Large Eddy Simulation for Incompressible Flows: An Introduction (Scientific Computation)

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

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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

Foreword to the Third Edition

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

Foreword to the Second Edition

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

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).

Foreword to the First Edition

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

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.

Preface to the Second Edition

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

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.

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

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

283 288 290 290 294 297 299 303

305 306 307 310 311 313 313 314 315 318

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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

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

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

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

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-

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

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).

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

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)

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

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

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

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.

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

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 =

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.

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

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

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 −∞

(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

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

φ=φ−∆

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

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)

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

−γ|x − ξ|2

(2.42)

∆

G(k) = exp

−∆ 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

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

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

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.

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

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]:

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=0

∆ 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!

∆ 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)

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

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

∆ /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)

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)| 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

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:

2. Formal Introduction to Filtering

dψ 1 (x) = dx ∆

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

∂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

Ak α(k) (ξ)∆ +

k=1,+∞

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)

∀ξ ∈ [α, β] ,

(k)

=
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

τ11 (x) = u21 (x) − (u1 (x)) ≤

G(x − ξ) (u1 (ξ)) d3 ξ < 0 .

(3.162)

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

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

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 =

p(0) + ∆ p(1) + ... ,

(3.168)

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)

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

−

(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

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)

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

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.

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

– 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 ,+∞

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

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.

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

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.

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

◦

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

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.

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

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)

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

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 .

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

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.

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

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

III

(5.14)

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.

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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

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].

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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,

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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

∞

1 νe (k|kc ) = 15

∞

∂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)

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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)

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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

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 =

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|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.

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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 ∆

(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 ,

∆ 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

(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

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

∂ C2 ∂xj

III

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

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 ∆

(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 =

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

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∆ |S|S ij + −2∆ F (∆)S ij −2∆

1+α

|F(u)|α (qc2 )

1−α 2

S ij

* −2∆

1+α

* 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 −

u[z(t ), t ]dt

(5.180)

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 =

−∞

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

−∞

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) =

−∞

JMM (x, 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

Cd βij , βij = −2∆ |S|S ij

(5.224)

(5.225)

|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

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

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

(5.240)

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

−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 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

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

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

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 =

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)

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

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

+ β 2 (∇u)d ⊗ (∇u)d

(5.282)

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

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:

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

(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

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

(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

/ 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

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

(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)

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

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

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)

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=

(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)

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)

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⎜ 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

−

∂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)

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

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: ; 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)

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:

(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:

(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

(λ/η)c

Logistic

√ −(2 + 2 2)1/2

Tent

-1/3

28.6

Sawtooth

-1/3

28.6

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)

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)

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)

= B(u2 , u2 ) I

266

7. Structural Modeling 2

+ B(u2 , v 1 ) + B(v1 , u2 ) + B(v 1 , v 1 )

II 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)

B(u2 , u2 ) IV 1

+ B(u2 , v 1 ) + B(v1 , u2 )

V 1

+ B(v 1 , v 1 )

(7.236)

VI 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

III

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) =

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.

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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.

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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 (

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)

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. Numerical Solution: Interpretation and Problems

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

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