the emphasis on buoyancy and stable stratiﬁcation eﬀects. Other physical models, such as Eulerian–Eulerian models for two-phase ﬂows, will not be considered. 15.3.1 Physical Model Buoyancy and stable stratiﬁcation eﬀects originate in the force to which a blob of ﬂuid is submitted when it is immersed in a ﬂuid with diﬀerent density. Therefore, a natural physical parameter to describe these eﬀects is the density. When very weak compressiblity eﬀects are taken into account, the density of “usual” ﬂuids is assumed to decrease when the temperature is increased. Thus, the temperature can also be used to represent the dynamics of these ﬂows. Both solutions are found in the literature, depending on the authors and speciﬁc purposes of the studies. In the following, the scalar ﬁeld θ will be related to the temperature ﬁeld. Using the Boussinesq approximation (see reference books for a detailed discussion of the range of validity of this model, e.g. [714, 708, 439]), the

15.3 The Active Scalar Case: Stratiﬁcation and Buoyancy Eﬀects

473

basic set of unﬁltered equations consists in the Navier–Stokes equations with a gravitational source term supplemented by a scalar equation identical to (15.1): ∂u + ∇(u ⊗ u) = −∇p + ν∇2 u + ∂t

θ−Θ g Θ0

,

(15.82)

feedback term

∇·u = 0 ,

(15.83)

∂θ + ∇ · (uθ) = κ∇2 θ , (15.84) ∂t where θ is potential temperature, deﬁned as the thermodynamic temperature increased by the normalized gravitational potential, g = (0, 0, −g) the gravity vector and Θ is the potential temperature ﬁeld associated to the hydrostatic equilibrium. The vector g is related to the gravitational acceleration. Θ0 is related to a reference value, which is assumed to be unique for the whole ﬂow domain. The molecular viscosity and the molecular diﬀusivity are assumed to be constant and uniform. The corresponding set of ﬁletered equations is very similar to those used in previous chapters, since the original set of equations diﬀers by only one source term: θ−Θ ∂u + ∇(u ⊗ u + τ ) = −∇p + ν∇2 u + g ∂t Θ0

,

∇·u = 0 ,

(15.85) (15.86)

∂θ + ∇ · (uθ + τθ ) = κ∇2 θ , (15.87) ∂t where it has been assumed that the scale separation operator perfectly commutes with all partial derivatives operator and that the mean ﬁeld Θ is varying slowly enough in space to have Θ = Θ. The subgrid ﬂuxes τ and τθ have exactly the same expressions as in the passive scalar case. A deeper insight into the two-way coupling is gained rewriting the evolution equations for the subgrid quantities. Assuming that the vertical direction is associated to x3 , the equations for the subgrid mometum ﬂuxes are: ∂τG (ui , uj ) ∂τG (ui , uj ) + uk ∂t ∂xk

∂uj ∂ui − τG (uk , uj ) ∂xk ∂xk ∂ui ∂uj ∂ 2 τG (ui , uj ) +ν − 2ντG , ∂xk ∂xk ∂xk ∂xk ∂p ∂τG (uk , ui , uj ) −τG uj , − ∂xi ∂xk g + (δi3 τG (uj , θ) + δj3 τG (ui , θ)) , Θ 0

= −τG (uk , ui )

coupling term

(15.88)

474

15. Coupling with Passive/Active Scalar

and the corresponding equations for the generalized subgrid kinetic energy is: 1 ∂τG (ui , ui ) ∂ ∂τG (ui , ui ) = τG (ui , ui , uj ) + τG (p, uj ) − ν ∂t ∂xj 2 ∂xj ∂ui ∂ui ∂ui − ντG , − τG (ui , uj ) ∂xj ∂xj ∂xj g + δi3 τG (θ, ui ) . (15.89) Θ0

coupling term

The subgrid scalar ﬂuxes are solutions of ∂τG (ui , θ) ∂τG (ui , θ) + uk ∂t ∂xk

∂θ ∂ui = −τG (uk , ui ) − τG (uk , θ) ∂xk ∂xk ∂ui ∂θ ∂ + , θ + κτG , ui ντG ∂xk ∂xk ∂xk ∂ui ∂θ ∂p −(ν + κ)τG , − τG θ, ∂xk ∂xk ∂xi g ∂τG (uk , ui , θ) + δi3 τG (θ, θ) , (15.90) − ∂xk Θ0

coupling term

while the generalized subgrid scalar variance equation is kept unchanged, since there is no gravitational source term in the scalar equation. Therefore, the subgrid variance is not directly aﬀected by the feedback eﬀect on the velocity ﬁeld, but it is sensitive to it via changes in the latter. 15.3.2 Some Insights into the Active Scalar Dynamics Flows governed by (15.82)–(15.84) exhibit a very wide and complex range of physical mechanisms, which originate in the coupling that may exists between convection, diﬀusion, stratiﬁcation and other features such as mean shear, rotation and boundary conditions. It is meaningless to try to give an exhaustive description of all these possibilities. The present section will focus on very simple cases and the emphasis will be put on results dealing with interscale transfers, which are of primary interest for subgrid modeling. The inﬂuence of the stratiﬁcation eﬀects on the subgrid kinetic energy is seen looking at the last term of (15.89), which can lead to an increase associated to buoyancy eﬀect or a decrease due to stable stratiﬁcation damping, depending on its sign. Its relative importance with respect to the production term is measured by the ﬂux Richardson number: Rif =

g Θ0 τG (u3 , θ) ∂ui τG (ui , uk ) ∂x k

,

(15.91)

15.3 The Active Scalar Case: Stratiﬁcation and Buoyancy Eﬀects

475

High values of Rif are associated to ﬂows in which stratiﬁcation eﬀects are dominant, while they can be neglected when Rif 1. Neglecting energy transport across the ﬂow, one obtains Rif 1 −

ε ∂ui −τG (ui , uk ) ∂x k

,

(15.92)

showing that destabilizing eﬀects are present if Rif > 1 (the denominator of the last term being assumed to be positive). Two basic cases are discussed in the present section: – The case of stable stratiﬁcation (p. 475), in which stratiﬁcation eﬀects tend to damp the subgrid kinetic energy. An important feature of ﬂows with stable stratiﬁcation is the existence of dispersive internal gravity waves. These waves are important in many applications dealing with meteorology and oceanology. They induce irrotational large-scale horizontal motions, whose interactions with small scale turbulence will be one of the most important issue discussed below. – The case of unstable stratiﬁcation (p. 480), where buoyancy eﬀects generate turbulence, a well known example being thermal plumes. Another relevant parameter is the group N =

g ∂θ Θ0 ∂z

.

(15.93)

Under the assumption that the subgrid heat ﬂux τG (θ, u) obeys a Fickianlike law, i.e. τG (θ, u) −C∇θ, where C is a positive constant, it is seen from (15.88), (15.89) and (15.90) that the local stablizing/destabilizing eﬀect of stratiﬁcation depends on the sign of N , i.e. on the local sign of the vertical resolved gradient ∂θ/∂z: 1. If ∂θ/∂z > 0, the eﬀect is stabilizing, since it appears as a sink term in the subgrid kinetic energy equation. In this regime, gravity waves are stable and their frequency is characterized by the Brunt–V¨ais¨ al¨ a frequency, N ≡ √ N. 2. If ∂θ/∂z < 0, gravity waves are unstable and break up into turbulence: the buoyancy eﬀects are seen to increase the production of√subgrid kinetic energy. The buoyancy time scale is estimated as Tb ≡ 1/ −N . Energy Transfers in Stably Stratiﬁed Flows. Energy transfers in stably stratiﬁed ﬂows have been addressed by many authors, and are observed to be case dependent. Therefore, no general description of the transfer across a cutoﬀ similar to what exists for the momentum equation is available, the main explanation for that being that such a general description is nearly impossible. The diﬃcult points which preclude such an analysis are the following

476

15. Coupling with Passive/Active Scalar

– These ﬂows are strongly anisotropic, and it has already been seen that kinetic energy transfers within hom*ogeneous anisotropic ﬂows cannot be described in a simple and general way, since they are case-dependent. – The description in terms of interscale transfers is not suﬃcient to obtain an accurate picture of the governing physical processes. It appears that the concept of modes must be introduced to achieve a meaningful analysis: the energy must be split into the vortex kinetic energy and the total wave energy, the latter being further decomposed into the potential energy and the wave kinetic energy [266]. The analysis of triadic interactions using this scheme leads to a global description which is too complex to yield results that can be straightforwardly utilized for subgrid parameterization. This modal decomposition results from a local decomposition of the velocity and temperature ﬁeld in the Fourier space: (k) = u 1 (k) + u 2 (k) , u

(15.94)

1 (k) and u 2 (k) are related to the vortex and internal gravity wave where u components, respectively, with 1 (k) = φ1 (k)e1 (k), u where e1 (k) =

2 (k) = φ2 (k)e2 (k) , u

k×g , |k × g|

e2 (k) =

k × e1 (k) |k × e1 (k)|

(15.95)

.

(15.96)

The two vectors (e1 (k), e2 (k)) generate an orthonormal basis for the plane perpendicular to k. The vortex kinetic energy spectrum, Φ1 (k), the wave kinetic energy spectrum, Φ2 (k), and the potential energy spectrum P (k) are deﬁned as follows: Φ1 (k) =

< 1 ; φ1 (k)φ1 (−k) , 2 1 P (k)(k) = 2N 2

< 1 ; φ2 (k)φ2 (−k) 2 ; < θ(−k) θ(k) , Φ2 (k) =

,

where N 2 is equal to the square of the Brunt-V¨ais¨ al¨ a frequency built on the mean temperature gradient: N2 =

g dΘ Θ0 dz

.

(15.97)

The total kinetic energy spectrum is recovered summing the vortex and wave contributions: E(k) = Φ1 (k) + Φ2 (k). As a consequence, this section will be devoted to the description of the main features of very simple cases which have been very accurately analyzed using both theoretical tools and direct numerical simulation.

15.3 The Active Scalar Case: Stratiﬁcation and Buoyancy Eﬀects

477

The most simple case is initially isotropic turbulence submitted the strong stabilizing eﬀects. The main observed phenomenon is the breakdown of isotropy, associated to a drastic restriction of motion along the mean stratiﬁcation direction. After some times, the ﬂow is organized in thin layers with a strong variability along the stratiﬁcation direction (pancakeshaped large-scale vortices), leading to the occurance of a velocity ﬁeld which is almost two-component4 Detailed investigations of energy transfers using Direct Numerical Simulation and EDQNM models have been carried out [235, 304, 513, 266, 689, 267]. The main results are the following – Several phases are observed, the number of which being dependent on the inital condition and the value of the Richardson number. The ﬁrst phase (if initial potential energy is large enough) corresponds to a ﬂow whose large scales are governed by stratiﬁcation eﬀects and inertial gravity waves and exhibit a N 2 k −3 kinetic energy spectrum, while small scales are not aﬀected by stratiﬁcation eﬀects and are governed by the usual isotropic turbulent dynamics. In the last phase, all scales are anisotropic and stratiﬁcation eﬀects drive the whole ﬂow. – During the ﬁrst phase (pre-collapse phase), the ﬂow dynamics is controlled by the potential energy being transferred from large to small scales at a higher rate than kinetic energy, leading to a higher total energy decay as in unstratiﬁed isotropic turbulence. The kinetic energy along the stratiﬁcation direction is converted into potential energy, feeding the irreversible forward cascade of potential energy. A consequence of the potential energy cascade is the existence of a persisting counter-gradient scalar ﬂux. Such a counter-gradient scalar ﬂux should be associated to a negative subgrid diﬀusivity at small scales, making all previous models irrelevant. But this counter-gradient ﬂux seems to play no major dynamical role on the smallest scales during the ﬁrst phase if the molecular diﬀusivity is high enough. – The ﬁnal phase (post-collapse phase) is associated to a scalar ﬂux collapse at large scales. The ratio of the kinetic energy along the the stratiﬁcation direction and the potential energy reach a constant value, while large-scale energetic motion in directions perpendicular to the mean stratiﬁcation direction arise from turbulence, forming the vortex part of the ﬂow. The small scale dynamics is dominated by stratiﬁcation eﬀects. An explanation is that, because of the collapse of large scales which reduces the rate of the forward energy cascade, the smallest scales are made sensitive to stratiﬁcation. – A detailed analysis of the transfers reveals that the pure vortex-vortex interactions (i.e. triadic interactions involving vortex modes only) are initially important among all triadic interactions, the resonant interactions involving at least one wavy mode being less important. The former are 4

But not two-dimensional, since its dynamics is very diﬀerent from the one of two-dimensional turbulence. In particular, no strong backward energy cascade is observed.

478

15. Coupling with Passive/Active Scalar

responsible for the isotropy breakdown and the blocking of a possible backward energy cascade. A very important feature, which limits the accuracy of the modeling of triadic transfers in the physical space, is that the energy transfers are strongly dependent on both the modulus and the direction of the three wave numbers. Oscillatory exchanges between wave kinetic energy and potential energy are also observed (their sum exhibiting a non-oscillatory behavior), associated to irreversible anisotropy creation, leading to angular variations (in the spectral space) of the vortex kinetic energy and the total wave energy. – The counter-gradient scalar ﬂux at small scales can inhibit mixing if the molecular diﬀusivity is too low, leading to a complex behavior of the scalar variance dissipation rate. These results show that the triadic transfer patterns are much more complicated than in unstratiﬁed ﬂows. The physical picture presented above is to be further complexiﬁed to account for additional physical mechanisms, such as turbulence generation by a forcing term, interaction with a mean shear, coupling with dynamics, etc. The main information retrieved from the previous analysis is that the forward kinetic energy cascade is decreased by stratiﬁcation eﬀects in the post-collapse phase. The analysis of the transfers across a cutoﬀ wave number during the pre- and post-collapse phase was carried out by M´etais and Lesieur [514] using numerical simulations. Using the orthogonality of the local reference frame in the Fourier space, the authors extend the analysis presented in Sect. 5.1.2 by decomposing the total kinetic energy transfer into a vortex and wave part, and introduce a spectral subgrid viscosity representation for each part: e,i Tsgs (k|kc ) , (15.98) νei (k|kc ) = − 2 2k Φi (k) where i = 1, 2 refers to the subgrid transfer associated with the corresponding mode in the modal decomposition. As in the unstratiﬁed case, these subgrid eﬀective viscosties can be normalized using the kinetic energy at the cutoﬀ, yielding ' νei (k|kc ) = νei+ (k|kc )

E(kc ) kc

.

(15.99)

The total eﬀective subgrid viscosity, νe (k|kc ), is obtained using the following relationship νe (k|kc )E(k) = νe1 (k|kc )Φ1 (k) + νe2 (k|kc )Φ2 (k) .

(15.100)

This last expression illustrates the high diﬃculty which arises in the modeling of the subgrid transfers via a single subgrid viscosity, since its value depends on both the energy cascade rate of the vortex and wave components and the repartition of the kinetic energy among these two modes. This double dependency shows that the total eﬀective viscosity must be sensitive to

15.3 The Active Scalar Case: Stratiﬁcation and Buoyancy Eﬀects

479

the initial conditions, which seems to preclude any general accurate deﬁnition. The subgrid potential temperature transfer is formally deﬁned as in the passive scalar case, since the evolution equations are the same. It is observed in the numerical simulations carried out by M´etais and Lesieur that – In the pre-collapse phase (see Fig. 15.6), νe1+ (k|kc ) and νe2+ (k|kc ) are almost identical and do not diﬀer from the eﬀective subgrid viscosity observed in unstratiﬁed isotropic turbulence: the same plateau (at low wave numbers) and cusp (near the cutoﬀ) behaviors are observed. The eﬀective subgrid diﬀusivity exhibits a plateau at low wave numbers but do not show any cusp at high wave numbers. – In the post-collapse phase (see Fig. 15.7), the cusp in the two subgrid viscosities is higher than in the pre-collapse phase because of the shifting of the spectrum maxima towards low wave numbers. The vortex-related viscosity, νe1+ (k|kc ), exhibits a plateau with constant value 0.09 at low wave numbers, while the wave mode related viscosity exhibits the same plateau value at intermediate scales but increases with decreasing wave number at low wave numbers. The cusp is also increased in the subgrid diﬀusivity. Its is important noting that despite this increase in the relative intensity of the transfer at small scales, the absolute level of the eﬀective

Fig. 15.6. Eﬀective subgrid viscosities νe1+ (k|kc ) (Solid line) and νe2+ (k|kc ) (Dashed line) and subgrid diﬀusivity (Dotted line) in the Lesieur–M´etais stablystratiﬁed case (pre-collapse phase).

480

15. Coupling with Passive/Active Scalar

Fig. 15.7. Eﬀective subgrid viscosities νe1+ (k|kc ) (Solid line) and νe2+ (k|kc ) (Dashed line) and subgrid diﬀusivity (Dotted line) and subgrid diﬀusivity (Dotted line) in the Lesieur–M´etais stably-stratiﬁed case (post-collapse phase).

subgrid viscosities, which scales like E(kc ), is decreased in the postcollapse phase, since stable stratiﬁcation inhibits the forward kinetic energy cascade, leaving less energy in the small scales. These results show that, depending on the considered ﬂow regime and initial conditions [287], both the intensity and the spectral shape of the global transfer operator vary, rendering its accurate modeling in the physical space nearly impossible. Energy Transfers in Buoyancy-Driven Flows. We now turn to the case were the buoyancy force acts as a turbulence generator, as in natural convection ﬂows. In these ﬂows, buoyancy-generated instabilities create unsteady motion which will lead to turbulence. Similarly to the case of stable stratiﬁcation eﬀects, buoyancy-driven ﬂows exhibit a wide range of physical mechanisms which prevent an exhaustive detailed analysis. As an example, let us consider the analysis of energy transfers in thermal plumes carried out by Baastians et al. [29]. Using an experimental database, these authors observe a time mean backscatter in the lower part of the plume and instantaneous backscatter in the total up-ﬂowing area. The energy transfer rate across the cutoﬀ exhibits large ﬂuctuations, instantaneous values being about ten times larger than the mean value. Forward and backward energy cascades are equally important in the mean in the plume region. These results

15.3 The Active Scalar Case: Stratiﬁcation and Buoyancy Eﬀects

481

show that a reliable subgrid model should be able to account for backscatter, and to distinguish between diﬀerent regions of the ﬂow, rendering the modeling task even more diﬃcult as in the stably stratiﬁed ﬂow case. But it is important to remark that, in the basic philosophy which underlies most of the developments in the ﬁeld of Large-Eddy Simulation, it is assumed that driving mechanisms must be directly captured. In the case of buoyancy-driven ﬂows, buoyancy eﬀects should therefore be directly simulated if they are the only source of turbulence production (free convection). In the case of mixed convection or forced convection, the requirement might be less stringent since other physical mechanisms are at play. This is why some authors use unmodiﬁed passive scalar models (and usual closures for the momentume equations) to treat Rayleigh–B´enard convection at medium Prandtl number. 15.3.3 Extensions of Functional Models We now discuss the most illustrative functional models in the physical space for the active scalar case. Since a two-way coupling exists, subgrid models for both the momentum and the scalar equations must be revisited to account for stratiﬁcation/buoyancy eﬀects. Modeling strategies based on the deﬁnition of scalar subgrid viscosity/diﬀusivity parameters are ﬁrst considered (p. 481). Dynamic models based on the Germano identity are presented in the second part of the section (p. 486). Scalar Subgrid Viscosity/Diﬀusivity Models. The brief review of results dealing with interscale energy transfers in ﬂows with active scalar given in the previous section obviously shows that they are strongly aﬀected in both stable and unstable stratiﬁcation cases. This is why most scalar models are derived considering a simpliﬁed kinetic energy balance equation which includes buoyancy eﬀects. The most popular simpliﬁed energy balance expression is obtained by neglecting all diﬀusive and convective eﬀects in (15.89), yielding an extended local equilibrium (production = dissipation) assumption: ε = −τ : S − b · τθ ,

b≡

1 g Θ0

,

(15.101)

where it is recalled that ε, τ and τθ are the subgrid kinetic energy dissipation rate, the momentum subgrid tensor and the potential temperature subgrid ﬂux vector, respectively. Now specializing Eq. (15.101) by inserting scalar functional models of the form τ = −2νsgs S,

τθ = −κsgs ∇θ

,

(15.102)

one obtains νsgs |S|2 − κsgs Nc2 = ε

,

(15.103)

482

15. Coupling with Passive/Active Scalar

where the local parameter Nc2 is deﬁned as Nc2 ≡ −b · ∇θ =

g ∂θ Θ0 ∂z

,