simulation and modelimg of turbulent flows_ lumley j.l

Simulation and Modelingof Turbulent Flows

ICASE/LaRC Series in Computational Science and Engineering

Series Editor: M. Yousuff Hussaini

Wavelets: Theory and Applications

Edited byGordon Erlebacher, M. Yousuff Hussaini, and Leland M. Jameson

Simulation and Modeling of Turbulent Flows

Edited byThomas B. Gatski, M. Yousuff Hussaini, and John L. Lumley


Edited byThomas B. Gatski

NASA Langley Research Center

M. Yousuff HussainiICASE

John L. LumleyCornell University

New York OxfordOxford University Press

1996

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Library of Congress Cataloging-in-Publication DataSimulation and modeling of turbulent flows / edited by Thomas B. Gatski,

M. Yousuff Hussaini, John L. Lumley.p. cm. — (ICASE/LaRC series in computational science and engineering)

Includes bibliographical references and index.ISBN 0-19-510643-1

1. Turbulence—Mathematical models. I. Gatski, T. B.II. Hussaini, M. Yousuff. III. Lumley, John L. IV. Series.

TA357.5.T87S56 1996532'.0527'015118—dc20 96-11573

1 3 5 7 9 8 6 4 2

Printed in the United States of Americaon acid-free paper

PREFACE

This book is based on the lecture notes of ICASE/LaRC ShortCourse on Turbulent Modeling and Prediction, held on March 14-18,1994. The purpose of the course was to provide the scientists andengineers with a knowledge of the state-of-the-art turbulence modeldevelopment including the latest advances in numerical simulationsand prediction of turbulent flows. The lectures focussed on topicsranging from incompressible, low-speed flows to compressible, high-speed flows. A key element of this short course was the systematic,rational development of turbulent closure models and related aspectsof modern turbulent theory and prediction.

The first chapter is based on the lecture by John Lumley writtenin collaboration with Gal Berkooz, Juan Elezgaray, Philip Holmes,Andrew Poje and Cyril Volte. It covers the basic physics pertainingto turbulent scales and spectral cascades for both equilibrium andnonequilibrium flows. It also includes a discussion of proper orthogo-nal decomposition and wavelet representation of coherent structuresin turbulent flows.

The next two chapters are on the numerical simulation of tur-bulent flows - the direct numerical simulation (DNS) by AnthonyLeonard, and the large-eddy simulation (LES) by Joel Ferziger. Bothhave been among the pioneers in this field. The chapter on DNSexplains the critical issues of numerical simulation, and discussesvarious solution techniques for the Navier-Stokes equations, in par-ticular divergence-free expansion techniques and vortex methods ofwhich the author has been a leading proponent. The chapter onLES examines the modeling issues, surveys the various subgrid-scalemodels, and describes some accomplishments and future prospects.

In the fourth chapter written in collaboration with I. Staroselsky,W. S. Flannery and Y. Zhang, Steven Orszag provides an introduc-tion and overview of modeling of turbulence based on renormaliza-tion group method, which he and his group has pioneered for over adecade. Although the application of RNG method to Navier-Stokesequations is by no means rigorous, this chapter illustrates its use-fulness by the quality of results from its application to a variety ofturbulent flow problems. A key feature is the emphasis on the greyareas which require further analysis.

vi Preface

The fifth chapter is based on the lecture by Charles Speziale onmodeling of turbulent transport equations, whereas the sixth chap-ter, by Brian Launder, on the prediction of turbulent flows usingturbulent closure models. Both the authors are well-known leadersin the area of turbulence modeling which is receiving ever increasingattention in national laboratories and industries.

We want to thank Ms. Emily Todd for her usual skillful attentionto detail and organization which resulted in a very smooth week oflectures, and to Ms. Barbara Stewart and Ms. Shannon Keeter whotyped or reformatted some of the manuscripts. Thanks also goes toMs. Leanna Bullock for her expert revamping of many of the figuresso that they could be electronically assimilated into the text.

Thomas B. GatskiM. Yousuff HussainiJohn L. Lumley

CONTENTS

PREFACE v

INTRODUCTION 1

FUNDAMENTAL ASPECTS OF INCOMPRESSIBLEAND COMPRESSIBLE TURBULENT FLOWS 5John L. Lumley

1 INTRODUCTION 51.1 The Energy Cascade in the Spectrum in Equilibrium

Flows 61.2 Kolmogorov Scales 91.3 Equilibrium Estimates for Dissipation 101.4 The Dynamics of Turbulence 112 EQUILIBRIUM AND NON-EQUILIBRIUM FLOWS 132.1 The Spectral Cascade in Non-Equilibrium Flows ... 132.2 Delay in Crossing the Spectrum 142.3 Negative Production 192.4 Mixing of Fluid with Different Histories 202.5 Deformation Work in Equilibrium and

Non-Equilibrium Situations 232.6 Alignment of Eigenvectors 252.7 Dilatational Dissipation and Irrotational Dissipation 262.8 Eddy Shocklets 283 PROPER ORTHOGONAL DECOMPOSITION AND

WAVELET REPRESENTATIONS 293.1 Coherent Structures 293.2 The Role of Coherent Structures in Turbulence

Dynamics 323.3 The POD as a Representation of Coherent

Structures . 333.4 Low-Dimensional Models Constructed Using

the POD 373.5 Comparison with the Wall Region 423.6 Generation of Eigenfunctions from Stability

Arguments 523.7 Wavelet Representations 67

1

viii Contents

3.8 Dynamics with the Wavelet Representation in aSimple Equation 68

4 REFERENCES 72

2 DIRECT NUMERICAL SIMULATION OFTURBULENT FLOWS 79Anthony Leonard

1 INTRODUCTION 792 PROBLEM OF NUMERICAL SIMULATION 803 SIMULATION OF HOMOGENEOUS

INCOMPRESSIBLE TURBULENCE 854 WALL-BOUNDED AND INHOMOGENEOUS FLOWS 865 FAST, VISCOUS VORTEX METHODS 916 SIMULATION OF COMPRESSIBLE TURBULENCE 1007 REFERENCES 104

3 LARGE EDDY SIMULATION 109Joel H. Ferziger

1 INTRODUCTION 1092 TURBULENCE AND ITS PREDICTION 1112.1 The Nature of Turbulence 1112.2 RANS Models 1122.3 Direct Numerical Simulation (DNS) 1153 FILTERING 1164 SUBGRID SCALE MODELING 1184.1 Physics of the Subgrid Scale Terms 1184.2 Smagorinsky Model 1194.3 A Priori Testing 1234.4 Scale Similarity Model 1254.5 Dynamic Procedure 1274.6 Spectral Models 1324.7 Effects of Other Strains 1354.8 Other Models 1375 WALL MODELS 1386 NUMERICAL METHODS 1417 ACCOMPLISHMENTS AND PROSPECTS 1438 COHERENT STRUCTURE CAPTURING 1468.1 The Concept 146

Contents ix

8.2 Modeling Issues 1489 CONCLUSIONS AND RECOMMENDATIONS . . . . 14910 REFERENCES 150

4 INTRODUCTION TO RENORMALIZATION GROUPMODELING OF TURBULENCE 155Steven A. Orszag

1 INTRODUCTION 1552 PERTURBATION THEORY FOR THE

NAVIER-STOKES EQUATIONS 1593 RENORMALIZATION GROUP METHOD FOR

RESUMMATION OF DIVERGENT SERIES 1624 TRANSPORT MODELING 1695 REFERENCES 182

5 MODELING OF TURBULENT TRANSPORTEQUATIONS 185Charles G. Speziale

1 INTRODUCTION 1852 INCOMPRESSIBLE TURBULENT FLOWS 1872.1 Reynolds Averages 1872.2 Reynolds-Averaged Equations 1892.3 The Closure Problem 1892.4 Older Zero- and One-Equation Models 1902.5 Transport Equations of Turbulence 1922.6 Two-Equation Models 1932.7 Full Second-Order Closures 2103 COMPRESSIBLE TURBULENCE 2203.1 Compressible Reynolds Averages 2213.2 Compressible Reynolds-Averaged Equations 2213.3 Compressible Reynolds Stress Transport Equation . . 2233.4 Compressible Two-Equation Models 2263.5 Illustrative Examples 2274 CONCLUDING REMARKS 2345 REFERENCES 236

Contents

AN INTRODUCTION TO SINGLE-POINTCLOSURE METHODOLOGY 243Brian E. Launder

1 INTRODUCTION 2431.1 The Reynolds Equations 2431.2 Mean Scalar Transport 2451.3 The Modeling Framework 2461.4 Second-Moment Equations 2471.5 The WET Model of Turbulence 2532 CLOSURE AND SIMPLIFICATION OF THE

SECOND-MOMENT EQUATIONS 2552.1 Some Basic Guidelines 2552.2 The Dissipative Correlations 2572.3 Non-Dispersive Pressure Interactions 2582.4 Diffusive Transport dij, dio 2732.5 Determining the Energy Dissipation Rate 2752.6 Simplifications to Second-Moment Closures 2782.7 Non-Linear Eddy Viscosity Models 2813 LOW REYNOLDS NUMBER TURBULENCE NEAR

WALLS 2843.1 Introduction 2843.2 Limiting Forms of Turbulence Correlations

in the Viscous Sublayer 2863.3 Low Reynolds Number Modelling 2883.4 Applications 2994 REFERENCES 302

INDEX 311

x

6

LECTURERS

Joel H. FerzigerThermosciences DivisionMechanical Engineering

DepartmentStanford UniversityStanford, CA 94305-3030(415) 723-3615

John L. LumleySibley School of Mechanical

and Aerospace EngineeringUpson and Grumman HallsCornell UniversityIthaca, NY 14853-7501(607) 255-4050

Brian E. LaunderDepartment of Mechanical

EngineeringUMISTP.O. Box 88Manchester M60 1QDENGLAND(44) 161 200 3701

Steven A. OrszagProgram in Applied

and Computational Mathematics218 Fluid Dynamics Research CenterForrestal CampusPrinceton UniversityPrinceton, NJ 08544-0710(609) 258-6206

Anthony LeonardGraduate Aeronautical

LaboratoriesCalifornia Institute of

TechnologyPasadena, CA 91125(818) 356-4465

Charles G. SpezialeDepartment of Aerospace

and Mechanical EngineeringBoston University110 Cummington StreetBoston, MA 02215(617) 353-3568

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Introduction

The aim of this book is to provide the engineer and scientist with thenecessary understanding of the underlying physics of turbulent flows,and to provide the user of turbulence models with the necessary back-ground on the subject of turbulence to allow them to knowledgeablyassess the basis for many of the state-of-the-art turbulence models.

While a comprehensive review of the entire field could only bethoroughly done in several volumes of this size, it is necessary to focuson the key relevant issues which now face the engineer and scientistin their utilization of the turbulent closure model technology. Theorganization of this book is intended to guide the reader through thesubject starting from key observations of spectral energy transfer andthe physics of turbulence through to the development and applicationof turbulence models.

Chapter 1 focuses on the fundamental aspects of turbulence phys-ics. An insightful analysis of spectral energy transfer and scalingparameters is presented which underlies the development of phe-nomenological models. Distinctions between equilibrium and non-equilibrium turbulent flows are discussed in the context of modifica-tions to the spectral energy transfer. The non-equilibrium effects ofcompressibility are presented with particular focus on the alterationto the turbulent energy dissipation rate. The important topical issueof coherent structures and their representation is presented in thelatter half of the chapter. Both Proper Orthogonal Decompositionand wavelet representations are discussed.

With an understanding of the broad dynamic range covered byboth the turbulent temporal and spatial scales, as well as their modalinteractions, it is apparent that direct numerical simulation (DNS) ofturbulent flows would be highly desirable and necessary in order tocapture all the relevant dynamics of the flow. Such DNS methods,in which all the important length scales in the energy-containing

1

2 Introduction

range and in the dissipation range are accounted for explicitly ispresented in Chapter 2. Emphasis is on spectral methods for incom-pressible flows, including the divergence-free expansion technique.Vortex methods for incompressible bluff body flows are describedand some techniques for compressible turbulent flow simulations arealso discussed briefly.

Unfortunately, while the utopic desire is to perform simulationsof tubrulent flows without recourse to models of any type, the realityof the broad spectral range of such flows, discussed and exemplifiedin the first two chapters, precludes such calculations in most flowsof engineering interest. Thus, in computational studies using thelarge eddy simulation (LES) method, the largest scales of motionare represented explicitly, and the small scales are approximated ormodeled. A chapter on LES of reasonable length can not possi-bly be comprehensive. However, Chapter 3 updates earlier reviewsand provides a relatively comprehensive, yet succint discussion onthe subject. It begins with some cryptic remarks on the nature ofturbulence and the prediction methods. Then the stage is set forthe discussion of LES with a brief overview of methodologies for theReynolds-aver aged Navie-Stokes equations and DNS. This overlapsnicely with the preceding and following chapters which deal in somedetail with the DNS and phenomenological modeling of turbulencerespectively, and explains their relation to LES. It contains a longsection on subgrid-scale modeling which is a distinguishing feature ofLES, and further includes a section on numerical methods in practicefor solving the relevant equations. A key feature is the discussion ofaccomplishments and exploration of the feasible boundaries for LESapplications. It concludes with a discussion of modeling issues andauthor's retrospect and prospect.

As the reader can assess from both the DNS and LES formula-tions, the attempt is to directly compute either all (or most) of theturbulent scales, or just the large turbulent scales, explicitly. In theLES approach, the concept of turbulence modeling makes its first ap-pearance. The remainder of the book indeed focuses on the presentstate-of-the-art approaches to the turbulence modeling problem. Un-like the DNS and LES approaches, however, no attempt is made toexplicitly represent any of the turbulent scales through direct com-putation. Rather, the modeling approach is to represent the effectof the turbulence on the mean flow in toto. This can be done in avariety of ways, but the emphasis in the remaining chapters will be

Introduction 3

on the development of transport equations for turbulent single pointsecond moments, such as the turbulent Reynolds stresses or turbu-lent kinetic energy, solved in conjunction with a suitable turbulentscale equation.

The concept of renormalization group (RNG) applied to the de-velopment of turbulent closure models has been shown to be rathersuccessful. Chapter 4 discusses the basis for the RNG method andits application to a variety of flow problems. Once again, an under-standing of the spectral structure of turbulent flows, specifically, thefundamental assumption of the universality of the small scales, playsa key role in the application of the RNG technique. RNG theorythen provides a description, or model, of the small scales which canbe used to isolate the large scales. This leads to equations of mo-tion for the large scales, and turbulence models for the prediction oflarge-scale flow properties.

A more common formulation in turbulent flow prediction meth-ods is the utilization of modeled transport equations for the turbu-lent Reynolds stresses and/or turbulent kinetic energy. In Chapter5, the theoretical foundations of Reynolds stress models in turbu-lence are assessed from a basic mathematical standpoint. It is shownhow second-order closure models and two-equation models with ananisotropic eddy viscosity can be systematically derived from theNavier-Stokes equations for incompressible turbulent flows that arenear equilibrium and only weakly inhomogeneous. Properly cali-brated versions of these models perform extremely well in the pre-diction of two-dimensional mean turbulent flows that are not toofar from equilibrium. The development of reliable Reynolds stressmodels for more complex turbulent flows, particularly those involv-ing large departures from equilibrium or high-speed compressible ef-fects, presents greater difficulties. In regard to the latter flows, re-cent progress in the modeling of compressible dilatational terms isdiscussed in detail. The central points of the chapter are illustratedby a variety of examples drawn from compressible as well as incom-pressible turbulent flows.

Higher-order models, such as the Reynolds stress or second-mo-ment closures, are being used more frequently in solving complex tur-bulent flows. Chapter 6 presents both the methodology and specificclosure proposals for modeling the turbulent Reynolds stresses. Par-ticular attention is given to second-moment closure in which evolu-tion equations are solved for the Reynolds stresses themselves. While

4 Introduction

the capabilities of 'the basic model', used in current CFD software,are briefly reviewed, most attention is given to a new non-linearclosure based rigorously on realizability constraints. This schemepermits many types of near wall flows to be handled without 'wall-reflection' corrections. Also presented is a new non-linear eddy vis-cosity model that links the turbulent stresses explicitly to strain andvorticity tensors up to third order. This leads to a scheme with a bet-ter sensitivity to secondary strains that is possible at the quadraticlevel.

As the reader can readily see, these chapters provide both a thor-ough introduction and state-of-the-art assessment of predicting tur-bulent flows through simulations or transport equation modeling.With this overall view of the field, the reader can begin to get aclearer understanding of the focus of turbulent modeling research atthis time, and become sensitized to the important link between soundmathematical and physical analysis underlying the development ofwell-posed turbulence models.

Chapter 1

FUNDAMENTAL ASPECTSOF INCOMPRESSIBLE ANDCOMPRESSIBLETURBULENT FLOWS

John L. LumleyGal Berkooz, Juan Elezgaray, Philip Holmes

Andrew Poje, Cyril Volte

1 INTRODUCTION

Turbulence generally can be characterized by a number of lengthscales: at least one for the energy containing range, and one fromthe dissipative range; there may be others, but they can be expressedin terms of these. Whether a turbulence is simple or not dependson how many scales are necessary to describe the energy containingrange. Certainly, if a turbulence involves more than one productionmechanism (such as shear and buoyancy, for example, or shear anddensity differences in a centripetal field) there will be more than onelength scale. Even if there is only one physical mechanism, say shear,a turbulence which was produced under one set of circumstancesmay be subjected to another set of circumstances. For example,a turbulence may be produced in a boundary layer, which is thensubjected to a strain rate. For a while, such turbulence will havetwo length scales, one corresponding to the initial boundary layer

5

turbulence, and the other associated with the strain rate to which theflow is subjected. Or, a turbulence may have different length scalesin different directions. Ordinary turbulence modeling is restrictedto situations that can be approximated as having a single scale oflength and velocity. Turbulence with multiple scales is much morecomplicated to predict. Some progress can be made by applyingrapid distortion theory, or one or another kind of stability theory,to the initial turbulence, and predicting the kinds of structures thatare induced by the applied distortion. We will talk more about thislater. For now, we will restrict ourselves to a turbulence that has asingle scale of length in the energy containing range. We may takethis to be the integral length scale:

where p(r) is the autocorrelation coefficient in some direction, say

where x — {xi, x^, £3}, a is an arbitrary direction, and we adopt theconvention of no sum on Greek indices. In just the same way, anyturbulence will have at least two velocity scales, at least one for theenergy containing eddies, and one for the dissipative eddies. Exactlythe same remarks apply here, and many flows of technological interestmay be expected to have more than one scale of velocity. Here,however, we will restrict our attention to flows that have a single scaleof velocity in the energy containing range. We will take as our scaleof velocity the r.m.s. turbulent fluctuating velocity, u =< u,-u,- >1//2,where we use the Einstein summation convention; that is, if an indexis repeated, we understand a sum over i = I — 3. Note that, in theabove, we are supposing that < . > is a long time average, or anaverage over the full .space, or an ensemble average.

1.1 The Energy Cascade in the Spectrum in EquilibriumFlows

Fourier modes are too narrow in wavenumber space to representphysical entities. In physical space, the corresponding statement isthat Fourier modes extend without attenuation to infinity, while weknow that the largest entity of any physical significance in a turbulent

6 J. L. Lumley et al.

Fundamental Aspects 7

flow has a size not much larger than /. Hence, we must seek anotherphysical entity. Traditionally, we have talked about "eddies," butthese have never been very well denned. We want to introduce herethe wavelet. We will talk more later about wavelets, and how to usethem for a complete representation of the velocity field, and how toconstruct physical models using wavelets. For now, we want to usethe simplest properties of wavelets. If we consider a clump of Fouriermodes, as the band in Fourier space becomes wider, the extent inphysical space becomes narrower. A reasonable size appears to be aband lying between about 1.62/c and K/1.62; the numerical value canbe obtained by requiring that KO, — K/a = K, from which a = 1.62.That is, the bandwidth is equal to the center wavenumber. Thisresults in a wavelet in physical space that is confined to a distanceof about a wavelength.

We can now discuss how these wavelets interact in Fourier space.Reasoning in the crudest way, a wavelet exists in the strain ratefield of all larger wavelets. This strain rate field induces anisotropyin the wavelet, which permits it to extract energy from the largerwavelets. This energy extraction process is associated with vortexstretching: when a vortex is stretched by a strain rate field, thestrain rate field does work on the vortex, increasing its energy, andlosing energy in the process. This process, of extracting energy fromthe larger wavelets, and feeding it to the smaller wavelets, is known asthe energy cascade. In Tennekes and Lumley (1972), it is shown thatthe cascade is not particulatly tight — a given wavelet receives half itsenergy supply from the immediately adjacent larger wavelet, and theother half from all its neighbors. Similarly, of all the energy crossing agiven wavenumber, three quarters goes to the next adjacent smallerwavelet, and the remainder is distributed to all the even smallerwavelets. Nevertheless, we can to a crude approximation considerthat the energy enters the spectrum at the energy containing scales,and then is passed from wavelet to wavelet across the spectrum untilit arrives at the dissipative range, where it is converted to heat.

There is some discussion in the literature of what is called "back-scatter." This is an unfortunate choice of words, since it suggests thatenergy that started in one direction, is turned around and ends upgoing in the other direction. This is not at all what is meant. Rather,this refers to transfer of energy in the spectrum in the direction fromsmall to large wavelets. Now, there is no question that, taking av-erages over long times or large regions of space, the energy transfer

8 J. L. Lnmley et al

in the spectrum of three-dimensional turbulence is from large scalesto small. In two-dimensional turbulence, however, the energy cas-cade goes in the opposite direction, since the mechanism is totallydifferent. There is no vortex stretching in two dimensional turbu-lence; instead, vortices coalesce to form larger vortices, and this isthe mechanism for energy transfer. Now, it is perfectly possible ina three dimensional turbulence, if one considers short time averages,or averages over only a small region of physical space, to have co-alescing vortices, and hence locally, temporarily, energy transfer inthe "wrong" direction. Some initial instabilities are two-dimensional,and for a while the energy transfer will surely be in the wrong di-rection, until the flow is thoroughly three-dimensionalized, and theenergy transfer can proceed in the usual direction. Probably manyflows of technological importance, which are young — that is, notfully developed, have highly anisotropic remnants of initial instabil-ities, and turbulent structures that are highly anisotropic, and maywell have for limited times or over limited regions, energy transferin the "wrong" direction. It should not be difficult to build such aprocess into a model of the energy transfer, using the idea that it isprobably associated with two-dimensionality.

Now, let us consider the transfer of energy from one wavelet toanother. If V(CLK) is the velocity typical of a wavelet with center wave-number CLK, which has a size of roughly 2?r/aK, then the energy (perunit mass) in this wavelet is v2(aK), and the rate typical of the energytransfer will be set primarily by the strain rate of the wavelet at K/a,which is f (K/a)/(27ro//v). Hence, the rate of energy transfer shouldbe approximately v2(aK,)v(K/a)/2Ka/K). If the Reynolds number islarge, and if the turbulence is in equilibrium, then we may expect thisquantity to be approximately constant across a considerable range ofwavenumbers in the middle of the spectrum, and it must be equal tothe dissipation s. Now, since a is not a large number, we may prob-ably approximate this by •y3(K)/(2?r/K) = e. We should probablyinclude a constant to be on the safe side, but we would expect it tobe of order one, since physically the two sides should be of the samemagnitude. In fact, it turns out experimentally that the left handside is equal to about 0.3£. This, in fact, gives the classical form ofthe high Reynolds number, inertial subrange equilibrium spectrum,E ~ ae2/3/^"5/3, where we interpret EK — v2(K,), since the width ofthe wavelet in Fourier space is K. We are approximating the integralof E from K/a to OK by K£.


1.2 Kolmogorov Scales

In 1941, Kolmogorov suggested that, as the energy was passedfrom wavelet to wavelet, it would lose detailed information aboutthe mechanism of energy production. If the number of steps in thecascade was sufficiently great, we could presume that all informationwould be lost. The small scales would know only how much energythey were receiving. They might be expected to be isotropic (havinglost all information about the anisotropy of the energy-containingscales). Note that this state of isotropy would exist only at infiniteReynolds number (infinitely many steps in the cascade). At anyfinite Reynolds number, the small scales would be expected to beless anisotropic than the energy containing scales, but still somewhatanisotropic.

Note that these ideas cannot be applied directly to the spectrumof a passive scalar. It can be shown fairly easily that, in the presenceof a mean gradient of the scalar, velocity eddies can produce sharpfronts or interfaces between quite different scalar values; these sharpjumps correspond to very high wavenumber processes, and the ori-entation of the jumps appears to be determined by the orientation ofthe mean gradient. Hence, serious anisotopy is introduced into thesmallest scales.

We may also mention at this point that there is permanent an-isotropy even in the smallest scales of the velocity spectrum of a shearflow. This is discussed in Lumley (1992). The anisotropy exists be-cause the energy from the mean flow is fed into one component, andmust be redistributed to the other two. However, while the amountof anisotropy remains fixed as the Reynolds number increases, itsteadily decreases when considered as a proportion of the total meansquare velocity gradient. Hence, it is still correct to say (from at leastone point of view) that the velocity spectrum becomes increasinglyisotropic in the small scales as the Reynolds number increases.

With these reservations in mind, at very high Reynolds numberthe smallest scales in the velocity spectrum will be aware only ofthe amount of energy they receive, e (in an equilibrium situation).Hence, we can make scales dependent only on v and £, and these areknown as the Kolmogorov scales:

If we adopt the Kolmogorov 1.962 position, and consider £ r, averaged


over a sphere of radius r, then of course we can define scales rf andvr based on the local value of er. If we consider that within eachmaterial domain there is a cascade, depending on the local value of£ r , producing a Kolmogorov spectrum locally, then averaging thisspectrum (with certain assumptions on the distribution of £ r) willproduce slight changes in the power of K.

1.3 Equilibrium Estimates for Dissipation

Now, it is a slight stretch to apply the ideas of section 1.2 inthe energy containing range, since it is not fair to assume that therate of energy transfer is determined entirely by the strain rate ofthe next largest wavelet, since we are in a range of wavenumberswhere the wavelets are under the direct influence also of the meanflow. However, we can certainly ask whether u2u/l is constant, andpossibly equal to £. In fact, it turns out that u3/I = e to within about10%, which is at first surprising. More mature consideration suggeststhat it is probably only true in flows that are in equilibrium, which isto say, those in which the rate associated with the energy containingeddies u/l is equal to the rate associated with the mean flow Uij. Weexpect any turbulent flow to try to equilibrate all these rates, andultimately they will all be equal (or at least evolve proportionately).

We can use £ = u3/l to generate convenient forms for the variousscales. For example,

We can use these ideas to obtain an upper bound on the variationof £ as a function of Reynolds number. Suppose that the dissipationis so unevenly distributed that it is all in one tiny region. Call thismaximum value em. This region must have a size of order rjm, basedon £m. Hence, the total dissipation must be given by

where we are assuming that the average dissipation < £ > is deter-mined by averaging over regions of the size of the integral scale. Thisis true in most flows - i.e.- the integral scale is of the order of the sizeof the flow. Using the definition of r)m, we can easily obtain

where RI — ul/v.


1.4 The Dynamics of Turbulence

With this prelude about turbulent scales and the turbulent spec-trum, we can now turn to a discussion of the energetic dynamics ofthe turbulence. In what follows, we are paraphrasing Tennekes andLumley (1972), where all the missing details can be found.

We first split all quantities into a mean and a fluctuating part,e.g. Ui = Ui + Ui, where < Ui >= £/,-, and < U{ >= 0, and wedesignate the instantaneous total velocity by -u,-, with q2 given byU{U{. The equation for the turbulent fluctuating kinetic energy canbe written as:

where < . > denotes some kind oi averaging (possibly time; one,two or three-dimensional space; phase; or ensemble), but if we donot indicate otherwise, we will take the average to be an ensembleaverage. For the moment we will consider the flow to be incompress-ible, Uiti = 0, and later we will consider what modifications we willhave to make for the case of compressibility. We are also designatingd ( . ) / d x j = (.),j. Then, the mean strain rate 5,-j is = ([/,-j + Ujii)/2,and e is the dissipation of turbulent fluctuating kinetic energy. Prop-erly speaking e = 1v < SijSjj >, where Sij is the strain rate of thefluctuating motion; however, at high Reynolds number this can bewritten as v < UijUij >. We have neglected a number of other termsin equation (1.4.1) which can be shown to be small at high Reynoldsnumbers. These are all of the form of transport terms - that is, theycan be written as divergences of something, and hence, if integratedover a closed region, contribute nothing to the net turbulent kineticenergy budget, but simply move kinetic energy from place to place.These neglected terms are of importance in the neighborhood of thewall, for example, where the local Reynolds number is low.

Recall that we are supposing that < . > is a long time average,or an average over the full space, or an ensemble average. In aninhomogeneous flow (where, of course, we must use either a timeaverage or an ensemble average), such a quantity may be a functionof position, or in an instationary flow (where we must use either aspace average or an ensemble average) it may be a function of time,but it will be a smooth function, and will not be a random variable.This was the kind of average envisioned originally in the early workof Kolmogorov (1941). There are, of course, other possibilities. For


example, one could average over a sphere of radius r, centered at x,and designate such an average as v < u^jU^j > r= er(x). This isnow a random variable that varies erratically from time to time andfrom point to point in physical space; how much it varies dependson the value of r. This was the point of view taken by Kolmogorovin 1962. The 1962 point of view results in slight (but sometimessignificant) changes in the conclusions from the 1941 point of view.We will generally take the 1941 position, unless otherwise noted.

The first term on the right of equation (1.4.1) is called the tur-bulent kinetic energy production, or production for short. It canalso be identified as the deformation work, that is, the work doneto deform an arbitrary volume against the stresses induced by theturbulence. Such a term appears as a drain in the equation for themean flow kinetic energy, and, of course, the work done against thesestresses goes into the mechanism responsible for the stresses, theturbulence. There is a similar term describing deformation workagainst the stresses induced by the molecular motions, but it is easyto show that this work is small compared to the deformation workdone against the turbulent stresses, at high Reynolds numbers. Thismeans that, at high Reynolds numbers, the energy flow is from themean flow to the turbulence, and then to the molecular motion.

The second term on the right is the transport of fluctuating en-thalpy p / p + q2/I. Equation (1.4.1) can be regarded equally as theequation for the mean fluctuating enthalpy, < p / p + q2 /2 >, since< p >= 0. The transport of any quantity <j> by the turbulence can bewritten as — < (f>Ui >,;. That is, — < <j>U{ > is the flux of <j> per unittime into a surface with a positive normal in the z-direction. (p is theconcentration of <^>-stuff per unit volume, while —Uj is the volumeper unit time per unit area entering the surface. It was hoped atone time that the term < pui > /p would be small, largely becauseit was difficult to measure and hard to predict. However, it is nowrealized that it is probably of about the same magnitude as the otherterm, and may be of the opposite sign. At least one model for thisterm suggests that < pui > /p = —C < (g2/2)-u; >. DNS resultssuggest that this may not be a bad model in nearly homogeneouscircumstances.

Homogeneous turbulence is observed to be approximately Gaus-sian in the energy containing scales (turbulence is never Gaussianin the small scales, due to the spectral transport, but more aboutthat in the next section). A Gaussian distribution has all zero third


moments, and hence all fluxes of the form < U{U3u^ > will be zero,and hence < (g2/2)u; > will vanish. In a homogeneous flow, alltransport vanishes, since the transport terms are of the form of adivergence, and spacial derivatives are all zero in a homogeneous sit-uation. More than this, however, by crude physical reasoning, weexpect that probably all fluxes will vanish in a homogeneous flow - ifeverything is statistically the same everywhere, there is no reason foranything to flow from one place to another. Hence, non-zero fluxes,and thus transport, are associated with a departure from a Gaus-sian probability density, specifically with the appearance of skewnessin the density, associated with inhomogeneity. A formal expansionhas been developed for the case of weak inhomogeneity, relating thethird moments with the gradients (Lumley, 1978), which does notwork badly in practice (Panchapakesan and Lumley, 1993).

Finally, the last term is the dissipation of turbulent kinetic energyper unit mass discussed previously. This is, technically, the rate atwhich turbulent fluctuating kinetic energy per unit mass is convertedirreversibly to heat (to entropy). We will see later, however, that inan equilibrium situation, this is also the rate at which kinetic energyis removed from the energy containing scales, and the rate at whichkinetic energy is passed from scale to scale across the spectrum. Ina non-equilibrium situation, of course, these three quantities are notnecessarily equal to each other.

2 EQUILIBRIUM AND NON-EQUILIBRIUM FLOWS

2.1 The Spectral Cascade in Non-Equilibrium Flows

In this section we describe briefly a model which was first pre-sented in Lumley (1992). First, in a steady turbulent flow, we believethat the level of dissipation is determined by the rate at which turbu-lent kinetic energy is passed from the energy-containing eddies to thenext size eddies; that is, by the rate at which the turbulent kineticenergy enters the spectral pipeline, eventually to be consumed byviscosity when it reaches the dissipative scales. Hence, this quantityis only secondarily dissipation, and maybe should be called some-thing like spectral consumption. This picture has the consequencethat the level of dissipation should be independent of Reynolds num-ber at infinite Reynolds number; a change in the turbulent Reynolds


number should change only the wavenumber where the dissipationtakes place. These are all ideas of Kolmogoiov (1941).

There appears to be not much question that the level of dissipa-tion in a steady state is determined by scales of the energy containingrange. Much more to the point, however, is the behavior of the dis-sipation under changing conditions, both spacially and temporally.We need a dynamical equation for the dissipation. Of course, we canwrite down an exact equation for the dissipation, but as has beendetailed elsewhere (Tennekes and Lumley, 1972) this is expressed en-tirely in terms of the small scales. To first order this is a balancebetween two large terms, representing the stretching of fluctuatingvorticity by fluctuating strain rate, and the destruction of fluctuatingvorticity by viscosity. At the next order, these terms are slightly outof balance, and the imbalance, of course, is governed by the energycontaining scales, but in ways that we do not fully understand. Atthe moment, the entire equation must be modeled phenomenologi-cally (see Lumley, 1978) - by analogy with energy, we suspect thatthere must be production of dissipation and destruction of dissipa-tion, and we suppose that production of dissipation should keep pacewith the production of energy, while the destruction of dissipationshould keep pace with the destruction of energy, and these conceptsgive us a model that works reasonably well. We believe everyone whouses this model, however, is uneasy.

2.2 Delay in Crossing the Spectrum

We can place this model on a much sounder physical footing.The quantity that appears in the equation for the turbulent kineticenergy is the true dissipation, as opposed to the rate at which en-ergy is lost from the energy containing wavenumbers to the nextsmaller wavenumbers. In a steady state, of course, these are equal,but in an unsteady situation they may not be (see figure 2.2.1).It seems likely that there will be a lag in the development of thetrue dissipation, corresponding to the time it will take the energylost to the energy containing wavenumbers to be reduced in sizeto the dissipative wavenumbers. We can compute this time lag,using the model suggested in Tennekes and Lumley (1972). Ifwe divide the spectrum logarithmically into eddies centered at OK,a3K, ... where a = (1 + 51/2)/2 (fig. 2.2.2), so that the eddieshave the same width in wavenumber space as their center wavenum-


Figure 2.2.1. Impressions of the distribution of spectral flux againstwavenumber in steady state (a) and in unsteady state (b).

ber, then the time required to cross the spectrum is something likeI(O,K) / U(O,K,) + l(a3K,)/u(a3K) + l(a5K)/u(a5K) + ... + l(aN K,}/u(aN K),where K is at the peak of the energy containing range, say nl = 1.3,and aNKr/ = 0.55, to place this at the peak of the dissipation range.Note that these eddies, which we introduced in 1972 are, in fact,wavelets, as pointed out by Sreenivasan (Zubair et al. 1992). Addingthis up (supposing that all terms are within the inertial subrange)we find that the total time T = 2(//u)(l - 1.29JRJT

1/2), where / and uare scales characteristic of the energy containing range. Bear in mindthat we should not pay too much attention to the numerical valuesof the coefficients in the expression for T; probably the only thingthat is significant is the general form, and the value of the exponent.Note also that, for low Reynolds number, the time shrinks to a verysmall value, since the energy is dissipated at essentially the samewavenumber where it is produced, while for high Reynolds numberit goes to 2l/u. Now, the idea of a simple lag suggests that the cas-cade is tight, that all the energy must pass through each wavenumber


in order to arrive at the dissipative scales, as we have suggested infigure 2.2.2. This would produce a hyperbolic behavior, a sort oftelegraph equation, like diffusion with a finite velocity (only in timeinstead of space - see Monin and Yaglom 1971). We know, however,from the discussion in Tennekes and Lumley (1972) that at each step,although most of the energy is passed to the next wavenumber, a di-minishing fraction is passed to all higher wavenumbers (see figure2.2.3). Thus, the dissipative wavenumbers receive a small amount ofenergy almost immediately, and increasing amounts as time goes on,finally receiving it all in a time of the order of T.

Figure 2.2.2. Simplified view of spectral flux being passed from eddyto eddy.

Figure 2.2.4 is reproduced from Meneveau et al. (1992) (their fig-ure 11). The results are obtained from direct numerical simulationof forced isotropic turbulence. The figure represents the time evolu-tion of the energy in logarithmic wavenumber bands after a pulse ofenergy was added to the first band. Band n represents the energyin wavenumbers 2™~1 < k < 2n, normalized to its value at t = 0.Band 1 would be represented by a horizontal line at a value of 2.The figure completely supports our speculation regarding the energytransfer (at the end of the last paragraph).

The fractions and times involved are known (Tennekes and Lum-ley, 1972) and it is, in principle, possible to work out as a function oftime, the energy received at the dissipative wavenumbers resultingfrom a step in. input at the energy containing wavemtmbers. It is alsopossible to determine this from exact simulations, using the codes ofDomaradzki et al. (1990). This approach is rather complicated, how-

Figure 2.2.3. More realistic view of spectral liux. Now the fluxcrossing the wavenumber K goes mostly to eddy &K, but a decreasingfraction goes to a3K, a5K, etc. In its turn (at the second step), thatwhich had gone to a,K is redistributed to a3K, a5K, etc., at the sametime that which had gone to a3K is redistributed to &5K, etc., andthat which had gone to a5K is redistributed to a7K and a9K, and . ..This is only the second step. On the third step, each packet mustagain be redistributed.

ever, and it seems likely that this can be modeled satisfactorily by anexponential. We would thus expect that dissipation would be givenby something like

We are ignoring, for simplicity, the possibility that T is a functionof time, which it will be in general. Hence, e(i) would be governedby the following differential equation:

T here can now be a function of time. All of these considerationsrelate to a given mass of fluid of energy-containing scale, carriedalong by the mean flow and the energy- containing velocities. Hence,we must add to this equation an expression for the advection by themean flow and the turbulence. From the point of view of turbulencemodeling, this equation is of no use, since we need a value for £ inorder to determine a value for /. Hence, we do not have a value for /.(Note that it would be satisfactory to use e/g2 for u/l in T, whereasit would not on the right hand side of the £ equation, since its use


18 J. L. Lnmley et al.

Figure 2.2.4. From Meneveau et al. (1992). The time evolution ofthe energy in logarithmic wavenumber bands after a pulse of energywas added to the first band. Band n represents the energy in wave-numbers 2n~1 < k < 2n, normalized to its value at t = 0. Band 1would be represented by a horizontal line at a value of 2.

there would vitiate the equation). The most likely candidate as amodel for u3/l is the turbulent energy production, —Uij < UiUj >,although this is not quite right for several reasons, among them thatsome of the turbulent energy extracted from the mean flow goes toincrease the kinetic energy, and some is transported, and in factthe difference between the production and the transport and rate ofincrease in kinetic energy is the dissipation, and not the rate at whichenergy enters the spectral pipeline. However, it has been the best wecould think of, and has become standard, giving as an equation

This is the equation that has been used to obtain £ in turbulencemodeling essentially since the beginning, with the slight modificationthat our time scale T is now a weak function of Reynolds number.


We have introduced a constant c, presumably of order unity, sincethe turbulent energy production is only an approximation for therate at which energy enters the spectral pipeline.

We see that the second term (on the right) does not represent de-struction of dissipation, but rather reflects the presumed exponentialrise of the energy arriving at the dissipative wavenumbers in responseto a step input to the spectral pipeline. It is an approximation tothe extent that the rise has been modeled as exponential. We cansee two ways to improve the equation: first, we can examine moreclosely (on the basis, for example, of the simple model in Tennekesand Lumley, 1972) the rate of arrival of energy at the dissipativewavenumbers; it will surely be possible to develop successive approx-imations, improving on the simple exponential behavior. We do notthink this is likely to make much difference.

2.3 Negative Production

Second, we can try to find a better approximation for the rateat which energy enters the spectral pipeline. This is a lot harderto do, and it is not at first clear where to search. Let us restrictour attention to isothermal shear flows; if we can successfully handlethese, we can later consider other flows, buoyantly driven, for exam-ple. The turbulent kinetic energy production is not a.bad guess inmany circumstances; it is usually estimated as u3//, but, of course,this estimate is an equilibrium estimate, assuming the production isapproximately equal to dissipation, and dissipation is approximatelyequal to the rate at which energy enters the spectral pipeline. Thecurrent model suggests that in equilibrium situations production willapproximately equal dissipation, which is true at the edges of wakes,jets and shear layers, for example. We are interested here, however,in precisely those situations that are not in equilibrium. For exam-ple, there are regions in many turbulent flows where the turbulentkinetic energy production is zero (or even negative in small regions),partly due to the vanishing of the mean strain rate, but also due tothe vanishing of the Reynolds stress resulting from advection by thefluctuating velocity field of material with different strain histories.The centerlines of wakes and jets are examples where the produc-tion goes to zero. However, energy will still be entering the spectralpipeline in these regions. Hence, we may expect our model usingthe turbulent kinetic energy production to fail in these regions. We


might hope to build a slightly better model if we take the magni-tude of the mean strain rate as an estimate of u/l = [ S i j S i j ] 1 / 2 , andwrite u3/I = [SijSij]l/2q2/3. This has a similar problem, in that thestrain rate vanishes at extrema of the mean velocity profile, whereenergy is certainly still entering the spectral pipeline. It also suggeststhat at equilibrium the (inverse) time scale based on e/q2 is equalto that based on [ S i j S i j ] 1 ' 2 , which is probably only true in regionsfar from extrema where most of the fluid has been subjected to thesame strain history during living memory.

2.4 Mixing of Fluid with Different Histories

To get a model that avoids these difficulties, we must considerwhat it is that determines the time scale of a material region. Ifa material region remains subject to the same sign and magnitudeof strain rate during its entire lifetime (that is to say, for times ofthe order of l/u), then we expect that the characteristic time scaleof the region, l/u, will become equal to the magnitude of the (in-verse) strain rate [ S ^ j S i j ] ' 1 / 2 . In an inhomogeneous flow, however,at a given point, as time passes, fluid will arrive from many differentregions, where the sign and magnitude of the strain rate are quite dif-ferent (figure 2.4.1). This is why the Reynolds stress on the centerlineof a wake is zero - each packet of fluid brings its own value of Rey-nolds stress, depending on its history, and since the Reynolds stressis both positive and negative, and on the centerline equal quantitiesof fluid are seen, that have spent their lifetimes in regions of positiveand negative strain rate, the net value is zero. Hence, we shouldconsider averaging over the magnitude of the strain rates in the ar-eas from which fluid is advected to the point in question; ideally,backward along the mean trajectory though the point in question,with a growing Gaussian (in first approximation) averaging volume,with a fading memory (figure 2.4.2). We need not actually writethis as an integral in our equation; we can rather write an auxil-iary equation for the quantity. We might also consider averagingthe production itself (instead of the strain rate magnitude), whichwould give a different weighting. However, the rate at which energyenters the spectral pipeline should be determined by the local valueof the energy, and the local value of the time scale, the latter beingdetermined by history.


Figure 2.4.1. Zone of influence in a turbulent flow. The point shownis influenced by the histories of material points arriving (at differenttimes) along the trajectories indicated.

If we consider an auxiliary quantity, an inverse time scale, or rate,which we may call S, we need an equation of the type

where VT — c'q4/e, and T = ciq2/£, where c' and c\ are constantsof order unity. This will give just such a spreading Gaussian averagewith fading exponential memory of scale 7~ back along the meanstreamline. The exact values of c' and c\ probably are not too critical,since they only determine the exact size of the region over which theaveraging is done. Then we can write


Figure 2.4.2. The real process of figure 2.4.1 replaced by a model,in which the point shown is influenced by an integral back alongthe mean streamline through the point, with a spreading zone ofinfluence and a fading memory.

where c" is another constant of order unity. Note that this is nowa non-local theory. Stan Corrsin (1975) pointed out some years agothat the k — e model was also non-local for similar reasons.

If we apply this model to grid turbulence we may obtain somerelations among the constants. We should identify S with (propor-tional to) the value of u/l determined by whatever mechanism. Ina grid turbulence, the initial value of u/l is determined by the grid;thereafter, there is no further input to determine the value of u/l(since Sjj is identically zero) and it simply relaxes, or decays. Inan equilibrium homogeneous shear flow (which may not exist), Swill take on the value [SijSij]1/2 asymptotically, which will have thevalue f/'/A/2 = u/l^/2. If we identify the initial value S0 (at time


t 0 ] of S in the decaying grid turbulence as the value of u/l\/2 att0, we have S0t0/3 — n/2\/2, where q2 oc t~n. In order for similar-ity to hold, we require cj = n/2, and c" = \/2 — C22\/2(ft + l)/ft,where clearly we require c2 < n/2(n + 1). A typical value of n isabout 1.25, although there is a weak variation with Reynolds num-ber that should be investigated. From our reasoning we expect that

1 /o

c2 oc (1 — c^Rl ). It seems reasonable to require both c\ and c2

to have this behavior with Reynolds number, and hence to requirethat n should have this behavior also. We should examine the dataof Comte-Bellot and Corrsin (1966) for confirmation.

2.5 Deformation Work in Equilibrium andNon-Equilibrium Situations

We can construct a simple picture of the turbulence production—Sij < UiUj >. Imagine a velocity field that consists of randomlyoriented vorticity with equal amounts in all directions. If we writethe production in principal axes of the mean strain rate, it becomes

We presume that the 1-direction is the direction of maximum pos-itive strain rate, while the 3-direction is the direction of maximumnegative strain rate. The strain rate in the 2- direction is intermedi-ate in value; it may be positive or negative, but it will be smaller inmagnitude that the other two. Now, the vorticity in the 1-directionwill be stretched and intensified, while the vorticity in the 3-directionwill be shrunk and attenuated. Associated with the 1-direction vor-ticity is 2- and 3-direction velocity, which will be intensified, whileassociated with the vorticity in the 3-driection is velocity in the 1-and 2-direction, which will be attenuated. We can make a chart:

where a horizontal arrow indicates no change, while an up arrowindicates an increase, and a down arrow a decrease. As a result of

24 J. L. Lumley et a].

£22, W2 may go up or down somewhat, and there will be consequentsmall changes in < u\ > and < u§ >, but these will be smallerthan the changes due to wj and u^. We have designated these as "nochange" with a horizontal arrow, for simplicity. If the initial vorticityis more-or-less uniformly distributed, it is evident that there will bea net decrease in < u\ > and a net increase in < u| >, while < u\ >will remain essentially unchanged. As a result, < u\ > — < u? >< 0,while < w§ > - < u\ » 0. Since 5n > 0, and 633 < 0, both termsof the expression for the production will be positive.

What we have described is an equilibrium situation. That is, wehave described a situation in which the anisotropy of the turbulencehas been generated by the strain rate field. The anisotropy of theturbulence is consequently in equilibrium with the strain rate field.In the real world, of course, this sometimes happens; however, it isalso quite likely that the turbulence will have been generated by onemechanism, will have lived its entire lifetime under this mechanism,and will be in equilibrium with that mechanism, in the sense thatthe time scales will have equilibrated, and the principal axes willhave taken on an equilibrium orientation, and that this turbulencewill then be subjected to a distortion of a wholly different nature.For example, the boundary layer formed on the pressure surface ofa leading edge slat on an aircraft wing (in take-off configuration) issuddenly subjected to the strain rate due to passage through thegap between the slat and the wing. Under these circumstances, theproduction can take on both positive and negative values. A goodexample of this is the wall jet, which has a maximum which is notsymmetric. In this vicinity, material is sometimes swept past whichhas come to equilibrium with the strain rate field on the wall side ofthe maximum, and sometimes from the other side of the maximum,where the strain rate has the opposite sign. Because of the lack ofsymmetry, the amounts of the two types of material are not equal,and hence the net value of the Reynolds stress does not vanish at thepoint where the strain rate vanishes. There is, thus, a narrow regionin which the production is negative. A more impressive situationcan be generated in a special wind tunnel. Let the flow be in the x\direction, and the tunnel be arranged to produce a positive strainrate along the x? axis, and a negative strain rate along the #3 axis.If the turbulence is subjected, to this distortion for long enough, theturbulence structure will have come to equilibrium with the strainrate. At this point, reverse the direction of the strain rate - that is,


let the positive strain rate be in the x3 direction, and the negativestrain rate be in the x-i direction. This is easy to arrange in practice -the direction that had been shrinking now begins to expand, and viceversa. Experimentally, it is found that the production immediatelybecomes negative throughout the tunnel and remains so until theanisotropy of the turbulence can adjust itself to the new value of thestrain rate field, which takes some time. Any model for turbulencewhich hopes to deal with non-equilibrium situations must take thisinto account - this means, in practice, that a separate equation mustbe carried for the Reynolds stress; only in equilibrium situations canthe value of the Reynolds stress be related directly to the strain ratefield.

2.6 Alignment of Eigenvectors

The question of whether the turbulent field is in equilibriumwith the mean strain rate field arises also in connection with theeigenvectors. This is really just another way of looking at the samequestion, perhaps a more enlightening way. In a shear flow, sayUi : Ui(x2),0,0, the principal axis of positive strain rate is at 7T/4,while the principal axis of negative strain rate is at 37T/4. In a purestrain (i.e. with no rotation), the principal axes of the Reynolds stresswould be aligned with those of the strain rate field. In our shear flow,however, we have rotation as well as strain rate. A material regionis being continually rotated clockwise in this flow. Hence, althoughthe strain rate tries to align the principal axes of the Reynolds stresswith its own principal axes, the material with its principal axes isrotated clockwise. Hence, we expect to find the principal axes ofthe Reynolds stress rotated clockwise from those of the strain rate.The relaxation time of the turbulence is of the order of l/u, andthe mean angular velocity is (\/1)dU\/dx^. The net angle throughwhich the axes will rotate is perhaps half of this, to give somethinglike ( l / 4 ) ( l / u ) d U i / d x 2 = 1/4, or some 15°. By picking the value of1/2, we are trying to account for the fact that the relaxation is goingon continually - we are replacing the real process with an artificialone which does not relax at all until l/2u, and then relaxes com-pletely. In fact, the principal axes of the Reynolds stress are foundat 30° and 120°, rotated clockwise exactly 15° from the axes of themean strain rate. The agreement with our crude calculation is toogood to be true, but at least the direction and order of magnitude


are both correct and believable.In turbulence modeling of the k — £ type, the Reynolds stress is

parameterized as being proportional to the mean strain rate, whichmeans that the two tensors are forced to have the same principal axes.The relationship is calibrated to give the correct value of the off-diagonal stress, which means that the diagonal stresses, the turbulentintensities, must be wrong. In calculating simple shear flows, thisdoes not matter much, since the normal stresses are not used. Inmore complex situations; however (such as separated flows), whereany of the components may be the important one, it matters, slit isalso not clear in such situations how to calibrate the relationship.

In non-equilibrium situations, it is essential that the equation forthe Reynolds stress be used, so that the rotation and relaxation ofthe principal axes of the Reynolds stress can gradually accommo-date to the mean strain rate history, being at any instant probablymisaligned with those of the mean strain rate.

2.7 Dilatational Dissipation and Irrotational Dissipation

Up to this point, we have ignored compressibility. In fact, in theboundary layer at low Mach number U/u* is about 30 (between 25and 35) between a length Reynolds number of 107 and 5 X 10s. Thespeed q is about 2.65u*. Hence, the mean velocity'is about 11 timesthe turbulent speed. As the Mach number rises, the wall temper-ature rises, and the density and, hence, the skin friction drops, sothat this ratio increases somewhat. This means that the fluctuatingMach number is of the order of 1/11 or smaller of the mean flowMach number. As a result, unless the mean flow is hypersonic, witha Mach number in the neighborhood of 12-15, the turbulent Machnumber will not be anywhere close to one. Hence, at moderate meanflow Mach numbers (say, below 5) we may expect the effects of com-pressibility on the turbulence to be relatively small. An exception, ofcourse, is interaction with a shock wave; boundary layer turbulencemay be essentially incompressible, but if it passes through a shockcompressibility effects will be felt. Note that the relative fluctuationsin the isentropic speed of sound are of order (7 — l)m2/4, where mis the fluctuating Mach number. Hence, even at a fluctuating Machnumber of unity, the fluctuations in the isentropic speed of sound areof order 10%, and can be ignored in the definition of the fluctuatingMach number.


Let us consider the dissipation. Let us begin by considering thestress in a compressible flow. This can be written as

where 9 = Ui^,s'- is the deviatoric strain rate, and p is the thermody-namic pressure, that is, p = (7 — l)/oe, where e is the internal energy.[iv is the bulk viscosity. This is zero in a monatomic gas, and is ofthe order of Q.66/J, in Nitrogen (Sherman, 1955). The (negative of)the average normal stress —r,-;/3 is not equal to the thermodynamicpressure because there is a lag between the rotational temperatureand the translational temperature. Under compression (9 negative),all temperatures are rising, but the rotational temperature is laggingbehind the translational temperature, so that the translational tem-perature is a little higher than the thermodynamic relation predicts.Thus the actual (negative) average normal stress is higher than thethermodynamic pressure (see, for example, Light hill, M. J., 1956).From (2.7.1) we can see that the viscous stress depends on the di-latation only through the bulk viscosity. In a monatomic gas therewould be no dependence.

If we now form the equation for the turbulent fluctuating energyin a homogeneous flow, we can write

The first term, of course, is the pressure-dilatation correlation, or therecoverable work. The remainder is the entropy production.

Now, we may write (if 5,-j = (uij + Uj^/2, r\,j = (uitj - -Uji,-)/2),

The second term may be written as < u^jUj^ > — < O2 > (in a homo-geneous flow), where 9 = w t i , . Hence, we can write (in a homogeneoussituation)

In addition, we may write < TIJTIJ >=< Wj-u;; > /2, and < SijSij > =< s'-s'- > + < O2 > /3, so that we may write

28 J. L. Lumley et ad.

Finally, the total entropy production may be written as

Let us consider the Helmholtz decomposition. This purely kine-matic decomposition states that any vector field Uj can be writtenuniquely as the sum of two components, say Vi + W{ , where «,- issolenoidal (but rotational), while w^ is irrotational (but compress-ible). Viewed from this perspective, Ui is associated only with thesolenoidal component v,-, while 0 is associated only with the irrota-tional component w^. It thus seems fairly safe to identify // < u;;uz- >as the conventional (solenoidal) dissipation, and (fiv + 4///3) < #2 >as the dilatational dissipation. Note that, in an inhomogeneous situ-ation, there are other terms that vanish only as the Reynolds numberbecomes infinite.

Both Zeman (1990) and Sarkar (1992) have developed modelsfor the compressible component of the dissipation. In the compress-ible mixing-layer flow (Sarkar and Lakshmanan, 1991), this com-pressibility correction is essential for predicting the spreading ratecorrectly. In an extension of the dilatational dissipation model-ing, Sarkar (1992) has also examined contributions to the pressure-dilatation using an analogous decomposition of the pressure field intoincompressible and compressible parts. The proposed model hasbeen compared to results from DNS of compressible homogeneousshear flow (Sarkar et a/., 1991).

2.8 Eddy Shocklets

We tell our classes, and we believe, that in incompressible turbu-lence the level of the dissipation is controlled by the rate at whichenergy is fed into the spectral pipeline at the large scale end. Now,even when the fluctuating Mach number is above unity, we expectto find relatively incompressible turbulence separating a distribu-tion of randomly oriented shocklets, which are relatively thin, andhence correspond to high wavenumbers. Thus, we expect the energycontaining range (characterized by motions which, though energetic,change only slowly in space) to be relatively incompressible and therate at which energy is transferred into the spectral pipeline in theusual way should be unchanged from the incompressible case. Hence,the ordinary dissipation should be unchanged (Zeman, 1990). Pre-sumably in a compressible turbulence, the compressibility can be


characterized by the spectrum of < $2 >, which might be expectedto rise with wavenumber, so that the compressibility would occurprimarily at the larger wavenumbers. This supports our physicalargument, and is consistent with the shocks being thin.

Consider the dilatational dissipation from a physical point ofview. Energetic eddies, which locally exceed m = 1, form shock-lets, and the passage through these shocklets removes energy fromthe eddy. The dilatation is almost completely confined to the shock-let, hence to high wavenumbers. Somehow, the energy is gettingfrom the low wavenumbers to the high wavenumbers, where it is dis-sipated. This certainly does not seem to be by the usual cascadeprocess of vortex stretching. It seems rather, that in the passagethrough the shock, the eddy is compressed in the direction normalto the shock, and this reduces its scale, and increases its wavenum-ber. In addition, the existence of the shock alters the velocity fieldapproaching it; the flow tries to avoid the shock, turning aside to goaround it if possible, and this may also result in a reduction in scale.Certainly, it results in a transfer of energy from one component toanother. This is presumably < p9 > at work. There are indicationsfrom the work of Zeman (1991) that < p9 > stores energy duringpassage through the shock, returning it to the vortical mode down-stream of the shock, causing an increase in the turbulent intensityon the downstream side. On the other hand, certain simple modelssuggest that under some circumstances the term < p9 > will havethe form of an additional dissipation. It probably does all of these.

Incidentally, it is clear from our discussion above that < p9 >and [iv < d2 > are just two parts of the same thing: < (—r,-,-/3p)# >,the work that is done by the normal stress during the compression,not all of which is recoverable.

3 PROPER ORTHOGONAL DECOMPOSITION ANDWAVELET REPRESENTATIONS

3.1 Coherent Structures

If we examine pictures of various turbulent flows, we will dis-cover that the proportion of organized and disorganized turbulencein each flow is different. For example, if we look at mixing layers


from undisturbed initial conditions (with only thin laminar bound-ary layers on the splitter plate), we find that there is an energet-ically large organized component, which only relatively slowly be-comes three-dimensional and disorganized, although the nearly two-dimensional organized structures have from the beginning a stochas-tic component, so that their occurrence is not precisely periodic,nor are their strengths equal. On the other hand, if we examine amixing layer from quite disturbed initial conditions, with a thick,turbulent boundary layer on the splitter plate, we find that the pro-portion of organized component is considerably less - although theorganized component is still visible, it is no longer dominant. Thus,in the same type of flow, we find that the initial conditions changethe relative strength of the organized and disorganized components.We examine a different flow, for example a jet, we find the samedifference - that is, if the flow from the orifice is initially undis-turbed - thin laminar boundary layers on the inner surface of thenozzle - then there is initially a laminar instability which graduallybecomes three-dimensional and undergoes transition, leaving in thedownstream development of the turbulence the remnants of the in-stability structure. On the other hand, if the boundary layers on theinside of the nozzle are initially thick and turbulent, there is no ini-tial laminar instability, and there are no visible organized remnantsin the turbulent motion. More than this, however, there is evidentlya substantial difference between this flow and the mixing layer. Evenin the undisturbed state, the organized component of this flow is avery great deal weaker than that in the mixing layer under the samecircumstances. It is bearly discernible; when the initial conditions aredisturbed, the organized structure becomes essentially undetectable.Thus, we can conclude that different flows, even under similar condi-tions, have different relative strengths of organized and disorganizedcomponents.

So far it seems that the organized structures, to the extent thatthey are present, are the remnants of initial laminar instability. Letus look at the wake of a flat plate normal to the stream. If thewake is visualized close to the plate, we see one kind of organizedstructure, which evidently is the remnant of the initial laminar in-stability. However, if the wake is visualized far from the body, wesee a somewhat, different, though similar organized structure. Ev-idently the organized structures initially present decayed, and newdistinct, though similar, structures arose. If we examine the wake of


a plate (still normal to the stream) which is sufficiently porous so asnot to give rise to an initial instability, we find at first a turbulentwake without organized structures. However, after a time the or-ganized structures present in the late part of the initially disturbedwake spontaneously appear in this initially undisturbed wake (we sayundisturbed, but of course the wake is initially turbulent, and hencedisturbed; however, organized structures are not initially present.)It seems reasonable to conclude that the organized structures thatappear in the late wake are a type of instability of the developed tur-bulent flow, drawing on the mean velocity profile to obtain energy,and giving up energy to the turbulent transport. The precise formthen, will be a function of the mean velocity profile, as well as of thedistribution of the turbulent stresses. The energy budget for such ex-isting organized structures is complex, because they have reached anon-linear energetic equilibrium, and their transport is modifying themean velocity profile, as well as the turbulent stresses. Their initialgrowth is complicated also, since they must be imagined to grow froma mean velocity profile (and a profile of turbulent stresses) existingwithout the organized structures. What these profiles are is moot.We will return to these ideas later when we discuss the prediction ofthese organized structures.

Hence, in any given situation we may expect to find organizedstructures the relative strength of which are a function of the initialconditions of the flow, the type of flow, where we are in the flow (howfar downstream), and which may be remnants of initial instabilitiesor may be a new instability of the turbulent flow.

If we restrict our attention to narrow two-dimensional shear flows(jets, wakes, mixing layers) we will find that the organized structuresoccur with more-or-less the same orientation and distance from theflow centerline each time; that, is, their orientation and positionin the cross-stream, inhomogeneous direction is largely fixed by theboundaries of the flow; in the streamwise, or homogeneous direction,the location is more random; the existence of one structure seems tosuppress the presence of another, but as soon as we are sufficientlydistant from a structure, another one appears.

Organized structures are much more difficult to find in homo-geneous flows, mostly because it is not clear where to look. Thereis nothing in the flow to pin them down to a particular location.Consider, for example, the homogeneous shear. In direct numeri-cal simulations, Moin and his coworkers have found hairpin vortices


throughout the flow. The orientation of these hairpin vortices isdetermined by the direction of the mean velocity shear, but theirlocation in three dimensions is random. They are thought to arisefrom a type of non-linear instability connected with the same in-stability that produces Langmuir cells in the ocean surface mixedlayer, or that produces streamwise rolls in a turbulent boundarylayer. Both of these flows are inhomogeneous in the direction normalto the surface, and hence the location of the organized structures isfixed by this inhomogeneity. The instability mechanism depends ontransverse vorticity associated with the mean shear being deflectedvertically (in the direction of the mean gradient), and then beingtransported in the streamwise direction at different rates at differentheights, due to a gradient in the Stokes drift, resulting in a stretchingand intensification of the streamwise component of vorticity.

We may probably conclude that any turbulent flow will have amore-or-less organized component, the strength of which will be acomplex function of the type of flow, the age of the flow, and theinitial conditions, and which may, depending on the situation, berandom in orientation and location in up to three dimensions.

3.2 The Role of Coherent Structures in TurbulenceDynamics

Whether it is necessary to take into account the presence of orga-nized structures in a turbulent flow, when considering the dynamicalbehavior, will have a different answer in different situations. Whenwe study a turbulent flow for practical purposes, we are seldom in-terested in more than the Reynolds stress. This is not a very so-phisticated property of a turbulent flow; it is uninfluenced by subtlechanges in the structure of the flow, and tells us little about the flow.Often, if the coherent structures in the flow scale in the same way asthe disorganized motions, they can all be lumped together and theevident differences ignored. For example, in the turbulent mixinglayer, if both the turbulence and the organized structures have thesame origin, and have been growing together since the origin, thenthey will scale in the same way, and need not be considered sepa-rately. See Shih et al. (1987), where a compressible turbulent mix-ing layer was successfully predicted using a second order turbulencemodel that completely ignored the presence of coherent structures.Many of the features of the mixing layer that are thought to be as-


sociated with the coherent structures (e.g. the asymmetry of theentrainment at the two sides of the mean velocity profile) are in factmandated by the dynamics of the situation and must be produced bywhatever physical mechanism is doing the transporting of momen-tum, whether organized or disorganized. However, if a turbulenceproduced under one set of circumstances, and consequently havinga given set of scales, is subjected to a different set of conditions (anew strain rate field, for example) it is quite likely that the new con-ditions will give rise to a new instability of the turbulent profiles,giving rise in its turn to a new organized structure. Until the scaleshave had a chance to equilibrate, we will have a situation consist-ing of background disorganized turbulence with one set of scales, onwhich are growing organized structures with a different set of scales.The transport produced by this combination will be quite difficultto predict unless explicit account is taken of the organized structure.We must also consider that the initial turbulence may already havean organized structure with which it is in equilibrium.

3.3 The POD as a Representation of CoherentStructures

The extraction of deterministic features from a random, finegrained turbulent flow has been a challenging problem. Zilbermanet al. (1977) write: "there are no consistent methods for identifica-tion which are independent of the techniques and the observer" and"we cannot unambiguously define the signature of an eddy without apriori knowledge of its shape and its location relative to the observa-tion station and cannot map such an eddy because we do not have aproper criterion for pattern recognition." In contrast, Lumley (1967)proposed an unbiased technique for identifying such structures. Themethod consists of extracting the candidate which is the best corre-lated, in a statistical sense, with the background velocity field. Thedifferent structures are identified with the orthogonal eigenfunctionsof the proper orthogonal or Karhunen-Loeve decomposition theoremof probability theory (Loeve, 1955). This is thus a systematic wayto find organized motions in a given set of realizations of a randomfield. The method applied here is optimal in the sense that the seriesof eigenmodes converges more rapidly (in quadratic mean) than anyother representation.

The use of these modes for a low dimension dynamical system


study requires a very fast convergence of the series. The methodwe propose here is limited in application to certain types of flowsin which large coherent structures contain a major fraction of theenergy. It has been demonstrated that axisymmetric turbulent jetmixing layers (Glauser et a/., 1985) and wall regions of turbulentboundary layers (Moin, 1984; Herzog, 1986) belong to this group.

Specifically, we will develop a model for the wall region of theboundary layer (from x2+ = 0 to x%+ = 40 in wall units (Tennekesand Lumley, 1972)), using the proper orthogonal decomposition ofLumley (1967, 1970, 1981) in the direction normal to the wall, inwhich the flow is strongly non-homogeneous. In the stream wise andspanwise directions the flow is essentially homogeneous, and Fouriermodes will suffice. Used in conjunction with Galerkin projection,the proper orthogonal decomposition yields an optimal set of basisfunctions in the sense that the resulting truncated system of ODEscaptures the maximum amount of kinetic energy among all possibletruncations of the same order. The method has obvious advantagesover a priori decompositions, based on linear normal modes, but itdoes not appear to have been used before due to the difficulty ofcomputing the proper orthogonal modes. For this one requires threedimensional autocorrelation tensors averaged over many realizationsof the flow in question, data only obtainable from lengthy experi-ments and analyses or from detailed numerical simulations. In ourcase complete data is only available from experimental work in a glyc-erine tunnel (Herzog, 1986), although Moin (1984), has derived two-dimensional orthogonal modes from large eddy simulations. How-ever, as we shall see, knowledge of the autocorrelation tensor, anduse of the Navier-Stokes equations, does allow one to uniquely deter-mine the unsteady flow, in contrast to Cant well's (1981) expectation.Lumley (1967) proposed a method of identification of coherent struc-tures in a random turbulent flow. An advantage of the method is itsobjectivity and lack of bias. Given a realization of an inhomoge-neous, energy integrable velocity field, it consists of projecting therandom field on a candidate structure, and selecting the structurewhich maximizes the projection in quadratic mean. In other words,we are interested in the structure which is the best correlated withthe random, energy-integrable field. More precisely, given an ensem-ble of realizations of the field, the purpose is to find the structurewhich is the best correlated with all the elements of the ensemble.Thus, we want to maximize a statistical measure of the magnitude of


the projection, which can be given by the mean square of its absolutevalue. The calculus of variations reduces this problem of maximiza-tion to a Fredholm integral equation of the first kind whose sym-metric kernel is the autocorrelation matrix. The properties of thisintegral equation are given by the Hilbert Schmidt theory. There isa denumerable set of eigenfunctions (structures). The eigenfunctionsform a complete orthogonal set, which means that the random fieldcan be reconstructed. The coefficients are uncorrelated and theirmean square values are the eigenvalues themselves. The kernel canbe expanded in a uniformly and absolutely convergent series of theeigenfunctions and the turbulent kinetic energy is the sum of theeigenvalues. Thus, every structure makes an independent contribu-tion to the kinetic energy and Reynolds stress. The most significantpoint of the decomposition is perhaps the fact that the convergenceof the representation is optimally fast since the coefficients of theexpansion have been maximized in a mean square sense. The meansquare of the first coefficient is as large as possible, the second isthe largest in the remainder of the series once the first term hasbeen subtracted, etc. We have described here the simplest case, thatof a completely inhomogeneous, square-integrable, field. If the ran-dom field is homogeneous in one or more directions, the spectrum ofthe eigenvalues becomes continuous, and the eigenfunctions becomeFourier modes, so that the proper orthogonal decomposition reducesto the harmonic orthogonal decomposition in those directions. SeeLumley (1967, 1970, 1981) for more details.

The flow of interest here is three dimensional, approximately ho-mogeneous in the stream wise direction ( x \ ) and span wise direction(23), approximately stationary in time (/), inhomogeneous and ofintegrable energy in the normal direction (x^).

We want a three dimensional decomposition which can be sub-stituted in the Navier-Stokes equations in order to recover the phaseinformation carried by the coefficients. We have to decide whichvariable we want to keep. Time is a good candidate since we areparticularly interested in the temporal dynamics of the structures.Such a decomposition is possible and we do not need a separation ofvariables in the eigenfunctions of the type <f>(x,u) = A(ui)tl>(x} (assuggested by Glauser et a/., 1985) if we do not use any decomposi-tion in time and choose the appropriate autocorrelation tensor. Theidea is to measure the two velocities at the same time and determine< Ui(xi, £2, £3, t~)uj(x'i, x'2, £3,2) >= Rij. Since the flow is quasista-


tionary, Rij does not depend on time and nor do the eigenvalues andeigenfunctions. The information in time is carried by the coefficientsa(") which are still "stochastic," but now evolve under the constraintof the equations of motion. Thus the decomposition becomes

and we have to solve equation (3.3.2) for each pair of wave numbers(&i, £3). 4>ij now denotes the Fourier transform of the autocorrelationtensor in the #1,3:3 directions.

Our second change to the decomposition is a transformation ofthe Fourier integral into a Fourier series, assuming that the flow isperiodic in the x^ and x3 directions. The periods L\,Lz are de-termined by the first non zero wave numbers chosen. Finally, eachcomponent of the velocity field can be expanded as the triple sum

In this case, a "structure" is denned by:

and the entire velocity field is recovered by the sum of all the struc-tures (over n).

The candidate flow we are investigating is the wall region (whichreaches x2+ = 40) of a pipe flow with almost pure glycerine (98%) asthe working fluid (Herzog, 1986). The Reynolds number based on thecenterline mean velocity and the diameter of the pipe is 8750. Thecorresponding Reynolds number based on the shear velocity UT is 531.From this data the autocorrelation tensor at zero time lag (t — t' = 0)between the two velocities, RIJ(X\ — x ' l , X 2 , x ' 2 , X 3 — x'3)t_ti-0, was ob-tained and the spatial eigenfunctions were extracted by numericalsolution of the eigenvalue problem. The results show that approxi-mately 60% of the total kinetic energy and Reynolds stress is con-tained in the first eigenmode and that the first three eigenmodescapture essentially the entire flow field as far as these statistics are


concerned. This very fast convergence of the decomposition in thenear wall region is in good agreement with Moin's results (1984).From a large eddy simulation data base, Moin uses the proper or-thogonal decomposition successively in one and two dimensions inthe wall region (up to :r2+ = 65). His first structure contains 60%of the total kinetic energy and 120% of the Reynolds stress (thisapparent paradox occurs because the contribution of higher orderstructures to the Reynolds stress is negative). Ninety percent of thekinetic energy is captured by the first three terms.

3.4 Low-Dimensional Models Constructed Usingthe POD

We decompose the velocity - or the pressure - into the mean(defined using a spatial average) and fluctuation in the usual way.We substitute this decomposition into the Navier-Stokes equations.Taking the spatial average of these equations we obtain, in the quasistationary case, an approximate relation between the divergence ofthe Reynolds stress and the mean pressure and velocity.

is achieved by use of the complete set of eigenfunctions <fi(n> 's in aninfinite sum:

This is substituted in the Navier-Stokes equations, giving an equationfor the fluctuating velocity. Equation (3.4.1) may be solved to givethe mean veloctity U in terms of the Reynolds stress < u\u^ >in a channel flow in a manner which gives some feedback to thesystem of equations as the fluctuation varies. We will see that thisfeedback is necessarily stabilizing for the first structure (according tothe experimental results) and increases as the Reynolds stress getsstronger. In other words, this term controls the intensity of the rolls,by reducing the mean velocity gradient as the rolls intensify, thusweakening the source of energy.

The expansion of the Fourier transform U{ of the fluctuating ve-locity Ui, defined by


Since we want to truncate this sum, we use a Galerkin projectionwhich minimizes the error due to the truncation and yields a setof ordinary differential equations for the coefficients. After takingthe Fourier transform of the Navier-Stokes equations Ni = 0 andintroducing the truncated expansion, we apply Galerkin projectionby taking the inner product

Finally we obtain a set of ordinary differential equations of the form:

where A and B are matrices. Here A is the identity matrix (sincethe complete set of eigenfunctions is orthogonal) and N.L. are nonlinear terms. The non linear terms are of two sorts: quadratic andcubic. The quadratic terms come from the non linear fluctuation-fluctuation interactions and represent energy transfer between thedifferent eigenmodes and Fourier modes. Their signs vary. The roleof the Reynolds stresses < U{Uj > on these terms should be men-tioned. They vanish for all wave number pairs except for (ki,ks) =(0,0) for which they exactly cancel the quadratic term. Thereforethey prevent this mode from having any kind of quadratic interac-tions with other Fourier modes. Since the cubic terms are zero too,the (0,0) mode just decays by action of viscosity and does not par-ticipate in the dynamics of the system.

The cubic terms come from the mean velocity-fluctuation inter-action corresponding to the Reynolds stress < u\u?. > in the meanvelocity equation (the other part of this equation leads to a linearterm). Since the streamwise and normal components of the firsteigenfunction have opposite signs, they make a positive contributionto the turbulence production and hence provide negative cubic termswhich are thus stabilizing. We remark that this is not necessarily thecase for higher-order eigenfunctions.

By use of the continuity equation and the boundary conditions

and


it can be seen by integration by parts that the pressure term woulddisappear if the domain of integration covered the entire flow volume.Since this is not the case (rather the domain is limited to X£ — 40),there remains the value of the pressure term at the upper edge Xiof the integration domain which represents an external perturbationcoming from the outer flow.

The exact form of the equations obtained from the decomposi-tion, truncated at some cut-off point (&ic, k^c, nc), does not accountfor the energy transfer between the resolved (included) modes andthe unresolved smaller scales. The influence of the missing scales willbe parameterized by a simple generalization of the Heisenberg spec-tral model in homogeneous turbulence. Such a model is fairly crude,but we feel that its details will have little influence on the behaviorof the energy-containing scales, just as the details of a sub-grid scalemodel have relatively little influence on the behavior of the resolvedscales in a large eddy simulation. This is a sort of St. Venant'sprinciple, admittedly unproved here, but amply demonstrated ex-perimentally by the universal nature of the energy containing scalesin turbulence in diverse media having different fine structures anddissipation mechanisms (see Tennekes and Lumley, 1972 for a fullerdiscussion). The only important parameter is the amount of energyabsorbed.

We begin by defining a moving spatial filter which removes fromthe total field the unresolved modes. The details of the definitionare not important - it is sufficient to conceive of the possibility ofsuch a filter. This filter is also an averaging operator. The velocityfield may now be divided into the resolved and unresolved field byusing this filter. The Reynolds stress of the unresolved field may nowbe defined as the average using our filter operator, of the product ofthe unresolved velocities; this acts on the resolved field. We supposethat the deviator of this Reynolds stress is proportional to the strainrate of the resolved field. We neglect the Leonard stresses, whichessentially supposes that there is more of a spectral gap than reallyexists. This is what is done in the Heisenberg model, without illeffect. The way in which we are treating the effect of the unresolvedmodes on the resolved ones is very much like what is done in largeeddy simulation, and is called sub-grid scale modeling; our modelwould probably be called a Smagorinsky model (there are minor dif-ferences in the definition of the equivalent transport coefficient). Letus agree to designate the resolved field as ut-< and the unresolved

40 J. L. Lumley et aL

field as tij>, while an average of <j>, say, over the unresolved modes(the filtering process) can be designated as < d> >^. Thus:

with

and

Here < denotes the sum over all the modes (ki,k3,n) such thatk\ < kic,ks < &3C, n < nc and > denotes the sum over all themodes (ki,kz,n) such that k\ > k\c or ^3 > k%c or n > nc, where(k\c, ksc, nc) is the cut-off mode. The characteristic scales of theparameter Vf are those of the higher modes. We have introducedan explicit dimensionless parameter ai, and will exclude adjustableconstants from i>x- By observation that the energy decreases rapidlywith increasing n and fc, we assume that these relevant scales aregiven by characteristic scales of the first neglected modes. This isprobably a good approximation as far as the eigenmodes are con-cerned since they are separated by large gaps in the spectrum and itis a reasonable assumption for the Fourier modes since the steps ofour Fourier series are also large.

Finally, the parameter VT is taken equal to

(where u> and /> are characteristic scales of the neglected modes).This can be expressed in terms of the eigenvalues and eigenfunctionsof the first neglected modes in the three directions (see figure 3.4.1).

We will refer to «i as a Heisenberg parameter. We will adjust a\upward and downward to simulate greater and smaller energy lossto the unresolved modes, corresponding to the presence of a greateror smaller intensity of smaller scale turbulence in the neighborhoodof the wall. This might correspond, for example, to the environmentjust before or just after a bursting event, which produces a large


Figure 3.4.1. Inhomogeneous Heisenburg model applied to the spe-cific truncation discussed in the text. Legend: • resolved modes, xfirst neglected modes which are considered for the computation ofthe characteristic scales of the Heisenberg model.

burst of small scale turbulence, which is then diffused to the outerpart of the layer.

A term — l/3£,-j(< Uk>Uk> >> — « Ufc>tifc> >>>) appears inthe equation for the resolved field. This term could be combinedwith the pressure term and would not have any dynamical effect ifthe integration domain covered the entire flow volume. In our case,it needs to be computed since, like the pressure term, it leads to aterm evaluated at X^- We assume that the deviation (on the resolvedscale) in the kinetic energy of the unresolved scales is proportional


to the rate of loss of energy by the resolved scales to the unresolvedscales. This pseudo-pressure term gives some quadratic feed-back.The rate of loss of energy from the resolved scales to the unresolvedscales is 2aiVTSi]<Si:j<, so that a free parameter appears also in thisterm. Because this approximation involves a further assumption,and to give ourselves greater flexibility, we call this parameter a2 ,although in all work presented in this paper, we have set a.\ = a2-

Thus the Heisenberg model introduces two parameters in the sys-tem of equations, one, ai, in the linear term, the other one, 0*2, inthe quadratic term. The equations therefore have the following form:

where L and L' represent the linear terms, Q the direct quadraticterms, Q' the quadratic pseudo-pressure term and C the cubic termsarising from the Reynolds stress.

3.5 Comparison with the Wall Region

This study has two conflicting requirements. On one hand wewish to keep as few modes as possible in order to obtain a low-dimensional system, permitting us to apply the techniques of dy-namical system analysis. On the other hand, we would like to retainat least a qualitatively correct dynamical representation of the turbu-lence production phenomenon. A necessary condition for the secondrequirement consists of including as much of the energy and Reynoldsstress as possible in our system. This is already satisfied in the inho-mogeneous direction by the proper orthogonal decomposition itself,since the first structure is the most energetic. Given the energeticgap between the two first eigenfunctions (Figure 3.5.1), keeping onlyone structure seems quite reasonable. The choice of wave numbers isnow of great importance, especially as far as the Reynolds stress isconcerned (an important part of the Reynolds stress is contained inthe higher modes). The best selection is probably the one for whichthe cut-off modes correspond to the experimental measurement cut-off. Since we retain only a few modes, the steps of our Fourier seriesare large. However, provided that the periodic length is larger thana few integral scales, the Fourier transform of the autocorrelationtensor is unchanged and the eigenfunctions are still the same.

We now seek a minimum truncation. The experimental resultsshow that the ratio between the streamwise and spanwise character-


Figure 3.5.1. Convergence of the proper orthogonal decompositionin the near-wall region (x^~ = 40) of a pipe flow according to experi-mental data. Turbulent kinetic energy in the first three eigenmodes,A(n)(n =1,2,3) function of: the spanwise wave number (from Herzog,1986).

istic length scales is of order ten (see figure 3.5.1.). Our first approx-imation therefore neglects streamwise variations. We need at leastthree terms in the spanwise direction (see below). Thus the minimumtruncation consists of one eigenmode, one streamwise wave number(&! = 0) and three (i.e. two active) spanwise wave numbers (0, k, 2k).In this paper, up to six spanwise wavenumbers (0,..., 5&) will be con-sidered. In this case k has the value 3 X 10~3 and the lengths of theperiodic box are L\ — Ly, — 333. Even the model having six spanwiseFourier modes is still very crude, although the truncation seems tobe the one which contains an optimally large amount of the total


energy among those of the same dimension. The zero cut-off modein the streamwise direction in particular is a very rough approxima-tion. Such a truncation causes a drop of the spanwise and normalroot mean square values of the velocity which is particularly signifi-cant in the upper half of the layer. For this reason, we do not expectour velocity field reconstructed without rescaling to have more thanqualitative significance.

As a preliminary approach, we study the set of equations for atruncation limited to the first eigenfunction (n = p = q = r = 1),the zero streamwise wavenumber fci = 0 and up to six spanwisewavenumbers k$ = 0, k, ...5k, for a suitably chosen k.

When only the zero streamwise wavenumber is considered, theequations become much simpler. Indeed, because of the symme-tries of the eigenfunctions (Herzog, 1986) in the (&i, £3) wavenumberplane, the first and second components are purely real and the thirdcomponent is imaginary on the k<3 - axis (i.e. for k\ — 0). Using thesesymmetries, the equations for the complex modal coefficients e41=0 ks

can be readily derived. Letting ^3 take the values jk,j = 0,1, ...,5and writing a^. '_0 k __-k = Xj+i j/j, a typical equation takes the form:

In these equations the zero Fourier mode a(l)oo = XQ + iyo hasbeen removed since its imaginary part is identically zero and its realpart decays to zero under the influence of the viscosity (dxo/dt =aox0;ao < 0). The coefficients of the quadratic and cubic terms areeach the sums of two quantities: c^^-k'i Ck-k',k and d k t k ' , d k , - k ' - Theinfluence of the Heisenberg model appears in the linear and quadraticterms as follows:

where a\ and 0*2 are the proportionality (Heisenberg) coefficientsalready mentioned. The computation of the nondimensionalizedtransport coefficient i>x for this particular truncation gives the valueI/Rex = 6.28.


Numerical integrations of 3, 4, 5 and 6 mode models have beencarried out, but we shall only report in detail on the 6 mode (5 ac-tive mode) simulations here. We summarize here the behavior of thesystem. For a > 1.61 a unique circle of globally attracting stablefixed points exists in the 2/4 subspace (with a window of instabil-ity between 2.0 and 2.3): the /l, 7*3,7-5 components, along with thetrivial TQ component, are zero. As a increases, the magnitude of ther-2 and r4 components decreases until, at a ~ 2.409, this non-trivialcircle of fixed points coalesces with the origin, which for a > 2.409is the unique and globally attracting fixed point for the problem.For 1.37 < a < 1.61, an 51-symmetric family of globally attractingdouble homoclinic cycles F exists, connecting pairs of saddle pointswhich are TT out of phase with respect to their second (x^^y^) com-ponents. The points r + , r~ discussed above are typical members ofthis family. The existence of the cycles F implies that, after a rela-tively brief and possibly chaotic transient, almost all solutions entera tubular neighborhood of F and thereafter follow it more and moreclosely. As they approach F, the duration of the "laminar" phase ofbehavior (in which r2 and r4 remain non-zero and almost constantand ri^r3,r5 grow exponentially in an oscillatory fashion) increaseswhile the bursts (in which TI, 7-3 and r$ collapse) remain short. In anideal, unperturbed system, the laminar duration would grow with-out bound, but small numerical perturbations, such as truncationerrors, presumably prevent this occurring in our numerical simula-tions. More significantly, the pressure perturbation will limit thegrowth of the laminar periods. Thus, there is an effective maximumduration of events, which is reduced as a is decreased from the crit-ical value aj ~ 1.61.

For the present study the intermittent behavior exhibited by thesix mode model for a between 1.35 and 1.61 appears to be of great-est interest, since it corresponds in a fairly clear way to the physicalinstability, sweep and ejection event observed in boundary layer ex-periments. We now describe this intermittent behavior, starting withsome general remarks.

In the theory of dynamical systems, three types of intermittencyhave been distinguished (Pomeau and Manneville, 1980). They areassociated with solutions repeatedly passing close to a weakly unsta-ble fixed point or periodic orbit. The solution spends a long "lami-nar" phase near the point or orbit until it reaches a critical amplitudeand a brief turbulent "burst" ensues, in which it travels far and fast


in phase space before returning. (These terms were appropriated bythe dynamical systems community and have been used in a predomi-nantly metaphorical fashion thus far). The laminar phase is governedby the linearized dynamics near the fixed point or periodic orbit, butthe burst and return are associated with a "global reinjection mecha-nism," usually a homoclinic orbit or heteroclinic cycle (Tresser et al.(1980), Silnikov (1965, 1968, 1970), Tresser (1984), Sparrow (1982),Guckenheimer and Holmes (1983, §6.5).)

While evidence of intermittency has been detected before in fluidsystems and models (Pomeau and Manneville (1980), Berge et al.(1980), Maurer and Libchaber (1980), Dubois et al, 1983) there hasbeen little evidence of type II, associated with a subcritical Hopf bi-furcation, which is what we observe in the present model at a = 1.61.We will, therefore, explore the connection between this dynamicalphenomenon and its analogue in the turbulent boundary layer insome detail. Our model displays a "regular" intermittency, in con-trast to the chaotic intermittency of Pomeau and Manneville (1980),in which event durations are distributed randomly.

We describe here the reconstructions of the velocity field fromthe expansion, using the computed values of xt- and y,-. Probably themost interesting sets of solutions are those exhibiting intermittency,obtained for 1.3 < a < 1.61. In the flow field, the rapid event whichfollows the slowly growing oscillation and the repetition of the processreminds one of the bursting events experimentally observed (Kline etal. (1967), Corino and Brodkey (1969), etc.). For that reason we callit a "burst." We will analyze its effect on the streamwise vortices.

In Figure 3.5.3 we show an enlargement of the time histories ofthe modal coefficients during a burst for a = 1.4 (cf. Figure 3.5.2).A description of the motion of the eddies during the burst is given inFigure 3.5.4 for a •= 1.4 by plotting u? and 113 at the different timesindicated on Figure 3.5.3. Before and after the event, two pairs ofstreamwise vortices are present in the periodic box, a structure verysimilar to that previously obtained for 1.61 < a < 2.41. Nonetheless,pictures 1 and 14 are shifted in the span wise direction by 333+/4,corresponding to the phase shift A#2 = T, A#4 = 2?r. The burstingevent leads to variations of positions and amplitudes of the basicstreamwise rolls and formation of other vortices. The oscillation,death and rebirth of vortices make the streak spacing vary, in agree-ment with experimental results (Kline et a/., 1967). However, sincethe intermittent solution is always very close to the real subspace or


a rotation of it, the vortices remain symmetric and paired. Moreoverit is possible to adjust the value of the viscosity parameter (a ~ 1.5)so that the bursting period is 100 wall units as experimentally ob-served. It is found that, in this case, the "burst" lasts 10 wall unitswhich is also the right order of magnitude.

Figure 3.5.2. Time histories of the real (x;) and imaginary (y,-) partsof the coefficients for a value of the Hewisenberg parameter a = 1.4

The behavior of the eddies corresponding to a chaotic solution(1.0 < a < 1.3) is shown in Figure 3.5.5, as before, by plottingthe u2, u3 velocity components at specific times. The behavior isless regular and isolated vortices sometimes emerge. This is consis-tent with the flow visualization experiments of Smith and Schwarz(1983) who observed a significant number of solitary vortices amongthe predominant vortex pairs. These patterns are also very rich in


Figure 3.5.3. Intermittent solution corresponding to an Heisebergparameter a = 1.4. Time history of the real and imaginary parts ofthe model coefficients Xj , y; ( i =1 1,2,.. .5).

dynamics. We intend to study the regime further in future work.Due to the truncation of our model to a single eigenmode and

one streamwise Fourier component, the absolute and relative energyof the various velocity components is affected. First, the choice ofstreamwise length scale affects the absolute values of all the compo-nent energies, but not their relative values. We chose a streamwiselength scale of 333. Keeping only the Fourier mode at zero stream-wise wavenumber, this is equivalent to supposing that the distribu-tion of the first eigenvalue with wavenumber is flat to 333, and dropsto zero. Comparison with the true distribution of the first eigenvaluewith streamwise wavenumber indicates that this should overestimatethe energy in this eigenvalue. A more reasonable length scale wouldbe between 400 and 500, a value determined by the requirement thatthe area between it and the origin is the same as the true area under


Figure 3.5.4. (left) Intermittent solution corresponding to an Heisen-berg a = 1.4. Vector representation of u.2, 113 in an X2-X3 plane attimes indicated on figure 3.5.3.

Figure 3.5.5. (right) Chaotic solution for Heisenberg parameter a =1.2. Vector representation of U2, us in an X2-xs plane at uniformlyspaced times (close enough to resolve the motion).

the distribution of the first eigenvalue with streamwise wavenumber.However, examination of the equations indicates that changing thevalue of LI leaves the equations invariant. Hence all phenomena re-ported here would be unchanged, with simply a decrease in energylevel of all the velocity components. This would decrease the over-estimated streamwise energy contained in our system but would notnecessarily improve our model since it would also decrease the energy,


already deficient, in the normal and cross-stream components.The truncation also affects the relative energy in the various ve-

locity components. We have examined the distribution with x2+ ofthe energy in the three velocity components for truncations of 3, 6,and 17 cross-stream wavenumbers, 1, 2 and 7 stream wise wavenum-bers, and 1 and 3 eigenmodes. It is clear that the energy in the u^and 11,3 components is relatively low (compared to the u\ component)when only one streamwise wavenumber is included. This is partic-ularly true in the upper part of the layer. While inclusion of morethan 6 cross-stream wavenumbers has no effect, addition of stream-wise wave numbers does make a difference. In particular, addition ofanother streamwise wavenumber helps considerably. We must con-clude that there is energy in cross-stream and normal motions ofhigher streamwise wavenumbers, that makes little relative contri-bution to the streamwise velocity component. As a consequence,our predicted rolls are somewhat weak compared to reality, and werescaled them. This rescaling will not be necessary in future work,when we will add streamwise wavenumbers.

One naturally asks if the addition of streamwise modes will dras-tically change the behavior of our model. In this respect, we observethat the same symmetries, inherited from invariance of the Navier-Stokes equations to spanwise translation and reflection, and stream-wise translation, operate at any order of truncation. In particular,this implies that the subspace spanned by basis functions with zerostreamwise wavenumber is invariant and that the coefficients of theequations restricted to this subspace remain unchanged as stream-wise components are added. Thus, the model studied in this paper,together with its dynamics, exists unchanged, as a subsystem in thelarger set of differential equations, much as the 2-4 subsystem existsin the present model. This raises the interesting possibility that the2-4 branch may be destabilized by streamwise modes as well as bythe 1, 3, 5 "intersticial" spanwise Fourier modes.

We do not have present in our modal population a mechanism torepresent the production of higher wavenumber energy when an in-tense updraft is formed, presumably as a result of a secondary insta-bility. Thus, although our eddies are capable of exhibiting the basicbursting and ejection process, the labor is in vain - there is no sequel,no production of intense higher wavenumber turbulence. A contri-bution is made only to the low wavenumber part of the streamwisefluctuating velocity and the Reynolds stress. We could easily sim-


ulate the production of this high wavermmber turbulence, althoughwe do not expect its inclusion to change the dynamics of our systemqualitatively. However, we have held the value of our Heisenberg pa-rameter constant, whereas its value should rise and fall with the levelof this intense higher wavenumber turbulence in the vicinity of thewall. The transport effectiveness of the intense higher wavenumberturbulence would be expected to damp the system, suppressing theinteresting dynamics until the higher levels are either blown down-stream or lifted and diffused to the outer edge of the boundary layer.The production of higher values of the Heisenberg parameter couldeasily be parameterized by a single time constant first order equa-tion tied to a measure of the amplitude of the coefficients. Themajor effect would simply be to cut off each burst more rapidly thanat present. We plan to study the behavior of such a model in thefuture.

The occurrence of the bursts for values of the Heisenberg pa-rameter between 1.4 and 1.6 appears to be pseudo-random. This isessentially due to round-off error in the calculations. The transitionsfrom one solution to the other and back are extremely sensitive to theprecise solution trajectory, and a minute change can make a consider-able change in the time at which the next transition occurs. Initiallywe did not exercise the pressure term. Recall that the pressure termappeared due to the finite domain of integration. It represents theinteraction of the part of the eddy that we have resolved with thepart above the domain of integration, which is unresolved. The or-der of magnitude that we estimated for this term was small, andfor that reason we at first neglected it. It has, however, an impor-tant effect, while not changing the qualitative nature of the solution.The term has the form of a random function of time, with a smallamplitude. This slightly perturbs the solution trajectory constantly;away from the points r+ and r- this has little effect, but when thesolution trajectory is very close to these points, the perturbation hasthe effect of throwing the solution away from the fixed point, so thatit need not wait long to spiral outward. This results in a thoroughrandomization of the transition time from one solution to the other,while having little effect on the structure of the solution during aburst. While in the absence of the pressure term (and round-off er-ror), the interburst time tends to lengthen as the solution trajectoryis attracted closer and closer to the heteroclinic cycle (this effect isvisible in figure 3.5.2), with the pressure term, the mean time stabi-


lizes (see figure 3.5.6). The mean interburst times for various valuesof the pressure signal are shown in figure 3.5.7. Probably the mostsignificant finding of this work is the identification of the etiology ofthe bursting phenomenon. That is, the bursts appear to be producedautonomously by the wall region, but to be triggered by pressure sig-nals from the outer layer. Whether the bursting period scales withinner or outer variables has been a controversy in the turbulence lit-erature for a number of years. The matter has been obscured by thefact that the experimental evidence has been measured in boundarylayers with fairly low Reynolds numbers lying in a narrow range, sothat it is not really possible to distinguish between the two types ofscaling. The turbulent polymer drag reduction literature is particu-larly instructive, however, since the sizes of the large eddies, and thebursting period, all change scale with the introduction of the polymer(see, for example, Kubo and Lumley, 1980, Lumley and Kubo, 1984).The present work indicates clearly that the wall region is capable ofproducing bursts autonomously, but the timing is determined by trig-ger signals from the outer layer. This suggests that events during aburst should scale unambiguously with wall variables. Time betweenbursts will have a more complex scaling, since it is dependent on thefirst occurrence of a large enough pressure signal long enough aftera previous burst; "long enough" is determined by wall variables, butthe pressure signal should scale with outer variables.

3.6 Generation of Eigenfunctions from StabilityArguments

Ideally, one would like to apply the POD approach to a wide rangeof flows where coherent structures are known to play an importantrole in the dynamics. The POD procedure, however, requires thetwo-point velocity autocorrelation tensor as input, thus necessitatingcomplete documentation of the flow before the analysis can proceed.For flows with very high Reynolds numbers or complicated geome-tries this can be prohibitively expensive given current computationaland experimental capablities. In this section we will describe ananalytic procedure for extracting basis functions (structures) whichapproximate those given by the POD but which requires much lessa priori statistical information about the flow.

The method presented is based on energy stability considerationsput forth by Lumley (1971). First, the instantaneous flow field is de-


Figure 3.5.6. Pressure perturbations: solutions for a = 1.6 withpressure term.

composed into three components in order to isolate the large scalestructures. Evolution equations can then be written for the coherentvelocity field and the coherent kinetic energy. A procedure can thenbe formalized to search for the structures which maximize the in-stantaneous growth rate of coherent energy, the rationale being thatthe structures which on average have the largest growth rates willcompare well with the structures which contribute the most to theaverage turbulent kinetic energy (POD eigenfunctions).

As an example we consider turbulent channel flow assumed sta-tistically homogeneous in both the downstream (zj) and crosstream(x3) directions. In order to extract spatial structures from the totalvelocity field, we avoid traditional Reynolds averaging and insteaddecompose the instantaneous velocity field into three components:the spatial mean (U), the coherent field (i;) and the incoherent back-

54 J. L. Lumley et al

Figure 3.5.7. Mean interburst duration T as a function of eigenvalueA+ and pressure signal amplitude £. Solid curves indicate predictionof Aubry et al. (1988), open squares indicate results of numericalsimulations.

ground turbulence (it').

The spatial mean is an average over the x\ — 23 plane; we will indicateit by [...]. We introduce a second averaging procedure, denoted by< ... >, which eliminates the small-scale turbulence while leaving the


coherent field intact.

Practically this can be accomplished in several ways (Reynolds andHussain, 1972; Gatski and Liu, 1980; Liu, 1988; Brereton and Kodal,1992; Berkooz, 1991). We will refer to this average as a phase aver-age. For our purposes here it is sufficient that the phase average andspace average commute, and that the cross correlations be negligible.

Given these averaging procedures, we can manipulate the Navier-Stokes equations to arrive at evolution equations for the coherentvelocity field.

where D/Dt denotes the mean convective derivative, Sij the meanrate of strain, and v the kinematic viscosity. TJJ represents the rec-tified effects of the small scale fluctuations on the coherent field andis defined by

This can be thought of as a perturbed Reynolds stress, which is un-known and will ultimately require modeling. In the limit of a com-pletely random turbulence containing no structure (i.e. < ... >= 0)this quantity is equal to the usual Reynolds stress. In the case whenthe turbulence is completely structured so that < ... >= [...], T{J isidentically zero.

We now follow classical energy method stability analysis for thecoherent field. First, the growth rate of the volume averaged coherentenergy E is defined as a functional of the coherent velocity field.

Integration by parts and continuity are used to eliminate the nonlin-ear convective and pressure terms. We seek the solenoidal velocityfield which maximizes A. Application of the calculus of variationsthen gives the Euler equations for the maximizing v field in the formof an eigenvalue relation.


We consider coherent fields which are periodic in the homogeneousdirections. This allows a decomposition into poloidal and toroidalcomponents which satisfy continuity exactly (Joseph, 1976)

The two scalar functions are then expanded in normal modes in thestreamwise and spanwise directions.

Substituting the above into equation (3.6.7) and eliminating the pres-sure TT results in two coupled equations, forming a differential eigen-value problem.

In order to precede we need to specify a mean velocity field anda model for the unknown stress terms (see figure 3.6.1.).

Figure 3.6.1. Model inputs; (a) mean velocity and mean gradient;(b) eddy viscosity and Reynolds stress.

We have investigated two different models for the unknown stressterms appearing in the eigenvalue relation. It should be noted that,modulo the modeled terms, equation (3.6.7) is linear in the coher-ent velocities, providing an inexpensive means of determining basisfunctions. This linearity is an essential advantage of the method andfor this reason we will constrain any stress model to be both linearand homogeneous in the v field insuring that the governing equation


remains a regular eigenvalue problem. Tensorially this requires

The nature of the averaging procedure implies that the scales of thecoherent field and the background turbulence are different. Assum-ing that the background turbulence evolves on much shorter timeand length scales then the structures, it seems plausible that a New-tonian stress-strain relationship like that for the molecular stresseswill provide the basis for a model. We set

Due to the inhomogenity of the turbulence in the wall normal direc-tion, we specify vt as a function of x?. corresponding to experimentallydetermined values of the traditional eddy viscosity. We will refer tothis basic model as the isotropic eddy viscosity model.

Using the basic stress model and an analytic expression for thefully turbulent mean profile (Reynolds and Tiederman, 1967) we havesolved the resulting equations numerically. Figure 3.6.2. shows com-parisons between the calculated eigenvectors and the POD resultsof Moin and Moser (1989) obtained from a numerical data base.Although there are qualitative similiarities in the shape of the struc-tures, the modes predicted by the stability method fall off muchmore rapidly away from the wall than do the POD functions. Theeigenvalue spectrum clearly shows that the stability analysis favorsmodes which have a much higher wavelength than the maximumenergy modes of the POD.

Although there may be a number of reasons for this discrepancy,we choose to first examine more closely the closure model. Since boththe POD analysis and the stability method favor modes which areinfinitely long in the streamwise direction, we examine our equationssetting k\ = 0. We find that the isotropic eddy viscosity modelcreates no coupling between the different components of the coherentvelocity. When there is no streamwise variation of the coherent fieldthe only coupling terms in the equations are those multiplying themean gradient. For realistic mean profiles, regions of high shear areconfined to thin regions near the wall, and the structures predictedmay be expected to fall off as quickly as the shear.

We now seek to develop a stress model that allows for some an-isotropy in the eddy viscosity, and thus couples the component equa-tions through the stress terms. We begin with the evolution equation


Figure 3.6.2. Isotropic eddy viscosity model: •, POD; , isotropicmodel; (a) k3 = 6.00, (b) k3 = 9.00, (c) fc3 = 12.00, (d) k3 = 15.00.

for the Reynolds stresses, where we are obliged to model a numberof terms to obtain a closed system. We use standard second-orderturbulence models. We model the pressure-strain correlation bya return-to-isotropy term and an isotropization-of-production term(Naot et al., 1970); we use an isotropic dissipation. We assume thestresses are in local equilibrium: D[uiUj]/Dt — 0. This reduces theevolution equation for the Reynolds stress to an algebraic expression.

Now we set up a perturbation expansion in terms of mean fieldquantities, taking the coherent field as an order e perturbation tothe spatial mean. On physical grounds, we argue that the perturbedstress field is due entirely to the presence of the structures and conse-quently we restrict the model to include only production due directlyto coherent velocity gradients. This is in agreement with a cascade


analogy for the complete flow: the coherent structures are fed energydirectly by the mean gradients while the small scale turbulence is inturn fed by gradients of the coherent field. If we identify the Othorder stresses with an eddy viscosity tensor, then the closure modelcan be written as

where the tensor viscosity has the following structure in this specificcase: z/13 = 1/33 = 0, 1/33 = 1/22-

Despite the absence of mean production terms, this model is stilla major improvement over the isotropic eddy viscosity formulation.In the simple model the effects of the mean field have been neglectedentirely. Here we have allowed for modulation of the perturbationstresses by the mean field through the 0th order stresses appearingin the production terms. Also we have unconstrained the modelin an important way since the tensorial form of the eddy viscosityallows the pricipal axes of the stress tensor to be unaligned withthe axes of the rate of strain. This is more realistic considering thethree-dimensionality of the coherent field. This model leads to theexpected cross coupling of the equations through the stress terms.

Figure 3.6.3. shows eigensolutions for several values of k3. Theresults compare well with the POD eigenvalues, especially for wave-numbers at or below the peak in the POD spectrum. The improve-ment with decreasing wavenumber is expected given the modelingconsiderations. The separation of scales between the backgroundturbulence and the coherent structures increases as the wave numberdecreases adding to the expected accuracy of the stress model. Thecomparison of the two models indicates significant improvements inthe results given by the anisotropic eddy viscosity form. The energymethod procedure with the more refined closure model appears ca-pable of extracting structures which closely approximate those givenby the POD at least at the energy containing scales of motion.

Despite the general improvement, it is still clear that more needsto be done. From Figures 3.6.2. and 3.6.3., it is evident that the eigenspectrum produced by solution of Equation (3.6.1),while improvedby the use of the anisotropic closure model, still predicts structureswith maximum growth rate that are a factor of 2 smaller than thosecontaining the most energy (as given by the POD). We next con-sidered the effect on the spatial mean velocity field of the growingcoherent perturbation (see figure 3.6.4.).

Figure 3.6.3. Anisotropic eddy viscosity model: •, POD; iso-tropic model; - - - anisotropic model: (a) k3 = 6.00, (b) £3 = 9.00,(c) fc3 = 12.00, (d) k3 = 15.00.

At this point we consider the role of the mean velocity in thetwo methods. The POD structures are derived from solutions tothe non-linear Navier-Stokes equations which allow for complicatedinteraction between the different scales of motion. The structuresevolve in a mean velocity field that is changing due to the presenceof the structures themselves. Conditionally averaged mean profilesclearly show the evolution of the local shear in the presence of co-herent structures (see Figures 3.6.3, 3.6.4). We see that structures,in the relatively long period before bursting, act to erode the shearthat they see. The POD eigenfunctions are given by averages ofcontributions from different mean profiles.

The stability method on the other hand docs not allow for any



Figure 3.6.4. Comparison of the POD spectrum with the eigenspec-trum of the stability problems.

interaction between the mean and the coherent field. The mean flowis imposed and the resulting structures are calculated. The meanprofiles we have used are time averages which mask any contributionfrom the coherent field. As such the stability analysis predicts thatthe highest growth modes are those which can best extract energyfrom the time averaged mean shear which is concentrated in the smallnear-wall region. Since the structures have an aspect ratio of about1, the narrow region of high imposed shear leads to a peak in theeigen spectrum at a large wave number.

To allow the mean field to evolve under the influence of the coher-ent field, we follow Liu (1988) and write time evolution equations forthe energy density of the coherent field. We allow the mean profile todepend on the coherent velocity as it does in reality. We expect thatequilibrium solutions for the energy density as a function of crossstream wave number will approximate the average energy content asgiven by the POD spectrum.

We assume that the coherent field is given by the eigenvalues of


the stability problem, but we now allow them to vary in time

where V>i = vexp{ikx3}, i/>2 = ikil}exp{ikx3}, -03 = —Difjex.p{ikx3}.By examining the evolution equation for r,-j =< M;UJ > — [t^Wj], theforcing terms are of the form < UiUj > v^j. Consequently, we assumethe perturbation stresses also to be a product:

Since we have used an eddy viscosity in obtaining the coherent formsWP fnrt.Vipr assume tlia.t'

where z/n = 1/22 = ^33 = ^r,^i2 = ^21 = Ar^is = "23 = 0. All thatremains is to model the mean profile. For this we adopt the quasi-steady model used in Aubry et al. (1988). This allows the mean torespond to growing structures providing the necessary feedback tothe evolving modes. Using the friction velocity, ur and the channelhalf height, a, the scaled equation for the mean gradient is:

The rate of dissipation of turbulent energy is given by a simple modeladopted from second order closures schemes.

Substituting these various models into the energy equations resultsin a set of three, coupled ODEs for the temporal evolution of theenergies and dissipation.

In order to evaluate the integrals appearing in these equations,we need to assume the spatial form of the averaged turbulence quan-tities Bij and the dissipation D(x-i). For this simple model we haveassumed that while the intensity of the turbulence varies its spa-tial dependence remains unchanged. We use experimental data forfully developed turbulent channel flow to determine both B and D.The coherent structures are found by energy stability analysis, asdescribed above.


Figure 3.6.5. Temporal evolution of coherent energy density for pa-rameter values corresponding to different wavenumbers.

We show in figure 3.6.5. the temporal evolution of the coher-ent energy density for parameter values corresponding to differentwavenumbers.

In order to quantify comparisons of the single mode evolutionmodel and the POD eigenspectrum, we define an average of the life-time of an individual structure. Taking A2(ti) = A 2 ( t f ) , where /,• isthe initial time and tt is the final time, we designate

where the integral is over t{ to fy, and T = tj — £,-.Comparison between {Ai}(k^} and the spectrum of the POD is

shown in figure 3.6.6. The single mode model, while not capturingthe shape of the POD spectrum exactly, gives a good indication ofwhich wavenumbers are the most energetic. The discrepancy at highwavenumbers is not surprising when we consider the simplificationsinvolved in the model. Our equations describe only the interactionsof a single mode with the local mean and the background turbulence.The effects of interactions between modes may be negligible for the

64 J. L. Lumley et a].

large scale structures, but must become important for the smallerscales. We will consider such interactions below.

Figure 3.6.6. Comparison of ensemble averaged energy content withPOD spectrum.

In order to estimate the effects of larger scale motions on theevolution of the smaller scales, we examine the interaction betweena fundamental wave disturbance (small scale) and its subharmonic(large scale). The coherent field is then made up of two components,vi = v'j + vi'i, where the fundamental u" is periodic in x$ with awavelength A/2 and the subharmonic v[ has a wavelength A.

Since the periods of the disturbances have been artificially pre-scribed, two phase averages can be denned to decompose the velocityfield analytically:

In this way the coherent contribution from the instantaneous velocityis given by


and the fundamental can be separated from the subharmonic usingthe « . » average:

The volume averaged kinetic energy equations for the two compo-nents of the coherent field are:

As before, the coherent velocities are taken as the eigenfunctions ofthe stability problem:

and the mean velocity model now contains contributions from boththe fundamental and the subharmonic:

New coupling terms appear in the evolution equations for the coher-ent energy densities due to the interaction between the different sizemodes:

/|2 and /I1 represent the effect on the mean shear production offundamental (subharmonic) coherent energy due to the presence ofthe subharmonic (fundamental). /4 is a measure of the direct energytransfer between the two modes due to the working of the sunhar-monic stresses against the fundamental rate of strain. This quantityappears with opposite sign in each equation.

As an example, we have calculated the interaction integrals usingthe mode £3 = 20 as the fundamental. The results of integratingequations (3.6.28) for these coefficient values are shown in figure3.6.7. The mode-mode dynamics are dominated by terms due to themean velocity feedback model. The direct interaction term J4 is much


smaller. The effect of the large scale subharmonic on the behaviorof the small scale fundamental is dramatic. Although J^1 is twiceas large as I^, the slowly growing subharmonic mode is practicallyunaffected by the presence of the fundamental. The fundamental,however, is quickly damped by the larger scale motion. This agreeswith our intuitive picture of the physics. The small scale motions seenot only the mean shear, but the strain rate due to all larger scales.

Figure 3.6.7. Temporal evolution of coherent energy densities show-ing the effect of mode-mode interaction.

While the situation analyzed here is admittedly artificial, sincea real turbulent flow will contain structures of all sizes at any giveninstant, there is no reason not to expect a qualitatively similar effecton the smaller scale motions in the real flow. Since the larger eddiesare relatively unaffected by the smaller scales, the arguments pre-sented earlier and the small wavenumber results shown in the figureswill carry over to the case of manyn interacting eddies. The spuriousslow fall-off in the spectrum of {^42}, however, would be eliminated.


3.7 Wavelet Representations

All the flows of interest to us have one or more homogeneous di-rections. We are accustomed to use in these directions the Fouriertransform, which is the homogeneous equivalent of the POD. How-ever, the Fourier transform is not nearly so appropriate in the ho-mogeneous case as is the POD in the inhomogeneous case. This isbecause the Fourier modes are not confined to a neighborhood, butextend to infinity without attenuation. All disturbances in a fluid,and coherent structures in particular, are localized. There is there-fore considerable motivation to find another representation that ismore appropriate.

In Tennekes and Lumley (1972) it was suggested that a more ap-propriate quantity would be the energy surrounding a wavenumberK, say from K/O, to UK, where a = 1.62. In physical space, this packetwith appropriate phase relations is confined to a region, essentiallydropping to zero in about lit/'K from the origin. Tennekes and Lum-ley called these "eddies," but they are an example of what are nowcalled wavelets.

While wavelets appear to make more physical sense, we mightworry because we would be discarding the optimality of the Fourierrepresentation; would convergence be much slower, so that we wouldneed many more terms, or would we lose considerable energy if weused the same number of terms? A main result of a recent paper(Berkooz et al, 1992) is that very little energy is lost when usinga wavelet basis instead of a Fourier basis. Although wavelets arephysically appealing, it would also be nice to have reassurance fromcalculations that physical behavior would be preserved in a waveletrepresentation. To set our minds at rest on this point, Berkooz etal. (1992) also display a relatively low-dimensional wavelet model ofthe Kuramoto-Sivashinsky equation that shows dynamical behaviorsimilar to the full equation.

Without getting involved with mathematical details, an orthonor-mal wavelet basis is constructed by starting with a function, say t^(x],similar to the eddy suggested by Tennekes and Lumley (1972). Fromthis, construct a set •0j,o(;r) = V'(a;2j), j = 1 , 2 , . . . . Each of these isshrunk affinely, but is geometrically similar to the original function.Now consider the translates of V^o : ^j,k(x} — ^j,o(x — k2j~),k =1,2 , . . .

Berkooz et al. (1992) consider a periodic, homogeneous stochastic

68 J. L. Lumley et si.

process. It is then obvious that the POD decomposition becomesidentical to the Fourier decomposition. Now if, for a given £, weneed N ( e ) POD modes in order to satisfy

then, if the {&;} are the coefficients in a wavelet basis, we get forsome constant C, depending only on the process (and not on e):

for some -V(e) > N(e) slightly bigger than -/V(e) (the precise state-ment is given in Berkooz et al. 1992).

3.8 Dynamics with the Wavelet Representation in aSimple Equation

Berkooz et al. (1992) wished to apply these ideas to a simplesituation. The three-dimensional, three component Navier-Stokesequations are too complicated for a first effort.

The one-dimensional, scalar Kuramoto-Sivashinsky (K-S) equa-tion appears in a variety of contexts, such as quasi-planar fronts,chemical turbulence, etc . It shares some properties with Burgers'equation and the Navier-Stokes equations, but is much easier to dealwith. UT> to some rescaling, this equation can be written as:

u periodic on [0, L], where L, the length of the spatial domain, is theonly free parameter in the problem.

Although the dynamical behavior for small values of L is fairlywell understood (see Hyman et a/., 1986 for an overview), manyopen questions remain concerning the limit L —» oo (Zaleski, 1989;Pomeau et al, 1984). As can be seen from the numerical simulations,for L —> 30, a chaotic regime involving both space and time disorderoccurs (see Figure 3.8.1, where we plot a space-time representationof a typical solution, L = 400, 0 < t < 100).

In order to check the estimate (3.7.2), we compare the energyresolved by a given number of modes using either a Fourier (POD)


Figure 3.8.1. Space-time representation of a solutionequation for L = 400, 0 < t < 100.

or wavelet basis. Note that, to compare the Fourierbases, all the translates in the wavelet basis must beHere are some results (see Fig. 3. 8. 2):

No. of modes wavelets (m — 6) wavelets (m = 8)64(j = 6) 70.84% 71.5%

96(j = 6,5) 79.1% 79.43%127(0 < j < 6) 84.1% 84.9%255(0 < j < 7) 99.9% 99.9%

The scale j = 6 which captures most of the energy on

to the K-S

and waveletconsidered.

Fourier72.2%83.3%89.7%99.9%

the average,


Figure 3.8.2. Energy resolved by a given number of modes using aFourier or wavelet basis.

corresponds to a characteristic length 2~6i = Q"1, which is also thelength scale associated with the most unstable wavelength qm. Inagreement with the general shape of the energy spectrum, the scales0 < j < 5 are shown to capture more energy than the scales inthe dissipative range (j > 7). The above figures show that (forsufficiently smooth splines) the wavelet projection captures almostthe same amount of energy as the Fourier (= POD) decomposition(within 5%).

These results prompted us to conclude that from an average en-ergy point of view wavelets are a reasonable candidate for a modal de-composition of the K-S equation. The localized nature of the waveletsmay give us a unique view of the spatial attributes of the coherent


structures. We outline our approach. We conjecture the existenceof a dynamically relevant length scale LC such that interactions be-tween physical regions of distance greater than LC are dynamicallyinsignificant (a dynamical St. Venant principle). Determining the va-lidity of this conjecture is part of our study. We use this conjectureto remove terms in a wavelet-Galerkin projection that correspond tointeractions between regions of distance greater than LC- To studywhether the dynamics of coherent structures are indeed locally de-termined we construct truncations corresponding to a small box sizeLB in the larger box of size L.

We need to address the role of unresolved physical space (i.e.modes located outside the box of size LB}- It is obvious that theDirichlet type of boundary condition imposed will create a bound-ary layer which will affect the dynamics, especially in small boxeswhich are of interest to us. There are two plausible approaches to re-move this effect. One approach uses a stochastic boundary condition(which is hard to implement numerically and treat analytically).The other approach appeals to the conjecture on the existence ofLC- One takes a box of size LB greater that 2Lc so that one canperiodize the small model using resolved relatively distant modesinstead of unresolved ones. We opted for the second approach.

We present some preliminary results of the integration of onesuch model. We resolved a box of size LB = 50 (this is 1/8 of theoriginal box). Figure 3.8.3 shows the spatio-temporal evolution ofthe full system, with a Fourier basis. Figure 3.8.4 shows the spatio-temporal evolution of a (rescaled) model with LC — 50 X 3/8, whichis in excellent qualitative agreement with the dynamics of the fullsystem. If LC is too small, after a long initial transient, the systemeventually settles down to a periodic oscillatory state, in which nointeraction between the localized structures is observed. This canbe avoided by an increase in LC, which adds significant non-linearinteractions between relatively distant physical space locations. Itmight also be avoided by a stochastic boundary condition.

ACKNOWLEDGMENTS

Supported in part by Contract No. F49620-92-J-0287 jointlyfunded by the U. S. Air Force Office of Scientific Research (Controland Aerospace Programs), and the U. S. Office of Naval Research, in

Figure 3.8.3. (left) Spatio-temporal evolution of the full system usinga Fourier basis.

Figure 3.8.4. (right) Spatio-temporal evolution of a (rescaled) modelwith Lc = 50 x 3/8.

part by Grant No. F49620-92-J-0038, funded by the U. S. Air ForceOffice of Scientific Research (Aerospace Program), and in part bythe Physical Oceanography Programs of the U . S . National ScienceFoundation (Contract No. OCE-901 7882) and the U. S. Office ofNaval Research (Grant No. N00014-92-J-1547).

Parts of these Notes previously appeared in: Aubrey, N., Holmes,P.J., Lumley, J.L. and Stone, E. f990. The behavior of coherentstructures in the wall region by dynamical systems theory. In Near-Wall Turbulence eds. S.J. Kline and N. H. Afgan, pp. 672-691.Washington, DC: Hemisphere; Berkooz, G., Elezgaray, J., Holmes,P., Lumley, J. and Poje, A. 1993 The proper orthogonal decompo-sition, wavelets and modal approaches to the dynamics of coherentstructures. In Eddy Structure Identification in Free Turbulent ShearFlows, ed. J. P. Bonnet and M. N. Glauser, pp. 295-310. Dordrechtetc.: Kluwer; and Lumley, J. L. 1992. Some comments on turbulence.The Physics of Fluids, A 4(2): 203-211.

4 REFERENCES

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Berkooz, G., 1991. "Turbulence, coherent structures and low di-mensional models," Ph.D. Thesis, Cornell University.

Berkooz, G., Elezgaray, J. and Holmes, P., 1992. "Coherent struc-tures in random media and wavelets," Physica D. 61 (1-4):pp. 47-58.

Brereton, G.J. and Kodal, A., 1992. "A frequency domain filteringtechnique for triple decomposition of unsteady turbulent flow,"J. Fluids Engineering 114 (1): pp. 45-51.

Cantwell, B.J., 1981. "Organized motion in turbulent flow," Ann.Rev. Fluid Mech. 13: pp. 457-515.

Corino, E.R., and Brodkey, R.S., 1969. "A visual investigationof the wall region in turbulent flow," J. Fluid Mech. 37 (1):pp. 1-30.

Comte-Bellot, G. and Corrsin, S., 1966 "The use of a contractionto improve the isotropy of grid-generated turbulence," J. FluidMech. 25: pp. 657-682.

Corrsin, S., 1975. Private communication.

Domaradzki, J.A., Rogallo, R.S. and Wray, A.A., 1990. "Interscaleenergy transfer in numerically simulated homogeneous turbu-lence," Proceedings CTR Summer Program, Palo Alto: Stan-ford.

Dubois, M., Rubio, M.A. and Berge, 1983. "Experimental evidenceof intermittencies associated with a subharmonic bifurcation,"Phys. Rev. Lett. 51, pp. 1446-1449.

Gatski, T.B. and Liu J.T.C., 1980. "On the interactions betweenlarge-scale structure and fine-grained turbulence in a free-shearflow III. A numerical solution," Phil. Trans. Roy. Soc. 293(1403): pp. 473-509.

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Glauser, M.N., Leib, S.J. and George W.K., 1985. "Coherent struc-ture in the axisymmetric jet mixing layer," Proc. of the 5thSymp. of the Turbulent Shear Flow Con/., Cornell University.(Springer selected papers from TSF).

Guckenheimer, J. and Holmes, P.J., 1983. "Nonlinear oscillations,dynamical systems and bifurcations of vector fields," Springer-Verlag, New York. Corrected second printing, 1986.

Herzog, S., 1986. "The large scale structure in the near-wall regionof turbulent pipe flow," Ph.D. thesis, Cornell University.

Hyman, J.M., Nicolaenko, B. and Zaleski, S., 1986. "Order andcomplexity in the Kuramoto Sivashinsky model of weakly tur-bulent interfaces." Physica D, 18: p. 113.

Joseph, D.D., 1976. "Stability of Fluid Motion," Springer Tracts inNatural Philosophy, Berlin.

Kline, S.J., Reynolds, W.C., Schraub, F.A. and Rundstadler, P.W.,1967. "The structure of turbulent boundary layers," J. FluidMech. 30 (4): pp. 741-773.

Kubo, I. and Lumley, J.L., 1980. "A study to assess the potentialfor using long chain polymers dissolved in water to study tur-bulence," Annual Report, NASA-Ames Grant No. NSG-2382,Ithaca, NY: Cornell.

Kolmogorov, A.N., 1941. "Local structure of turbulence in an in-compressible fluid at very high Reynolds numbers," DokladyAN SSSR 30, pp. 299-303.

Kolmogorov, A.N., 1962. "A refinement of previous hypothesesconcerning the local structure of turbulence in viscous incom-pressible fluid at high Reynolds number," J. Fluid Mech. 13(l) ,pp. 82-85.

Lighthill, M. J., 1956. "Viscosity effects in sound waves of finiteamplitude," in Surveys in Mechanics, eds. G. K. Batchelorand R. M. Davies, Cambridge, UK: The University Press, pp.250-351.


Liu, J.T.C., 1988. "Contributions to the understanding of largescale coherent structures in developing free turbulent shearflows," Advances in Applied Mechanics, 26: pp. 183-309.

Loeve, M., 1955. "Probability Theory," Van Nostrand, New York.

Lumley, J.L., 1967. "The structure of inhomogeneous turbulentflows," Atmospheric Turbulence and Radio Wave Propagation,A. M. Yaglom and V. I. Tatarski, eds. p. 166. Moscow: Nauka.

Lumley, J.L., 1970. "Stochastic tools in turbulence," AcademicPress, New York.

Lumley, J.L., 1971. "Some comments on the energy method," De-velopments in Mechanics 6, L.H.N. Lee and A.H. Szewczyk,eds. Notre Dame, IN: N. D. Press.

Lumley, J.L., 1978. "Computational modeling of turbulent flows,"Advances in Applied Mechanics 18, edited by C.S. Yih, p. 123,New York: Academic Press.

Lumley, J.L., 1981. "Coherent structures in turbulence," Transitionand turbulence, edited by R.E. Meyer, Academic Press, NewYork: pp. 215-242.

Lumley, J.L., 1992. "Some comments on turbulence," Phys. FluidsA 4 (2): pp. 203-211.

Lumley, J.L. and Kubo, I., 1984. "Turbulent drag reduction bypolymer additives: a survey," The Influence of Polymer Addi-tives on Velocity and Temperature Fields, IUTAM SymposiumEssen 1984, Ed. B. Gampert, pp. 3-21, Springer Berlin/Heidel-berg.

Maurer, J. and Libchaber, A., 1980. "Effects of the Prandtl numberon the onset of turbulence in liquid helium," J. Phys. ParisLett. 41, L-515.

Meneveau, C., Lund, T.S. and Chasnov, J., 1992. "On the localnature of the energy cascade," Proceedings of the Summer Pro-gram, Center for Turbulence Research. Stanford/NASA Ames:CTR.


Mom, P., 1984. "Probing turbulence via large eddy simulation,"AIAA 22nd Aerospace Sciences Meeting.

Moin, P. and Moser, R.D., 1989. "Characteristic-eddy decomposi-tion of turbulence in a channel," Journal of Fluid Mechanics200: pp. 471-509.

Monin, A.S. and Yaglom, A.M., 1971. "Statistical fluid mechan-ics: mechanics of turbulence," edited by J.L. Lumley, 1, Cam-bridge, MA: MIT Press.

Naot, D., Shavit, A., and Wolfshtein, M., 1970. "Interaction be-tween components of the turbulent velocity correlation tensor,"Israel J. Tech., 8, 259.

Panchapakesan, N.R. and Lumley, J.L., 1993. "Turbulence mea-surements in axisymmetric jets of air and helium, Part I: AirJet and Part II: Helium Jet." ,7. Fluid Mech. 246: pp. 197-223,pp. 225-247.

Pomeau, Y. and Manneville, P., 1980. "Intermittent transition toturbulence," Comm. Math. Phys. 74, pp. 189-197.

Pomeau, Y., Pumir, A. and Pelce, P., 1984. "Intrinsic stochasticitywith many degrees of freedom," /. Stat. Phys 37: pp. 39-49.

Reynolds, W.C. and Hussain, A.K.M.F., 1972. "The mechanics ofan organized wave in turbulent shear flow. Part 3," Theoreticalmodels and comparisons with experiment, J. Fluid Mech. 54:pp. 263-287.

Reynolds, W.C. and Tiederman, W.G., 1967. "Stability of turbu-lent channel flow with application to Malkus' theory," J. FluidMech. 27: pp. 253-272.

Sarkar, S. & Lakshmanan, B., 1991. "Application of a Reynolds-stress turbulence model to the compressible shear layer," AIAAJ. 29 (5): pp. 743-749.

Sarkar, S., 1992. "The pressure-dilatation correlation in compress-ible flows," Phys. Fluids .A 4 (12): pp. 2674-2682.

Sarkar, S., Erlebacher, G., & Hussaini, M.Y., 1991. "Direct sim-ulation of compressible turbulence in a shear flow," Theoret.Comput. Fluid Dynamics 2: pp. 319-328.


Sarkar, S., Erlebacher, G., Hussaini, M.Y. & Kreiss, H.O., 1991."The analysis and modeling of dilatational terms in compress-ible turbulence," J. Fluid Mech. 227: pp. 473-493.

Sherman, F.S., 1955. "A low-density wind tunnel study of shock-wave structure and relaxation phenomena in gases," NACA TN3298.

Shih, T.-H., Lumley, J.L. and Janicka, J., 1987. "Second ordermodeling of a variable density mixing layer," J. Fluid Mech.180: pp. 93-116.

Silnikov, L.P., 1965. "A case of the existence of a denumerable setof periodic motions," Soviet Math. Dokl 6: pp. 163-166.

Silnikov, L.P., 1968. "On the generation of a periodic motion fromtrajectories doubly asymptotic to an equilibrium State of Sad-dle type," Math. U.S.S.R. SbornikG, pp. 427-438.

Silnikov, L.P., 1970. "A contribution to the problem of the structureof an extended neighborhood of a rough equilibrium state ofsaddle-focus type," Math U.S.S.R Sbornik 10 (1), pp. 91-102.

Smith, C.R. and Schwarz, S.P., 1983. "Observation of streamwiserotation in the near-wall region of a turbulent boundary layer,"Phys. Fluids 26 (3): pp. 641-652.

Sparrow, C.T., 1982. "The Lorenz equations: bifurcations, chaosand strange attractors," Springer-Verlag, New York, Heidel-berg, Berlin.

Tennekes, H. and Lumley, J.L., 1972. "A first course in turbulence,"Cambridge, MA: MIT Press.

Tresser, C., 1984. "About some theorems by L.P. Silnikov," Annde L'Inst. H. Poincare, 40: pp. 441-461.

Tresser, C. Coullet, P. and Arneodo, A., 1980. "On the existenceof hysteresis in a transition to chaos after a single bifurcation,"J. Physics (Paris) Lett. 41, L2 43-246.

Zaleski, S., 1989. "Stochastic model for the large scale dynamics ofsome fluctuating interfaces," Physica D 34: p. 427


Zeman, O., 1990. "Dilatation dissipation: the concept and appli-cation in modelling compressible mixing layers," Phys. FluidsA. 2: pp. 178-188.

Zeman, 0., 1991. "On the decay of compressible isotropic turbu-lence," Phys. Fluids A 3: pp. 951-955.

Zilberman, M., Wygnanski, I. , and Kaplan, R.E., 1977. "Transi-tional boundary layer spot in a fully turbulent environment,"Phys. Fluids Supp. 20 (10): S258-S271.

Zubair, L., Sreenivasan, K. R. and Wickerhauser, V. 1992. "Char-acterization and compression of turbulent signals and imagesusing wavelet-packets," In The Lumley Symposium: Recent De-velopments in Turbulence, edited by T. Gatski, S. Sarkar, C.G. Speziale, Berlin: Springer, pp. 489-513.

Chapter 2

DIRECT NUMERICALSIMULATION OFTURBULENT FLOWS

Anthony Leonard

1 INTRODUCTION

The numerical simulation of turbulent flows has a short history.About 45 years ago von Neumann (1949) and Emmons (1949) pro-posed an attack on the turbulence problem by numerical simulation.But one could point to a beginning 20 years later when Deardorff(1970) reported on a large-eddy simulation of turbulent channel flowon a 24x20x14 mesh and a direct simulation of homogeneous, iso-tropic turbulence was accomplished on a 323 mesh by Orszag andPatterson (1972). Perhaps the arrival of the CDC 6600 triggeredthese initial efforts. Since that time, a number of developments haveoccurred along several fronts. Of course, faster computers with morememory continue to become available and now, in 1994, 2563 sim-ulations of homogeneous turbulence are relatively common with oc-casional 5123 simulations being achieved on parallel supercomputers(Chen et al., 1993) (Jimenez et a/., 1993). In addition, new algo-rithms have been developed which extend or improve capabilities inturbulence simulation. For example, spectral methods for the simu-lation of arbitrary homogeneous flows and the efficient simulation ofwall-bounded flows have been available for some time for incompress-ible flows and have recently been extended to compressible flows. In

79

80 A. Leonard

addition fast, viscous vortex methods and spectral element methodsare now becoming available, suitable for incompressible flow withcomplex geometries. As a result of all these developments, the num-ber of turbulence simulations has been increasing rapidly in the pastfew years and will continue to do so. While limitations exist (Rey-nolds, 1990; Hussaini et a/., 1990), the potential of the method willlead to the simulation of a wide variety of turbulent flows.

In this chapter, we present examples of these new developmentsand discuss prospects for future developments.

2 PROBLEM OF NUMERICAL SIMULATION

We consider an incompressible flow whose time evolution is givenby the Navier-Stokes equations for the velocity, u (x, t), and thepressure, p (x, t) as

along with appropriate initial and boundary conditions. It is assumedthat the density = 1.

The character of the solution depends on the Reynolds number ofthe flow, Re = UL/V, where U and L are a characteristic velocity anda characteristic length of the large scales and v is the kinematic vis-cosity. For small Reynolds numbers, one obtains a laminar flow thatis smoothly varying in space and time; for large Reynolds numbers,one obtains a turbulent flow. Turbulent flows have been described asrandom, chaotic, vortical, three-dimensional, and unsteady, and theyare known to contain a wide range of scales. It is the combinationof all these attributes that makes the numerical simulation of suchflows extremely challenging.

In turbulent pipe flow, for example, we estimate, according touniversal equilibrium theory (see, e.g. Batchelor, 1967), the smallestimportant scale of turbulence to be proportional to the dissipation,or Kolmogorov, length, rj = (i/3/e)1//4, where e is the energy dissipa-tion rate per unit mass, and the largest important scale to be somemultiple of the pipe diameter. Using the volume-aver aged s given by

Direct Numerical Simulation 81

where U is the mean velocity and D is the pipe diameter, we findthat,

where Re = UD/z/ and f is the friction factor,

and UT is the wall shear velocity given by

The friction factor, given implicitly by the formula (Hinze, 1975),

is only weakly dependent on Reynolds number so that the requirednumber of mesh points on a three-dimensional grid would be propor-tional to (D/??)3 oc Re3'4. Figure 1 shows the energy spectrum mea-surements of Laufer (1954) for high-Reynolds-number (Re = 500,000)pipe flow. The pipe diameter is 25.4 cm. The wave number corre-sponding to the Kolmogorov length, k^ = 27r/?y, is seen to be wellbeyond the measured data. To simulate reliably the dissipation ofturbulence energy, the grid spacing must be somewhat smaller thanthe length scale corresponding to the peak in the dissipation spec-trum. If isotropy of the small scales is assumed, the dissipationspectrum is proportional to kfEi(kj) . In Laufer's experiment thispeak, away from the wall, corresponds to a length of 150 77 or 0.03Dor k?7 = 0.04.

Figure 2 shows energy spectra in a high speed boundary layermeasured by Saddoughi and Veeravalli (1994). Note again that thepeak in the energy spectrum occurs at k?y well below 2?r, this timenear kij K 0.06. Thus we expect that a resolution of the fine scalessuch that kmax7j = 1 should be sufficient and this, indeed, seems tobe the case (Huang and Leonard, 1994).

Therefore, as an estimate of the mean spacing between grid pointsA, required in the direct, simulation of turbulent pipe flow, we takeA = 3r/. Table I gives corresponding estimates of the number ofmesh points required for several Reynolds numbers, assuming that

Figure 1. Longitudinal energy spectra measured in pipe flow at Re= 500,000, r' is the distance from the pipe wall. The pipe diameter,D, is 25.4 cm (Laufer, 1954).

the computational domain extends 10 diameters in the streamwise di-rection. (This estimate could be off by a factor of 3 either way. Somemeasurements and their interpretation suggest correlation lengths of20D, others correlation lengths of 2D; see Coles(1981).) It appearsthat the two lowest Reynolds number cases would be accessible topresent day supercomputers (1010 floating point operations per sec-ond, 109 words storage). In fact, Kim et al. (1987) previously per-formed a direct simulation of plane channel flow at Re = 3300, basedon channel half width, using 4 X 106 grid points, roughly correspond-ing to the Re = 5000 case for the pipe as given in Table I.

In addition it should be verified that the spacing A = 377 is

82 A. Leonard


Figure 2. Longitudinal and transverse energy spectra measured in aturbulent boundary layer at a momentum thickness Reynolds num-ber, Re<? of 370,000 or a Taylor scale Reynolds number, RA of 1400.Measurements taken at y+ w 16,000. Solid lines are fits to the data(Saddoughi, 1994).

Reynolds

Number

5 X 103

1 X 104

5 X 104

1 X 105

5 x 105

7?/D

0.00450.00280.000930.000580.00019

^wall units

1.61.82.42.83.8

N = ^(ff

3.1 X 106

1.3 X 107

3.6 x 108

1.5 x 109

4.2 x 1010

For Re from 5,000 to 500,000, A+ ranges from 4.8 to 11.4 (see TableI). This spacing would be marginally sufficient resolution to repro-duce all important wall-layer structures (such as streamwise streaks),which have characteristic lengths of 50-100 wall units with somestructures down to 20 units in size.

The number of time-steps, Ns, required to follow one realizationfor a time T and obtain reasonable statistics also depends on Rey-nolds number. The time-step At is roughly limited to

Using the above estimate for A and (2/f)1 '4 = 3 we find that

84 A. Leonard

Table I. - Mesh-Point Requirements

Number ofKolmogorov length mesh points

(A = 37?)

sufficiently small to allow resolution of all important turbulence phe-nomena near the wall. The grid spacing measured in wall units isgiven by

n+

103


And if T = 100 D/U, then

or 6,500 steps for Re = 5,000.

3 SIMULATION OF HOMOGENEOUSINCOMPRESSIBLE TURBULENCE

A variety of homogeneous turbulent flows can be treated by writ-ing the velocity field u as the sum of a mean component and a tur-bulent romnonfiTit.

and assuming that the components of U have the form

where repeated indices are summed (see Rogallo, 1981). By trans-forming to coordinates, x, moving with the mean flow we obtainmomentum and continuity equations for u' that contain no explicitdependence on x. Then assuming that the turbulent component ofthe homogeneous flow is periodic in x-space, with period Lm in di-rection m, no further boundary conditions are required, and spatialderivatives can be computed accurately by Fourier interpolation.

Thus. u'(x, t) is represented by

Here the mth component of k is

and I ranges over -N/2 + 1 < I < N/2 - 1. The Navier-Stokes equa-tions become a 3(N — I)3 system of ordinary differential equations(ODEs),

/here k2 = knkn, and

86 A. Leonard

(In the above, zero mean flow, Umn = 0, has been assumed in orderto simplify the presentation.) To avoid explicit evaluation of theconvolution sums u^un requiring 0(N6) operations per step, fastFourier transforms (FFTs) are used to return to physical space wherethe required products are formed and then transformed (by FFTs)back to Fourier space. Consequently, only 0(N3logN) operationsper step are required.

Suppose the tensor U^m is decomposed into a symmetric (R) andan antisymmetric (ft)tensor; then the only constraint on R is that itbe traceless Rnn = 0, but 17, related to the vorticity, must satisfy theevolution equation

Besides zero mean flow, four examples are: plain strain, TJ22 =—Uss = const; axisymmetric strain, 1)22 — Uss = —(l /2)Un =const; shear, Uia = const; and rotation, TJis = -Uai = const. SeeRogallo (1981) and Rogallo and Moin (1984) for more discussion andapplication.

4 WALL-BOUNDED AND INHOMOGENEOUS FLOWS

The direct simulation of wall-bounded and other inhomogeneousflows presents a new set of difficulties. For wall-bounded flows, partof the problem is due to the small-scale physical processes takingplace near the wall. The thin shear layer next to the wall is con-tinually breaking up via three-dimensional (3-D) inertial instabilitiesresulting in a violent 3-D wrinkling of the vortex layer. In the span-wise direction, the scale of the breakup is of the order of the thicknessof the layer or tens of wall units and, perhaps, somewhat more in thestreamwise direction.

Another part of the problem is algorithmic and is due either tothe no-slip boundary condition at the wall or the presence of semi-infinite domains of fluctuating potential flow in the case of free shearflows or turbulent boundary layers. One can no longer use Fourier se-ries for the spectral expansion in inhomogeneous directions. Rather,to obtain rapid convergence independent of boundary constraints,one should employ global polynomials related to the eigenfunctionsof a singular Sturm-Liouville problem (Orszag, 1980). Chebyshev


and Legendre polynomials are popular choices for channel flow but,for example, in the case of pipe flow, other choices may be preferablebecause of special conditions of the problem at hand. Whatever thechoice may be, this change in basis functions complicates the impo-sition of, for example, the no-slip condition and the satisfaction ofthe continuity constraint. These two conditions become global con-straints on the expansion which are generally difficult or costly to im-pose simultaneously. By contrast, in the simulation of homogeneousflows using Fourier expansions in all three directions, the bound-ary conditions are built into the expansion and the divergence-freeconstraint is satisfied by a simple projection which is local in wave-number space.

In the following, we will describe a technique for overcoming thealgorithmic difficulties described above, at least for flows in simplegeometries. The technique consists of expanding the velocity fieldin terms of a set of divergence-free vector functions satisfying theappropriate boundary conditions. First we write the Navier-Stokesequations in rotational form:

Here o> — V x u '1S the vorticity. The boundary condition at awall is u|waii = 0. Other boundary conditions, such as periodicity orfreestream conditions at infinity, are imposed as appropriate.

The role of the pressure in incompressible flows is to enforce thecontinuity condition. This may be expressed formally be recallingthat an arbitrary vector field f may be uniquely decomposed into asum of a divergence-free field satisfying tangency at the boundaryand the gradient of a potential.

Let <p be the projection operator that accomplishes this decomposi-tion, that is.

applying the projection operator p to equation (18a) we obtain

88 A. Leonard

eliminating the dynamic pressure (Chorin and Marsden, 1979). Theabove equation is the starting point for the numerical scheme de-scribed below.

Vector expansion method:

We write u as the exoansion

where each if>n satisfies

and the homogeneous boundary condition on u. We need to derivea system of ODEs for the coefficients an(t) (n = 1, 2, . . . , N). Wedo so by substituting the expansion (22) into (18a) and taking theinner product of the result with a set of weight vectors £m (m = 1,2, . . . , N) satisfying

and

If the £m form a complete set and N -> oo, this operation is equiva-lent to applying the projection operator because

for an arbitrary scalar field (f>. The result is the following system ofODEs:

where

and gm is quadratic in the an s,

Thus, each evaluation of the an's requires the solution of a linearsystem and the computation of the nonlinear term. The choice of the


vector functions -0n and £m is crucial to the success of the method.Mathematically, of course, the sets {ifin} and {Cm} must be com-plete in appropriate spaces of functions satisfying (23) plus bound-ary conditions and (24) and (25), respectively. From a computationstandpoint we would like (i) rapid convergence of our expansions ofu; (ii) minimum (and ordered) coupling of the modes through thetime-dependent and viscous operators, (for example, banded struc-ture for A and B); (iii) efficient construction of the matrices A andB; and (iv) efficient computation of the nonlinear term g.

In practice, the index n ranges from one to three representingeach spatial direction. The matrices A and B are diagonal in thehomogeneous directions where Fourier series may be used but non-diagonal in the direction normal to the wall. For example, in theapplication to pipe flow (Leonard and Wray, 1983), A and B arebanded with the same number of bands. Thus, implicit treatment ofthe viscous term is possible at no extra cost.

Equation (27) is a complete statement of the dynamics. No extraequations are required to enforce boundary conditions or continuity,and no fractional time-steps are needed. In addition, the numberof equivalent grid points in the computation is N/2. Thus, onlytwo unknowns per mesh point are required because of the constraint(23) on expansion vectors, allowing considerable savings in memoryrequirement.

As a relatively simple example, consider two-dimensional channelflow with the velocity field assumed to be periodic in the streamwisedirection x with period L. The velocity u(x, y,t) is expanded as fol-lows:

where k = 0, ±2?r/L, ±4?r/L,... and y is the coordinate perpendicularto the channel walls located at y = ±1. The condition that eachexpansion function be divergence free requires that

Boundary conditions and completeness of the expansion with rapidconvergence are satisfied if we choose for k ^ 0,

where In(y) 1S the Lhebychev polynominal.

90 A. Leonard

The divergence condition (30) demands that

If k = 0 we choose

Table II shows the exponential convergence of the method when ap-plied to the eigenvalue problem for the Orr-Sommerfeld equation.The Table gives the real part of the time eigenvalue A using theexpansion functions for even v(y) : (1 - y2)2T2n-2(y)?n = 1,2, ...,N.

Table II. Channel Flow Eigenvalues

Re= 104,kx = l,kz = 0

N Real (A)15 .00372 . . . .20 .0037398 . . . .25 .003739669 . . .30 .003739670616.35 .003739670622740 .003739670622345 .003739670621650 .0037396706222

In treating vorticity-containing layers of finite thickness, boundedby a semi-infinite domain of unsteady potential flow, it is desirableto include, in the expansion, terms that represent the potential flowexplicitly. This is so because potential flow contributions, in general,decay much more slowly, in the direction normal to the layer, thanthe vortical contributions (Spalart et ai, 1991). Thus, for boundarylayer flows, for each (kx, kz) one adds an extra vector to the expansion


given by

such that

as y —> oo. The vortical expansion functions behave as

for large y, where y0 is on the order of the boundary layer thickness.Two applications that incorporate the above technique for finite-

thickness vorticity layers are Spalart's (1989) simulations of turbu-lent boundary layers and Moser and Rogers' (1991) study of planemixing layers. Other efforts that have used the general strategyof divergence-free vector expansions include studies of transitionalpipe flow (Leonard and Wray, 1982) (Leonard and Reynolds, 1988),curved channel flow (Moser et al, 1983), vortex rings (Stanaway etal, 1988) and spherical couette flow (Dumas and Leonard, 1994).

5 FAST, VISCOUS VORTEX METHODS

Traditionally, vortex methods have been used to model unsteady,high Reynolds number incompressible flow by representing the fluc-tuating vorticity field with a few tens to a few thousand Langrangianelements of vorticity. Now, with the advent of fast vortex algorithms,bringing the operating count per timestep down to 0(N) from 0(N2)for N computational elements, and recent developments for the accu-rate treatment of viscous effects, one can use vortex methods for high-resolution simulations of the Navier-Stokes equations. Their classicaladvantages still hold — (1) computational elements are needed onlywhere the vorticity is nonzero, (2) the flow domain is grid-free, (3)rigorous treatment of the boundary conditions at infinity is a nat-ural byproduct, and (4) physical insights gained by dealing directlywith the vorticity field - so that vortex methods have become aninteresting alternative to finite difference and spectral methods forincompressible turbulence simulation.

To describe vortex methods, we start with the three-dimensionalvorticity equation for the vorticity field w in a constant density, in-compressible flow,

92 A. Leonard

Combining the incompressibility condition for the velocity field u, Vu = 0, and the definition of vorticity, y x u — u>, we find that

We will consider vortex methods in the context of bluff-body flowsand so we are interested in solutions to (37) and (38) correspondingto no-slip at the surface of a rigid body moving with velocity Ub,

and free-stream conditions at infinity

To simplify the description of the method, we restrict the discussionto non-rotating bodies. The solution to (38) satisfying (39) and (40)is given in terms of the infinite-medium Green's function

In our numerical approach described below, it is important to recog-nize that the no-slip boundary condition is maintained by a continualflux of vorticity from the body surface into the fluid. Mathematically,this flux is such that co is nonzero only in the fluid (i.e., external tothe body) and that (41) gives the result that

where t is any tangent vector at the body surface. That (41) and(42) imply (39) may be shown as follows. Let -0, the stream functionfor the imaginary fluid within the body, be given by

Thus, within the body B,

and


because all vorticity is external to B. Now we use Green's identity

with u = v = ib[. Here n is in the outward normal direction. From(42) and (44),

at every point on the surface <9B for any tangent vector t. Thus,

~y~ — 0, so that (46) reduces to

Hence, -0' = const and

In the present method, the vorticity field is represented by a densecollection of N moving, vector-valued computational elements or par-ticles.

where the particle vector amplitudes, 0:̂ , have units of circulationtimes length and each particle has a radially symmetric spatial dis-tribution defined by

where a is the effective core radius of the particle, and £ has unitvolume integral. See Leonard (1985) and Winckelmans and Leonard(1993) for further discussion including choices for the function £. Forexample, the distribution for the gaussian particle is given by

To satisfy the inviscid components of motion of the fluid, each par-ticle moves with the local velocity

94 A. Leonard

and its vector a^ is stretched and rotated according to

The velocity field u is obtained by substituting (50) into (41) toobtain

The function q is given by

For a variety of simple choices of the distribution function £ (e.g.,gaussian) the truncation error is O(cr2) if the typical interparticledistance, h, satisfies h < <r. Higher order schemes are possible. SeeLeonard (1980) and Beale and Majda (1982) for more details.

Although equation (54) for the inviscid evolution of the x^ seemsmost natural, other possibilities, based on the fact that

also yield convergent, accurate schemes as discussed by Winckelmansand Leonard (1993). For viscous diffusion we need to update thevorticity field following the equation

with the boundary condition implied by (39). The random walkmethod, a discussed by Chorin (1973), has been applied widely fora number of years but has the disadvantage of being slowly conver-gent. More recently, the technique of particle strength exchange (seeDegond and Mas-Gallic (1989)) has been proposed. This method hasgood convergence properties but requires an occasional remeshing ofthe particles as discussed below. In this algorithm we approximatey2 as an integral operator and discretize the integration over theparticles. Thus, we use


where, for example, with gaussian particle distributions,

Consider first the application to an infinite domain. The approxima-tion (59) is good to second order in cr, as we can see by taking theFourier transform of both sides,

Using (59) and the particle representation, we find that (58) can bewritten

Note now that by approximating the integral on the right hand sideof (62) over the particles and using (60) we obtain the evolutionequation for the problem (58) in an infinite domain,

where v is the particle volume.The above algorithm conserves total vorticity exactly, that is,

E-^dctj/dt = 0. Furthermore, it is found that if (63) is integrated intime with Euler timestepping and

the results are superaccurate (Pepin 1990, Leonard and Koumout-sakos 1993). This phenomenon may be explained by noting that theexact solution to (58) in an infinite domain at time t + At is givenby

For wall-bounded flows we enforce the no-slip condition in the form(42). In addition, however, for a substep of the diffusion step we use(63), that is we do not enforce du/dn — 0 at the wall during thissubstep (Koumoutsakos et a/., 1994). This leads to an additional

96 A. Leonard

spatially varying flux of vorticity from the surface. This effect alongwith the pressure gradient produces a wall slip (Usijp) at time t + At.We compute this slip as an average over computational panels onthe wall directly from the Green's function integral for the velocitypotential, with a form analogous to (41), using the fast algorithm.Thus the total flux to be emitted into the flow for the other substepof the diffusion process is given by

This flux is then distributed to the particles by discretizing theGreens' integral for the inhomogeneous Neumann problem for thediffusion equation. See Koumoutsakos et al. (1994) and Winckel-mans et al. (1995).

Rerneshing:

In order for the numerical simulation to be accurate, the particlesmust, to a certain degree, be uniformly distributed. This is requiredfor accuracy in the convection step as well as the diffusion step.On the other hand the local strain rate following a particle maygenerate a substantial contraction or crowding of particles in one ortwo directions accompanied by an expansion in the other directions,similar to the situation at a hyperbolic stagnation point in steadyflow. When remeshing is deemed necessary, we overlay a regulargrid (the new particles) over the old particles, keeping the averageparticle density constant, and interpolate the old vorticity onto thenew particle locations. We use a 27-point scheme to distribute theold vorticity to the new mesh. Specifically, away from a wall, theith old vortex with circulation I\ and location (xi ,yi ,Zj) contributesAFj(i circulation to new mesh point (xj ,y j ,Zj) , according to

where the interpolation kernel A is given by

If the old particle is less then a distance h from a wall the interpo-lation is modified to maintain the same conservation properties. See


Koumoutsakos (1993) and Hockney and Eastwood (1981) for furtherdiscussion. This scheme conserves the circulation, linear and angularmomentum.

Fast Vortex Methods:

The straightforward method of computing the right-hand sides of(53) and (54), using (55) for every particle, requires O(N2) operationsfor N vortex elements. This precludes high resolution studies of bluffbody flows with more than say 50,000 elements.

Recently, fast methods (see e.g., Barnes and Hut (1986), Green-gard and Rohkin (1987), and Salmon et al. (1994)) have been devel-oped that have operation counts of 0(N log N) or 0(N) dependingon the details of the algorithm. The basic idea of these methods is todecompose the element population spatially into clusters of particlesand build a hierarchy of clusters (tree) - smaller neighboring clus-ters combine to form a cluster of the next size up in the hierarchyand so on. Figure 3 shows an example of particle clustering in twodimensions.

The contribution of a cluster of particles to the velocity of agiven vortex can then be computed to desired accuracy if the particleis sufficiently far from the cluster in proportion to the size of thecluster and a sufficiently large number of terms in the expansion istaken. This is the essence of the particle/box (PB) method, requiring0(N log N) operations. One then tries to minimize the work requiredby maximizing the size of the cluster used, while keeping the numberof terms in the expansion within a reasonable limit, and maintaininga certain degree of accuracy.

The box-box (BB) scheme goes one step further as it accounts forbox-box interactions as well. These interactions are in the form ofshifting the expansions of a certain cluster with the desired accuracyto the center of another cluster. Then those expansions are usedto determine the velocities of the particles in the second cluster.This has as an effect the minimization of the tree traversals for theindividual particles, requiring only 0(N) operations.

Applications:

To illustrate the use of the viscous vortex method in two dimen-sion, we show in Figure 4 the vorticity distribution in acceleratingflow past a flat plate at 90° angle-of-attack. (P. Koumoutsakos, pri-vate communication 1994). For similar applications to cylinder flow,

Figure 3. Example of particle clustering for an elliptical spiral dis-tribution of 1000 particles

see Koumoutsakos and Leonard (1995). A three-dimensional exam-ple is shown in Figure 5. Shown is the deformation of an initiallyspherical vortex sheet corresponding to potential flow past a sphere.

Spectral Element Method:

A powerful alternative approach for complex geometries is thespectral element method (Patera, 1984) (Karniadakis, 1989) in whichlocal spectral (polynomial) expansions are used over quadrilateralsubdomains in two dimensions or hexahedral subdomains in threedimensions. See Chu and Karniadakis (1992) for the application toturbulent flow over streamwise grooves or riblets. Recently, Hender-son and Karniadakis (1994) extended the method to a nonconforming

98 A. Leonard


Figure 4. Vorticity contours for accelerating freestrearn flow past aflat plate at 90° incidence. Viscous vortex method. Reynolds numberbased on acceleration, a, = VaL3/z/ = 1,296. (a) Nondimensionaltime = t<i/a/L — 2.5, number of vortex particles ~ 210,000; (b)time = 3.0, number of vortex particles « 290,000 (P. Koumoutsakos,private communication 1994).

100 A. Leonard

Figure 5. Evolution of a spherical sheet of vorticity correspondingto potential flow past a sphere. Three-dimensional, inviscid vortexparticle method with 81920 particles (Salmon et a/., 1994).

grid structure so that local mesh refinement is possible. In addition,Sherwin and Karniadakis (1994) have extended the technique to tri-angular (2D) and tetrahedral (3D) subdomains.

6 SIMULATION OF COMPRESSIBLE TURBULENCE

The previous sections have considered the direct simulation ofincompressible turbulence for constant density flows. To set themesh spacing in this case for a fixed-grid computation, one need


only take into account the generation of small-scale vorticity struc-tures brought about by the straining of the vorticity field. Meshrequirements are dictated by the smallest of these scales that sur-vive viscous diffusion, and the timestep scales with the mesh widthdivided by a convection velocity. In addition, the pressure is justa Lagrange multiplier whose purpose is to keep the velocity fielddivergence-free (recall Eq.(21) above for a succinct, pressure-free de-scription of incompressible fluid mechanics). In compressible turbu-lence, the pressure is a thermodynamic variable and has dynamicalsignificance. It is, in principle, computed from the equation of state,knowing the density and the internal energy. These variables, inturn, are determined from the mass conservation equation,

and the total energy equation, respectively. We can convert theenergy equation to an evolution equation for the pressure. For aperfect gas, the resulting pressure equation is

These two equations now join the momentum equation

and the equation of state

to give a complete system of equations for the dynamics. In theabove, the density, pressure p, and temperature T have been madenondimensional by their respective reference values pa,pa,and Ta,while distance, time and velocity are nondimensionalized using areference velocity, U, and length, L. The absolute viscosity fj, and thethermal conductivity K are also normalized by reference values fia

and Ka respectively. In addition,

is the viscous stress tensor (with the bulk viscosity assumed zero),L is the identity tensor,

102 A. Leonard

is the viscous dissipation function, 7 is the ratio of specific heats,M = U/V^RTa is the reference Mach number (R is the universalgas constant), Re = paUL/Ma is the Reynolds number, and Pr =MaCp/Ka is the Prandtl number (Cp is the specific heat at constantpressure).

With the additional thermodynamic degrees of freedom, thereare other modes besides vorticity modes that can become active. Inparticular, Kovasznay (1953) identified acoustic modes and entropymodes that evolve independent of the vorticity modes and indepen-dent of each other in linearized flow. Acoustic modes can, of course,be in the form of sound waves and are present, for example, at highamplitudes in the case of an eddy shocklet discussed below. Entropymodes can arise even in the incompressible limit, in which the ther-modynamic pressure is constant and no acoustic modes are present.In this limit, density, and hence entropy, fluctuations travel with thelocal fluid velocity and interact with the vorticity field through thenonlinear baroclinic source term for vorticity,

which arises when one takes the curl of the momentum equation toobtain the vorticity transport equation.

The initial conditions for, say, isotropic, homogeneous turbulenceare, of course, more varied for compressible flows. The initial ve-locity field may be decomposed (Helmholtz decomposition) into asolenoidal or "incompressible" component, u1 and a dilatational or"compressible" component, uc, i.e.,

where

Similarly the pressure may be expressed as

where p1 is the incompressible pressure given by


Thus, for initial conditions and in addition to considerations of initialspectra, one can have a varying initial fraction of compressible kineticenergy

and an arbitrary initial compressible potential energy, | < pc2 >/72, where <> denotes volume average.

Also, initial density fluctuations may be imposed independently.However, linearized analysis of low-Mach number flows (Erlebacheret al. 1990, Sarkar et al. 1991) indicates that the compressiblepotential energy and the compressible kinetic energy come into equi-librium rapidly, i.e. on an acoustic timescale that is 0(M) times theturbulence timescale.

Direct simulations of isotropic turbulence (Sarkar et al. 1991, Leeet al. 1991) have substantiated this equilibrium process as well asshowing the importance of the turbulent Mach number, defined by

(where c = speed of sound = \/7pa//>a) in determining, for exam-ple, the level of compressible dissipation. When the turbulence Machnumber is sufficiently large and when the Reynolds number is rea-sonably large, Lee et al. (1991) have observed the formation of eddyshocklets, local regions in the flow that have all the proper jumpconditions of a shock wave. Simulations of homogeneous shear tur-bulence (Sarkar 1994) have shown the importance of gradient Machnumber Mg = S l/c, (where S is the mean dimensional shear rateand £ is a representative integral length scale of the turbulence) aswell as turbulent Mach number. Studies of inhomogeneous turbu-lence include the efforts of Sandham and Reynolds(1991) to studyvariations in the structure of three-dimensional large-scale eddies incompressible plane mixing layers as the convective Mach number ischanged. In such flows, the convective Mach number is given by

where Ui and U2 are the freestream speeds on either side of the layer,and GI and C2 are the corresponding speeds of sound. It is known fromexperiment that compressible layers have growth rates significantly

104 A. Leonard

lower than their incompressible counterparts, apparently dependingon the convective Mach number.

7 REFERENCES

Barnes, J. E., and Hut, P., 1986. "A hierarchical 0(N log N) forcecalculations algorithm," Nature, 324, pp. 446-449.

Batchelor, G. K., 1967."The theory of homogeneous turbulence,"Cambridge University Press, Cambridge.

Beale, J. T., and Majda, A., 1982. "Vortex methods II: High orderaccuracy in two and three dimensions," Math. Comp. 39, pp.29-52.

Chorin, A. J., and Marsden, J. E., 1979. "A mathematical intro-duction to fluid mechanics," Springer-Verlag, New York.

Chorin, A., 1973. "Numerical study of slightly viscous flow," J.Fluid Mech., 57, pp. 380-392.

Chu, D. C., and Karniadakis, G. E., 1992. "A direction numeri-cal simulation of laminar and turbulent flow over streamwisealigned riblets," J. Fluid Mech., 25, pp. 1-42.

Coles, D., 1981. "Prospects for useful research on coherent structurein turbulent shear flow," Proc. Indian Acad. Sci. (Engg. Sci.),4, pp. 111-127.

Deardorff, J. W., 1970. "A numerical study of three-dimension-al turbulent channel flow at large Reynolds numbers," /. FluidMech., 41, pp. 453-480.

Degond, P., and Mas-Gallic, S., 1989. "The weighted particle methodfor convection-diffusion equations, Part I: the case of isotropicviscosity, Part II: the anisotropic case," Math. Comp. 53, pp.485-526.

Dumas, G., and Leonard, A., 1994. "A divergence-free spectralexpansions methods for three-dimensional flows in spherical-gap geometries," J. Comput. Phys., Ill, pp. 205-219.


Emmons, H. W., 1949. "The numerical solution of the turbulenceproblem," in Proceedings of Symposia in Applied Mathematics,1, McGraw-Hill, New York, pp. 67-71.

Erlebacher, G., Hussaini, M. Y., Kreiss, H. 0., and Sarkar, S., 1990."The analysis and simulation of compressible turbulence," The-oret. Comput. Fluid Dyn., 2, pp. 73-95.

Greengard, L., and Rohklin, V., 1987. "A fast algorithm for particlesimulations," J. Comput. Phys., 73, pp. 325-348.

Henderson, R. D., and Karniadakis, G. E., 1994. "Unstructuredspectral element methods for turbulent flows," J. of Comput.Phys., submitted.

Hinze, J. 0., 1975. "Turbulence", 2nd ed., McGraw-Hill Inc., NewYork, p. 722.

Hockney, R. W., arid Eastwood, J. W., 1981. "Computer simula-tions using particles,", McGraw- Hill New York.

Huang, M.-J., and Leonard, A., 1994. "Power-law decay of homo-geneous turbulence at low Reynolds numbers," Phys. Fluids,6, pp. 3765-3775.

Hussaini, M. Y., Speziale, C. G., and Zang, T. A., 1990. "Thepotential and limitations of direct and large eddy simulations,"Whither Turbulence? Turbulence at the Crossroads, Springer-Verlag, Berlin, pp. 354-368.

Jimenez, J., Wray, A. A., Saffman, P. G., and Roggalo, R. S., 1993."The structure of intense vorticity in isotropic turbulence", J.Fluid Mech., 255, pp. 65-90.

Karniadakis, G. E., 1989. "Spectral element simulations of laminarand turbulent flows in complex geometries," App. Num. Math.,6, p. 85.

Kim, J., Moin, P., and Moser R., 1987. "Turbulence statistics infully developed channel flow at low Reynolds number," J. FluidMech., 177, pp. 133-166.

Koumoutsakos, P., and Leonard, A., 1995. "High resolution simu-lations of the flow around an impulsively started cylinder usingvortex methods,", J. Fluid Mech. (to appear).

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Koumoutsakos, P., Leonard, A., and Pepin, F., 1994. "Bound-ary conditions for viscous vortex methods," J. Comput. Phys.,113, pp. 52-61.

Koumoutsakos, P., 1993. "Large scale direct numerical simulationsusing vortex methods." Ph.D. thesis, Caltech.

Kovasznay, L. S. G., 1953. "Turbulence in supersonic flow,", Jour-nal of Aeronautical Sciences, 20, pp. 657-682.

Laufer, J., 1954. "The structure of turbulence in fully developedpipe flow," NACA 1174.

Lee, S., Lele, S. K., and Moin, P., 1991. "Eddy shocklets in decayingcompressible turbulence," Phys. Fluids A, 3, pp. 657-664.

Leonard, A., and Chua, K., 1989. "Three-dimensional interactionsof vortex tubes," Physica D, 37, pp. 490-496.

Leonard, A., 1985. "Computing three-dimensional incompressibleflows with vortex elements," Ann, Rev. Fluid Mech., 17, pp.523-529.

Leonard, A., and Koumoutsakos, P., 1993. "High resolution vortexsimulation of bluff body flows," J. Wind Eng. and Indust.Aero., 4G&47, pp. 315-325.

Leonard, A., and Reynolds, W. C., 1988. "Turbulence researchby numerical simulation," Perspectives of Fluids Mechanics,Lecture Notes in Physics, Vol. 320, Springer-Verlag.

Leonard, A., and Wray, A., 1982. " A new numerical method for thesimulation of three- dimensional flow in a pipe," Proceedingsof the 8th International Conference on Numerical Methods inFluids Dynamics, June 28-July 2, 1982, Aachen, W. Germany,Springer-Verlag, New York, pp. 335-342.

Leonard, A., 1980. "Vortex methods for flow simulation," J. Com-put. Phys., 37, pp. 289-335.

Moser, R. D., and Rogers, M. M., 1990. "Mixed transition and thecascade to small scales in a plane mixing layer," Phys. FluidsA, 5, pp. 1128-1134.


Moser, R. D., Moin, P., and Leonard, A., 1983. "A spectral numer-ical method for the Navier-Stokes equations with applicationsto Taylor-Couette flow," J. Comput. Phys., 52, p. 524.

Orszag, S. A., and Patterson, G. S., Jr., 1972. "Numerical simula-tion of three-dimensional homogeneous isotropic turbulence,"Phys. Rev. Lett., 28, pp. 76-79.

Orszag, S. A., 1980. "Spectral methods for problems in complexgeometries," J. Comput. Phys., 37, p. 70.

Patera, A. T., 1984. "A spectral element method for fluid dynamics;laminar flow in a channel expansion," J. Comput. Phys., 54,pp. 468-488.

Pepin, F., 1990. "Simulation of the flow past an impulsively startedcylinder using a discrete vortex method," Ph.D. thesis, Caltech.

Reynolds, W. C., 1990. "The potential and limitations of directand large eddy simulations," Whither Turbulence? Turbulenceat the Crossroads, Springer-Verlag, Berlin, pp. 313-343.

Rogallo, R. S., 1981. "Numerical experiments in homogeneous tur-bulence," NASA TM-81315.

Rogallo, R. S., and Moin, P., 1984. "Numerical simulation of tur-bulent flows," Ann. Rev. Fluid Mech., 16, pp. 99-137.

Saddoughi, S. G., and Veeravalli, S. V., 1994. "Local isotropy inturbulent boundary-layers at high Reynolds-number," J. FluidMech., 268, pp. 333-372.

Salmon, J. K., Warren, M. S., and Winckelmans, G. S., 1994. "Fastparallel tree codes for gravitational and fluid dynamical N-bodyproblems," Int. J. Supercomputer Applications, 8, pp. 129-142.

Sandham, N. D., and Reynolds, W. C., 1991. "Three-dimensionalsimulations of large eddies in the compressible mixing layer,"J. Fluid Mech. 224, pp. 133-158.

Sarkar, S., 1994. "The stabilizing effect of compressibility in tur-bulent shear flow," NASA Contractor Report 194932, ICASEReport No. 94-46, pp. 1-36.

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Sarkar, S., Erlebacher, G., Hussaini, M. Y., and Kreiss, H. 0., 1989."The analysis and modelling of diltational terms in compress-ible turbulence," J. Fluid Mech., 227, pp. 473-493.

She, Z.-S., Chen, S.-Y., Doolen, G. D., Kraichnan, R. H., andOrszag, S. A., 1993. "Reynolds-number dependence of iso-tropic Navier-Stokes turbulence," Phys. Rev. Letters, 70, pp.3251-3254.

Sherwin, S. J., and Karniadakis, G. E., 1994. "A triangular spec-tral element method: algorithms and flow simulations," Pro-ceedings of the 14th International Conference on NumericalMethods in Fluids Dynamics, Bangalore, 11-15 July.

Spalart, P. R., 1987. "Direct simulation of a turbulent boundarylayer up to R<? = 1410," J. Fluid Mech., 187, pp. 61-98.

Stanaway, S. K., CantweU, B. J., and Spalart, P. R., 1988. AIAAPaper 88-0318.

von Neumann, J., 1963. "Recent theories of turbulence," 1949 re-port to the Office of Naval Research, reprinted in John vonNeumann, Collected Works, 6, A. H. Taub, ed., MacmillanCo., New York, pp. 437-472.

Winckelmans, G. S., Salmon, J. K., Warren, M. S., and Leonard,A., 1995. "The fast solution of three-dimensional fluid dynam-ical N-body problems using parallel tree codes: vortex elementmethod and boundary element method," Seventh SIAM Conf.on Parallel Processing for Scientific Comp., Feb. 1995, SanFrancisco.

Winckelmans, G. S., and Leonard, A., 1993. "Contributions to vor-tex particle methods for the computation of three-dimensionalincompressible unsteady flows," J. Comput. Phys., 109, pp.247-273.

Chapter 3

LARGE EDDY SIMULATION

Joel H. Ferziger

1 INTRODUCTION

Over a decade ago, the author (Ferziger, 1983) wrote a review ofthe then state-of-the-art in direct numerical simulation (DNS) andlarge eddy simulation (LES). Shortly thereafter, a second review waswritten by Rogallo and Moin (1984). In those relatively early daysof turbulent flow simulation, it was possible to write comprehensivereviews of what had been accomplished. Since then, the widespreadavailability of supercomputers has led to an explosion in this field so,although the subject is undoubtedly overdue for another review, it isnot clear that the task can be accomplished in anything less than amonograph. The author therefore apologizes in advance for omissions(there must be many) and for any bias toward the accomplishmentsof people on the west coast of North America.

In the earlier review, the author listed six approaches to theprediction of turbulent flow behavior. The list included: corre-lations, integral methods, single-point Reynolds-averaged closures,two-point closures, large eddy simulation and direct numerical sim-ulation. Even then the distinction between these methods was notalways clear; if anything, it is less clear today.

It was possible in the earlier review to give a relatively completeoverview of what had been accomplished with simulation methods.Since then, simulation techniques have been applied to an ever ex-panding range of flows so a thorough review of simulation results is

109

110 J. H. Ferziger

no longer possible in the space available here. Simulation techniqueshave become well established as a means of studying turbulent flowsand the results of simulations are best presented in combination withexperimental data for the same flow. There is also a danger that thesuccess of simulation methods will lead to attempts to apply themtoo soon to flows which the models and techniques are not ready tohandle. To some extent, this is already happening.

Direct numerical simulation (DNS) is a method in which all ofthe scales of motion of a turbulent flow are computed. A DNS mustinclude everything from the large energy-containing or integral scalesto the dissipative scales; the latter is usually taken to be the viscousor Kolmogoroff scales. For any reasonable Reynolds number, thisrequires a large number of grid points or modes and is very costly.Despite the cost, the ability of the DNS to yield all of the flow vari-ables at a large number of spatial locations for many instants of timehas made it a valuable tool for investigating the physics of turbu-lence. Indeed, for a number of simple flows, it is now the methodof choice. This subject is covered in greater detail in Chapter 2 byLeonard and will be dealt with only briefly here.

By large eddy simulation (LES), we mean an approach in whichthe largest scales of motion are represented explicitly while the smallscales are treated by some approximate parameterization or model.Large eddy simulations are three dimensional and time dependent,and thus, expensive although, but as we shall see, they may be muchless costly than a DNS of the same flow.

As a result, LES has come to occupy a kind of middle ground.Whenever DNS is feasible for a given flow, it should be the methodof choice, especially when a study of the detailed physics of the flowis the goal. However, cost often makes DNS infeasible. Use of LESis a good choice for investigating flows that are too complex to becomputed economically by DNS; in practice, with today's comput-ers, this means any flow which is inhomogeneous in more than onedirection. LES is now becoming powerful enough to be worthy ofconsideration as a method to be employed selectively by the work-ing (as opposed to the research) engineer; we shall comment on thisbelow.

In this article, we describe the methods employed in LES; in thisarea, there is a great deal of overlap with DNS. We shall concentrateon the differences, principally the models that need to be employedto represent the small scales and numerical methods that can be

Large Eddy Simulation 111

used in more complex geometries. First, brief overviews of Reynoldsaveraged methods and direct numerical simulation will be given toset the stage for the discussion of LES. We shall then explore theboundaries of what is feasible with LES today. We begin with someremarks on turbulence and prediction methods for practical flows.

2 TURBULENCE AND ITS PREDICTION

2.1 The Nature of Turbulence

It is difficult even for expert researchers to agree on a definitionof turbulence. Required elements that are generally agreed uponinclude three dimensionality, unsteadiness, strong vorticity, and abroad-banded spectrum. Beyond that, the issue becomes difficult.Most would agree that turbulent flows are highly random and/ornoisy; the term chaotic could be used but it has been given a strictermeaning in recent years.

However, there is more to turbulence than randomness. It is gen-erally agreed that so-called coherent structures exist in nearly all tur-bulent flows; we say 'so-called' because, despite considerable discus-sion, an agreed definition of this term does not yet exist. Moreover,although the coherent structures account for only a small fraction ofthe turbulent energy, they are apparently responsible for more thantheir fair share of the transport of properties such as species, mass,momentum and energy. The coherent structures of a particular floware not identical and do not appear regularly in either time or space.The remainder of the turbulent energy is apparently due to truly ran-dom motion (which may be the remains of old coherent structures)and probably causes the irregularity of the coherent structures.

This picture helps explain why turbulence is such a difficult prob-lem. If it were random, statistical methods would probably havesolved the problem by now. If it were purely deterministic, com-puter simulation might have solved the problem. In fact, turbulenceis sufficiently incoherent that the signal-to-noise ratio of the coherentstructures is very low; at the same time, the lack of a clear definitionof a structure makes eduction of a pattern from noisy data nearlyimpossible.

When we add to this picture the probability that the coherentstructures are different in each flow, we see that the likelihood of

112 J. H. Ferziger

finding a simple method for predicting all flows is exceedingly small.The search for a single universal method capable of predicting allturbulent flows has gone on for along time and, while it has producedmany useful results, the goal remains a long way off.

2.2 RANS Models

In Reynolds averaged approaches to turbulence, all of the un-steadiness is averaged out; this means that all unsteadiness is re-garded as part of the turbulence. The non-linear terms in the Navier-Stokes equations give rise to the Reynolds stress term in the Reynolds-averaged Navier-Stokes (RANS) equations. This term must be mod-eled if the equations are to be closed. The complexity of turbulencemakes it unlikely that any single model will be able to represent allturbulent flows. Thus, turbulence models should be regarded as en-gineering correlations or approximations rather than scientific laws;this interpretation allows one to 'tune' models, hopefully in sensibleways, for particular features that may arise in individual turbulentflows. Experience with RANS-based turbulence models has yieldedboth successes and failures. It is the lack of consistency that has ledto interest in new approaches, such as large eddy simulation.

Although the subject of this work is large eddy simulation it ishelpful to note a few things about RANS models. The mean velocityfield may be defined by ensemble, time, or spatial averaging; in anycase, the RANS equations are:

where [/,- =< ul >, the brackets denoting one of the averages listedabove, and T;J is the Reynolds stress tensor,

(actually, a stress divided by the density) and needs to be modeled.In this paper, we shall assume that the flow is incompressible exceptwhere noted otherwise (see Section 4F). We also use the conventionthat any repeated index is summed over.

In laminar flows, energy dissipation and transport of mass, mo-mentum, and energy normal to the streamlines are all mediated bythe viscosity, so it is natural to assume that the effect of turbulence


can be represented as an increased viscosity. This leads to the eddyviscosity model:

where VT is the eddy viscosity.In the simplest description, turbulence can be characterized by

its kinetic energy, k, or equivalently, a velocity q = \/2&, and a lengthscale, L. The eddy viscosity, which carries dimensions Iength2time~1,must be:

In mixing length models, k is determined from the mean velocityfield using q = LdU/dy and L, which should be the integral scaleof the turbulence, is prescribed in terms of a physical length scaleor a shear layer thickness. Accurate prescription of L is possiblefor simple flows but not for separated or highly three dimensionalboundary layers. The simplicity of mixing length models allows themto be easily modified to account for pressure gradients, curvature,transpiration, etc.

Two-equation models retain the Boussinesq eddy viscosity con-cept but use a partial differential equation for the turbulent kineticenergy k to determine the velocity scale. To obtain the dissipationand the length scale L, we note that in so-called equilibrium turbu-lent flows i.e. ones in which the rates of production and destructionof turbulence are in near-balance, the following relation among" thedissipation, e, and k and L:

may be used. Eq. (2.5) allows one to use an equation for the dissi-pation as a means of obtaining both e and L. No constant is used inEq (2.5) because this constant combines with others in the completemodel.

An exact equation for the dissipation can be derived from theNavier-Stokes equations and has a form similar to the energy equa-tion. The modeling applied to the dissipation equation is so severethat it is probably best to regard the entire equation as a model initself. Difficulties associated with the dissipation equation (or any

114 J. If. Ferziger

other equation used to determine the length scale) are the most dif-ficult ones in two-equation modeling.

Some of the significant deficiencies of models based on Eq. (2.3)are direct consequences of the eddy viscosity relationship itself. Inthree dimensional flows, the eddy viscosity may no longer be a scalar;measurements and numerical simulations show that it becomes highlyanisotropic, i.e. it is actually a tensor quantity.

The effect of an eddy viscosity can be interpreted in anotherway. Enough viscosity is added to the equations to assure that thecomputed flow is stable i.e., the solution of the RANS equations isan effective laminar flow with a velocity field that is the mean (in theReynolds sense) of the turbulent velocity field. In two dimensions, itis always possible to define a spatially dependent eddy viscosity thatproduces the correct mean flow. In general, it is not possible to findthis eddy viscosity without knowledge of the solution but it is usefulto know that it exists in principle. In three dimensional flows, theeddy viscosity may be a tensor of either second or fourth rank.

Anisotropic or tensor models have been proposed. If a modelis to be applicable to a wide range of flows, it should possess in-variance properties, i.e. it must give the same results independentof the coordinate system is used in the calculation. Many earlyanisotropic models were not properly invariant. Recently, invarianttensor and/or non-linear models have been proposed, for example,see Speziale (1987) and Horiuti (1990). These models take the form:

where Sij is

Although these models contain more constants than the scalar eddyviscosity model, some of them are fixed by requiring invariance. Adetailed discussion of these types of closures can be found in Chapters5 and 6.

The most complex models in common use today are Reynoldsstress models which are based on dynamic equations for the Rey-nolds stress tensor itself. As these are complicated, and becausetheir application as subgrid scale models in large eddy simulation


has been limited, we shall not describe or discuss them here. A de-tailed discussion of these types of closures can be found in Chapter5 and 6.

With few exceptions, RANS models cannot be applied to flownear a surface without modification. Special near-wall versions of themodels, especially the k — c model, have been developed and workquite well for attached boundary layers. One of the most recent ofthese models was proposed by Rodi and Mansour (1992), who includereferences to other models.

2.3 Direct Numerical Simulation (DNS)

The most exact approach to turbulence simulation is to solve theNavier-Stokes equations without averaging or approximation. Theresult is a single realization of a flow and is equivalent to a short-duration laboratory experiment; this approach is called direct nu-merical simulation (DNS).

It is important to recognize that the considered domain must beat least as large as the largest turbulent eddy; from a practical pointof view this means that the linear dimension of the domain must beat least a few times the integral scale L. On the other hand, for asimulation to capture all of the dissipation, which occurs on the smallscales on which viscosity is active, the grid must be no larger thanthe viscously determined Kolmogoroff scale, 77. For homogeneousturbulence, the simplest type of turbulence, there is no reason touse anything other than a uniform grid. In this case, the number ofgrid points in each direction must be at least L/r); it is easily shown(Tennekes and Lumley, 1976) that this ratio is proportional to .Re3/4.Since this number of points must be employed in each of the threecoordinate directions, and the time step is related to the grid size,the cost of a simulation scales as Re3.

This means that direct numerical simulation can be carried outonly at relatively low Reynolds numbers. For homogeneous turbulentflows, the Reynolds number of interest must be based on the turbu-lent velocity and length scales. As these scales are typically an orderof magnitude or more smaller than the corresponding macroscopicscale, the ability to compute flows with turbulent Reynolds numbersof 100 actually allows DNS to reach the low end of the range of Rey-nolds numbers of engineering interest. For further details of DNS seeChapter 2.

Having described the methods that bracket it, we now turn to

116 J. H. Ferziger

the principal subject of this paper, large eddy simulation.

3 FILTERING

We now begin the description of large eddy simulation (LES).The idea is to simulate the larger scales of motion of the turbulencewhile approximating the smaller ones. One can think of it as ap-plying DNS to the large scales and RANS to the small scales; it isa compromise between the two approaches, a concept that we shallexplore in more detail later. The justification for such a treatmentis that the larger eddies contain most of the energy, do most of thetransporting of conserved properties, and vary most from flow toflow; the smaller eddies are believed to be more universal and lessimportant and should be easier to model. It is hoped that univer-sality is more readily achieved at this level than in RANS modelingbut this assertion remains to be proven.

As in the RANS case, it is essential to define the quantities to becomputed precisely. To do this it is essential to define a velocity fieldthat contains only the large scale components of the total field. Thisis best done by filtering (Leonard, 1974); the large or resolved scalefield is essentially a local average of the complete field. We shall useone dimensional notation for convenience; the generalization to threedimensions is straight-forward. The filtered velocity is defined by:

where G(x, x'), the filter kernel, is a localized function or a functionwith compact support i.e., one which is large only when x and x' arenot far apart. Filter kernels which have been applied in LES include:

• Gaussian. The Gaussian has the advantage of being smoothand infinitely differentiable in both physical and Fourier space. Infact, its Fourier transform is Gaussian in wavenumber space.

• Box. This is simply an average over a rectangular region. Itis a natural choice when finite difference or finite volume methodsare used. Two versions of this filter have been used and ought to bedistinguished:

In the moving box filter, the average is taken over a region ofspace surrounding any chosen point. According to this definition,


Ui(x) is a continuous function of x. This filter is similar in manyways to the Gaussian.

A filter which is an average over a grid volume of a finite differenceor finite volume mesh is tied more closely to the numerical method.According to this definition, u is a piecewise constant function of x(Schumann, 1973).

• Cutoff. This filter is defined in Fourier space and eliminates allof the Fourier coefficients belonging to wavenumbers above a partic-ular cutoff. It is natural to use this filter in conjunction with spectralmethods as it leaves more energy in the large scale field than the fil-ters defined above. However, it is difficult to apply to inhomogeneousflows.

When the Navier-Stokes equations are filtered one obtains a setof equations very similar in form to the RANS equations.

Of course, the definitions of the velocities appearing in Eqs (2.1)and (3.2) differ but the closure issues are very similar. Since

a modeling approximation for the difference between the two sidesof this inequality,

must be introduced. In the context of large eddy simulation, r,-j iscalled the subgrid scale (SGS) Reynolds stress. It plays a role in LESsimilar to the role played by the Reynolds stress in RANS modelsbut the physics that it models is different. The SGS energy is amuch smaller part of the total flow than the RANS turbulent energyso model accuracy may be less crucial in an LES than in RANScomputations.

Subgrid scale modeling is the most distinctive feature of largeeddy simulation and is the subject of the next section, the longestone in this work.

118 J. If. Ferziger

4 SUBGRID SCALE MODELING

4.1 Physics of the Subgrid Scale Terms

The models used to approximate the SGS Reynolds stress (3.4)are called subgrid scale (SGS) models. This nomenclature is derivedfrom the kind of LES in which one applies a finite volume approx-imation directly to the Navier-Stokes equations; the filter is thenclosely connected to the grid used to discretize the equations. Thistechnique was used in the earliest large eddy simulations and thenomenclature has stuck. More generally, there need not be a con-nection between the filter and the grid used in the solution methodso this nomenclature is more restrictive than necessary but it is toolate to change it.

One important difference between filtering and Reynolds aver-aging is that, in general, filtering a field a second time does notreproduce the original filtered field:

The exception is the cutoff filter for which equality does hold. Thedifference between the two sides of this inequality will be exploitedfor modeling purposes later.

For now, we note that the difference represented by Eq (4.1)means that the both the physics and modeling of the subgrid scaleReynolds stresses (SGSRS) may be more complicated than for theRANS Reynolds stresses. By using the kind of decomposition ofthe velocity field used in RANS modeling i.e. writing the completevelocity field as a combination of the filtered field and a subgrid scalefield, we can decompose the SGSRS into three sets of terms:

which can be ascribed physical significance. In particular, theseterms represent the following physics:

• The first term, which can be computed explicitly from the fil-tered velocity field, u, represents the interaction of two resolved scaleeddies to produce small scale turbulence. It has been called theLeonard term and, sometimes, the outscatter term.

• The second term represents the interaction between resolvedscale eddies and small scale eddies. This term, also called the cross


term, can transfer energy in either direction but, on the average,transfers energy from the large scales to the small ones.

• The third term represents the interaction between two smallscale eddies to produce a large scale eddy and is called the truesubgrid scale term; as it produces energy transfer from the small tothe large scales, it is also called the backscatter term; as noted above,the cross term may produce backscatter as well.

In the past, it was thought that, as each term represents a differ-ent physical phenomenon, it ought to be modeled separately. How-ever, modeling (either SGS or RANS) is far from exact and the un-certainty in the modeling defeats any attempt at precision. Con-sequently, in the recent past, it has become common to model theentire subgrid scale Reynolds stress as a single unit.

It should also be noted that the subgrid scale Reynolds stressis a local average of the small scale field. This means that modelsfor it should be based on the local velocity field or, perhaps, on thepast history of the local fluid. The latter can be accomplished byusing a model that solves partial differential equations to obtain theparameters needed to determine the SGSRS.

We next look at subgrid scale models in some detail.

4.2 Smagorinsky Model

By far the most commonly used subgrid scale model is the oneproposed by Smagorinsky (1963). It is an eddy viscosity model thatcan be thought of as an adaptation of the Boussinesq concept of Eq(2.3) to the subgrid scale. It is:

This model can also be derived in a number of other ways. Theseinclude heuristic methods, equating production and dissipation ofsubgrid scale turbulent kinetic energy, and via turbulence theoriessuch as the direct interaction approximation (DIA) (Leslie, 1973),the eddy damped quasi-normal Markovian (EDQNM) approximation(Lesieur, 1992), and renormalization group theory (RNG) (Yakhotand Orszag, 1986).

The form of the subgrid scale eddy viscosity can be derived bydimensional arguments. We shall present a heuristic argument but

120 J. H. Ferziger

it can also be derived from theories including Kolmogoroff-like argu-ments (Lilly, 1965) and the theories mentioned above. The followingargument contains some elements of these approaches.

At high Reynolds numbers, the dissipation in a turbulent flowtakes place at very small scales while energy is introduced at thelargest scales. Between these is a regime in which there is neithersignificant production nor dissipation of turbulent energy, the iner-tial subrange. In this range, only inviscid mechanisms are activeand energy is transferred from large to small scales. Since it is thenon-linear (advective) term in the Navier-Stokes equations that isresponsible for the energy transfer, the rate of transfer to the smallscales may be estimated as the magnitude of the contribution of thisterm to the kinetic energy equation, which is (!/2}d(uiUiUj}/dxj.As the large energetic scales supply the largest contribution to thisterm, the magnitude scales as:

where Q is a velocity scale for the energetic eddies and L is theintegral scale of the turbulence.

Let us further assume that the largest subgrid scales are far re-moved from the viscous scales. A repeat to the above argument thenshows that:

where q is a velocity typical of the subgrid scale field (most of whichresides in the largest subgrid scales) and A is the size of the largestsubgrid eddies, the length scale associated with the filter.

In a large eddy simulation, the large or resolved scales lose energyby transferring it to the subgrid scales. From the point of view of thelarge scale eddies, this appears to be dissipation i.e., it is energy lostnever to be recovered. A model of the eddy viscosity type representsthis energy transfer as effective viscous dissipation. Since the modelmost affects the smallest resolved scales (which are of size A), themagnitude of the effective dissipation may be estimated as:

Equating (4.5) and (4.6) shows that the eddy viscosity must takethe form:


which could have been derived via dimensional arguments. We nowfind q by using Eqs. (4.4) and (4.5) and substitute it into Eq. (4.7)to obtain:

Finally, estimating Q as:

and inserting a model parameter to produce equality, we have:

As noted above, this result can be derived in a number of other ways.The theories provide estimates of the constant as well as the form ofthe model.

The presence of the integral scale L in the formulation for theeddy viscosity makes the model difficult to use. Computing the in-tegral scale, especially in inhomogeneous flows, could require a greatdeal of effort. For this reason, the substitution:

is often used, leading to the usual form of the Smagorinsky model:

Other derivations lead directly to this form of the model.As noted, the theories also predict the value of Cs, which is more

appropriately called a parameter than a constant. Most of thesederivations are truly valid only for isotropic turbulence but they allagree that Cs ~ 0.2. LES of isotropic turbulence also shows thatthis value of the parameter is optimum; varying it by about tenor fifteen percent produces acceptable results. It should be noted,however, that the range of Reynolds numbers studied is relativelynarrow so this may not be a very severe test of the model. Indeedthe substitution (4.11) used to produce the standard version of theSmagorinsky model may mean that the parameter, Cs, is not a trueconstant, but rather a function of A/Z< which is, in turn, a function

122 J. H. Ferziger

of Reynolds number. We should not be surprised if we find that theparameter Cs needs to be a function of Reynolds number or othernon-dimensional parameters or is different in different flows.

The Smagorinsky model, although relatively successful, is notwithout problems. For example, it has been found that, to simulatechannel flow, several modifications are required. The first is thatthe value of the parameter Cs in the bulk of the flow has to bereduced from 0.2 to approximately 0.065 which amounts to reducingthe eddy viscosity by almost an order of magnitude. Secondly, inthe region close to the surface, the value has to be reduced evenfurther. A recipe that has been found to be successful is the vanDriest damping that has long been used to reduce the near-wall eddyviscosity in RANS models:

where y+ is the distance from the wall in viscous wall units (y+ =yur/v, where ur is the shear velocity (r//?)1 '2 and T is the shear stressat the wall) and A+ is a constant usually taken to be approximately25. Although this modification produces the desired results, it isdifficult to justify in the context of LES. The SGS model shoulddepend solely on the local properties of the flow and it is difficultto see how the distance from the wall qualifies as an appropriateparameter in this regard.

The purpose of the van Driest damping is to reduce the subgridscale eddy viscosity near the wall; it is generally believed that VT ~ y3

in this region and models ought to respect this property. It followsthat an alternative to van Driest damping is a subgrid scale modelwhich reduces the magnitude of the viscosity in the proper mannerwhen a subgrid scale Reynolds number (the obvious one is |5|A2/!/)becomes small. Models of this kind were suggested by McMillan andFerziger (1980) and by Yakhot and Orszag (1986): the latter usedrenormalization group theory to derive their model. It is necessary topoint out that these issues focus on the fully turbulent flow. Applica-tion to transitional flows present further problems and modificationswhich have been addressed initially by Piomelli et al.

A further problem is that, near a wall, the structure of the flowis very anisotropic. Regions of low speed fluid (streaks) are created.They have dimensions of approximately 1000 viscous units in thestrearriwise direction and perhaps 100 viscous units in the spanwise


and normal directions. Resolving these streaks requires a highlyanisotropic grid and the question arises: what is the appropriatelength scale to use in the SGS model in this region? The usual choiceis (A1A2A3)1/3 but another possibility is (Af + A?, + A^)1/2 and stillothers are easily constructed. It is possible that, with a proper choiceof length scale, the damping (4.13) would become unnecessary. Analternative model for the near-wall region was proposed by Schumann(1973); this model employs the horizontally averaged velocity andthus does not satisfy the condition that a model should be basedonly on the local velocity field. These issues were discussed in somedetail by Moin and Kim (1982) and Piomelli et al. (1989).

It has been found that, in a stably stratified fluid, it is againnecessary to reduce the value of the parameter in the Smagorinskymodel. Stratification commonly occurs in geophysical flows, wherethe practice is to make the parameter a function of the Richardsonnumber, a non-dimensional parameter that represents the relativeimportance of stratification and shear. Similar effects occur in flowsin which rotation and/or curvature play significant roles.

Thus, there are many difficulties with the Smagorinsky model. Itmay be that the principal reason why this model has been relativelysuccessful is that most of the flows for which accurate results havebeen obtained are relatively simple low Reynolds number cases; anexception is buoyancy-driven flows for which good results have beenobtained with the Smagorinsky model even at relatively high nondi-mensional parameter values. In such flows, the energy in the subgridscales and the rate of energy transfer to these scales are both rela-tively small. Then the model may not need to reproduce the actualsubgrid scale Reynolds stress very accurately to produce acceptableresults; it may suffice to simply dissipate energy at the proper overallrate. However, if we wish to simulate more complex and/or higherReynolds number flows, it may be important that the model be moreaccurate in detail.

4.3 A Priori Testing

The traditional test of a model consists of applying it to the so-lution of a problem and comparing the prediction with experimentalresults for the same flow. This is an obvious and practical way oftesting models which we shall call the a posteriori approach.

The availability of direct numerical simulations (and, possibly, in

124 J. H. Ferziger

the future, detailed experimental data) makes another kind of testingpossible. Let us accept that the results of a DNS represent an exactrealization of a turbulent flow. Having DNS results, one can ask howLES would fare for the same realization. In particular, it becomespossible to evaluate exactly those terms which must be modeled inthe LES and, at the same time, the model estimates of them. Letus see how this might work; for simplicity, we assume that the flowunder consideration is homogeneous.

Given the exact velocity field at an instant, u,-, it is straight-forward to filter it to obtain the large eddy component of that field,Ui. It is then not difficult to compute the subgrid scale Reynoldsstress tensor, u^Uj — U{Uj. Finally, one can compute the model esti-mate of the Reynolds stress. We thus have data on the exact Rey-nolds stress (R) and its model representation (M) at essentially everypoint in the flow. To test the accuracy of the model, one need merelycompare the two. Two popular methods of doing so are by computinga correlation coefficient:

and by producing a scatter plot i.e., a plot of the exact values ofa quantity vs. the corresponding model values. In this way, anunambiguous test of a model can be produced. Examples of suchscatter plots are given in Fig. 1; it presents the Reynolds stress andthe Smagorinsky estimate of it at approximately 4000 points. If themodel was exact, all of the points would fall on a single straightline. It is clear that the model is far from perfect. The correlationcoefficient of the data shown in this figure is approximately 0.35;as the square of the correlation coefficient represents the fraction ofthe data predicted by the model, this means that the model onlyrepresents about ten percent of the data!

In a similar way, by comparing the magnitudes of the model andexact values, one can obtain a value of the model parameter. The val-ues obtained in this way are in good agreement with those obtainedby other means (cf. Clark et a/., 1979).

It is thus clear that, in the precise sense that a priori testingprovides, the Smagorinsky model is not very accurate. Clark et al.(1979) used the a priori test to show that the problem does not arisefrom the value of the parameter but rather from the fact that theSGS Reynolds stress tensor and the strain rate of the resolved field,

Figure 1. Scatter plot of the Smagorinsky model prediction of thesubgrid scale Reynolds stress and the exact value obtained from adirect numerical simulation. From Bardina et al., 1980.

which the Smagorinsky model assumes to be proportional, actuallyhave little relation to each other. In particular, the principal axesof the two tensors (which need to be identical for proportionality tohold) are not well correlated.

A problem with the a priori method is that it is in some waysakin to an in vitro biological test; great differences may be foundwhen in vivo testing is performed. Similarly, using a model in anactual LES may give results that differ from what the a priori testfinds. The velocity field computed will differ from the large scale partof the DNS field used in the test. Indeed, despite the poor ratingthe Smagorinsky model receives in a priori tests, it seems to performreasonably well in LES.

4.4 Scale Similarity Model

The concept that the small scales of a simulation can be used tostudy modeling has a number of interesting extensions. One leadsto an alternative model for the subgrid scales, the scale similaritymodel (Bardina et al., 1980). A more important extension will bepresented in the following section.

The idea behind the scale similarity model is that the importantinteractions between the resolved and unresolved scales involve thesmallest eddies of the former and the largest eddies of the latteri.e., eddies that are a little larger or a little smaller than the length


126 J. H. Ferziger

scale, A, associated with the filter. These can be extracted from thevelocity field in the following manner.

From the complete velocity field, v,j, we can compute the resolvedor large scale field ¥,• by filtering and the small or subgrid scalefield u\ — U{ — U{ by subtraction. From these we can construct afurther subdivision. The very largest resolved scales may be definedby filtering a second time to obtain ¥; so the smallest resolved scalesare defined by Uj — u,-. The largest unresolved scales are defined byu\, A simple calculation shows that these are identical. This leadsto the following possibility as a subgrid scale model:

No constant is used because it can be shown (Reynolds, pri-vate communication, and Speziale, 1983) that Galilean invariancedemands that the constant be unity. From its construction, it is notsurprising that this model correlates very well with the actual SGSReynolds stress in a priori tests, see Fig. 2. The argument of thepreceding suggests that, in essence, it is an identity. When applied ina large eddy simulation, it is found that this model hardly dissipatesany energy and thus cannot serve as a 'stand alone' SGS model. Itdoes transfer energy from the smallest resolved scales to the largerresolved scales, which is useful.

Figure 2. Scatter plot of the mixed scale similarity - Smagorinskymodel prediction of the subgrid scale Reynolds stress and the exactvalue obtained from a direct numerical simulation. From Bardina ataL, 1980.


To correct for the lack of dissipation, it is necessary to combinethe Smagorinsky and scale similarity models to produce the 'mixed'model. This model does indeed improve the quality of the simulationsas one can see from Fig. 3.

Figure 3. Comparison of the spectra obtained from a large eddysimulation of decaying homogeneous isotropic turbulence with andwithout the mixed scale similarity - Smagorinsky model. From Bar-dina et a/., 1980.

When the large scales are defined by the cutoff filter, ¥; = ¥,-,and the scale similarity model produces nothing i.e., Eq. (4.15)evaluates to zero. This difficulty can be removed by noting that, forthe Gaussian filter, filtering twice is equivalent to a single filteringwith A replaced by \/2A. This is easily mimicked for the cutoff filterby defining the double overline filter as a cutoff filter correspondingto this width; however, the correlation is not as good as it is whenthe Gaussian filter is used, cf. Bardina et al. (1980).

4.5 Dynamic Procedure

The concepts of the preceding sections can be taken one stepfurther, leading to the concept of a dynamic model, an idea originally

128 J. H. Ferziger

proposed by Germane et al. (1990). It might better be called aprocedure than a model as it takes one of the models described aboveas its basis. Perhaps the simplest way to explain the concept isthe following. Suppose we are doing a large eddy simulation on arelatively fine grid. We could regard it as a DNS and use its velocityfield as the basis for a priori estimation of the subgrid scale modelparameter. This can be done at every spatial point and time step.The scales used in such a test are, of course, the smallest resolvedscales of the LES. If we assume, as we did in constructing the scalesimilarity model, that the behavior of these scales is very similar tothat of the subgrid scales, the parameter so obtained can be appliedin the subgrid scale model of the large eddy simulation itself. In thisway, a kind of self-consistent subgrid scale model is produced.

The actual procedure of Germano et al. is a bit more formal thanwhat we have just suggested. We now present this formal procedure.The subgrid scale Reynolds stress that must be modeled in the actualLES is:

The second or test filter (the one used to determine the parame-ter) is similar to the second filter used in the scale similarity modelbut is denoted by a tilde (~) to make explicit the idea that the orig-inal and test filters need not be identical. The subgrid scale stressthat must be modeled in the test-filter level LES is:

Now let us define the large scale component of the SGS Reynoldsstress at the test filter level. This is the portion that is directlycomputable from the LES field by filtering:

and is essentially the Leonard stress associated with the test filter.Now it follows directly from these definitions that:

This is a mathematical identity that is a consequence of the defini-tions given above; it has come to be known as Germano's identityand provides the basis of the dynamic model.


The basic assumption that leads to the dynamic model is thatparticular model applies on both filter levels with the same value ofthe parameter(s). We shall use the Smagorinsky model as an examplebut there is no reason why other models cannot be used; indeed, themixed model has been used as a base model (Zang et al., 1993). Onthe original LES level, the Smagorinsky model is:

On the test filter level the Smagorinsky model is:

Now we substitute the last two equations into Eq (4.18) to obtain:

Everything on both sides of this equation is computable fromthe velocity field computed in the LES, ¥;. This means that it canbe used to compute the constant, C. However, as (4.22) representsfive independent equations, C is overdetermined. Germane et al.suggested that the scalar product of Eq (4.22) with Sij be taken. Animprovement was made by Lilly (1992), who suggested computingthe optimum in the least squares sense. If we call the right hand sideof Eq (4.22) CM;j, one can show that this is equivalent to taking thescalar product of Eq (4.22) with M.^ and yields:

Thus, the model parameter can be computed, at every spatialgrid point and at every time step, directly from results produced bythe LES itself. In other words, we have a kind of self-consistent or,as it is more commonly known, dynamic model.

Although this concept is very appealing, there are significantproblems with the resulting model. First of all, it was assumed in de-riving Eq (4.22) that the model parameter is a constant; this allowedit to be removed from the test filter and evaluated. The resultingexpression for C, (4.23), is a function of the spatial coordinates andtime, violating an assumption made in the derivation. Furthermore,in actual simulations, C is found to be a very rapidly varying functionwhich takes on large values of both signs, leading to eddy viscosities

130 J. H. Ferziger

of both signs. Although negative eddy viscosity is not prohibited (itmay be considered a way of representing backscatter), if the eddyviscosity remains negative over too large a spatial region or for toolong a time, numerical instability may result and the simulation mustbe stopped; this occurs in actual simulations. Clearly, a cure for thisproblem needs to be found.

The negative eddy viscosities occur because the numerator ordenominator of Eq (4.23) may become negative. Similarly, largevalues are a generally a consequence of the denominator being small.In turn, a small value of the denominator means there is relativelylittle energy in the highest wavenumbers resolved in the LES. Thisfurther implies that there is not much energy in the subgrid scalesand therefore that the eddy viscosity should be small. One cure forthe problem is thus to simply set any eddy viscosity wj < —z/, themolecular viscosity, equal to —v. This has been used successfully butis not very satisfying so other methods have been developed; someof these are discussed below.

One useful alternative is to employ averaging. For a homogeneousflow, we may average the scalar product of Eq (4.22) with M,-j overthe entire domain prior to computing C. A more satisfactory deriva-tion of this result is to apply the least squares method to L±j over afinite spatial region. This leads to the replacement of Eq (4.23) with

where the brackets (<>) represent an average over the spatial re-gion to which the least square method was applied. This techniqueproduces excellent results; it has been used to compute a variety ofhomogeneous turbulent flows, fully developed channel flow, and tran-sitional channel flow, all with excellent results. This version of thedynamic model removes many of the difficulties described earlier:

• It was noticed that, in shear flows, the required value of theSmagorinsky model parameter is much smaller than in isotropic tur-bulence. The dynamic model produces this modification automati-cally.

• Near walls, the value of the model parameter has to be reducedeven further, for example by using van Driest damping (4.13). Inchannel flow, the averaging in Eq (4.24) is usually averaging overplanes parallel to the wall. When this is done, the model automati-cally decreases the parameter near the wall.


• The definition of the length scale is unclear when the filter isnot isotropic. This issue becomes moot with the dynamic modelbecause, if the length scale is incorrect, the model compensates bychanging the value of the parameter. Essentially, the model actuallycomputes the eddy viscosity, not the model parameter.

To overcome the problem created by the large negative eddy vis-cosities generated by the simpler forms of the dynamic model severalcures have been suggested. Two were already presented above—averaging over homogeneous directions and limiting the magnitudeof negative eddy viscosities. These are successful but the former isavailable only in flows with some degree of homogeneity and the lat-ter is not satisfying from an esthetic point of view. This has led to asearch for other methods of dealing with the problem.

One such approach is to use a combination of local spatial andtemporal averaging which are available in any flow (cf. Piomelli,1992). These have proven successful so long as one can find a spatialregion large enough to smooth out the parameter variation but smallenough that it does not contain significant inhomogeneity.

Another approach is based on the recognition that part of theproblem arises from the removal of the model parameter from thefilter. In order to do so, it was assumed that the parameter is con-stant but the resulting values are far from constant, invalidating theassumption. Instead, one can regard Eq. (4.22) (with the param-eter inside the filtering operation) as an integral equation for theparameter. This integral equation is then solved for the parameter.It turns out that this removes some, but not all, of the variation ofthe parameter and thus does not completely cure the problem andincreases the computational effort somewhat.

A further improvement is obtained by subjecting the integralequation referred to in the last paragraph to a constraint that the to-tal viscosity (eddy plus molecular) be everywhere non-negative. Theresulting problem can then be solved only in a least squares sense,leading to a constrained optimization problem. This can produceexcellent results at a cost of some increase in computer time and hasbeen called the dynamic localization model (Moin et a/., 1994).

Finally, we mention that the arguments on which the dynamicmodel is based are not restricted to using the Smagorinsky modelas the base model. One could, instead, use the mixed Smagorinsky-scale similarity model or one of the models presented below. Themixed model was used in this regard by Zang et al. (1993) with

132 J. H. Ferziger

considerable success. However, the flow to which this method wasapplied is a transitional flow and it is not known whether the findingsextend to fully developed flows. Ghosal et al. (1994) applied thedynamic procedure to a one equation (or turbulent kinetic energy)subgrid scale model, obtaining good results.

4.6 Spectral Models

The Smagorinsky and scale similarity models are not the onlyones that have been used to represent subgrid scale turbulence. Forguidance as to how improved models might be constructed, one canturn to turbulence theories. In order to deal with the distributionof turbulent energy over a range of length scales, in most turbulencetheories the principal variables are the Fourier transforms of the ve-locity components:

or, more frequently, its squared amplitude, the energy spectrum:

where the integral is over all wavevectors A; on a sphere of radiusk and, as usual, a sum over the index i is implied. Use of a spec-tral representation of the velocity field implies that these theoriesare applicable only to homogeneous turbulence and, usually, onlyto isotropic turbulence. Despite these limitations, they can provideconsiderable insight into the issues and extensions to more complexflows are possible.

A number of turbulence theories exist; most produce similar re-sults. Let us begin with an observation. From the Fourier transformof the Navier-Stokes equations, one can derive a dynamic equation forthe energy spectrum. In this equation, the advective terms, the onlynon-linear terms in the equations, transfer energy from one wave-number to another but neither produce nor destroy total energy.The pressure terms disappear entirely; their function is to transferenergy from one component of the turbulence (say wf(fc) ) to another(say u\(k)} and so play no role in the energy equation. The viscousterm is responsible for the dissipation:


which acts as an energy drain on the turbulence. For further detailsof these theories and the roles of the various terms, it is recommendedthat the reader consult the book by Lesieur (1992).

Results derived from turbulence theories make it possible to de-fine an effective or spectral eddy viscosity. As just noted, the non-linear term transfers energy from one wavenumber to another. Onecan imagine doing a large eddy simulation (because spectral theoriesare formulated in terms of spectra, only the cutoff filter is normallyused) and ask how much energy is transferred from a given wave-number k to wavenumbers above the cutoff. If we call this energytransfer rate T>(k) and think of it as a dissipation at wavenumberk, we can define an effective spectral eddy viscosity by:

An example of such an eddy viscosity, taken from Lesieur (1992), isgiven in Fig. 4. The decrease in the effective viscosity at low wave-numbers is of little consequence because little energy is transferredfrom these wavenumbers to the small scales. The rise at the highwavenurnbers is due to the local nature of the interactions in tur-bulence; simulations have shown that incorporating the rise in theviscosity into the subgrid scale model is capable of producing simu-lations in which the spectrum maintains an inertial subrange shapeup to the cutoff wavenumber.

Although the rise in the eddy viscosity at high wavenumber doesnot follow any simple law, if spectral computational methods areused, a fit to the results can be constucted and used. Alternatively,the curve can be approximated by a constant plus a term propor-tional to some power of k. This suggests that using the Smagorinskymodel (which roughly approximates the constant component of theeddy viscosity) together with a hyperviscosity (a dissipative term-containing even-order velocity derivatives of order higher than sec-ond) might be a good choice for a subgrid scale model. The simplestchoice to implement is a so-called fourth order viscosity which intro-duces a term proportional to the fourth derivative into the Navier-Stokes equations. Two possibilities for the added terms, which weshall call r^4 , are:

Figure 4. Spectral eddy viscosity computed from eddy damped quasi-normal Markovian theory (EDQNM). From Lesieur, 1992.

and

These become identical if the eddy viscosity is constant. If we wishto model these terms in the spirit of the Smagorinsky model, dimen-sional analysis suggests that an appropriate expression for the fourthorder viscosity might be:

The introduction of such a term into the model increases theorder of the partial differential equation and raises the possibility thatadditional boundary conditions might be needed. This unpleasant

134 J. H. Ferziger


possibility can be avoided if the new viscosity 2/4 vanishes rapidlyenough in the vicinity of the wall.

Another means of using turbulence theories is to simulate thelarge scales in the usual LES manner and use a theory to describethe subgrid scale motions statistically. A method that used EDQNMfor the subgrid scales was developed by Aupoix (1987). He obtainedgood simulations of high Reynolds number flows, but only for isotro-pic turbulence, and at an order of magnitude increase in cost relativeto the Smagorinsky model. Clearly, if this method is to be practical,it will need to be simplified to make the cost more reasonable.

Other suggestions for models have been made. We shall not covermost of these here because they were constructed for use in Fourierspace and cannot be easily converted to physical space models. Theabsence of such a conversion possibility renders a model almost im-possible to use with finite difference or finite volume discretizations,restricting their usefulness. One attempt in this direction was madeby Metais and Lesieur (1992) who devised what they called a struc-ture function model; in practice, this model is very similar to theSmagorinsky model.

We note that it is possible to use spectral eddy viscosity modelsin the dynamic context. Although, to the author's knowledge, thishas not been attempted, it seems an interesting possibility.

We end with a brief note on another approach to turbulence sim-ulation. A number of simulations have been made which claimed tobe direct numerical simulations of complex flows; for one of manyexamples, see Kawamura and Kuwahara (1985). A brief analysiswill quickly convince one that these cannot possibly be DNSs in thesense defined earlier. These simulations use third order upwind ap-proximations to the spatial derivatives which produce fourth ordererror terms similar to the fourth order viscosities presented above.So these simulations can be re-interpreted as large eddy simulationswith a fourth order subgrid scale model. The danger in this approachis that the eddy viscosity is determined by the grid and the solutionmay therefore depend on the grid used.

4.7 Effects of Other Strains

The models described above have been designed for flows without'extra strains' ( e . g . , rotation, compressibility and curvature); despitethis, we have seen that the dynamic model can handle some of these

136 J. H. Ferziger

without problems.Meteorologists and oceanographers who predict global circulation

deal with flows that are nearly two-dimensional; an eddy viscosityis used to represent the unresolved motions. At the smallest scales,three-dimensional equations may be used; simulations are routinelydone on several levels. A single model (with a single parameter)that can account for phenomena at all the various scales probablydoes not exist. A systematic approach is needed to build a firmfoundation for modeling in these areas. The task is difficult andprogress may come slowly. A possible approach is the following. Atthe lowest level, one can simulate the small-scales e.g., the planetaryboundary layer or the ocean mixed layer and use the data producedto construct parameterizations that to be used represent motionsthat belong to the subgrid scale on the next larger scale, perhapsthe regional scale. To assure that all possibilities are included, arange of cases containing all physically possible phenomena must besimulated to ensure that the full range of parameters are includedin the database. By bootstrapping in this way, and allowing two-way interaction between simulations at different scales, it may bepossible to develop methods that allow phenomena on all scales tobe predicted. It should be obvious that there are difficulties in thisscenario for which solutions are yet not available.

Extra strains can be roughly divided into two classes. Some, suchas rotation, curvature, and stratification, affect the large scales morestrongly than the small scales. In these cases, SGS models designedfor incompressible flows without extra strains can probably be usedwithout major modification. For example, large eddy simulations ofa stable planetary boundary layer performed with the Smagorinskymodel (Mason and Derbyshire, 1990) agree very well with both directsimulations (Colernan et a/,, 1989) and field data.

On the other hand, for 'strains' whose action is principally in thesmall scales, the situation is less clear. Compressible turbulence atlow Mach numbers can be treated with incompressible models. Athigher Mach numbers, small shock waves ('eddy shocklets') develop,and the flow behavior can be quite different (Blaisdell et a/., 1991,Lee et a/., 1993). If one is to do large eddy simulations of such flows,there are two possibilities. The first approach is similar to standardSGS modeling. In the absence of shocks, SGS models applicable tothe Favre-filtered equations can be developed in a manner analogousto the incompressible case (Erlebacher et a/., 1992; Speziale et a/.,


1988). We note that, as a real shock is too thin to be resolved bythe grid, the viscosity and thermal conductivity need to be increasedso that the simulated shock becomes thick enough to be resolved;this approach has been used in many 'shock capturing' methods inaerodynamics. The second approach is to replace the actual curvedshock by a straight one, using a subgrid scale model to account forthe larger dissipation of the curved shock; this is akin to the 'shockfitting' approach to aerodynamics.

In combusting flows, flames are normally thin with respect toeven the smallest scales of the turbulence and LES is again very dif-ficult. Again, one can imagine two types of LES that are similarto the ones described for shocks above. The first is applicable onlywhen the chemistry is simple enough to be characterized by one ortwo constants. In this case, one can modify the reaction rate and dif-fusivity so as to increase the thickness of the flame while maintainingthe flame speed constant. In the second approach, the flame is ide-alized as an infinitely thin sheet. The function of the SGS model isthen to account for the 'wrinkles' that occur on scales that are notresolved by increasing the local reaction rate. Such a suggestion hasbeen made by Ashurst et al. (1988). A different approach based onthe use of probability density functions (which are commonly used incomputing reacting flows) was suggested by Gao and O'Brien (1993).

4.8 Other Models

It is possible to use more complex models for the subgrid scale.Any model used in RANS calculations can be modified and adaptedas an SGS model. In particular, models based on solving partialdifferential equations may be used but this has been done only a fewtimes.

In RANS, the next step beyond a mixing length model (which cor-responds to the Smagorinsky model in LES) is a two-equation modelwhich requires equations that determine the turbulence velocity andlength scales. The model length scale is not normally an issue in SGSmodeling because a natural length scale, the filter width, is available,so two equation models see little service in LES. One equation mod-els can fill this role. In such a model, a partial differential equationfor the subgrid scale kinetic energy is constructed and solved. Thereis only a little experience with such models. Schumann (1978) foundthat it provided no significant improvement over the Smagorinsky

138 J. H. Ferziger

model for fully developed channel flow. This is no surprise because,as we noted earlier, the major deficiency of the Srnagorinsky model isthat the principal axes of SGS Reynolds stress and rate of strain ten-sors are not aligned and this is not addressed by the model. On theother hand, some benefit was obtained in transitional flow; however,the dynamic model performs as well in transitional flows so the needfor the more complex model has not been demonstrated. It is worthmentioning that Ghosal et al. (1994) have constructed a dynamicmodel that includes a differential equation for the turbulent kineticenergy equation that appears promising.

The most complex RANS models in use today are Reynolds stressmodels in which a set of equations is derived for the Reynolds stressand the various terms are modeled. It is, of course, possible to deriveequations for the SGS Reynolds stress components as well; they are abit more complicated that the corresponding RANS equations due tothe properties of the filtering operator. LES with an SGS Reynoldsstress model has been tried only once, and that in a relatively earlysimulation of the atmospheric boundary layer by Deardorff (1974).He found a huge increase in the cost but almost no improvement inthe results.

In both the turbulent kinetic energy and Reynolds stress subgridscale models, the constants were taken from RANS models. This isprobably inappropriate as the physics of subgrid scale turbulence isdifferent from that for all the turbulence; the relative importance ofthe various terms is probably different in the two cases. Unfortu-nately, at present, there is little guidance for improving the models.DNS data could probably be used to guide the development of modelsbut this has not yet been attempted.

5 WALL MODELS

Another issue of great importance is modeling of the flow in thevicinity of a wall. This question receives less attention than SGSmodeling because it is not as amenable to theoretical treatment, butit is at least as important. Before discussing the wall models, weshall review some results of experiments and direct and large eddysimulations with no-slip wall boundary conditions.

Shear flows near solid boundaries contain alternating thin streaksof high- and low-speed fluid. If they are not adequately resolved, the


turbulence energy production near the wall (which is a large fractionof the total energy production) is underpredicted (Kim and Moin,1986), resulting in reduction of the Reynolds stress and the skinfriction.

Simulations suggest that wall-region turbulence and the region farfrom the wall are relatively loosely coupled. Chapman and Kuhn's(1986) simulation with an artificial boundary condition imposed atthe top of the buffer layer (y+ = 100) displayed most of the charac-teristics of the wall layer. Thus, accurate prediction of the flow nearthe wall does not require accurate simulation of the outer flow.

On the other hand, Schumann (1973) and Piomelli et al. (1987)among others have shown that, relatively crude lower boundary con-ditions can represent the effect of the wall region in a simulation ofthe central part of a channel flow. Thus, details of the flow in thewall region need not be known in order to simulate the outer region,i.e. either region can be well-simulated if given a reasonable approx-imation of the conditions at the interface between it and the otherzone.

These results suggest that useful simulations can be done withoutresolving the entire flow. This is important because a fine grid isrequired to resolve the wall region. If it can be represented via amodel, huge savings are possible. For rough walls, one has littlechoice but to use a model to represent the wall region.

DeardorfPs (1970) original model contained weaknesses that wereremedied by Schumann (1973). The latter's model, with modifica-tions, is still widely used. It assumes that the instantaneous velocityat the grid point nearest a wall is exactly correlated with the shearstress at the wall point directly below it:

where y\ is the height of the first grid point, < TW > is the mean wallshear stress, and Ui(yi) is the mean velocity at y\.

Mason and Callen (1986) assumed that the logarithmic profilefor the mean velocity in the buffer region, holds locally and instanta-neously. This assumption is incorrect but their boundary conditionis often used by meteorologists. Piomelli et al. (1987) found it to beinadequate for engineering applications. This is an example of howthe differing needs of two disciplines can lead to opposite conclusionsabout the effectiveness of a model.

140 J. R. Ferziger

Piomelli et al. (1987) used direct simulation results to test walllayer models including Schumann's model and two new models. Thefirst of these is based on the idea that Reynolds-stress producingevents do not move vertically away from the wall but, rather, at asmall angle to it. This leads to the so-called shifted model,

where As — 3/1/cos8° is a spatial shift, 8° being the observed meanangle of event trajectories.

The second model notes that significant Reynolds-stress contain-ing events involve vertical motion, so the vertical component of thevelocity rather than the horizontal should be correlated with the wallshear stress:

Both of these models give improved agreement with experiments anddirect simulations for channel flow, including cases with transpirationand high Reynolds number flows.

In fully developed channel flow at Reynolds number 15,000, useof these conditions reduced the time of a simulation from 100 hoursto 10 hours (Piomelli et a/., 1989) so their value is unquestionable.

Finally, we mention and interesting proposal by Bagwell et al.(1993). They used a linear estimation method developed by Adrianto determine the best estimate of the skin friction given the velocitydistribution at some distance from the wall. They found that the skinfriction estimate could be improved (relative to those given above)by using a weighted average of the velocity on the computationalplane closest to the wall. A disadvantage of this method is thatthe two point correlation, which becomes a complicated function ofthe coordinates in complex flows, and especially near separation andreattachment, is required.

All of the models of this section have been applied only to flows onflat walls with mild pressure gradients. They are almost certainly in-adequate for separated flows (with or without reattachment), or flowsover complex-shaped walls. They may work in three-dimensionalboundary layers because the direction of the horizontal componentof the velocity changes slowly with distance from the wall in thelower part of the flow. Because experimental data are scarce and


lack detail, the development of trustworthy methods for simulatingthese flows will probably require simulations with no-slip conditions.

6 NUMERICAL METHODS

A wide variety of numerical methods have been employed in largeeddy simulation. Almost any method of computational fluid dynam-ics can be used. Because these methods are adequately describedelsewhere (see, for example, Ferziger and Peric, 1993), we shall notdescribe particular methods here. Instead, a few generalities aboutissues peculiar to LES will be discussed.

The most important requirements on numerical methods for LESarise from the goal of producing an accurate realization of a flowthat contains a wide range of length scales. The need to producea time history means that techniques used for steady flows must berejected. Time accuracy requires a small time step and it is importantto know whether the time-advance method is stable for the time stepdemanded by accuracy. This is generally the case so most simulationsuse explicit time advance methods. An exception occurs close towalls where fine grids must be used in the normal direction andinstability may arise from the viscous terms; in this case, only theviscous terms involving derivatives normal to the wall are treatedimplicitly. The numerical methods most commonly used in LES areof second to fourth order; Runge-Kutta methods have been used mostcommonly but others, such as Adams-Bashforth and leapfrog havefound application.

The need to handle a wide range of length scales means renderssome concepts of computational fluid dynamics relatively unimpor-tant. The most common means of describing the accuracy of a spatialdiscretization method is its order, a number that describes the rateat which the discretization error decreases as the grid size goes tozero. To see why this is not applicable in LES, it is useful to think interms of the Fourier decomposition of the velocity field. A discreteversion of Eq (4.25) is

The highest wavenurnber k that can be resolved on a grid of sizeAx is 7T/A:r, so it is sufficient to restrict the range of k to {0, TT/Ao:}.

142 J. H. Ferziger

The derivative of (6.1) may be taken term by term and, if we ignorethe effect of boundary conditions, it is sufficient to consider the ef-fect of the discretization on a single Fourier mode, elkx'. The exactderivative is, of course, ikelkx. All discrete approximations replacethis by ikejje

lkx where kejf is called the effective wavenumber. Forexample, the central difference approximation:

when applied to e , gives:

so

for this method. For small &, the Taylor series approximation:

shows the second order nature of the approximation. A plot of keffis given in Fig. 5 which shows that the Taylor series approximationis useful only for k < 7r/2Ax, the first half of the wavenumber rangeof interest. Other discretizations give different expressions for theeffective wavenumber. Upwind approximations give effective wave-numbers that are complex, reflecting the dissipative nature of thediscretization error for these schemes. A similar treatment of the er-rors in approximations for the second derivative is easily constructedbut will not be described here.

The problem in LES is that the spectrum of the solution (itsdistribution over wavenumber) covers a significant part of the wave-number range {0,?r/Aa;}. The order of the method is no longersufficient to define the accuracy of a scheme. A better measure ofthe error is:

Again, similar expressions can be given for the second derivative.Using the measure (6.6), Cain e.t al. (1981) found, for a spectrum


Figure 5. Effective wavenumber of the central difference approxima-tion to the first derivative.

typical of isotropic turbulence, that a fourth order method had halfthe error of a second order method, much more than most peoplewould have anticipated.

The final point is that the methods and step sizes in time andspace need to be chosen together. The errors made in the spatial andtemporal discretizations ought to be as nearly equal as possible i.e.they should be balanced. This is not possible in detail but, if thisis not done, one is using too fine a step in one of the independentvariables and the simulation could be made with little loss of accuracyat lower cost.

7 ACCOMPLISHMENTS AND PROSPECTS

Large eddy simulation has been applied to a range of flows toolarge to be covered in a single paper; there no need to do so. The pur-pose of many simulations was to study the physics and modeling offlows and the results produced by DNS and LES are often treated asexperimental data. For that reason, it makes more sense to considerthe results together with experimental data on a flow-by-flow basis.We shall, therefore, give only a short overview of the kinds of flows

144 J. H. Ferziger

that have been treated with LES, a snapshot of the state-of-the-art,and a discussion of what may be possible in the next few years.

In the early days (in the 1970's), LES was used, at least in en-gineering, for investigating simple flows in order to understand thephysics of turbulence and the accuracy of RANS models. Flowsthat were treated in this way included all the homogeneous flows,plane channel flow, and free shear flows that are inhomogeneous inone direction (2D mixing layers, wakes, and jets). Computers havenow become sufficiently large and fast that these flows can be dealtwith by direct numerical simulation. Since DNS does not have anyuncertainty arising from subgrid scale modeling, it is the preferredtechnique for this kind of investigation and should be used when-ever possible. In the early and mid-1980's LES almost fell into dis-use; however, interest was rekindled (Hussaini et al., 1990; Reynolds,1990), and at the outset of the 1990's application of the method wason the rise.

LES is now being applied to flows that remain beyond the reachof DNS. A very important engineering issue is that of flow separationand reattachment, phenomena that occur in many technological flowsand, with few exceptions, are not well predicted by RANS methods.They are also, at present, outside the reach of DNS. The simplestseparating flow is the backward facing step in which a plane channelflow encounters a sudden expansion on one wall of the channel. DNSof this flow was performed by Le and Moin (1993) and althoughgood results were obtained, 1100 hours were required on a singleprocessor Cray-YMP. An LES of this flow by Akselvoll and Moin(1993) using the same computer required only about 30 hours. Thelatter figure, while much more than a working engineer would, careto pay for a simulation, brings the cost to a point at which it maybe sensible to do a simulation occasionally to check the validity ofRANS results and/or the models used in RANS calculations. Otherflows of this kind which have been simulated recently include the twodimensional obstacle (Yang and Ferziger, 1993) and flow over a cube(Mauch, 1991; Shah and Ferziger, 1994). The former introduces aflow-determined separation not found in the backward facing stepflow. The latter introduces three dimensional separation.

For the near future, a sensible role for LES to play is as a check onthe validity of RANS turbulence models and predictions for complexflows. It will be possible to perform large eddy simulations of someflows of engineering interest but the method will remain too costly


for routine engineering use for a long time to come. Further, LEScan and will be used as a complement and partial substitute forexperimental testing. An occasional LES can be compared to RANSpredictions to test the adequacy of a design and/or the methods usedto develop it. It will also continue to be used to directly test theaccuracy of RANS models, a role that it has played with distinctionthroughout its history. By using LES to tune RANS models, it shouldbe possible to obtain most of the benefits of LES at a small fractionof the cost.

Flows that may be good candidates for LES in the near futureinclude the turbine blade passage and the internal combustion enginecylinder. These are both relatively low Reynolds number flows andof obvious technological importance. Both of these flows also containmany extra strains that renders the development of RANS modelsfor them exceedingly difficult, making the possibility of using LESdirectly in the design process and interesting one.

A word of caution is necessary. LES and its subgrid scale modelshave been validated only for relatively simple flows at fairly low Rey-nolds numbers. In these flows, most of the energy is in the resolvedscales and, even if the subgrid scale model is not very accurate, its ef-fect on the results may not be too important. If one uses the successof these simulations as a justification for applying LES to much morecomplex flows, although reasonable looking results may be obtained,placing one's trust in them may be risky. The leap is simply too greatto allow expectation of success in this kind of endeavor. Simulationsof this kind have been made but, in the author's opinion, they havebeen premature and their value is questionable.

It is also important that the goal of a simulation be defined inadvance. Doing a simulation merely to show that it can be done isof limited value. It is known in advance that it can be done and, ifenough resources are deployed, good results will be obtained. Thevalue is in learning about the physical nature of the flow, how it maybe modeled, or, perhaps, in making a contribution to the improve-ment of a design.

It would be very valuable to have models that eliminate the needto specify no-slip conditions at a wall. Boundary conditions of thiskind exist for attached flows and were discussed above. What isnot known is whether conditions of this kind can be constructed forseparated flows. Doing so for RANS models has proved exceedinglydifficult and there is no reason not to expect the task to be at least

146 J. H. Ferziger

as difficult for LES.In flows in complex geometries, it is impossible to construct opti-

mum grids prior to the calculation, even for steady flows; one simplycannot know in advance where the maximum resolution is required.To compensate for the absence of this information, methods whichmodify the grid as the solution procedure converges have been devel-oped. The author's favorite such method is one developed by Bergerand Oliger (1984). This method has been adapted for elliptic flows byCaruso et al. (1985). It also combines well with the multigrid solu-tion procedure, one of the best methods for solving elliptic problems(Thompson and Ferziger, 1989).

It is difficult to specify the grid requirements for the flows men-tioned above at this time. The numbers will depend on whethertechniques of the kinds described in the last two paragraphs can bedeveloped. All that is certain is that they will need to be largerthan those now in use. The large parallel machines now comingonto the scene will allow simulations to be done on grids containing512 X 512 X 512 points (or other grids containing roughly the samenumber of points). It is conceivable that simulations of turbine bladepassages and engine cylinders can be done with these grids.

8 COHERENT STRUCTURE CAPTURING

8.1 The Concept

Up to the present, researchers have attempted to build LES fromthe ground up. The idea is to start with simple flows (preferably onesthat can be treated with DNS), use them to learn about SGS mod-eling, and then go on to increasingly more complex flows. To date,most researchers have taken care to simulate only flows in which alarge fraction of the energy of the turbulence can be resolved. Theimportance of the model is thereby minimized and good results havebeen achieved. This success does not assure equal success for LES ofcomplex flows; this might be the case if sufficient computer resourceswere available but, for most flows of technological interest, there islittle possibility this will happen in the foreseeable future. Further-more, the objective is usually to obtain just a few selected propertiesof the flow at minimum cost. That being the case, LES and DNS arebest not used as everyday tools.


A better choice for the near term is to perform LES and/or DNSon 'building block' flows, i.e. flows that are structurally similar tothe ones of actual interest. From the results of such simulations,RANS models that can be applied to the more complex flows canbe validated and improved. RANS computations can then be usedas the everyday tool. LES need be performed only when there aresignificant changes in the design or as an occasional check on thevalidity of the RANS results.

As noted earlier, there have been attempts at large-eddy anddirect numerical simulation of complex flows. Unfortunately, in mostof these, the subgrid scale model was uncontrolled and the results areof uncertain value. This appears to be a case of reaching too far toofast; we shall not present examples here.

Since answers to questions involving technologically significantflows are required, the following questions arise. Is there a methodthat will enable more complex flows to be simulated on availablemachines? Are there flows of importance that are good candidatesfor simulation via LES in the relatively near future?

The answer to these questions appear to be a qualified yes. Otherthan the flows mentioned earlier, particularly good candidates areflows in which there are a small number of important, energetic, andeasily identified coherent structures. In all the cases that have beensuggested, the large structures are vortices. Let us consider two suchcases.

Flows over bluff bodies usually produce strong vortices in theirwakes. The vortices produce strong fluctuating forces on the body inboth the streainwise and spanwise directions whose prediction is veryimportant in many applications. The latter include buildings (windengineering) and ocean platforms, among others. If the vortices aresufficiently larger than the bulk of the motions that constitute the'turbulence' it should be possible to construct a filter that allows thevortices to be retained in the resolved field while removing all of thesmaller scale motions. We have called a method that accomplishesthis 'coherent structure capturing' or CHC (Ferziger, 1993). The au-thor and others earlier called it very large eddy simulation or VLES,a term we now find less descriptive. Methods of this kind were sug-gested a long time ago (Ferziger, 1983) but deliberate simulationsof this type do not appear to have been attempted other than a fewcases which apparently gave unsteady results when a steady flow wasexpected. An exception to this might be the recent work of Orszag

148 J. H. Ferziger

discussed in Chapter 4.The cylinders of internal combustion engines provide another ex-

ample. This flow is inherently unsteady, so there is no possibility ofmodeling it as a steady flow. Several interesting issues arise whichlead to the following questions. What does RANS mean in such aflow and how should RANS results be compared with experimentalresults? Since the flow is unsteady, LES can only produce a single re-alization; can such a simulation provide sufficient information aboutthe flow? A partial answer to the first question is that the RANSmean velocity should probably be defined as an average over manycycles and the turbulence as the deviation from the multi-cycle aver-age. LES should simulate a single cycle. To see what the differencesare consider the following. After the intake stroke, the flow containsa strong vortex whose location, size, and strength varies from cycleto cycle; this vortex is important to engine behavior. In a RANScalculation, the result should contain an average vortex, one that isrelatively large and of average circulation. In LES, the vortex shouldbe smaller, of similar circulation, but its location should vary fromrealization to realization, it should be possible to construct a filterthat can separate the vortex from the rest of the turbulence field.

8.2 Modeling Issues

The models to be used in CHC should be different from thoseused in both RANS and LES. Indeed one needs to be very carefuland considerable experience is probably required before this kind ofsimulation can be trusted.

According to the Smagorinsky model, the length scale to be usedin LES is the filter width, A. But, in CHC, the filter width maybecome quite large; it may indeed become larger than the lengthscale used in RANS models, which is an approximation to the integralscale, L. If this is the case, the LES viscosity could exceed the RANSviscosity, in violation of the concept that the RANS viscosity shouldbe large enough to remove all of the unsteadiness (at least all of itthat is considered turbulence) from the flow. This situation shouldnot be allowed to arise. To prevent it from occurring, it may becomenecessary to introduce an equation for the length scale to be usedin the subgrid scale model and LES may then inherit many of thedifficulties that RANS models have with length scale modeling. Theonly thing that is clear is that considerable effort will be needed to


make this approach work.

9 CONCLUSIONS AND RECOMMENDATIONS

After years of being regarded as a method of second choice rela-tive to direct simulation, LES is receiving increased attention. Theprincipal reasons are dissatisfaction with the performance of RANSturbulence models on the one hand and the inherent limitations andcost of direct simulation on the other.

Improved models for both the small-scale turbulence and the walllayer are also needed if LES is to become a useful engineering tool.The dynamic model offers promise of removing many of the difficul-ties that have plagued LES and to give it an important advantagewith respect to RANS modeling. Improved models for the wall re-gion, especially for separating and reattaching flows, are needed justas badly and are an important subject for future research.

However, if LES is to prove useful in truly complex high Reynoldsnumber flows, a great deal more work may be needed. For the nearfuture, it is probably best to use LES to understand the physicalnature of the flow and to tune RANS turbulence models in a waythat will allow them to produce more accurate predictions.

Some 'extra strains,' namely those that mainly affect the largescales, appear to be relatively easy to incorporate into large eddysimulations; little, if any, modification of the SGS models is required.Others, which act on scales smaller than the Kolmogoroff scale, forexample, compressibility, may require significant changes in the SGSmodel.

ACKNOWLEDGMENTS

The author has been active in this field for over twenty years andthe list of people who have helped him is too long to be covered in ashort acknowledgment. I will, therefore, limit myself to mentioning afew people who have influenced my thinking in this area in the pastfew years. These include my colleagues: Profs. Peter Bradshaw, Jef-frey Koseff, Parviz Moin, Stephen Monismith and William Reynolds,and my students Matthew Bohnert and Kishan Shah. The supportreceived from a number of agencies over the year has also been very

150 J. H. Ferziger

important; these include NASA, the Office of Naval Research andthe Air Force Office of Scientific Research.

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Bagwell, T.G., Adrian, R.J., Moser, R.D. and Kim, J., 1993. "Improved approximation of wall shear stress boundary condi-tion for large eddy simulation," in Near Wall Turbulent Flows(R.M.C. So, C.G. Speziale and B.E. Launder eds.), Elsevier.

Bardina, J., Ferziger, J.H., and Reynolds, W.C., 1980. "Improvedsubgrid models for large eddy simulation," AIAA paper 80-1357.

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Clark, R.A., Ferziger, J.H., and Reynolds, W.C., 1979. "Evalua-tion of subgrid scale turbulence models using a fully simulatedturbulent flow," J. Fluid Mech., 91, p. 92.

Coleman, G.N., Ferziger, J.H. and Spalart, P.R., 1990. "A nu-merical study of the stratified turbulent Ekman layer," Rept.TF-48, Thermosciences Div., Dept. of Mech. Engr., StanfordUniv.

Deardorff, J.W., 1970. "A numerical study of three-dimensionalturbulent channel flow at large Reynolds number," J. FluidMech., 41, p. 452.

Deardorff, J.W., 1974. "Three dimensional numerical modeling ofthe planetary boundary layer," Boundary Layer Meteorology,1, p. 191.

Erlebacher, G., Hussaini, M.Y., Speziale, C.G., and Zang, T.A.,1992. "Toward the large-eddy simulation of compressible tur-bulent flows," J. Fluid Mech., 238, p. 155.

Ferziger, J.H., 1983. "Higher level simulations of turbulent flow," inComputational Methods for Turbulent, Transonic, and ViscousFlows, J.-A. Essers, ed., Hemisphere.

Ferziger, J.H., 1993. "Simulation of complex turbulent flows: recentadvances and prospects in wind engineering," in ComputationalWind Engineering 1, S. Murakami ed., Elsevier.

Ferziger, J.H. and Peric, M., 1994. "Computational methods forincompressible flow," in Computational Fluid Dynamics, (M.Lesieur and J. Zinn-Justin eds.), Elsevier.

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Gao, F. and O'Brien, 1993. "A large eddy simulation scheme forturbulent reacting flows," Phys. Fluids A 5, p. 1282.

Germane, M., Piomelli, U., Moin, P., and Cabot, W.H., 1990. "Adynamic subgrid scale eddy viscosity model," Proc. SummerWorkshop, Center for Turbulence Research, Stanford CA.

Ghosal, S., Lund, T.S., Moin, P., and Akselvoll, K., 1994. "A dy-namic localization model for large eddy simulation of turbulentflows," submitted to /. Fluid Mech..

Horiuti, K., 1990. "Higher order terms in anisotropic representationof the Reynolds stress," Phys. Fluids A 2, p. 1708.

Hussaini, M. Y., Speziale, C. G., and Zang, T. A., 1990. "Thepotential and limitations of direct and large eddy simulations,"Whither Turbulence? Turbulence at the Crossroads, Springer-Verlag, Berlin, pp. 354-368.

Kawamura, T., and Kuwahara, K., 1985. "Direct simulation ofa turbulent inner flow by a finite difference method," AIAApaper 85-0376.

Le, H., 1993. "Direct numerical solution of turbulent flow over abackward facing step," Dissertation, Dept. of Mech. Engr.,Stanford Univ.

Lee, S., Lele, S.K., and Moin, P., 1993. "Simulation of spatiallyevolving turbulence and the applicability of Taylor's hypothesisto compressible flow," Phys. Fluids A 5, pp. 1521-1530.

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Lesieur, M., 1991. "Turbulence in fluids," Second ed., Kluwer, Dor-drecht.

Leslie, D.C., 1973. "Theories of turbulence," Oxford U. Press.

Lilly, D.K., 1965. "On the computational stability of numerical so-lutions of time-dependent, nonlinear, geophysical fluid dynamicproblems," Mon. Wea. Rev., 93, p. 11.

Lilly, D.K., 1992. "A proposed modification of the Germano subgridscale closure method," Phys. Fluids A 4, p. 633.


Mason P.J., 1989. "Large eddy simulation of the convective atmo-spheric boundary layer," /. Atmos. Sci., 46, p. 1492.

Mason, P.J., and Calien, N.S., 1986. "On the magnitude of the sub-grid scale eddy-coefficient in large eddy simulation of turbulentchannel flow," J. Fluid Mech., 162, p. 439.

Mason, P.J., and Derbyshire, S.H., 1990. "Large eddy simulationof the stably stratified atmospheric boundary layer," Bound.-Layer MeteoroL, 53, p. 117.

Mauch, H., 1991. "Berechnung der 3-D umstroemung eines quadr-erfoermigen koerpers in kanal," Dissertation, Univ. Karlsruhe.

McMillan, O.J. and Ferziger, J.H., 1980. "Tests of new subgridscale models in strained turbulence," AIAA paper 80-1339.

Metais, O. and Lesieur, M., 1992. "Spectral large eddy simulationsof isotropic and stably stratified turbulence," J. Fluid Mech.,239, p. 157.

Moin, P. and Kim, J., 1982. "Large eddy simulation of turbulentchannel flow," J. Fluid Mech., 118, p. 341.

Moin, P., Carati, D., Lund, T., Ghosal S., and Akselvoll, K., 1994."Developments and applications of dynamic models for largeeddy simulation of complex flows," 74th AGARD Fluid Dy-namics Panel, Chania, Greece, April.

Piomelli, U., Ferziger, J.H. Moin, P., and Kim, J., 1989. "Newapproximate boundary conditions for large eddy simulations ofwall-bounded flows," Phys. Fluids A, 1, p. 1061.

Piomelli, U., Zang, T.A., Speziale, C.G. and Hussaini, M.Y., 1990."On the large-eddy simulation of transitional wall-boundedflows," Phys. Fluids A, 2(2), p. 257.

Piomelli, U., 1991. "Local space-time averaging in the dynamicsubgrid scale model," Bull. Amer. Phys. Soc., Vol. 35, No.10.

Reynolds, W. C., 1990. "The potential and limitations of directand large eddy simulations," Whither Turbulence? Turbulenceat the Crossroads, Springer-Verlag, Berlin, pp. 313-343.

154 J. H. Ferziger

Rodi, W. and Mansour, N.N., 1992. "Modeling the dissipation ratewith the aid of direct simulation data," Studies in Turbulence,(T. Gatski et al. eds.), Springer.

Rogallo, R.S. and Moin, P., 1984. "Numerical simulation of turbu-lent flows," Ann. Revs. Fluid Mech., Annual Reviews.

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Shah, K. and Ferziger, J.H., 1994."Simulation of flow over a wall-mounted cube," in preparation.

Smagorinsky, J., 1963. "General circulation experiments with theprimitive equations, part I: The basic experiment," Mon. Wea.Rev., 91, p. 99.

Speziale, C.G., 1987. "On nonlinear k — I and k — e models ofturbulence," /. Fluid Mech., 178, p. 459.

Speziale, C.G., Erlebacher, G., Zang, T.A. and Hussaini, M.Y.,1988. "The subgrid-scale modeling of compressible turbulence,"Phys. Fluids, 31, p. 940.

Tennekes, H., and Lumley, J.L., 1976. "A first course in turbu-lence," MIT Press.

Thompson, M.C. and Ferziger, J.H., 1989. "A multigrid adaptivemethod for incompressible flows," J. Comp. Phys., 82, p. 94.

Yakhot, V. and Orszag, S.O., 1986. "Renormalization group meth-ods in turbulence," J. Sci. Comp., 1, p. 3.

Yang, K.S. and Ferziger, J.H., 1992. "Large eddy simulation ofturbulent flow with a surface-mounted obstacle," Fifth AsianCong. Fluid Mech., Dae Jon, Korea.

Zang, Y., Street, R.L. and Koseff, J.R., 1993. "A dynamic mixedsubgrid scale model and its application to turbulent recirculat-ing flows," Phys. Fluids, 5, p. 3186.

Chapter 4

INTRODUCTION TORENORMALIZATIONGROUP MODELING OFTURBULENCE

Steven A. OrszagI. Staroselsky, W. S. Flannery, Y. Zhang

1 INTRODUCTION

The renormalization group (RNG) and related e-expansion meth-ods are a powerful technique that allow the systematic derivation ofcoarse-grained equations of motion for turbulent flows and, in par-ticular, the derivation of sophisticated turbulence models based onthe fundamental underlying physics. The RNG method provides aconvenient calculus for the analysis of complex physical effects incomplex flows. The details of the RNG method applied to fluidmechanics differ in some crucial respects from how renormalizationgroup techniques are applied to field theories in other branches ofphysics. At the present time, the RNG methods for fluid dynamicsare by no means rigorously justified, so their utility must be basedon the quality and quantity of results to which they lead. In thispaper we discuss the basis for the RNG method and then illustrateits application to a variety of turbulent flow problems, emphasizingthose points where further analysis is needed.

155

156 S. A. Orszag et al.

The application of a field-theoretic method like the RNG tech-nique to turbulence is based on the fundamental assumption of uni-versality of small scales in turbulent flows. Such universal behaviorwas first suggested over 50 years ago in the seminal work of A. N. Kol-mogorov who argued that the small-scale spectrum of incompressibleturbulence is universal and characterized by two numbers, the rateof energy dissipation £ per unit mass and the kinematic viscosityv. In fact, Kolmogorov predicted that the energy spectral density ofturbulence has the universal form

where k is the wavenumber and

is called the dissipation wavenumber and defines a scale below whichturbulent eddies are directly affected by viscosity. Kolmogorov ar-gued that this form of the energy spectrum would apply at scalessmall compared to those characterizing the inhomogeneities of theaverage flow. It is implicit in the Kolmogorov theory that the rateof energy dissipation £ is essentially independent of the viscosity v]this first fundamental law of turbulence implies that £ is indepen-dent of Reynolds number (R = VL/v, where V is the rms velocityof the turbulence and L is an appropriate large scale) and therefore6 = O(l) as R —>• oo. If this first fundamental law of turbulence istrue, then (2) shows that kd = 0(#3/4).

The Kolmogorov universal spectrum has two interesting specialcases. First, if k < kd, E(k] = CK£2/3k~5/3, where CK = .F(O),so the energy spectrum is a power-law with the universal exponent—5/3. In this so-called inertial range the energy spectrum is inde-pendent of viscosity. Second, if k >• kd it can be argued that F(x)must be of the form Cixae~C2X, so the energy spectrum decays ex-ponentially fast with increasing k in this so-called far dissipationrange (Chen et al., 1993). The wavenumber k^ separates inertial-range scales for which viscosity is not important and dissipation rangescales.

Experimental data demonstrating that £ — O(l) is still very lim-ited. The situation is somewhat better for the Kolmogorov spectrum(1) where a variety of experimental measurements have shown that

Renormalization Group 157

(1) is at least approximately satisfied. Measurements of the energyspectrum in the inertial range demonstrate that deviations from Kol-mogorov's 5/3 power law are small, although these deviations havedominated turbulence research for over two decades now through thesearch for so-called intermittency corrections to Kolmogorov's the-ory. The RNG technique discussed in this paper is based on theuniversality of the Kolmogorov spectrum. In contrast to the fluctua-tion theory of phase transitions, the RNG theory of turbulence is notused to calculate scaling exponents (like -5/3), but rather to calcu-late amplitudes (like C/<). If corrections to perfect Kolmogorov scal-ing are established at some future time, it will be possible to redo theRNG theory account for these effects in a systematic way. The RNGtheory of turbulence does not prove the validity of the Kolmogorovtheory; rather, the RNG theory provides quantitative predictions ofthe behavior of turbulent flows assuming that Kolmogorov's theoryholds.

The goal of a statistical theory of incompressible turbulence is togive a probabilistic description of a solenoidal velocity field v(x, t)governed by the incompressible Navier-Stokes equations (NSE):

supplemented by the boundary conditions that the velocity v at aboundary or interface matches the velocity of the boundary or inter-face. Ideally, such a statistical description of turbulence would yieldthe multi-point probability distribution function, which is equiva-lent to knowledge of all correlation functions of the system. A morelimited goal would be to obtain only certain velocity correlation func-tions, say, two-point correlation functions, which include informationon the energy spectrum as well as enough information for nearly allturbulence modeling requirements. Unfortunately, whereas the NSEare an explicit, well-defined system of partial differential equations,it is difficult to develop an efficient workable formalism to performstatistical averages on them that allows the development of predic-tive equations for the statistical properties of turbulence. In additionto the formal problem of developing efficient and effective averagingtechniques, it is necessary to build a formalism that reflects the uni-versality of the small scales of the flow; it is necessary to find some


way to de-emphasize the non-universal aspects of turbulence at largescales imposed by boundary and initial conditions. This is not atrivial matter because it is intuitively clear that boundary and initialconditions are directly responsible for turbulent energy production.

The goal of the RNG theory is, as mentioned above, the develop-ment of a quantitative description of small scales in turbulent flows.With such a quantitative theory of small scales available, it is pos-sible to "remove" the small scales from the turbulence dynamics,thereby deriving effective equations of motion for large scales and,in particular, turbulence models for the prediction of large-scale flowproperties. At finite Reynolds numbers, the 5/3 law holds in thelimited wavenumber range 1/L <C k <C k^. This range is limited be-cause the boundary conditions are imposed at a finite scale of orderL and the Reynolds number is finite.

To pursue the RNG analysis we first attempt to remove the con-straint due to boundary and initial conditions at the finite scale Lby, in effect, taking the limit L —>• oo. We do this by mimicking theeffect of initial and boundary conditions by imposing an artificialrandom force f in the NSE:

Once an appropriate force that represents random forcing at in-finitely large scales is imposed, the RNG theory proceeds to describethe resulting infinitely long inertial range. In particular, the RNGmethod is based on the assumption that the inertial range dynamicsis invariant under rescaling transformations, an assumption justifiedby the infinite extent of this range and its independence of viscousdissipation effects. As part of the RNG analysis, it is assumed thatthe turbulent dynamics in the inertial range can be described by a fi-nite number of effective transport coefficients, like eddy viscosity formomentum and eddy diffusivity for heat and mass transport. Whilethe rigorous basis of these steps is individually still open to question,the results to which they lead appear to have a remarkably closeagreement with reality.

Renorm&lization Group 159

2 PERTURBATION THEORY FOR THENAVIER-STOKES EQUATIONS

In this Section we introduce statistical perturbation methods toorganize the solution of the Navier-Stokes equations as an expansionin powers of effective Reynolds number. We do this in the context ofan unbounded flow in d dimensions but, for analytical convenience,we can equivalently view this flow as being confined to a cube hav-ing side £ on which periodic boundary conditions are imposed andthen the implicit limit £ —> oo being taken at the conclusion of allcalculations. The Fourier transform of the Navier-Stokes equations(3') is

where k = (k,w), u(fc) [f(A;)j is the space-time Fourier transform ofv(x,*)[f(x,i)],

and dq denotes the (d+ 1)-dimensional integral over the wavevectorand frequency components of q. In (5), AQ = 1 is a parameter thatallows us to conveniently determine the order of perturbation theoryto which we are working. The projection operator P;j(k) ensures theincompressibility of the flow; the term kikj/k2 in P,-j represents thepressure gradient Vp.

It is convenient to adopt a symbolic notation that absorbs boththe wavenumber and vectorial components into a single numericalsymbol, e.g., •u(l) = Uj1(ki) and where summation over repeated in-dices and integration over wavevectors is implied. Then (5) becomes

where the coupling operator g is defined as

The formal solution of the nonlinear stochastic partial differentialequation (7) is given by the Neumann series obtained by iterating(7):


Equation (9) expresses the fluctuating velocity u in terms of a func-tional power series in the random force /.

The goal of statistical perturbation theory is to obtain correla-tion functions of the random field u by averaging the series (9) overfluctuations of the random noise /. Of particular interest is the two-point correlation function (7(12) =< u(l)w(2) > which is directlyrelated to the energy spectrum and the nonlinear Green's (impulseresponse) function C?(12) =< <!m(l)/<5/(2) > which characterizes thefull nonlinear system response to small perturbations. The symbol<> indicates an ensemble average over all realizations of the randomfield /. The properties of the random noise / are chosen to facilitatethe evaluation of the terms in the expansions of U and G. Indeed, itis convenient to choose the force / to be zero-mean, divergence-free,white-in-time, and gaussian of the form /(I) = P,-1)t-2p(2) where p is arandom noise with the multi-point probability distribution function

The two-point correlation function of the random noise / is then

where

and the ^-function accounts for translational invariance in time andspace:

With this choice of gaussian random noise, multi-point averagesreduce to sums over products of pair correlations (10). In this case,the solution to the linearized Navier-Stokes equations leads to thetwo-point velocity correlation function

By applying these rules for averaging over the gaussian random noise/ to the velocity field and the nonlinear Green's function G, we ob-tain the full perturbation series for U and. G. The terms in these

Renormaliz&tion Group 161

series are usually classified based on the number of vertex opera-tors g, which is also twice the number of integrations over space-time wavenumbers. These integrations may or may not be indepen-dent, depending upon the detailed structure of the term. Certainterms may be identified as vanishing or non-existent, such as thosecontaining 0(123)^(23) oc Pili2ia(ki)6(l + 2 + 3)A'(23)<S(2 + 3) ocP t- l i 2i3(fci)tf(l) = 0, or 0(123)G(23), or any "closed cycle" of Green'sfunctions of the type G(12)G(23)...G(nl). Most generally, the fullseries can be conveniently represented using diagram methods [Wyld,1961; Kraichnan, 1961].

The lowest order nonvanishing terms appearing in the series forU and G are those of second order

It should be emphasized that these elementary terms already providebasic information about large-scale properties of the system. Analyz-ing the divergence of the associated integrals in wavenumber space, itis immediatedly found, for example, that the lowest order nonlinearcorrection to the Green's function is (omitting tensorial indices)

In order to describe fully developed turbulence, the forcing wouldhave been chosen at the largest spatial scales, i.e. D(q) a £d(q).This would immediately yield a formally infinite result ~ / 6(q)dq/q^.However, an infinite series of formally infinite terms may well givefinite and meaningful results, if summed properly. This motivates usnot to deal with the delta-functions directly but rather to introducean auxiliary power-law stirring force D(k] — Dok~y (see Orszag etal., 1993a for more details). Then (14) becomes

Equation (15) hints at the importance of the parameter e = 4 + y — dwhich controls the infrared divergence of the perturbation series.

162 S. A. Orsza,g et al.

3 RENORMALIZATION GROUP METHOD FORRESUMMATION OF DIVERGENT SERIES

Whereas there are methods of direct resummation of perturba-tion series of logarithmically diverging terms, there is no generalmethod to handle series with power-law divergences. However, thereis a different route, viz. RNG, that is based on ideas of scaling anduniversality. Suppose that at very large scales the effective couplingbecomes weak so that we can solve the problem perturbatively. Onan intuitive level, the main physical assumption is that there ex-ists an "eddy viscosity" which yields an effective (scale dependent)Reynolds number Re/f which is O(l) even as the "bare" Reynoldsnumber R, based on i/0, approaches infinity as the scale L —*• oo.In comparison with techniques of direct summation of perturbationseries, RNG methods are more robust and more appealing to ourphysical intuition about the significance or insignificance of eddy dy-namics at large scales.

The technical approach involved in RNG is that, rather thantrying to sum up the entire perturbation series, we treat the problemexplicitly using perturbative methods. In other words, we considerall expansions as performed in a true small coupling parameter g. Itwill be shown that, as R —> oo, the resummed perturbation seriesself-similarly approaches a limiting form, called a fixed point. Themain assumption of the RNG theory is that, at this fixed point, thenonlinear effective coupling (or the effective Reynolds number Reff)is small enough that useful results can be obtained by a perturbationexpansion in powers of Re/f. The logic of the RNG weak couplingapproach is checked by assuming it to hold, computing Reff andother quantities based on this assumption, and then checking the self-consistency of the results. It will be shown below that Reff oc A/£where e = 4 + y — d was introduced above. The case of (. — 0 will beshown to lead to zero coupling at the largest scales, so that Re/f —> 0as e —>• 0.

The basic idea of our approach is to iteratively remove narrowbands of wavenumbers, <5A, from the dynamic equations and therebyobtain new equations for the remaining variables. This is done as fol-lows. Suppose that at some stage of this renormalization process thedynamic variables involve wavenumbers from the band 0 < k| < A.When we begin this renormalization scheme, we choose A = kj butat later stages the moving cutoff A satisfies A < kj. We remove the


dynamic variables w>(^) in the band A — 6A < |k| < A by using theNeumann series (9) to obtain new expressions for the remaining vari-ables u<(A;), 0 < |k| < A — £A, expressing the dynamic variables forA — <*>A < |k| < A in terms of the random force in this narrow band,and then averaging over the corresponding subensemble of randomforces. At each stage of this process, the use of the Neumann series(9) is justified because <5A is small. This perturbation technique in <5Agenerates so-called recursion relations for the terms in the reduceddynamical system satisfied by n<(A;). The next step of the RNG pro-cedure is to solve these relations to obtain new dynamic equationsin the case where many narrow bands of modes are eliminated. Ofcourse, the errors in the reduced dynamical system that result fromsolving these relations need no longer be small because we are accu-mulating many small contributions. However, under the assumptionthat our Reff is small we proceed to use the dynamical system thatresults from the recursion relations. The justification for this laststep has not been given for turbulence, so it is only possible to judgethe validity of the RNG equations by the accuracy of the results towhich they lead. The advantage of the present explanation of theRNG technique is that the e—expansion (Re/f <C 1) is required onlyto eliminate high-order nonlinearities in the dynamical equations fortt<(A;) and not to evaluate constants appearing in the lowest orderdynamical equations for u<(A;).

When the velocity field is decomposed as

No approximations have yet been invoked. Equations (16) and (17)exactly represent the fluctuating Navier-Stokes dynamics in the cor-responding wavenumber bands. Computation of the Green's func-tion G<(12) =< <5it<(l)/(5/<(2) > involves averaging these stochas-tic PDEs over the subensemble of realizations />(A;) based upon therule (12) applied to the "<"-band equation resulting from the Neu-mann series (9):

(7) generates equations for the "<"-band and the '>"-band:


Upon taking the functional derivative < <5/<5/<(6) > in (18) andgaussian averaging over the subensemble in which the Fourier modesof / are held fixed for |k| < A — <5A, one obtains the correction to theGreen's function which is second-order in the coupling constant:

When explicit notations are introduced, say, (1) = (a,u — 0,k)and (6) = (/3,o; = 0,k'), (19) translates into the following analyticexpression for the correction to the viscosity:

where the symbol / indicates integration over the band A — <5A <q < A, and we have included the factor Pa;g(k) on the left side of(20) due to the incompressibility of u(k). Notice that this procedureof unconditional shell-averaging is asymmetric with respect to thearguments of the Green's function 6*0 and the correlation functionUQ] the wavenumber of UQ is restricted to the shell A — 6A < k < Awhile there is no similar constraint upon the wavenumber of GQ.

To compute the integral in (20) we proceed as follows. First,using (6) and performing the integral over the internal frequency %,noting that

we obtain


where

In the limit k/q —> 0, this latter integral has the asymtotic form

Using the following properties of the projection operators Pa/g(k) andPa^(k),

we find

This integral can be expressed in terms of the elementary angularintegrals

where Sd is the area of a d—dimensional unit sphere. Thus, weevaluate IIa/3(k, q, n) as

166 5. A. Orszag et al.

Substituting this result into (20/), we obtain the viscosity correction

where r = log (fcj/A). Using analogous methods one finds that thereis no "bare force renormalization", i.e. no contribution to the forcingD0k~y. However, a "thermal" noise proportional to k2 is generated,namely,

We will now rewrite this system in terms of scale-dependent non-linearities and thereby make contact with classical turbulence phe-nomenological concepts like local Reynolds numbers, effective viscos-ity, etc. We introduce two nondimensional "charges" of the theory:namely, the effective nonlinear coupling constant <?(A) = D0/z/3(A)Ac

and the 'thermal' coupling constant <7:r(A) = _DrAc'~2/i'3(A) whichis the dimensionless intensity of the generated 'thermal noise' DyA;2.The charge g(A) is proportional to [Reff(A.)}2. Indeed,

may be evaluated in terms of the random force correlation function< // > noting that w(A) oc G(A)/(A) where G(A) is of order of thecharacteristic response time o^A)"1 oc [z/(A) A2]"1:

where < // >oc D0A yui(A)Ad. Therefore, we obtain

The system dynamics can then be described in terms of the be-havior of 0,tfr). From (21), (22) it follows that


Once the autonomous system (23) is solved, the scale-dependentviscosity Z'(A) is determined from the differential equation:

that follows from (21).Observe that when d — 2 > 0, there is a fixed point of this system

in which QT decays rapidly to 0 for r increasing beyond (d — 2)"1.Therefore, for large r, or, equivalently, at very large spatial scales,r oc e"1, the equation for g? becomes redundant since g? is small.At these large r, the equation for g becomes

When r —>• oo, the RNG transformation given by (25) has a simplefixed point in which the coupling constant is indeed small:

as conjectured above. In the vicinity of this fixed point, the viscositybehaves as V(T) oc e~1 '3exp(—c/3r) and Reff ~ ^/^•

When d < 2, the constant g? quickly approaches a finite limitof order 1. There is no solution in which g and g? are small as€ —» 0. This means that, unlike the previous case, the applicability ofsecond order approximation is violated and there is no self-consistentasymptotic expansion in e — 4+y-d. The case d — 2 requires specialconsideration which is beyond the scope of this paper.

It is a key part of the RNG procedure to obtain simple reduceddynamical equations for «<(&) in which various terms in the formalperturbation expansion can be argued to disappear in true dynamicsas the integral scale £ -^ oo. An example of such an "irrelevantterm" is the self-generated thermal noise. It is easy to see that whene < 1 and d > 2,

which is small compared with the driving force of the system. Wecaution that as of yet there is no rigorous proof that all such termsare negligible.

168 S. A. Oisz&g et al.

These general RNG results can be used to analyze three-dimen-sional fully turbulent flow (d = 3, e = 4). In this case, there is noforce renormalization and the effective viscosity at the cutoff A isgiven by the solution of (25) with the initial condition f(A0) = i>0:

It should be emphasized at this point that since the effective Rey-nolds number is proportional to e1'2 to lowest order in €, all pertur-bation expansions used in the theory are valid at lowest order in efor e —> 0. It should also be noticed that the constant A^ is indepen-dent of e, in contrast to the result obtained by Yakhot and Orszag(1986). With the present approach, it is not necessary to e—expandAJ. to obtain the RNG result f(A). The difference from the earlierderivation lies in symmetry of the integral (20) with respect to thearguments of the Green's function and the correlation function. Inthe limit e —> 0, the symmetrized and unsymmetrized versions of (20)become the same.

The unknown amplitude D0 is determined by Dannevik et al.(1987) using the property of energy conservation by the nonlinearterms. Expanding to second order in e they recover the EDQNMequations introduced by Orszag (1977) which express the mean dis-sipation rate £ in terms of the lowest-order energy spectrum E(k)and the effective viscosity v(k}. Based on this representation, as wellas upon Equation (27), they obtain the relation

Using (28), one finds

The coefficients of (29) and (30) agree well with the experimentallyobserved parameters.


4 TRANSPORT MODELING

The methods described above have also been used to derive trans-port models in which all the scales up to the integral scale of tur-bulent flow are averaged over. The RNG transport model describesthe dynamics of the mean flow /7;, turbulence kinetic energy li', andenergy dissipation rate £ and defines the eddy transport coefficientsin terms of the latter. For example, the eddy viscosity in the RNGK — £ model is given by

with C/j, = 0.0845, in good agreement with the "standard" value.RNG theory gives this result by eliminating the length scale I be-tween the expression (27) for turbulent viscosity and the followingexpression for the total kinetic energy in isotropic inertial range ed-dies at scales smaller than I:

Here the constant 0.71 is obtained from the Kolmogorov constantCK by integrating the inertial range energy spectra to the cutoffscale / = 2ir/k. Another advantage of the RNG theory is that it alsoallows us to interpolate (31) into the low Reynolds number regionsand obtain more general expressions which are valid across the fullrange of flow conditions from low to high Reynolds numbers.

The high Reynolds number form of the RNG K — £ model isgiven by

where i/j° = veddy + vmo\ and the rate-of-strain term R given by


This latter term is expressed in the RNG K — £ model equations as

where 77 = SK/£ and S2 = ISijSij is the magnitude of the rate-of-strain. The RNG theory gives values of the constants Csl = 1.42and C'£2 = 1.68, and a = 1.39, in comparison with the "standard"values (7^ fy 1.4 and Cs2 « 1.9, and a = 1.

For turbulent heat convection problems, the temperature field isgoverned by:

The inverse turbulent Prandtl numbers a in (34)-(35) and (38) areobtained from the RNG scalar heat transport relation derived anal-ogously to (27):

where for the heat transfer problem a0 refers to the molecular inversePrandtl number.

There are two interesting properties of the RNG transport modeldescribed above which lead to reduced eddy viscosity. First, the re-duced value of (7f2 compared with the standard S -equation coefficient(^standard ^ ^<^ j^g ^g interesting consequence of decreasing boththe rate of production of K and the rate of dissipation of S, leadingto smaller eddy viscosities. In regions of small 77, the .R-term tendsto increase eddy viscosity somewhat, but it is still typically smallerthan its value in the standard model. In regions of very large 77,where strong anisotropy exists, R can become negative and reducesthe eddy viscosity even more. This feature of the RNG model isperhaps responsible of the marked improvement in anisotropic large-scale eddies. The /^-dependency of C'£2 makes it possible to have aspatially as well as a temporally varying balance between productionand dissipation terms in the ^-equation. This feature of dynamicbalance, based on the RNG theory, is unique. Secondly, in regionsof low turbulence Reynolds numbers the RNG eddy viscosity has acut-off below which the eddy viscosity is zero.

The RNG K — £ model has been successfully applied to a numberof difficult turbulent flow problems. Below, we present some results


to illustrate the ability of this model to capture complex flow phe-nomena such as transitional behavior in turbomachinery heat trans-fer, and time-dependent behavior of large-scale structures.

Turbomachinery Heat Transfer

Two cases of predicting turbomachinery heat transfer will be dis-cussed, the so-called Langston cascade (Graziani et al., 1979) which islarge-scale and low-speed, and a transonic nozzle guide vane. For theformer case, the main difficulty is to accurately predict the patternof the heat transfer coefficient distribution over the suction surfacecorresponding to flow relaminarization and subsequent transition toturbulence. In the latter case, compressibility and an extremely highturning angle further complicate the subtle heat transfer behaviorover the surface.

In Figs. 1-2 we plot heat transfer data compared with the resultsof Graziani et al. as well as with results obtained from alternativeturbulence/transition models. The data is expressed in terms of theStanton number, Si = h/(pCpUo), where h is the heat transfer co-efficient defined as h = qgen/AT, qgen is the heat flux at the bladesurface, and p, Cp, and UQ are the values at the inlet of the density,specific heat, and velocity. In Fig. 1, we plot a comparison of theRNG results and the measured distribution of the Stanton number.Good qualitative agreement with the experimental data is observed.The quantitative agreement is excellent in the leading edge regionand good in the relaminarization region. In Fig. 2, we plot a com-parison of the Stanton number predictions for the RNG and standardK — £ models and for laminar heat transfer - in the laminar case, theflow field is the turbulent flow field determined by the RNG modelbut the turbulent heat transfer coefficients predicted by the RNGtheory are replaced by laminar heat transfer coefficients. Evidently,the standard K — £ model overpredicts the heat transfer for this caseby nearly a factor 2 and does not capture the transitional flow ef-fects observed near the leading edge on the suction side. On the otherhand, the laminar heat transfer model does do a good job near theleading edge, but does not account for the increase in heat transfernear the transition point. Overall, the RNG model has demonstrateda significant improvement over existing prediction methods, withoutthe use of an empirically fitted transition point.

Extensive parametric studies of heat transfer in this cascade werealso performed to investigate the effects of Reynolds number and in-

Figure 1. Distributions of Stanton number on the blade surface forthe Langston cascade: RNG-based computations versus experiment.

let turbulence intensity level on heat transfer patterns. Boundaryconditions for the heat transfer problem were the same for each runand corresponded to constant heat flux qgen — I.31kW/m? at theairfoil surface and ambient flow temperature at the inlet. The origi-nal experiment was done at a Reynolds number (based on axial chordand inlet velocity) of ReQ = 5.5 X 105 and inlet turbulence intensitylevel of 1%. In Fig. 3, we show the distributions of Stanton numberalong the airfoil surface corresponding to different Reynolds num-bers. The curves are labeled by the Reynolds number relative toRCQ. These data indicate that boundary layer transition occurs atRe between 3.3 X 10s and 3.6 X 105: the Stanton number distributionsat 3.3 X 105 and lower match the laminar pattern. Heat transfer pat-terns corresponding to Re0 = 5.5 x 10s and different inlet turbulenceintensity levels are shown in Fig. 4. Virtually no variation in theStanton number distributions was detected at turbulence intensitieslower than 0.5%.

The second case is a simulation of experiments performed in the



Figure 2. Comparison of computed distributions of Stanton numberon the blade surface for the Langston cascade.

von Karman Institute short duration Isentropic Light Piston Com-pression Tube facility CT-2 of a high pressure turbine nozzle guidevane (von Karman Institute for Fluid Mechanics Technical Note 174,1990). In Fig. 5 we plot the distribution of heat transfer coefficientin W/m^K as a function of the distance along the blade surface,with the suction side shown on the left side of the plot and pressureside on the right side. The accurate quantitative prediction of bladeheat transfer over almost the entire airfoil surface is readily observ-able. The two spikes in computed heat transfer coefficient representpurely numerical artifacts due to the junctions of the computationaldomains. By comparison the standard K — £ model overpredicts heattransfer by a factor of 2-5 due to significant overproduction of turbu-lence throughout the passage. At the same time, the distribution ofturbulent viscosity provided by the RNG model leads to an quanti-tative prediction of blade heat transfer over most of the entire airfoilsurface. Again, it should be emphasized that both the RNG andstandard model - based calculations were done using general pur-

Figure 3. Distributions of Stanton number over the blade surface asa function of Reynolds number. The curves are marked according tothe fraction of the nominal Reynolds number of 5 X 105 ( based oninlet velocity and axial chord ).

pose codes which contain no adjustable parameters characterizingthe turbine geometry.

Time-dependent Turbomachinery Computations

Some recent results from the computation of the flow over a com-pressor trailing edge indicate that steady-state computations proveinadequate in flows with large-scale unsteady structure. It will beseen that when these calculations have a steady-state solution it doesnot correspond to the time-averaged flow field. On the other hand,the RNG transport model, which has a higher effective Reynolds


Figure 4. Distributions of Stanton number over the blade surfaceas a function of inlet turbulence intensity. The curves are markedaccording the intensity at the inlet in percent (100 X u'/Uo).

number than the standard K—£ model, may be used to perform time-dependent, so-called Very Large Eddy Simulations (VLES). VLESrefers to the capability of the model to correctly predict the behav-ior of anisotropic eddies with a size of the order of the characteristicsize of the problem, in this case the thickness of the trailing edge.

The Reynolds number for the flow under consideration is 56,400based on the diameter of the half-cylinder trailing edge. The inlet,taken at 10.6 diameters upstream, is a symmetrical zero pressuregradient turbulent boundary layer set to match experimental con-ditions. A number of runs were computed including steady-stateand time-dependent cases, and a steady-state case with a splitter


Figure 5. Comparison of computed and experimental distributions ofheat transfer coefficient on the blade surface for a high-load turbinenozzle guide vane.

plate. First, steady-state computations of the trailing edge failed topredict the strength of the vortex in the wake yielding significantlyover-predicted base pressures in the wake. As seen in Fig. 6, the"symmetrizing" effect of the steady-state computation on the distri-bution of the pressure coefficient Cp is close to that obtained by theaddition of a splitter plate in the wake. The streamlines shown inFig. 7 also demonstrate that the size of the vortex is the same inboth cases.

Next, a time-dependent flow with the RNG K - £ model wascomputed. The computed Strouhal frequency is approximately 0.2 inreasonable agreement with experiment. In Fig. 8 the time-averageddistribution of Cp for the RNG K - £ run is plotted. The time-averaged VLES result is also in reasonable agreement with exper-iment and gives rise to the strong pressure minimum in the wake.The time-dependent run gives a strong vortex as indicated in Fig. 9,where both the magnitude and location of the velocity minimum a,rein reasonable agreement with the experimental values of-0.2 and 0.4


Figure 6. Pressure coefficient distributions over the surface and inthe wake of a compressor trailing edge for steady-state computationswith and without a splitter plate, and experimental splitter plateresults (Re = 56,400 based on freestream velocity and trailing edge(TE) thickness).

respectively.Finally, a time-aver aged calculation using the standard K — £

model was performed for comparison. A comparison of the time-averaged results for Cp with experiment is shown in Fig. 8. As withthe steady-state RNG computations, the base pressure in the wake issignificantly overpredicted by the standard model. Excessive turbu-lent production leads to large eddy viscosities and hence mean wakevortices which are too weak. It appears that the VLES RNG model,by producing less turbulence as a result of lower eddy viscosities, canpredict the time-dependent behavior of the (very) largest eddies.

Flapping-Hydrofoil Simulations

The flow over a two dimensional hydrofoil subject to vertical gustsat high reduced frequency has been simulated, and a comparisonwith the experiments performed in the MIT Variable Pressure WaterTunnel was done. A modified NAG A16 hydrofoil with chord length

Renormalization Group lj",

Figure 7. Computed steady-state streamlines in the wake of com-pressor trailing edge with and without a splitter plate.

of 18 inches and maximum thickness of 8.84% is mounted in thetest section on the centerline of the tunnel. Upstream there aretwo flappers operating at a reduced frequency K — 3.6 and withan amplitude of 6 degrees. The Reynolds number based on testfoil chord length and freestream velocity is 3.78 X 106. Turbulenceintensity was measured in the empty test section and was found tobe 1%. In the case of steady flow, measurement of velocities andpressure was taken on three nested boxes bounding the test foil.Surface pressure data were taken on eight locations and boundarylayer velocity data were taken in nine location on the suction sideand seven locations on the pressure side. There were also two sets ofwake data taken.

The computations were performed using the FLUENT code withthe RNG K — £ turbulence model. For the case reported below, thesimulation was for steady flow at an angle of attack of 1.18°. Thecomputational domain spans the tunnel height with the upstream in-let coinciding with the location where experimental measurement ofthe flow data is available. The geometry and computational domainare depicted in Fig. 10. A C-type grid was built around the test foiland there are 301x166 total grid points with approximately 35 gridpoints spanning the boundary layer. The upstream boundary con-ditions were taken from the experimental data, in the two small re-


Figure 8. Pressure coefficient distributions over the surface and inthe wake of a compressor trailing edge for time-aver aged computa-tions of RNG VLES models and Standard K — e with experiment(Re — 56,400 based on freestream velocity and trailing edge (TE)thickness).

gions between upper/bottom tunnel wall and the flapper wake whereexperimental data is not available, uniform velocity profiles were as-signed.

Two sets of runs were performed with the same grid and bound-ary conditions, but using different turbulence models. One used thestandard K — C model and the other used RNG K — £ model. Theskin friction coefficients are plotted in Fig 11. On the suction sidewith x/C > 0.4 where experiment data were available, the computa-tion with the RNG K — £ model produced excellent agreement whilethe computation results using the standard K — E model severelyoverpredicted the skin friction. The results from the RNG K — £model predicted the separation point (shown by the location whereskin friction first turns negative) at x/C = 0.968, while the exper-iment data showed the separation point between x/C — 0.972 andx/C — 0.990. The standard K — £ model, on the other hand, in-

Renorma,liza,tion Group 179


Figure 9. Comparison of the computed time-averaged centerline ve-locity in the near wake of the compressor trailing edge.

Figure 10. The geometry and computational domain for flapping-hydrofoil simulations.

dicates no separation. The RNG K — £ model computation alsopredicted transition regions at x/C ~ 0.09 on the suction side andat x/C ^ 0.13 on the pressure side. Since the experiment used atrip device to trigger transitions on the test foil, comparison is notavailable in these regions.

There now have been well over fifty substantially different testflows on which the RNG K — 8 equations have been successfullytested (e.g. Orszag et al, 1993b). This broad range of successes


Figure 11. Comparison of computed and experimental skin frictioncoefficients for the suction side of the hydrofoil.

provides, we believe, an important indirect verification of the ideasentering the RNG theory of turbulence. Indeed, turbulence modelsare very sensitive to their details and we believe it is quite unlikelythat a "wrong" theory would be likely to produce the breadth ofsignificant engineering results available so far. In effect, this meansthat we have a reasonable working theory of turbulence at least forengineering calculations. Much further work remains to be done toextend the range of applications, determine the limits of validity ofthe theory, and, hopefully, provide a rational basis for it as a physicaltheory of turbulence.

ACKNOWLEDGMENTS

This work has been supported by the ONR, NASA, and UnitedTechnologies Research Center. Discussions with Victor Yakhot havebeen a constant source of stimulation. We would also like to ac-knowledge discussions with Torn Barber and Dochul Choi of UTRC.Finally, we should like to acknowledge our collaborations with Fluent,Inc., especially Ferit Boysan, Nelson Carter, Dipankar Choudbury,


Joe Maruzewski, and Bart Patel.

5 REFERENCES

Chen, S., Doolen, G., Herring, J. R., Kraichnan, R. H., Orszag,S. A. and She, Z.-S., 1993. "Far dissipation range of turbu-lence," Phys. Rev. Letts. 70, p. 3051.

Dannevik, W., Yakhot, V., and Orszag, S. A., 1987. "Analyticaltheories of turbulence and the e—expansion," Phys. Fluids A30, p. 2021.

Graziani, "R. A., Blair, M. F., Taylor, J. R., and Mayle, R. E.,1979. "An Experimental Study of Endwall and Airfoil SurfaceHeat Transfer in a Large Scale Blade Cascade," ASME Paper79-GT-99.

Lurie, E. H., 1993. "Unsteady Response of a Two-Dimensional Hy-drofoil Subject to a High Reduced Frequency Gust Loading,"M.S. Thesis, Massachusetts Institute of Technology.

Kraichnan, R.H., 1961. "Dynamics of nonlinear stochastic sys-tems," J. Math. Phys. 2, pp. 124-148. Erratum: 3, p. 205,1962.

Orszag, S.A., 1977. "Statistical theory of turbulence," In: FluidDynamics 1973, Les Houches Summer School in Physics. Eds.R. Balian and J.-L. Peabe, Gordon and Breach, pp. 237—374.

Orszag, S.A., Staroselsky, I., and Yakhot, V., 1993a. "Some basicchallenges for large eddy simulation research," In: Large EddySimulation of Complex Engineering and Geophysical Flows. (B.Galperin and S.A. Orszag, eds). Cambridge University Press.

Orszag, S. A., Yakhot, V., Flannery, W. S., Boysan, F., Choudbury,D., Maruzewski, J., and Patel, B., 1993b. "Renormalizationgroup modeling and turbulence simulations," In: Near-WallTurbulent Flows. Eds. So, R. M. C., Speziale, C. G. andLaunder, B. E., Elsevier Science Publishers, pp. 1031-1046.

Paterson, R.W., 1984. "Experimental Investigation of a SimulatedCompressor Airfoil Trailing Edge Flowneld," AIAA Paper 84-0101.


Yakhot, V., and Orszag, S.A., 1986. "Renormalization group anal-ysis of turbulence. I. Basic theory," J. Sci. Comp., 1, pp. 3-51.

Wyld, H.W., 1961. Formulation of the theory of turbulence in anincompressible fluid," Ann. Phys. 14, pp. 143-165.

Chapter 5

MODELING OFTURBULENT TRANSPORTEQUATIONS

Charles G. Speziale

1 INTRODUCTION

The hlgh-Reynolds-number turbulent flows of technological im-portance contain such a wide range of excited length and time scalesthat the application of direct or large-eddy simulations is all but im-possible for the foreseeable future. Reynolds stress models remainthe only viable means for the solution of these complex turbulentflows. It is widely believed that Reynolds stress models are com-pletely ad hoc, having no formal connection with solutions of thefull Navier-Stokes equations for turbulent flows. While this belief islargely warranted for the older eddy viscosity models of turbulence,it constitutes a far too pessimistic assessment of the current gen-eration of Reynolds stress closures. It will be shown how second-order closure models and two-equation models with an anisotropiceddy viscosity can be systematically derived from the Navier-Stokesequations when one overriding assumption is made: the turbulenceis locally homogeneous and in equilibrium.

A brief review of zero equation models and one equation mod-els based on the Boussinesq eddy viscosity hypothesis will first beprovided in order to gain a perspective on the earlier approaches to

185

186 C. G. Spezide

Reynolds stress modeling. It will, however, be argued that since tur-bulent flows contain length and time scales that change dramaticallyfrom one flow configuration to the next, two-equation models con-stitute the minimum level of closure that is physically acceptable.Typically, modeled transport equations are solved for the turbulentkinetic energy and dissipation rate from which the turbulent lengthand time scales are built up; this obviates the need to specify thesescales in an ad hoc fashion. While two-equation models represent theminimum acceptable closure, second-order closure models constitutethe most complex level of closure that is currently feasible from acomputational standpoint. It will be shown how the former modelsfollow from the latter in the equilibrium limit of homogeneous turbu-lence. However, the two-equation models that are formally consistentwith second-order closures have an anisotropic eddy viscosity withstrain-dependent coefficients - a feature that most of the commonlyused models do not possess.

For turbulent flows that are only weakly inhomogeneous, fullReynolds stress closures can then be constructed by the additionof turbulent diffusion terms that are formally derived via a gradienttransport hypothesis. Properly calibrated versions of these modelsare shown to yield a surprisingly good description of a wide range oftwo-dimensional mean turbulent flows that are near equilibrium. Inparticular, the stabilizing or destabilizing effect of a system rotationon turbulent shear flows is predicted in a manner that is consistentwith hydrodynamic stability theory. However, existing second-orderclosures are not capable of properly describing turbulent flows thatare far from equilibrium and have major problems with wall-boundedturbulent flows.

High-speed compressible turbulent flows present a whole newrange of problems to Reynolds stress modeling. It has now becomeclear that the traditional approach of using variable density exten-sions of incompressible Reynolds stress models is not sufficient forthe reliable prediction of the high-speed compressible flows of aero-dynamic importance. Reynolds stress closures are needed where tur-bulent dilatational effects are accounted for in a physically consistentfashion. Recent efforts in the modeling of the dilatational dissipationand pressure-dilatation correlations have led to significant advancesin the description of free turbulent shear flows at high Mach num-bers. However, a variety of problems still remain. All of the centralpoints of the paper will be illustrated by examples and an assessment

Turbulent Transport Equations 187

will be made concerning future directions of research.

2 INCOMPRESSIBLE TURBULENT FLOWS

The governing equations of motion for incompressible turbulenceare:

Navier-Stokes Eq.

Continuity Eq.

where

Here, P is a solution of the Poisson equation

2.1 Reynolds Averages

The velocity and pressure are decomposed into mean and fluctu-ating parts, respectively, as follows (cf. Hinze 1975):

where an overbar represents a Reynolds average. This Reynolds av-erage can take a variety of forms for any flow variable 0:

Homogeneous Turbulence

188 C. G. Speziale

Statistically Steady Turbulence

General Turbulence

(f) = <j)(x,t) (Ensemble Average over N repeated experiments)

It is assumed that the Ergodic Hypothesis applies. In a homogeneousturbulence,

whereas in a statistically steady turbulence,

Reynolds Averaging Rules

Given that

then


2.2 Reynolds-Averaged Equations

The Reynolds average of the equations of motion (1) - (2) aregiven by:

Navier-Stokes Eq.

Continuity Eq

where

is the Reynolds stress tensor (note that dui/dxi = 0).

2.3 The Closure Problem

The Reynolds-averaged Navier-Stokes equation (10) is not closedunless a model is provided that ties the Reynolds stress tensor T{J tothe global history of the mean velocity Ui in a physically consistentfashion. In mathematical terms, r^ is assumed to be a functional ofthe global history of the mean velocity field, i.e.,

The oldest Reynolds stress closures are based on the Boussinesq eddyviscosity hypothesis:

where

For incompressible turbulent flows, the isotropic part of the Reynoldsstress tensor, given by |/f (here, T;J — \K&ij + DTij where K = |r,-,-is the turbulent kinetic energy), can simply be absorbed into themean pressure.

190 C. G. Spezi&le

2.4 Older Zero- and One-Equation Models

The eddy viscosity is given by

where to = turbulent length scale and to = turbulent time scale.

Zero-Equation Models

Here, both IQ and to are specified algebraically by empirical means.The first successful zero equation model based on the Boussinesqeddy viscosity hypothesis was Prandtl's mixing length theory formu-lated in the 1920's. In the mixing length theory of Prandtl (1925),

where IQ = Ky is the mixing length (K is the Von Karman constantand y is the normal distance from a solid boundary). This represen-tation is only formally valid for thin turbulent shear flows - near asolid boundary - where the mean velocity is of the simple unidirec-tional form v = w(j/)i.

Four decades later, this type of model was generalized to three-dimensional turbulent flows:

Smagorinsky (1963) Model

where

is the mean rate of strain tensor;

Baldwin and Lomax (1978) Model

where u> = Vxv is the mean vorticity vector. The former model hasbeen primarily used as a subgrid scale model for large-eddy simula-tions whereas the latter model has been used for Reynolds-averagedNavier-Stokes computations in aerodynamics (see Wilcox 1993).


One-Equation Models

The eddy viscosity in these models is typically given by

where K = \U{U{ is the turbulent kinetic energy which is obtainedfrom a separate modeled transport equation. Models of this typewere first proposed independently by Prandtl (1945) and by Kol-mogorov (1942) during the Second World War.

More recently, Baldwin and Earth (1990) and Spalart and All-maras (1992) proposed one-equation models wherein a modeled trans-port equation is solved for the eddy viscosity z/x- One-equation mod-els are superior, to zero-equation models in that the time scale to ofthe eddy viscosity is built up from turbulence statistics rather thanfrom the mean velocity gradients. However, both of these types ofmodels have limited predictive capabilities due to the fact that theturbulent length scale IQ has to be specified empirically. This is vir-tually impossible to do in complex three-dimensional turbulent flows.The turbulent length and time scales are not universal; they dependstrongly on the flow configuration under consideration.

It is thus argued that two-equation turbulence models - whereintransport equations are solved for two independent turbulence quan-tities that are directly related to IQ and to - represent the minimumacceptable level of Reynolds stress closure. These models shouldbe formulated with a properly invariant anisotropic eddy viscositymodel that is nonlinear in the mean velocity gradients. The stan-dard Boussinesq eddy viscosity hypothesis makes it impossible toproperly describe turbulent flows with:

(i) Body force effects arising from a system rotation or from stream-line curvature;

(ii) Flow structures generated by normal Reynolds stress anisotro-pies (e.g., secondary flows in non-circular ducts).

In the most common approach to two-equation models, 10 oc K3/2/e,to oc K/e and modeled transport equations for K and £ are solved(here e = ̂ f^ff1: is the turbulent dissipation rate). In lieu of e, theinverse time scale e/K has also been used (see Wilcox 1993).

Limitations in computer capacity, and issues of numerical stiff-ness, make second-order closure models - wherein transport equa-tions are solved for the individual components of T,-J along with a

192 C. G. Speziale

scale equation - the highest level of closure that is currently feasiblefor practical calculations.

2.5 Transport Equations of Turbulence

The transport equation for the fluctuating velocity U{ is obtainedby subtracting the Reynolds-averaged Navier-Stokes equation (10)from its unaveraged form (1) which yields:

Eq. (13) can be written in operator notation as JV"ti,- = 0.

Reynolds Stress Transport Equation

This equation is obtained by constructing the second moment

which is given by (cf. Hinze 1975)

where

The transport equation for the turbulent kinetic energy (K = ^TH)is obtained by a simple contraction of the Reynolds stress transportequation (14):

where


Dissipation Rate Equation

We construct the ensemble mean

which yields the dissipation rate equation

Both two-equation models and second-order closure models areobtained from the Reynolds stress transport equation as follows:

(i) Second-order closures are obtained by modeling the full in-homogeneous Reynolds stress transport equation (14);

(ii) Two-equation turbulence models with an anisotropic eddy vis-cosity are obtained algebraically from (14) by assuming thatthe turbulence is locally homogeneous and in equilibrium.

2.6 Two-Equation Models

Representation for the Reynolds stress tensor

For homogeneous turbulence, the Reynolds stress transport equationreduces to

194 C. G. Speziale

or

where (cf. Lumley 1978 and Reynolds 1987)

Since, from (3) - (4)

it follows that

Here,

which can be non-dimensionalized as follows:

It is assumed that an equilibrium state is reached where

achieve constant values that are independent of the initial condi-tions. With no loss of generality, it can then be shown that for two-dimensional mean turbulent flows in equilibrium (Speziale, Sarkar


and Gatski 1991):

where

For the moment, we will neglect the anisotropy of dissipation dijand the quadratic return term containing Cj. In many studies, theformer assumption is justified by invoking the Kolmogorov hypothesisof local isotropy for high Reynolds number turbulent flows. However,this assumption is somewhat debatable as we will see later.

The coefficients C\ — €4 were obtained by Speziale, Sarkar andGatski (1991) based on a calibration with homogeneous shear flowexperiments (see Table 1).

EquilibriumValues

(&ll)oo

(^22)00(£"12)00

(StfA)oo

LRR Model0.158-0.123-0.1875.32

SSG Model0.204-0.148-0.1565.98

Experiments0.201-0.147-0.1506.08

Table 1. Comparison of the predictions of the Launder, Reece andRodi (LRR) model and the Speziale, Sarkar and Gatski (SSG) modelwith the experiments of Tavoularis and Corrsin (1981) on homoge-neous turbulent shear flow.

For homogeneous turbulence in equilibrium, 6,-j achieves equilib-

196 C. G. Speziale

rium values that are independent of the initial conditions. Hence,

in equilibrium. Since,

it then follows from (24) that the Reynolds stress transport equationhas the equilibrium form

where (see Gatski and Speziale 1993):

By the use of integrity bases from linear algebra, Pope (1975) firstshowed that the general solution to the implicit algebraic stress equa-tion (26) is of the form:

where


are the integrity bases and {•} denotes the trace.

Three-Dimensional Solution

The resulting solution for G^ is given by (see Gatski and Speziale1993):

where the denominator D is given by

and

Taulbee (1992) derived a simplified, but highly special, three-dimensional solution for the degenerate case where €3 = 2. Theexplicit solution for two-dimensional mean turbulent flows simpli-fies considerably since only the first three integrity bases are linearlyindependent in this cases (see Pope 1975 and Gatski and Speziale1993):

where

198 C. G. Speziale

In more familiar terms, this expression is equivalent to

Here, 0:1,0:2 and 0:3 are not constants but rather are related to thecoefficients C\ — C$ and g. In mathematical terms, they are "projec-tions" of the fixed points of Aij and Mijkt onto the fixed points ofbij - quantities that can vary from one flow to the next. However,for two-dimensional mean turbulent flows, it appears that C\ — €4can be approximated by constants due to the linear dependence onb^ which allows us to use the principle of superposition. (For three-dimensional turbulent flows, the general representation for Mijkf isnonlinear in bij which leads to an inconsistency that will be discussedlater).

It should be noted that these models obtained via the algebraicstress approximation need to be regularized before they are appliedto complex turbulent flows. This can be accomplished via a Fadetype approximation whereby we take (Gatski and Speziale 1993):

This constitutes an excellent approximation for turbulent flows thatare close to equilibrium with a form that is regular for turbulentflows that are far from equilibrium (the original expression yields asingular or negative eddy viscosity for sufficiently large values of 77 -a feature that can lead to divergent computations).

It should be further noted that Yoshizawa (1984), Speziale (1987),Rubinstein and Barton (1990), and Zhou et al. (1994) have derivedmodels of this general form - that are tensorially quadratic with con-stant coefficients - via expansion techniques based, respectively, ontwo-scale DIA, continuum mechanics, e-RNG (see Yakhot and Orszag1986 and Yakhot et al. 1992), and recursion RNG techniques. How-ever, due to the constant coefficients, these models are not consistentwith second-order closures and they have dispersive terms that, growunbounded with the strain rate - a feature that can destabilize com-putations.


This explains why previous anisotropic corrections to eddy vis-cosity models have not fully succeeded:

(i) The coefficients should depend nonlinearly on the invariants ofthe rotational and irrotational strain rates.

(ii) Only the traditional algebraic stress models - based on thesolution of the implicit algebraic Eq. (26) such as the modelof Rodi (1976) - have had such a dependence and they areill-behaved!

Problem (ii) can now be overcome by a regularization procedurebased on a Fade type approximation as discussed above. In rotatingframes, Coriolis terms must be added to the r.h.s. of the Reynoldsstress transport equation (18). Gatski and Speziale (1993) showedthat this analysis exactly accounts for such non-inertial effects inrotating frames if the extended definition of W*, is used:

where fim is the angular velocity of the reference frame.If we have a separation of scales, then 77, £ <C 1 and, in the linear

limit, we recover the eddy viscosity model

which forms the basis for the standard K — e model of Launder andSpalding (1974). However, in practical turbulent flows we do nothave a separation of scales; T] and £ are of O(l). Nonetheless, fortwo-dimensional turbulent shear flows in equilibrium, the new modelof Gatski and Speziale (1993) yields

with

which is remarkably close to the value of CM = 0.09 used in thestandard K — £ model.

200 C. G. Spezi&le

The Dissipation Rate Model

In homogeneous turbulent flows, the turbulent kinetic energy is asolution of the transport equation

Hence, closure is achieved once a transport equation is provided fore in terms of T^J and dvi/dxj. The exact transport equation (17) fore in homogeneous turbulence can be written in the form:

where

This equation can be rewritten as

where

For isotropic turbulence,

The major contributions to this integral occur at high wavenum-bers where the energy spectrum E ( K , t ) oc E(Klk) given that Ik =


^3/4^1/4 jg ^e Kolmogorov length scale. This Kolmogorov scalingyields (Bernard and Speziale 1992):

Hence,

We then invoke the standard high-Reynolds-number equilibrium hy-pothesis whereby

Models for d^j and d^ can be obtained from an analysis of thetransport equation for the tensor dissipation which, for homogeneousturbulence, is given by (see Durbin and Speziale 1991):

where

In physical terms,

Nij = Production by Vortex Stretching - Destructionby Viscous Diffusion

A -If = Structure and Redistribution Term.

202 C. G. Spezia

Speziale and Gatski (1992) proposed the following models:

based on an expansion technique that makes use of tensor invariance,symmetry properties and the fact that dij is small so that only linearterms need to be maintained.

The standard equilibrium hypothesis is made where:

This leads to an algebraic system of equations analogous to thatobtained in the algebraic stress approximation discussed earlier. Fortwo-dimensional mean turbulent flows, the exact solution is:


where

(Speziale and Gatski 1992). The substitution of these algebraic equa-tions into the contraction of the e;; transport equation yields thescalar dissipation rate equation

where

It should be noted that the contraction of (40) yields dlydvi/dxj— adijdvi/dxj which was used to obtain this result. The constants03 and CE5 were evaluated using DNS results for homogeneous shearflow (Rogers, Moin and Reynolds 1986). The constant C£2 can beevaluated by an appeal to isotropic turbulence. This dissipation ratemodel predicts that the turbulent kinetic energy decays according tothe standard power law in isotropic turbulence (cf. Speziale 1991):

Based on this, we have taken Cei — 1.83 which yields an exponentof approximately 1.2 in agreement with the most cited experimentaldata (see Comte-Bellot and Corrsin 1971).

204 C. G. Speziale

For two-dimensional turbulent shear flows that are in equilibrium,

which is remarkably close to the traditionally chosen constant valueof Cei = 1.44. For more general two-dimensional homogeneous tur-bulent flows, this model takes the form:

where

In contrast to this result, the new dissipation rate model of Lumley(1992) takes the form

for near equilibrium turbulent flows where C\ is a constant. Themodel (49) obviously contains less physics than the model (48) de-rived herein since it does not depend on rotational strains. It has longbeen recognized that the dissipation rate is dramatically altered byrotations. The results presented in this paper clearly show that thiseffect can be incorporated by accounting for anisotropic dissipation.To the best knowledge of the author, this constitutes the first sys-tematic introduction of rotational effects into the scalar dissipationrate equation. Previous attempts to account for rotational effects(see Raj 1975; Hanjalic and Launder 1980; and Bardina, Ferzigerand Rogallo 1985) were largely ad hoc.

Applications

For weakly inhomogeneous turbulent flows that are near equilibrium,we can extend the K and s transport equations by the addition ofgradient transport terms that are obtained by a formal expansiontechnique:


where Uk and cre are constants that typically assume the values of1.0 and 1.3, respectively. These forms are assumed to be valid ap-proximations at high Reynolds numbers.

We will now consider several non-trivial applications of the two-equation model derived herein which can be referred to as an explicitalgebraic stress model (ASM) based on the SSG second-order closure.The first case that will be considered is homogeneous shear flow in arotating frame (see Figure 1). In this flow, an initially isotropic tur-bulence (with turbulent kinetic energy KQ and turbulent dissipationrate £Q) is suddenly subjected to a uniform shear (with constant shearrate 5) in a reference frame rotating steadily with angular velocityft. In Figures 2(a)-2(c), the time evolution of the turbulent kineticenergy predicted by this new two-equation model is compared withthe large-eddy simulations (LES) of Bardina, Ferziger and Reynolds(1983), as well as with the predictions of the standard K — e modeland the full SSG second-order closure. From these results, it is clearthat the new two-equation model yields the correct growth rate forpure shear flow (ft/5 = 0) and properly responds to the stabilizingeffect of the rotations ft/5 = 0.5 and ft/5 = —0.5. These resultsare remarkably close to those obtained from the full SSG second-order closure as shown in Figure 2. In contrast to these results,the standard K — £ model overpredicts the growth rate of the turbu-lent kinetic energy in pure shear flow (ft/5 = 0) and fails to predictthe stabilizing effect of the rotations illustrated in Figures 2(b)-2(c).Since the standard K — £ model makes use of the Boussinesq eddyviscosity hypothesis, it is oblivious to the application of a systemrotation (i.e., it yields the same solution for all values of ft/5). Thenew two-equation model predicts unstable flow for the intermediateband of rotation rates —0.09 < ft/5 < 0.53; this result is generallyconsistent with linear stability theory that predicts unstable flow for0 < ft/5 < 0.5.

In Figure 3, the prediction of the new two-equation model forthe mean velocity profile in rotating channel flow is compared withthe experimental data of Johnston, Halleen and Lezius (1972) for arotation number Ro — 0.068. It is clear from these results that themodel correctly predicts that the mean velocity profile is asymmetricin line with the experimental results - an effect that arises fromCoriolis forces. In contrast to these results, the standard K — emodel incorrectly predicts a symmetric mean velocity profile identicalto that obtained in an inertial frame (the standard K — £ model is

206 C. G. Speziale

Figure 1. Schematic of homogeneous shear flow in a rotating frame.

oblivious to rotations of the reference frame, as alluded to earlier).As demonstrated by Gatski and Speziale (1993), the results obtainedin Figure 3 with this new two-equation model are virtually as goodas those obtained from a full second-order closure. This is due to thefact that a representation is used for the Reynolds stress tensor thatis formally derived from a second-order closure (the SSG model) inthe equilibrium limit. It is now clear that previous claims that two-equation models cannot systematically account for rotational effectswere erroneous.

Two examples will now be presented that illustrate the enhancedpredictions that are obtained for turbulent flows exhibiting effectsarising from normal Reynolds stress differences. Here, we will showresults obtained from the nonlinear K — e model of Speziale (1987).For turbulent shear flows that are predominately unidirectional, withsecondary flows or recirculation zones driven by small normal Rey-


Figure 2. Time evolution of the turbulent kinetic energy in rotatinghomogeneous shear flow: Comparison of the model predictions withthe large-eddy simulations of Bardina et al. (1983). (a) ft/5 = 0,(b) ft/5 = 0.5 and (c) ft/5 = -0.5 (from Gatski and Speziale 1993).

nolds stress differences, a quadratic appoximation of the anisotropiceddy viscosity model discussed herein collapses to the nonlinear K—emodel (see Gatski and Speziale 1993). In Figure 4, it is demonstratedthat the nonlinear K — e model predicts an eight-vortex secondary

208 C. G. Speziale

Figure 3. Comparison of the mean velocity profile in rotating channelflow predicted by the new explicit ASM of Gatski and Speziale (1993)with the experimental data of Johnston, Halleen and Lezius (1972).

flow, in a square duct, in line with experimental observations; on theother hand, the standard K — e model erroneously predicts that thereis no secondary flow. In order to be able to predict secondary flowsin non-circular ducts, the axial mean velocity vz must give rise to anon-zero normal Reynolds stress difference rm — rxx (see Speziale andNgo 1988). This requires an anisotropic eddy viscosity (any isotro-pic eddy viscosity, including that used in the standard K — £ model,


yields a vanishing normal Reynolds stress difference which makes itimpossible to describe these secondary flows).

Figure 4. Turbulent secondary flow in a rectangular duct: (a) exper-iments, (b) standard K — e model, and (c) nonlinear K — e model ofSpeziale (1987).

In Figure 5, results obtained from the nonlinear K — £ modelare compared with the experimental data of Kim, Kline and John-

210 C. G. Speziale

ston (1980) and Eaton and Johnston (1981) for turbulent flow pasta backward facing step. It is clear that these results are excellent:reattachment is predicted at x/H sa 7.0 in close agreement with theexperimental data. In contrast to these results, the standard K — emodel predicts reattachment at x/H K 6.25 - an 11% underpredic-tion (see Thangam and Speziale 1992). This predominantly resultsfrom the inaccurate prediction of normal Reynolds stress differencesin the recirculation zone as discussed by Speziale and Ngo (1988).

2.7 Full Second-Order Closures

These more complex closures are based on the full Reynolds stresstransport equation with turbulent diffusion:

Full second-order closures are needed for turbulent flows with:

(i) Relaxation effects ;

(ii) Nonlocal effects arising from turbulent diffusion that can giverise to counter-gradient transport.

In virtually all existing full second-order closures for inhomoge-neous turbulent flows, II^ and peij are modeled by their homoge-neous forms. The pressure-strain correlation !!,•_,• is modeled as

as discussed earlier. In Section 2.6, the equilibrium limit of theSpeziale, Sarkar and Gatski (SSG) model was provided. For tur-bulent flows where there are mild departures from equilibrium, theSSG model takes the form (see Speziale, Sarkar and Gatski 1991)


Figure 5. Turbulent flow past a backward facing step: comparison ofthe predictions of the nonlinear K — e model with experiments, (a)Streamlines and (b) turbulent shear stress profiles.

where

The Launder, Reece and Rodi (1975) model is recovered as a specialcase of the SSG model when

212 C. G. Speziale

In most applications, at high Reynolds numbers, the Kolmogorovassumption of local isotropy is typically invoked where

(then, £,-j = §£<% and a modeled transport equation for the scalardissipation rate £ is solved that is of the same general form as thatdiscussed in Section 2.6). However, this assumption is debatable asdiscussed by Durbin and Speziale (1991). More generally, a repre-sentation of the form

can be used where the algebraic model of Speziale and Gatski (1992)discussed in Section 2.6 is implemented.

The only additional model that is needed for closure in high-Reynolds-number inhomogeneous turbulent flows is a model for thethird-order diffusion correlation Cijk- This is typically modeled usinga gradient transport hypothesis:

Some examples of commonly used models are as follows:

Hanjalic and Launder (1972) Model

Mellor and Herring (1973) Model

where Cs is a constant (« 0.11). When these models are used in afull second-order closure, counter-gradient transport effects can beaccounted for.

There is no question that, in principle, second-order closures ac-count for more physics. This is quite apparent for turbulent flowsexhibiting relaxation effects. The return to isotropy problem is aprime example where suddenly, at time t = 0, the mean strains in


a homogeneous turbulence are shut off; the flow then gradually re-turns to isotropy (i.e., 6,-j —>• 0 as t —> oo). In Figure 6, results for theReynolds stress anisotropy tensor obtained from the Speziale, Sarkarand Gatski (SSG) and Launder, Reece and Rodi (LRR) models arecompared with the experimental data of Choi and Lumley (1984) forthe return-to-isotropy from plane strain (here, T = Sot/Ko). It isclear from these results that the models predict a gradual return toisotropy in line with the experimental data. In contrast to these re-sults, all two-equation models — including the more sophisticated onebased on an anisotropic eddy viscosity derived herein - erroneouslypredict that at T = 0, fyj abruptly goes to zero.

It is worth noting at this point that while the SSG model wasderived and calibrated based on near equilibrium two-dimensionalmean turbulent flows, it performs remarkably well on certain three-dimensional, homogeneously strained turbulent flows. The predic-tions of the SSG and LRR models for the normal Reynolds stressanisotropies are compared in Figure 7 with the direct simulationsof Lee and Reynolds (1985) for the axisymmetric expansion (here,t* = Tt where F is the strain rate).

While the previous results are encouraging, it must be noted thatthe Achilles heel of second-order closures is wall-bounded turbulentflows:

(i) Ad hoc wall reflection terms are needed in many pressure-strainmodels (that depend on the distance y from the wall) in orderto mask deficient predictions for the logarithmic region of aturbulent boundary layer;

(ii) Near-wall models typically must be introduced that depend onthe unit normal to the wall - a feature that makes it virtuallyimpossible to systematically integrate second-order closures incomplex geometries (see So et al. 1991).

In regard to the former point, it is rather shocking as to whatthe level of error is in many existing second-order closures for thelogarithmic region of an equilibrium turbulent boundary layer, whenno ad hoc wall reflection terms are used. This can be seen in Table 2where the predictions of the Launder, Reece and Rodi (LRR), Shihand Lumley (SL), Fu, Launder and Tselepidakis (FLT) and SSG

214 C. G. Speziale

Figure 6. Time evolution of the anisotropy tensor in the return toisotropy problem. Comparison of the predictions of the SSG modeland Launder, Reece and Rodi (LRR) model with the experiment ofChoi and Lumley (1984) (from Speziale, Sarkar and Gatski 1991).

are compared with experimental data (Laufer 1951) for the log-layerof turbulent channel flow. Most of the models yield errors rang-ing from 30% to 100%. These models are then typically forced intoagreement with the experimental data by the addition of ad hoc wallreflection terms that depend inversely on the distance from the wall— an alteration that compromises the ability to apply the model incomplex geometries where the wall distance is not always uniquelydefined. Only the SSG model yields acceptable results for the log-


Figure 7. Time evolution of the anisotropy tensor in the axisymmet-ric expansion for eo/TKo = 2.45. Comparison of the predictions ofthe LRR model and SSG model with the direct simulations of Leeand Reynolds (1985).

layer without a wall reflection term. This results from two factors:(a) a careful and accurate calibration of homogeneous shear flow (seeTable 3) and (b) the use of a Rotta coefficient that is not too far re-moved from one (see Abid and Speziale 1993). The significance ofthese results is demonstrated in Figure 8 where full Reynolds stresscomputations of turbulent channel flow are compared with the ex-perimental data of Laufer (1951). It is clear that the same favorable

216 C. G. Speziale

trends are exhibited in these results as with those shown in Table 2which were obtained by a simplified log-layer analysis.

CHANNEL FLOW

Table 2. Comparison of the model predictions for the equilibriumvalues in the log-layer (P/£ — 1) with the experimental data of Laufer(1951) for channel flow.

HOMOGENEOUS SHEAR FLOW

Table 3. Comparison of the model predictions for the equilibriumvalues in homogeneous shear flow (P/e = 1.8) with the experimentaldata of Tavoularis and Karnik (1989).

The near-wall problem largely arises from the use of homoge-neous pressure-strain models of the form (53) that are only theo-retically justified for near-equilibrium homogeneous turbulence. Re-cently, Durbin (1993) developed an elliptic relaxation model thataccounts for wall blocking and introduces nonlocal effects in thevicinity of walls - eliminating the need for ad hoc wall damping func-

EquilibriumValues

611&12

&22

&33

SK/e

LRRModel0.129-0.178-0.101-0.0282.80

SLModel0.079-0.116-0.0820.0034.30

FITModel0.141-0.162-0.099-0.0423.09

SSGModel0.201-0.160-0.127-0.0743.12

ExperimentalData0.22-0.16-0.15-0.073.1

EquilibriumValues

&nbi2&22

&33

SK/e

LRRModel0.152-0.186-0.119-0.0334.83

SLModel0.120-0.121-0.1220.0027.44

FLTModel0.196-0.151-0.136-0.0605.95

SSGModel0.218-0.164-0.145-0.0735.50

ExperimentalData0.21-0.16-0.14-0.075.0


Figure 8. Comparison of full Reynolds stress calculations of chan-nel flow with the experimental data of Laufer (1951) (O) for Re =61,600. SSG model; FLT model; - • - LRR model; and

SL model, (a) bn component and (b) &i2 component (from Abidand Speziale 1993).

218 C. G. Speziale

tions. While this is a promising new approach, it does not alleviatethe problems that the commonly used pressure-strain models havein non-equilibrium homogeneous turbulence (the Durbin 1993 modelcollapses to the standard hierarchy of pressure-strain models givenabove in the limit of homogeneous turbulence). The failure of thesemodels in non-equilibrium homogeneous turbulence can best be illus-trated by the simple example shown in Figure 9. This constitutes arapidly distorted turbulent flow that, initially, is far from equilibriumsince SKo/eo = 50 (the equilibrium value of SK/e is approximately5). It is clear from these results that all of the models perform poorlyrelative to the DNS of Lee et al. (1990). Even the SSG model, whichdoes extremely well for homogeneous shear flow that is not far fromequilibrium, dramatically overpredicts the growth rate of the turbu-lent kinetic energy for this strongly non-equilibrium test case.

In the opinion of the author, it is a vacuous exercise to de-velop more complex models of the form (53) using non-equilibriumconstraints such as Material Frame-Indifference (MFI) in the two-dimensional limit (Speziale 1981, 1983) or realizability (Schumann1977 and Lumley 1978). While these constraints are a rigorous con-sequence of the Navier-Stokes equations, they typically deal withflow situations that are far from equilibrium (two-dimensional tur-bulence and one or two-component turbulence) where (53) would notbe expected to apply in the first place. Ristorcelli, Lumley and Abid(1995) - following the earlier work by Haworth and Pope (1986) andSpeziale (1989) - developed a pressure-strain model of the form (53)that satisfies MFI in the 2-D limit. Shih and Lumley (1985) at-tempted to develop models of the form (53) that satisfy the strongform of realizability of Schumann (1977). Reynolds (1987) has at-tempted to develop models of this form which are consistent withRapid Distortion Theory (RDT). All of these models involve com-plicated expressions for Mjjkt that are nonlinear in 6;r From itsdefinition,

which is linear in the energy spectrum tensor Eke(n,t). Since,


Figure 9. Comparison of the SSG, SL and FLT model predictionsfor the time evolution of the turbulent kinetic energy with the DNSresults of Lee, Kim and Moin (1990) for homogeneous shear flow(SKQ/£Q — 50) (from Speziale, Gatski and Sarkar 1992).

where

it follows that models for Mijke that are nonlinear in 6;j are alsononlinear in E^j. This is a fundamental inconsistency that doomsthese models to failure. It is clear that it is impossible to describe arange of RDT flows - which are linear - with nonlinear models (theprinciple of superposition is violated). Furthermore, Shih and Lum-ley (1985, 1993) unnecessarily introduce higher degree nonlinearitiesand non-analyticity to satisfy realizability. In the process of doing

220 C. G. Speziale

so, they arrive at a model that is neither realizable nor capable ofdescribing even basic turbulent flows (see Speziale, Abid and Durbin1994, Durbin and Speziale 1994 and Speziale and Gatski 1994).

Entirely new non-equilibrium models are needed for the pressure-strain correlation and the dissipation rate tensor. The former shouldcontain nonlinear strain rate effects and the latter should accountfor the effects of anisotropic dissipation and non-equilibrium vortexstretching (see Bernard and Speziale 1992 and Speziale and Bernard1992). Models of this type are currently under investigation.

3 COMPRESSIBLE TURBULENCE

The full equations of motion for an ideal gas will be considered:

Continuity

Momentum

Energy


wherep = mass density

Ui = velocity vectorp = thermodynamic pressure

[JL = dynamic viscosity<Tj-j = viscous stress tensorT = absolute temperatureK = thermal conductivityR = ideal gas constant

Cv = specific heat at constant volume$ = viscous dissipation

( \ • - d ( \\ A* = dxi\ )

3.1 Compressible Reynolds Averages

For any flow variable f, we can introduce the decomposition

where T is the standard ensemble mean, or the alternative decom-position

where

is the Favre (or mass-weighted ) average. Here,

3.2 Compressible Reynolds-Averaged Equations

The Reynolds-aver aged equations of motion for an ideal gas aregiven by (cf. Cebeci and Smith 1974 and Speziale and Sarkar 1991):

Continuity

222 C. G. Speziale

Momentum

Energy

where turbulent fluctuations in Cv have been neglected.

Molecular Diffusion Terms

In high-Reynolds-number turbulent flows, the molecular diffusionterms are dominated by the turbulent transport terms except in athin sublayer near walls. If we assume in this region that fluctuationsin the viscosity, thermal conductivity and density can be neglectedwe can then make the approximations:

The mean viscous dissipation is given by:

where £ = cr'-u^-fp is the turbulent dissipation rate ($ -> pe asRe -> oo).

Hence, in order to achieve closure, models are needed for:

(i) The Favre-averaged Reynolds stress, rt-j


(ii) The Favre-averaged Reynolds heat flux, Qi

(Hi) The turbulent mass flux, u"

(iv) The turbulent dissipation rate, e

(v) The pressure-dilatation correlation, p'u'^.

This makes the problem of compressible turbulence modeling farmore difficult than its incompressible counterpart.

3.3 Compressible Reynolds Stress Transport Equation

The Reynolds stress transport equation for compressible turbu-lent flows takes the form (cf. Speziale and Sarkar 1991)

where

are, respectively, the third-order turbulent diffusion correlation, thedeviatoric part of the pressure gradient-velocity correlation and thedissipation rate tensor.

224 C. G. Speziale

Transport Terms

Speziale and Sarkar (1991) proposed the following models for theturbulent transport terms:

where

More recently, from an analysis of the transport equation for theturbulent mass flux, it has been argued that

to the lowest order when an equilibrium hypothesis of the boundarylayer type is invoked (see Zeman 1993 and Ristorcelli 1993). Here,Mt EE (2K/~/RT)1/2 is the turbulent Mach number.

Dissipation rate model

The Kolmogorov assumption of local isotropy is typically used where

for turbulence Reynolds numbers Rt >• 1. The turbulent dissipationrate can be decomposed into solenoidal and compressible parts:


respectively, as proposed independently by Zeman (1990) and Sarkaret al. (1991) where

given that w,' is the fluctuating vorticity. Then, the Sarkar et al.(1991) model can be used for ec yielding

where

The solenoidal dissipation es is a solution of the modeled dissi-pation rate transport equation (see Speziale and Sarkar 1991):

where

and turbulent fluctuations in the molecular viscosity have been ne-glected.

Pressure-Dilatation Model

For turbulent shear flows, Sarkar (1992) proposed the following modelfor the pressure-dilatation correlation:

where

It should be noted that for turbulent flows with mean dilatation, aterm proportional to

226 C. G. Speziale

should be added (see Sarkar 1992).

Deviatoric Pressure Gradient- Velocity Correlation Iil}

A variable density extension of the pressure-strain model of Speziale,Sarkar and Gatski (1991) (the SSG model) is used for II,-j:

where

With this model, we now have a complete second-order closure forhigh-speed compressible turbulent flows.

3.4 Compressible Two-Equation Models

Two-equation models with an anisotropic eddy viscosity can besystematically derived from the full compressible second-order clo-sure presented in Section 3.3 via the same homogeneous equilibriumhypothesis used earlier for incompressible flows. This, to the lowestorder, yields


where C*, C*D and C*E are variable density extensions of the incom-pressible coefficients derived earlier, in Eq. (32), that depend onboth 77 and £.

This anisotropic eddy viscosity model can be solved in conjunc-tion with modeled transport equations for K and es that take theform

where

and the compressible terms ec, p'w''» and u" are modeled as before.

3.5 Illustrative Examples

A variety of examples of compressible homogeneous turbulentflows will first be presented based on the full compressible second-order closure provided in turbulent kinetic energy in compressibleisotropic turbulence are compared with the direct numerical simula-tions of Sarkar et al. (1991) for a variety of turbulent Mach numbers.The model is able to correctly capture the increase in the decay rateof the turbulent kinetic energy that arises from compressible effects- a feature that was traced to the compressible dissipation by Sarkaret al. (1991).

228 C. G. Spezi&le

Figure 10. The decay of turbulent kinetic energy in compressible iso-tropic turbulence for initial turbulence Mach numbers Mt,o = 0.1,0.3,and 0.4. (a) Direct numerical simulations and (b) Model predictions(from Speziale and Sarkar 1991 and Sarkar et al. 1991).


The second homogeneous flow that will be considered is the rapidcompression or expansion of isotropic turbulence. For this problem,the mean velocity gradient tensor is given by

where F is the expansion/compression rate which is constant. Themodel provided herein reduces approximately to the simple couplednonlinear ODE's:

for \T\K0/e0 > 1. The short-time solution to Eqs. (90) - (91) isgiven by:

where A = li'3//2/e is the integral length scale. These are identicalto the results obtained by Reynolds (1987) based on Rapid Distor-tion Theory (RDT). In contrast to these results, a variable densityextension of the commonly used second-order closures erroneouslypredicts that

(see Reynolds 1987). According to (95), the integral length scale willdecrease under an expansion (F > 0) and increase under a compres-sion (F < 0) - results that are clearly in error as first pointed out byReynolds (1987).

Finally, in regard to homogeneous turbulence, we will consider theproblem of compressible homogeneous shear flow. Here, an initiallyisotropic turbulence is subjected to a uniform shear rate S with thecorresponding mean velocity gradient tensor

230 C. G. Speziale

The time evolution of the turbulent kinetic energy and turbulent dis-sipation rate in compressible homogeneous shear flow obtained fromthe SSG model with the dilatational models of Sarkar are comparedwith the DNS results of BlaisdeU et al. (1991), for Mto = 0 andMt0 = 0.307, in Figure 11. It is clear from these results that themodel does an excellent job in reproducing the dramatic reductionin the growth rate that arises from compressible effects (withoutexplicit dilatational terms, all existing second-order closures drasti-cally overpredict the growth rate of the turbulent kinetic energy). InTable 4, the equilibrium values obtained using a variety of pressure-strain models - combined with the dilatational models of Sarkar -are compared with the DNS results of BlaisdeU et al. (1991). Here,the results are not so favorable: the normal Reynolds stress anisot-ropies are drastically underpredicted and the Reynolds shear stressanisotropy is overpredicted by more than 25%.

Table 4. Predicted equilibrium values for compressible homogeneousshear flow using the Launder, Reece and Rodi (LRR), Speziale,Sarkar and Gatski (SSG) and Fu, Launder and Tselepidakis (FLT)pressure-strain models with the dilatational models of Sarkar. (Takenfrom Speziale, Abid and Mansour 1994).

It can thus be concluded that with the addition of the newerdilatational models, existing second-order closures properly predictthe reduced growth rate of the turbulent kinetic energy in compress-ible homogeneous shear flow. However, the models are currently notcapable of predicting the equilibrium Reynolds stress anisotropies ac-curately. This deficiency arises from the use of variable density ex-tensions of incompressible pressure-strain models - a deficiency that

EquilibriumValues

bub\i&22

&33

Mt

LRRModel0.166-0.187-0.130-0.0360.65

SSGModel0.230-0.165-0.148-0.0820.60

FLTModel0.189-0.148-0.138-0.0510.65

DNSData0.424-0.118-0.236-0.1880.51


Figure 11. Time evolution of the turbulent fields in compressiblehomogeneous shear flow using the SSG model with the dilatationalmodels of Sarkar for Mtfl = 0.307; Mt,0 = 0; O DNS ofBlaisdell et al. (1991). (a) Turbulent kinetic energy and (b) turbulentdissipation rate (from Speziale, Abid and Mansour 1994).

232 C. G. Speziale

must be overcome in future research.The same kind of improved predictions for the reduction in the

growth rate of compressible homogeneous shear flow are obtained forthe spreading rate in the supersonic mixing layer (see Figure 12) withthis type of compressible model. In Figure 13, the spreading ratepredicted by a simple second-order closure both with and withoutthe dilatational models of Sarkar are compared with experimentaldata, as computed by Sarkar and Balakrishnan (1991). Withthe addition of the dilatational terms, it now becomes possible topredict the dramatic reduction in the spreading rate that arises fromhigh-speed compressible effects.

Figure 12. Schematic diagram of the supersonic mixing layer.


Figure 13. Normalized spreading rate as a function of the convectiveMach number Mc: Comparison of the model predictions with ex-perimental data for the supersonic mixing layer. Second-orderclosure with dilatational terms; Second-order closure withoutdilatational terms; o Experimental Langley curve.

Interesting enough, good results are obtained for the compressibleflat plate boundary layer for Mach numbers M^ as large as 8 andwall temperature ratios as low as 0.3 using properly calibrated two-equation models with no explicit dilatation terms, as shown by Zhanget al. (1993) (see Figure 14). In fact, it must be noted that thecompressible dissipation model of Sarkar causes a degradation of theskin friction predictions; the model is not formally correct in the log-layer of high-speed compressible boundary layers. This was probablyfirst pointed out by P. G. Huang (Huang 1990 and Huang et al.1994). Consequently, more research is also needed on the modelingof dilatational terms in wall-bounded turbulent flows.

234 C. G. Speziale

Figure 14. Comparison of the predictions of the K'—e model of Zhanget al. (1993) for the compressible flat plate boundary layer with theexperimental data of Kussoy and Horstman (1991) for M^ — 8.18and Tw/Taw — 0.3: (a) Mean velocity and (b) mean temperature.

4 CONCLUDING REMARKS

Some significant progress in turbulence modeling has been madeduring the past two decades. The following general conclusions can


be drawn about the current status of incompressible turbulence mod-els:

(1) There is a relatively sound theoretical basis for the existingtype of Reynolds stress models in two-dimensional mean tur-bulent flows that are close to equilibrium and only weakly in-homogeneous. A new generation of two-equation models andsecond-order closures has emerged that provides a surprisinglygood description of these flows.

(2) More research is needed to place Reynolds stress models onsolid theoretical grounds for three-dimensional, non-equilibriumturbulent flows. In addition, new methods for the integrationof second-order closures to a solid boundary in complex geome-tries are required. Until such methods are more fully developed,it is preferable to use two-equation models - with an anisotropiceddy viscosity systematically obtained from a second-order clo-sure - in complex wall-bounded turbulent flows. Fundamentalchanges in the way second-order closures are formulated maybe needed to overcome this problem.

Compressible Reynolds stress models are not as well developedas their incompressible counterparts. Significant progress has beenmade in the modeling of turbulent dilatational terms, however, muchmore research is still needed. Most importantly, fundamental new re-search on the compressible modeling of the pressure-strain correlationis needed. The modeling of turbulent dilatational terms in high-speedcompressible wall-bounded flows also needs more attention. Despitethese unresolved issues, there is still cause for optimism. With fur-ther progress, we should start to see Reynolds stress models make amajor impact on the computation of the complex turbulent flows oftechnological importance.

ACKNOWLEDGMENTS

Most of this work was funded by the National Aeronautics andSpace Administration under Contract NAS1-19480 while the authorwas in residence at ICASE. Partial funding by the Office of NavalResearch under Grant N00014-94-1-0088 (ARI on Nonequilibrium

236 C. G. SpezJale

Turbulence, Dr. L. P. Purtell, Program Officer) is also gratefullyacknowledged.

5 REFERENCES

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Bardina, J., Ferziger, J. H. and Reynolds, W. C., 1983. "Improvedturbulence models based on large-eddy simulation of homo-geneous, incompressible turbulent flows," Stanford UniversityTechnical Report No. TF-19.

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Bernard, P. S. and Speziale, C. G., 1992. "Bounded energy statesin homogeneous turbulent shear flow - An alternative view,"ASME J. Fluids Eng. 114, pp. 29-39.

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Cebeci, T. and Smith, A. M. O., 1974. Analysis of Turbulent Bound-ary Layers, Academic Press, New York.

Choi, K. S. and Lumley, J. L., 1984. "Return to isotropy of ho-mogeneous turbulence revisited," in Turbulence and ChaoticPhenomena in Fluids (T. Tatsumi, ed.), pp. 267-272, NorthHolland, New York.


Comte-Bellot, G. and Corrsin, S., 1971. "Simple Eulerian timecorrelation of full- and narrow-band velocity signals in grid-generated isotropic turbulence," J. Fluid Mech. 48, pp. 273-337.

Durbin, P. A., 1993. "A Reynolds stress model for near-wall turbu-lence," J. Fluid Mech. 249, pp. 465-498.

Durbin, P. A. and Speziale, C. G., 1991. "Local anisotropy instrained turbulence at high Reynolds numbers," ASME J. Flu-ids Eng. 113, pp. 707-709.

Durbin, P. A. and Speziale, C. G., 1994. "Realizability of second-moment closure via stochastic analysis," J. Fluid Mech. 280,pp. 395-407.

Eaton, J. K. and Johnston, J. P., 1980. "Turbulent flow reattach-ment: An experimental study of the flow and structure behinda backward facing step," Stanford University Report No. MD-39.

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Gatski, T. B. and Speziale, C. G., 1993. "On explicit algebraic stressmodels for complex turbulent flows," J. Fluid Mech. 254, pp.59-78.

Hanjalic, K. and Launder, B. E., 1972. "A Reynolds stress modelof turbulence and its application to thin shear flows," J. FluidMech. 52, pp. 609-638.

Hanjalic, K. and Launder, B. E., 1980. "Sensitizing the dissipationequation to irrotational strains," ASME J. Fluids Eng. 102,pp. 34-40.

Haworth, D. C. and Pope, S. B., 1986. "A generalized Langevinmodel for turbulent flows," Phys. Fluids 29, pp. 387-405.

Hinze, J. O., 1975. Turbulence, 2nd ed., McGraw-Hill, New York.

Huang, P. G., 1990. Private Communication.

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Huang, P. G., Bradshaw, P. and Coakley, T. J., 1994. "Turbulencemodels for compressible boundary layers," AIAA J. 32, pp.735-740.

Johnston, J. P., Halleen, R. M. and Lezius, D. K., 1972. "Effects ofa spanwise rotation on the structure of two-dimensional fully-developed turbulent channel flow," J. Fluid Mech. 56, pp. 533-557.

Kim, J., Kline, S. J. and Johnston, J. P., 1980. "Investigation of areattaching turbulent shear layer: Flow over a backward facingstep," ASME J. Fluids Eng. 102, pp. 302-308.

Kolmogorov, A. N., 1942, "The equations of turbulent motion inan incompressible fluid," Isv. Acad. Sci. USSR, Phys. 6, pp.56-58.

Kussoy, M. I. and Horstman, C. C., 1991. "Documentation of two-and three-dimensional shock wave/turbulent boundary layerinteraction flows at Mach 8.2," NASA TM-103838.

Laufer, J., 1951. "Investigation of turbulent flow in a two-dimen-sional channel," NAG A TN 1053.

Launder, B. E. and Spalding, D. B., 1974. "The numerical com-putation of turbulent flows," Comput. Methods Appl. Mech.Eng. 3, pp. 269-289.

Launder, B. E., Reece, G. J. and Rodi, W., 1975. "Progress in thedevelopment of a Reynolds stress turbulence closure," J. FluidMech. 68, pp. 537-566.

Lee, M. J. and Reynolds, W. C., 1985, "Numerical experiments onthe structure of homogeneous turbulence," Stanford UniversityTechnical Report TF-24.

Lee, M. J., Kim, J. and Moin, P., 1990. "Structure of turbulenceat high shear rate," J. Fluid Mech. 216, pp. 561-583.

Lumley, J. L., 1978. "Computational modeling of turbulent flows,"Adv. Appl. Mech. 18, pp. 123-176.

Lumley, J. L., 1992. "Some comments on turbulence," Phys. FluidsA 4, pp. 203-211.


Mellor, G. L. and Herring, H. J., 1973. "A survey of mean turbulentfield closure models," AIAA J. 11, pp. 590-599.

Pope, S. B., 1975. "A more general effective viscosity hypothesis,"J. Fluid Mech. 72, pp. 331-340.

Prandtl, L., 1925. "Uber die ausgebildete turbulenz," ZAMM 5,pp. 136-139.

Prandtl, L., 1945. "Uber ein neues formelsystem fur die ausge-bildete turbulenz," Nachr. Akad. Wiss. Gottingen Math.Phys. Kl 1945, pp. 6-19.

Raj, R., 1975. "Form of the turbulence dissipation equation asapplied to curved and rotating turbulent flows," Phys. Fluids18, pp. 1241-1244.

Reynolds, W. C., 1987. "Fundamentals of turbulence for turbulencemodeling and simulation," in Lecture Notes for Von KdrmdnInstitute, AGARD Lect. Ser. No. 86, pp. 1-66, NATO, NewYork.

Ristorcelli, J. R., 1993. "A representation for the turbulent massflux contribution to Reynolds stress and two-equation closuresfor compressible turbulence," ICASE Report No. 93-88, NASALangley Research Center.

Ristorcelli, J. R., Lumley, J. L. and Abid, R., 1995. "A rapid-pressure correlation representation consistent with the Taylor-Proudman theorem materially-frame-indifferent in the 2-Dlimit," J. Fluid Mech. 292, pp. 111-152.

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Rogers, M. M., Moin, P. and Reynolds, W. C., 1986. "The structureand modeling of the hydrodynamic and passive scalar fields inhomogeneous turbulent shear flow," Stanford University Tech-nical Report No. TF-25.

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Chapter 6

AN INTRODUCTION TOSINGLE-POINT CLOSUREMETHODOLOGY

Brian E. Launder

1 INTRODUCTION

1.1 The Reynolds Equations

The material presented in Section 1 will replicate that coveredin earlier chapters. It may nevertheless be helpful to include it tofamiliarize the reader both with the nomenclature adopted and theassumptions being made.

The instantaneous velocity field in a turbulent flow is describedby the continuity and the Navier-Stokes equations which in conser-vative form may be written:

Here /5,P,/i and F{ denote the density, pressure, viscosity and bodyforce per unit volume. The tildes appearing above all quantities serveas a reminder that each, potentially, will display fluctuations due toturbulence. Here, however, density and viscosity variations will be

243

244 B. E. Launder

assumed sufficiently small to have negligible direct effect on turbu-lence. A further limitation will be to flows which are incompressiblein the mean, i.e. ones where dUi/ dx{ = 0. The resultant somewhatsimpler pair of equations may then be written

We now proceed to average each term in the above equation.All except the second term on the left of eq (1.2b), the convectivetransport of a;,- momentum, contain only a single fluctuating quantity;so, the averaging results in the instantaneous value being replacedby the mean of the variable in question. In the exceptional case

Thus after some rearrangement, the averaged equations of motionare obtained as:

These equations, generally known as the Reynolds equations, dif-fer from those describing a laminar flow only by the presence of theterm containing averaged products of fluctuating velocities. The pro-cess it represents is the additional transfer rate of x,- momentum dueto turbulent fluctuations0. Habitually it is brought to the right side.

"For cases where density variations are large (requiring retention of densityfluctuations) the equations of motion are simplified by the adoption of massweighted velocities.

Single-Point Closures 245

Since the first term within the square brackets is the viscous stress,the second, -puiuj, has, naturally, been re-interpreted as a turbulentstress or, more usually, the Reynolds stress tensor. As the tensoris symmetric there are six independent Reynolds stress components.They are unknown elements in the averaged equations of motion andthe theme of these notes is that of obtaining a satisfactory approxi-mation to their magnitude.

1.2 Mean Scalar Transport

We consider the transport of some scalar property by the turbu-lent motion. Its instantaneous value 0 is assumed to be governed bythe equation:

where F is the appropriate molecular diffusivity and Sg is the rate ofcreation of the property per unit volume.

Besides conventional sources or sinks it is convenient to imaginethe term Sg absorbing any terms which, for a particular transportedscalar property, do not fit elsewhere. For example, the moleculardiffusion process may be governed by a more elaborate law thanthe simple gradient-diffusion relation supposed while, if © stands fortemperature, the time-dependent term in the equation should strictlybe multiplied by the specific heat at constant volume divided by thatat constant pressure, a difference that can be accounted for in Sg.

Applying an averaging to eq (1.5) leads to the following rateequation for the mean level of the scalar

Thus

The quantity pvivj is known as the mass-weighted Reynolds stress. Notice that,in contrast with conventional "volume" averaging, the number of turbulent cor-relations arising from averaging the convective terms is just one - as in a uniformdensity flow. This is the feature that has made the use of mass weighted quantitiesin variable-density flow so popular.

246 B. E. Launder

An overbar has been placed on Sg to serve as a reminder that thesource term may be a non-linear one and, in that event, its meanvalue may differ considerably from that obtained by inserting justthe mean values of the separate constituent terms. Having drawnattention to this possibility, we shall not consider it further in thepresent treatment, attention being limited to cases where the sourceor sink term is effectively zero. Just as with the averaged momen-tum equation, the additional convective flux of the scalar due tothe turbulent velocity fluctuations is conveniently interpreted as asupplementary diffusional process, so the term puj0 is transferredto the right side of the equation. Like the Reynolds stresses, thisturbulent scalar flux is an unknown and will correspondingly requireapproximation.

1.3 The Modeling Framework

In all practically interesting problems the mean momentum andcontinuity equations together, in many cases, with one or more equa-tion of the type (1.6) for transported scalars are to be solved numer-ically. The solving procedure will, in all probability, be of fairly gen-eral construction designed to cope with some class of flow problems:for example, two-dimensional thin shear flows, three-dimensional,confined, non-recirculating flows, etc. Ideally the turbulence model -the scheme for evaluating v^uj and u$ ~ should enjoy a range of ap-plicability comparable to that of the numerical procedure and shouldfit comfortably within it.

Schemes discussed in this article are single-point closures. In suchapproaches the only averaged products of fluctuating quantities thatappear are those in which the two or more quantities in question areevaluated at the same point in space. Mathieu and Jeandel (1984)and Leslie (1973) have contributed text books describing 'two-point'or 'spectral' schemes. Models of this type are seen more as helpingto reveal the underlying physics of turbulence than as models for usein engineering computational procedures.

Within the single-point framework there is a wide range of mo-delling methodology. At present most applied computational workon turbulent flows still adopts the idea that the turbulent fluxes and


stresses can be represented in terms of effective turbulent diffusioncoefficients for momentum, heat, chemical species, etc. Approachesof this type range from simple mean-field closures (where the tur-bulent viscosity is expressed in terms of the mean velocity field andflow topography) to the widely used two-equation models where theeffective diffusion coefficients are determined from local values of twoscalar properties of the turbulence (and which may or may not havea directly measurable physical significance). These in turn are obtai-ned from transport equations similar to those describing mean flowquantities save that source and sink terms always play an importantrole.

In some respects it makes sense to consider models based on theidea of a turbulent viscosity first before proceeding to more advancedsecond-moment treatments. On balance, however, it seems preferablehere to go directly to a more comprehensive treatment from which'turbulent viscosity' models emerge as special cases under particularcircumstances. The term 'second-moment' applies to models basedon the exact transport equations for the second moments, i.e. forU{Uj, UiO etc. These equations, while exact, are unclosed: they con-tain correlations that are not exactly determinable and which musttherefore be approximated in terms of quantities that are.

Section 2 presents briefly the most popular current approaches toclosure at this level. Before considering closure questions; however,it will be instructive to examine the exact second-moment equationsand, in particular, the processes causing these quantities to departfrom the levels found in isotropic turbulence. These are topics takenup below.

1.4 Second-Moment Equations

An exact equation describing the transport of the kinematic Rey-nolds stress u^Uj is formed by multiplying the momentum equationfor Ui (in which we now use k rather than j as the repeated suf-fix) by Uj and averaging, then adding to it the mirror equation inwhich suffices i and j are interchanged. After a certain amount ofmanipulation, the resultant equation may be written:

248 B. E. Launder

The left side of the equation expresses the total rate of increaseof the correlation ujuj for a small identified packet of fluid. The rateof change arises from an imbalance of the terms on the right. Herethe terms have been grouped, following well-established practice, soas best to allow a physical interpretation of the processes. One line isgiven to each process and, beneath each term, appears a shorthandsymbol for the process in question which we shall use to simplify laterequations. The first two processes represent rates of creation of UiUj,in one case by the effects of mean strain, P,-j, and in the other bybody forces,G;j. The first of these, comprising products of Reynoldsstresses and mean velocity gradients, can clearly be treated exactlyin a second-moment closure. If the body force is linear, as when oneexamines the flow in a rotating coordinate frame, that too can behandled without further approximation.

The correlation between fluctuating pressure and fluctuatingstrain, faj, is a very important one. We note that its trace is zero,since by continuity

The term thus makes no contribution to the overall level of turbulenceenergy but serves to redistribute energy among the normal stresscomponents (those for which i and j take the same value).

The terms comprising d{j are easily recognized as diffusive incharacter since we see from integrating them across a thin shear flowbounded by non-turbulent fluid that they make no contribution tothe average level ofuiUj at any section even though, within the shear-flow, the correlations themselves are non-zero. Their effect is thus


to promote a spatial redistribution. The last term of dij describesdiffusive transport due to molecular action; over all or nearly all theflow it will be entirely negligible.

Finally, term e,-j represents (very nearly) the destruction rate ofUiUj by viscous action. Unlike viscous diffusion, the dissipation termscannot in general be ignored. We may see this is so by contractingeq (1.7) to produce an equation for the transport of kinetic energy.Then, for the thin shear flow discussed above, dkk makes no contri-bution to the overall level of turbulence energy at any section while<f>kk vanishes identically at all points. The term P^k will be positiverepresenting the continual extraction of energy from the mean flowby the action of the Reynolds stress on the mean shear. Thus, if ekkwere negligible, there would be a limitless growth of the flow's tur-bulent kinetic energy. Such a scenario is contrary to both intuitionand observation. The crucial difference between e,-j and the viscousdiffusion terms of d^ is that the former comprises correlations offluctuating velocity derivatives and, in the finest scales of motionpresent, turbulent velocity derivatives are large.

Further consideration of <^-, d±j and Cjj is deferred to Section2. It will, however, be instructive to examine specific forms that theproduction tensors take in a few cases for, in most practical flows, theproduction terms firmly stamp the character of the resultant turbu-lent stress tensor. First, consider the case of simple shear, dUi/dx^— A , where A is assumed positive. From the upper row of Table 1.1,Pi2, the production rate of u\u^ is negative (since u\ is undoubtedlypositive); though short of a proof, the reader will accept the likeli-hood that uiu-2 is consequently negative (that is, of the same signas its production rate) which gives in turn the "sensible" result thatthe generation of u\ is positive. There is no direct production ofeither u\ or M§. This does not mean that there will be no turbulentfluctuations in the x-i and £3 directions for we have already notedthat the pressure-strain correlation fyj serves to redistribute energyamong the various normal stresses. Nevertheless, we_should expect— and this is amply confirmed by experiment - that u\ would be thelargest of the normal stress components. The self-sustaining charac-ter of turbulence in a simple shear is emphasized by the clockwiserotating arrows connecting the stresses in Figure 1.1: turbulent ve-locity fluctuations in the direction of mean-velocity gradient promotea growth in shear stress which, in turn, serves to augment the inten-

250 B. E. Launder

sity of streamwise fluctuations. Pressure interactions deflect someof this energy to fluctuations in the direction of the velocity gradi-ent - and so the sequence repeats itself. It is what we might callturbulence's eternal triangle.

Figure 1.1: Stress couplings in simple and curved shear flows:Primary strain; _ Secondary(curvature) strain; ~*-~ Pressure-strain effect.

The lower line in Table 1.1 shows the effects of superimposing onthe primary shear a weak secondary strain 6 (=dU2/dxi) which rep-resents a curving of the mean-flow streamlines. The shear stress andthe normal stress in the direction of the velocity gradient are directlymodified and their effects reinforce one another. That is to say, if6 is positive, the extra contribution to P\i will tend to enlarge the(negative) magnitude of u\ui while that for P22 leads to an enhance-ment of u| which in turn helps amplify u\U2 through the principalcontribution to P^. It is this mutual reinforcement property of P;J,represented in Figure 1.1 by the broken lines, that makes turbulentshear flows so sensitive to weak streamline curvature.

Buoyancy has an effect on turbulence generation that in somerespects is akin to streamline curvature. It is more complicated,however, for it involves a coupling of the Reynolds stress and scalarflux fields Ui& . The corresponding equation for u$ is obtained bymultiplying eq (1.5) by U{ then adding it to eq (1.2b) multiplied byand 0 averaging. The result may be expressed as


Pij Pll P"22 -Ps3 -Pi 2

Due to primary

shear, -lH^u^X 0 0 -u\X

^ = A9Z2

Due to curvature

perturbation, 0 — 2uiu^6 0 —u\d

QUi _ g9si

Table 1.1: Stress Production Rates due to Primary and SecondaryShear

The emerging equation is similar in structure to that describingthe transport of Reynolds stress. The generation terms P,# compriseproducts of second-moment correlations and mean-field quantitiesand will not require approximation. Diffusive transport of u$ (dio} iscaused by velocity and pressure fluctuations and by molecular trans-

252 B. E. Launder

port while pressure fluctuations also play a non-dispersive role (<fog)which we shall later see is of vital importance.

The unknown processes in the above equation will be approxi-mated in Section 2. Here we consider briefly the form taken by thegeneration terms under a simple temperature gradient dQ/dx^ in afluid moving in direction x\. If the background Reynolds stress fieldis isotropic (ufUk = \&ikUmu,m) the only direction in which a heatflux is generated is that of the temperature gradient x2 and the signof P-2B is opposite from that of the temperature gradient. This isalso the case in a non-isotropic stress field if the mean shear lies en-tirely in a plane normal to the temperature gradient (last line Table1.2). When the direction of mean velocity gradient coincides withthat of the temperature gradient; however, a stream wise scalar fluxis generated, contributions arising from both P{$\ and Pt'02- If we

accept the idea that the stresses and fluxes will be of the same signas the generation terms, we can see that the two contributions to thegeneration reinforce one another, both being of the same sign as theproduct

Table 1.2: Heat flux generation rates due to a temperature gradient8®/dx2

With the above information, it is now possible to infer the effectof a buoyantly stable stratification of fluid on the stress generation

Pie Pie Pie Pze

Due to a shear ~uiu2^ ~U2§^ ®

dU!/dx2 -^"g-

Duetoashear 0 ~ulj^ °

dUt/dxa


rates. Turbulent variations in density,yO, give rise to a fluctuatingbody force per unit pass —p' g/p in the vertical direction, g beingthe gravitational acceleration. Thus, in the Reynolds stress equa-tions (with #2 vertically up) we find non-zero buoyant generationcomponents as follows

In a stably stratified medium dQ/x^ is positive; to fix ideas let ussuppose dUi/dx? is also positive. So, P^Q and thus (we suppose) G^is negative. Likewise, from the above discussion, Pie and hence G\ewill be positive. Thus, there is a double-edged effect of buoyancyon the Reynolds shear stress. G-22 will tend to reduce u\ thus di-minishing PI 2 (see Table 1.1) while Gi2, being of the opposite signfrom P12 will also act to suppress the vertical transport of streamwisemomentum by turbulence.

1.5 The WET Model of Turbulence

The foregoing sub-section has obtained the exact transport equa-tion for UiUj and UiO and examined the form that the generationterms in these equations take in a few situations. As we have seen,these generation terms are exact.

Of course, before we can use the transport equations to find thestress levels, models must be devised for the unknown processes - thetask of Section 2. To round off the Introduction; however, here is amodel to avoid modelling. It is based on the simple economic ideathat

If the fluid in question is a perfect gas p'/p — —6/Q (where theorigin on the 0 scale is absolute zero) and so the generation ratesmay be re-written

As such it is a gross over simplification; yet it is still more truethan untrue that the more one earns the better off one is; and thatsomeone of 50, with his house paid for, will be wealthier than someone

254 B. E. Launder

of 25 just starting out on his career. Extrapolating this concept tothe second moments leads naturally to the proposition

Thus:

The choice of turbulent time scale Tt and how to compute it is de-ferred to the next section. The basic idea can, however, readily betested by looking at ratios of the scalar fluxes. Thus:

In a mildly heated shear flow without body forces in which dQ/dx^and dU\/dx2 are the only non-zero temperature and velocity com-ponents, one finds from experiment that the left side of eq (1.11) isapproximately -1.3 in a free flow, the right side about -1.6. Near awall the ratio of the turbulent heat fluxes is larger, about -2.2, as iscorrespondingly the ratio of the generation terms (about -2.1). Thus,there does indeed seem to be more than a casual connection bet-ween the left and right sides of eq (1.11). The success is particularlystriking when set against the background of results given by simpleeddy diffusivity models. Such schemes would predict that, becausethere are no x\ gradients in mean temperature, the turbulent flux inthat direction would be zero!

When considering the Reynolds stresses we need to apply theWET concept to departures from the isotropic state. (This is notinconsistent with eq (1.10); the isotropic value of a vector is of coursezero). Accordingly:

In fact, this formula, arrived at by a much less direct route, hasformed the basis of many successful predictions of turbulent freeshear flows.


2 CLOSURE AND SIMPLIFICATION OF THESECOND-MOMENT EQUATIONS

2.1 Some Basic Guidelines

Our aim is to mimic those processes not exactly determinableat second-moment level in terms of mean and turbulence propertiesthat are. At the practical level there is pressure to adopt the simplestclosure consistent with achieving the desired width of applicability.This naturally affects the importance that different workers haveattached to different closure principles.

Clearly, as a first requirement, any surrogate form must have thesame dimensions as the correlation it replaces; there is no controversyon this point. Next, we would also insist that the mathematicalcharacter of the model should conform in various respects with theoriginal. For example, if the process requiring approximation is asecond-rank, symmetric tensor with zero trace, the search for a modelshould be limited to forms possessing these properties. Althoughthis principle is usually adhered to at second-moment level, it isfrequently ignored in modelling the third moments. This can beregarded as the application of a further fundamental concept: theprinciple of receding influence. Broadly, the idea is that the nth-moment correlations have markedly less effect on the mean flow thanthose of (n — l)th order. So, rules that are held inviolate for secondmoments are sometimes dispensed with in the interests of algebraicand computational convenience in dealing with third moments. It isclearly a matter of taste and of the flows to be calculated how freelyone invokes this principle. Everyone developing models at this levelmakes some use of it; however, for it is that idea which ultimatelylegitimatizes second-moment closure.

Two further principles of mathematical physics have been in-voked in determining modelling approaches. It is generally acceptedthat the approximate forms should exhibit the same responses totranslations, accelerations and reflections of coordinate frame as thereal processes (e.g. Donaldson et a/., 1972). The second constraintis that the modelled set of transport equations should be renderedphysically incapable of generating impossible or 'unrealizable' valuessuch as negative normal stresses or correlation coefficients (such asuiuif^/(u\ ^2)) greater than unity, Schumann (1977). Here men-tion should also be made of the work of Andre et al. (1976) who

256 B. E. Launder

devised a scheme for overwriting or 'clipping' the values of triplemoments whenever they reached physically unattainable values incomparison with other double and triple moments. Although Schu-mann suggested ways of securing 'realizability', the interested readeris referred to the far more detailed treatment by Lumley (1978). Un-fortunately, although the principle of realizability is a sound one, itsadoption adds considerably to the complexity of the turbulence clo-sure and only the latest generation of modelling proposals employ'realizable' forms in computations of inhomogeneous flows, e.g. Shihand Lumley (1985), Fu et al (1987), Craft and Launder (1989).Models which in principle are capable of generating impossible val-ues may, in practice, only do so for flows which are of only academicinterest. Second-moment-closure studies of recirculating, swirling orother complex flows have been made with forms that do not guaran-tee realizability. Some would argue that to insist that all models befully realizable is rather like requiring that all new buildings in Parisbe as resistant to earthquake damage as those built in San Francisco.Of course, it could be the case that building in 'earthquake-prooftechnology brought other benefits to the overall design of the struc-ture that justified the expense. There are some indications thatthis might be the case provided a sufficiently capacious framework isadopted. For example, the UMIST group has found that employingfully realizable closures has enabled the observed reduction of tur-bulent mixing at high strain rates to be accommodated - a featurethat consistently eluded the simpler models that were not.

The next concept is one that blurs into intuition. The turbulencemodeller should always try to ensure that the proposed form is aphysically plausible substitute for the real process. This statementrelates to the choice of physical quantities; for example, whether themodel should comprise exclusively turbulence correlations or termscontaining mean-field elements - and the way they are combinedtogether. It is often helpful to explore different ways of expressingthe correlation of interest; one form might give much more insightthan another.

A considerable simplification to the task of turbulence modellingresults from applying the high-Reynolds-number hypothesis. Simplystated the proposal contains two complementary ideas:


1. that the large-scale interactions predominantly responsible formomentum and scalar transport are unaffected by the fluid'sviscosity;

2. that the fine-scale motions responsible for viscous dissipationare unaware of the nature of the mean flow and the large-scaleturbulence. Their structure is similar to that found in isotropicturbulence.

It is becoming increasingly evident that the fine-scale motion,particularly in regard to higher moment correlations, is not exactlylike isotropic turbulence (e.g. Wyngaard and Sundararajan, 1979),but nevertheless, if judiciously applied, both aspects of the high-Reynolds-number hypothesis are very useful in turbulence modelling.

2.2 The Dissipative Correlations

As noted in Section 1, the dissipative correlations c^j and e,-# arisefrom the fine-scale motion for it is in these eddies that gradients ofvelocity, temperature, etc. are steepest. We assume, therefore, fromthe high-Reynolds-number hypothesis, that the motions contributingpredominantly to e^ and e,-0 are isotropic. Now, in isotropic turbu-lence dui/dxkdd/dxf; changes sign if the coordinate direction x; isreversed; but the properties of isotropic turbulence are unaffectedby such reflections of axes. The only value that dui/dxkdO/dxk cantake, therefore, and be consistent with isotropic turbulence, is zero.That is:

Likewise etj must be expressible as proportional to the productof the contraction e(=v(duij'5xj)2) and the isotropic unit tensor Sij.Thus:

where the constant of proportionality is obtained by contracting eachside of the equation. An implication of (2.2) is that there is no viscoussink of shear stress. The process </>,-j is thus the only mechanism forpreventing the unlimited growth of the off-diagonal components ofUiUj.

There is still little agreement in the literature on whether thedissipative correlations can be adequately obtained from isotropic

258 B. E. Launder

relations. Direct measurements are not usually sufficiently accurateto allow conclusions to be drawn. Direct numerical simulations bythe Stanford group and others show significant and systematic de-partures from local isotropy (Reynolds, 1984) but inevitably thesesimulations are at relatively low Reynolds number. In practice, inapplying second-moment closure, it is difficult to disentangle depar-tures from equations (2.1) and (2.2) from errors in accounting for thepressure-strain and pressure-scalar gradient processes. So, a commonpractice (Lumley, 1978) is to adopt the isotropic relations for e,-j and€{g and to absorb any departure from isotropy in the dissipation pro-cesses into the turbulent parts of (f>ij and </>;# whose approximationis now considered.

2.3 Non-Dispersive Pressure Interactions

By taking the divergence of the Navier-Stokes equations (1.14)and subtracting its mean part, a Poisson equation is produced withthe fluctuating pressure as its subject:

We may regard terms on the right of eq (2.3) as sources of pres-sure fluctuations. There are evidently three agencies: non-linearinteractions between fluctuating velocities p^y a process involvingmean velocity gradients, p/2y and one arising from fluctuating bodyforces, p(3). It is to be expected, therefore, that a successful modelfor the turbulent correlations involving pressure fluctuations (0,-j andfag ) will contain terms corresponding to these different sources ineq (2.3). Thus, the pressure-strain process may be written


where the primes denote that the quantity in question is evaluated ata distance r from where faj is determined. It is convenient to writethe above equation in the short-hand form:

An analogous equation for fag^ may likewise be obtained and theresult written as

In principle, approximations to each of the constituent processesin (2.4) and (2.5) may be developed starting from the solution of thePoisson equation for p. Analyses of this type have been presentedby Naot et al. (1973), Lin and Wolfshtein (1979), and others. A lessformal approach is more often favoured, however. General surrogateforms are assumed for the different component parts of faj and fag.Then, by insisting that the model possess certain symmetry andcontraction properties of the original process and/or that it shouldcomply with some other physical constraint or experimental data,the various constants in the model are determined.

Mean-Strain Contributions fajzdiez

The strategy summarized in the above paragraph is well illus-trated by the 'quasi-isotropic' (QI) model for faj2. This approachassumes

where aikij is a fourth-rank tensor comprising Reynolds-stress ele-ments. Note that the postulated model, like p/2\, depends linearlyon the mean velocity gradient. The form of a^ij is unknown at theoutset so one begins by writing the most general form and then usesthe desired symmetry properties to reduce the number of unknownconstant coefficients. If attention is restricted to terms linear in theReynolds stress there are five independent coefficients.

260 B. E. Launder

Moreover, reference to eq (2.4a) shows that continuity requires thataiku = 0 while direct integration of the Poisson equation for thecontraction formed by setting j = k leads to the requirement that

Each of these constraints furnishes two relations that the coeffi-cients a, /3, etc. must satisfy; for example, the latter requires that

So, the model may be organized in a form with just one undeterminedcoefficient. After some algebra eq (2.6) may be expressed

where

Equation (2.7) has been invented and rediscovered many timesin the past twenty years e.g., Launder et al. (1973), Naot et al.(1973), Lumley (1975, 1978), Reynolds (1984). Different strategieshave been adopted by different workers in fixing the final constant, 6.The writer's group, like Naot et al. (1973), have chosen S to optimizethe relative stress levels found in simple shear (S = 2/55); Reynolds(1984) took 6 = 10/77 in order that the resultant expression shouldbe formally independent of the mean vorticity while Lumley (1978)takes 6 = -2/33. These vastly different values for S do not havea major effect on the first of the three groups in eq (2.7) (whichis the major term in components where P,-j is large) but give largevariations in the coefficients of the other two.

The behaviour produced in shear flows by either Reynolds' orLumley's value of 6 is quite unacceptable. To compensate, theseworkers introduce non-linear terms to a^ij (terms like u\u^ u,Uj/ketc.). An advantage of this elaboration is that, because of the ad-ditional unknown coefficients, one can satisfy the "two-component


limit." At a wall or an interface, such as a free surface, turbulentfluctuations normal to the surface vanish faster than the other com-ponents. This requires that (with xa the coordinate normal to theinterface) <t>aa2 should vanish as u^, goes to zero. If one retains allthe possible quadratic products of Reynolds stress in a^,-;- one findsthat the number of assignable coefficients exactly equals the num-ber of algebraic equations in satisfying the two-component limit aswell as the other constraints discussed above. The resultant form isconveniently written (Shih and Lumley, 1985; Fu et al, 1987):

where P is the generation rate of turbulence energy, | Pkk-

Eq (2.7b) has an interesting appearance. The leading term isthe same as in eq (2.7a) with a somewhat smaller coefficient. As weshall see later, the second term effectively introduces a dependenceon (P/e) into the return-to-isotropy coefficient of faji while the thirdintroduces more varied non-linear effects. This formulation, despitethe rigour and the respect for realizability implicit in the satisfactionof the two-component limit, does not lead to acceptable relative stresslevels in simple shear flows: with U\ — E/i(x2) it produces u\ >u| contrary to both experiment and direct simulations. Shih andLumley [4] (1985) overcame this problem by adding a correction term

^2

The parameter A is defined as [1 - 9/8 (A^ - AS)] where A^ andAS are the second and third invariants of the anisotropic part of theReynolds stress:

Az = a i jOfc i ; As = aikakmUmi where aij = (uTuJ — l / 3 6 i j U k U k ) / k .

It equals unity when the stress field is isotropic but its most usefulproperty (Lumley, 1978) is that it vanishes in two-component tur-bulence. It is thus tempting to refer to A as the "flatness factor",

262 B. E. Launder

despite the quite different meaning usually attached to that expres-sion.

In place of equation (2.7c) one may alternatively include third-rank products of Reynolds stress in aikij'. in that case there are intotal twenty independent groups. On applying the same kinematicconstraints as for the quadratic model one arrives, after considerablealgebra, at the following form (Fu et al., 1987):

where

The first two lines of this equation are identical to the quadraticmodel while the coefficients c-2 and c'2 of the cubic terms are freelyassignable. The quantities that these coefficients multiply are evi-dently of very different length, that associated with c'2 being particu-larly long. It has been found, however, that the behaviour of simplefree shear flows can be very satisfactorily mimicked by setting c% to0.6 and c'2 to zero, Fu et al. (1987) (see also Launder (1989)), thusreducing the algebra to more manageable proportions.

Notice that

where fij/t is the vorticity tensor (dUj/dxk — dUk/dxj). Thus, ifc2 is taken as zero, the cubic terms are identically zero in an irrota-tional deformation and they can be seen as purely a correction forrotationality.


Recently at UMIST there has been a tendency to include non-zero values of c'2. In fact, in his original study of homogeneous shearflows, Fu (1988) showed that the choice 02 = 0.55, c'2 = 0.6led to an improved prediction of the stress tensor in strong simpleshears6. Li (1993) (see Launder and Li, 1993, 1994) found that whenthese same values were adopted for near-wall inhomogeneous flowsthey were more successful in reproducing the increasing anisotropy asthe wall was approached than the earlier combination (0.6 and 0.0).Examples of predictions made with both sets are presented later.

The Isotropization of Production (IP) Model

In fact, to date, most computations of complex flows have beenmade with a simpler model of 0^2 than any of the above. Sometimescalled the Isotropization of Production model, it supposes that thenet effect of the mean-strain part of the pressure strain process iseffectively to reduce the anisotropy of the production tensor. Thusone postulates

</>i# = - C2 (Pij - \^Pkk). (2.8)

While the proposal was inspired by intuition (Naot et a/., 1970)it is interesting to note that if, as is often the case, the value of c%is taken as 0.6, in isotropic turbulence (uj/uj = 1/3^,-jupZfc"), oneobtains the exact result obtained by Crow (1968)

Equation (2.8) can be regarded as a simplification of equation(2.7d) in which simply the leading term - indeed the only linearterm of that equation - is retained. It has been widely applied in adiversity of flows and has been found to be conclusively better thanthe superficially more general linear form, eq (2.7a), given earlier, atleast when used in conjunction with the simple linear model of faji(discussed below).

The idea underlying the IP model is readily applied to the forcefield part of </>,j and, moreover, to the mean-strain contribution ofthe pressure-scalar gradient contribution fooi- Thus

bThe shear rate may be characterized by the dimensionless parameterS = k ( d U i / d x 2 ) / e . Local equilibrium is attained for S ~ 3.0 ; 'strong'shear relates to values of 5 of 5 or greater.

264 B. E. Launder

With the coefficient c2# set to about 0.5 this formula is distinctly moresuccessful than the linear quasi-isotropic form (i.e. the equivalentof eq (2.7a) for 0jj) which in t his case is obtained with no freecoefficient:

The latter form (Launder, 1973) is incompatible with predicting thecorrect ratio of heat fluxes down and at right angles to the meantemperature gradient in a simple shear flow. The fact that it has nev-ertheless been quite widely used in predicting boundary layer flowsreflects that turbulent heat fluxes in the streamwise direction areusually not important (being far outweighed by mean convection).The problems with eq (2.10) can be largely overcome by proceeding,within the 'quasi-isotropic' analysis, to include terms up to quadraticorder in the Reynolds stresses. The approximation level is then for-mally equivalent to that of eq (2.7d). The resultant form is of similarlength to that equation and is not reproduced here. Successful ap-plications of the model have, however, been reported by the UMISTteam (Craft and Launder, 1989; Cresswell et al. 1989) in a range offree shear flows.

Effects of Force Fields <pij3, 4*i93

Because of the considerable variety of possible 'force' fields andof source or sink terms in the transport equation for U{ and 0, nogeneral statement can be made on the effect that they bring to (^3and (f>ies- In engineering, the two most important force fields arethose due to buoyancy and to rotating the coordinate frame (Cori-olis forces) to facilitate, for example, the study of flows in rotatingpassages. In these cases, the QI model leads to a form identical tothat of the IP approach (Launder, 1975, 1976: Cousteix and Aupoix,1981; Hah, 1981), namely

The QI analysis gives a value of the coefficient 03 of 0.3 for buoyantflows though those adopting the IP approach have generally preferredto choose a value akin to that used for mean strain effects; a valueof 0.5 to 0.6 is typical. Correspondingly, for 4>ig3 we find


where c^g from a QI analysis equals 1/3 for buoyant flows while,again, most applications have tended to chose a value of about 0.5.The accuracy with which these simple models can hope to capturethe real processes hardly justifies a separate optimization for eachconstant. A choice in the range 0.3 to 0.5 would nearly always beappropriate.

In the case of Coriolis forces, there are grounds for taking the co-efficient €3 to be half that adopted for c%, a topic discussed extensivelyby Launder et al. (1987). To illustrate this point, an axisymmetricswirling jet can be analysed perfectly well in a stationary referenceframe but it can equally well be examined with the coordinate framerotating about the jet's axis. If, however, one takes any ratio of 03 : c2

other than one half, the predicted results are dependent on the rate ofrotation of the coordinate system. Obviously, the predictions shouldbe independent of the observer's frame of reference.

In fact, there are various ways of ensuring independence of theresults from the observer's motion as discussed by Fu et al. (1987).One is the practice noted above; another is to include the convec-tive transport tensor DUiUj/Dt - which is hereafter denoted dj -along with the production tensor and Coriolis tensors in applyingthe isotropization of production approach:

where Fij denotes the Coriolis tensor

Fu et al. (1987) found the use of eq (2.13) brought great improve-ments to their predictions of swirling jets while in non-swirling flowsthe net contribution of terms containing dj was small.

The Turbulence-Turbulence Part of faj and fag

To model the parts of 4>ij and fag arising from p^ we seek formscontaining only turbulent quantities. Experiments indicate that gridturbulence made strongly non-isotropic by passing it through a ductof rapidly changing cross-sectional shape will revert towards isotropyonce the mean strain is removed. In the absence of any alternative

266 B. E. Laundei

process, we must conclude that <$>(j\ (which, like the other parts o:(f>ij, is traceless) is the agency promoting this reversion. In mos1computations of complex flows Rotta's (1951) linear return mode'has been adopted:

More elaborate versions have been proposed (Lumley, 1978, Rey-nolds 1984, Fu et a/., 1987) which are conveniently expressed throughthe dimensionless anisotropic stress a,-j:

The second term in eq (2.15) produces an interesting asymmetry ofresponse. Suppose all the ajj's are zero except an = £ and a^i — —£(so A? = 2<52). If c( is positive, </>m takes a larger negative valuethan the (positive) value of ^221- Consequently u\ will tend to revertmore rapidly to isotropy from above than will u2. from below, whichin turn means that 0,33 will increase from its initial zero value.

The coefficients Cj and c" may be chosen so that the resultantexpression for </>,-ji exactly satisfies the two-component limit0. If, asbefore, we take u\ — k(l + <5), u\ = k(l — <5), the requirement that</>22i should vanish irrespective of the size of 6 leads to c^ = 3/2,c" = —3/4. While we have experimented at UMIST with thesevalues, a problem that their use introduces is that, for moderatelevels of anisotropy (A2 in the range 0.2 - 0.5, say), experiments andcomputer simulations on the return to isotropy of highly anisotropicstress fields indicate a faster proportionate rate of return than whenthe stress field is only weakly anisotropic. The negative value of c'{on its own produces the reverse of the desired effect. One can, inprinciple, offset the consequences of a negative c!{ by causing c\ toincrease rapidly with A2. No firm proposals of this type have so farbeen advanced, however. A simpler way of making </>,-ji vanish intwo-dimensional turbulence is to arrange that c\ should contain thefactor An. Lumley's group prefer this route and take the coefficientsc[ and c'{ as zero. At UMIST a non-zero value of c( is retained, thepresently adopted form being:

cClearly, Rotta's original form does not satisfy this limit since, if one compo-nent of Reynolds normal stress vanishes, its value of ai} is — 2/3 ; so (juj for thiscomponent equals +2/3 cie rather than zero.


The corresponding process fagi in the U{6 equation is convention-ally modelled, following Monin (1965), as:

More elaborate forms have been proposed (Samaaweera, 1978;Lumley, 1978; Craft and Launder, 1989)

while several workers (Launder, 1976; Elghobashi and Launder, 1983;Jones and Musonge, 1983) have suggested the partial or completereplacement of the dynamic time scale k/e by the scalar time scale1/2 ¥/Cg.

Optimum Choice of Coefficients in Basic Pressure-Strain Model

Equations (2.8) and (2.14) have been the basis of many differ-ent proposals and have been incorporated into several commercialsoftware packages. For this reason the pair of equations (including,where appropriate, wall-reflection terms as discussed below) is oftenreferred to as the basic model.

The question arises, however, as to what values should be as-signed to the coefficients c\ and c2. Values proposed for c\ rangefrom 1 to 5 while recommendations for c2 cover the range from zeroto 0.8, Fig 2.1. The range of different choices suggests, at first glance,that since such disparate values have been put forward the wholeapproach is worthless. Looked at with an engineer's eye, however,one might be tempted to fit a straight line through the 'data points'.Now, in the case of a simple shear flow in local equilibrium (i.e. whereturbulence generation and dissipation processes are in balance) it isreadily shown that with this basic model the resultant stress ten-sor depends not on the individual values of c\ and c2 but rather onthe single parameter (1 — c2)/ci. The line in Fig 2.1 is simply thatcorresponding to (1 — c2)/ci = 0.23 which evidently does rather agood job of fitting the various proposals. What we conclude is that,for equilibrium simple shear flows, the very different pairs of ci andc2 will lead to nearly the same results. In order to pick the 'best'pairing one needs to look at non-equilibrium cases. We have already

268 B. E. Launder

Figure 2.1: Map of proposals for coefficients in Basic Model of <j>ij.

noted that a value of c2 of 0.6 exactly describes the case of isotropicturbulence subjected to rapid distortion while direct simulations ofthe return of anisotropic turbulence towards isotropy suggest a levelof GI of from 1.5 to 2.0 if the level of stress anisotropy is similar tothat found in a typical free shear flow. The pairing usually adoptednowadays of 1.8 and 0.6 is fully compatible with these extreme casesand is marked by a triangle in Fig 2.1. Before leaving this topic,mention should be made of Younis' (1984) pairing since this is themost recent entry in the table. His proposal arose from the failure topredict swirling jets correctly when larger values of c2 were adopted.However, as has been earlier noted, PZJ is not an objective tensor. Ifthe model of <f>ij is made objective (as it should be) by including thestress convection tensor dj (eq (2.13)) then it is found that reason-ably satisfactory predictions of the swirling jet are obtained with thestandard values of GI and cj, Fu et al. (1987).


Wall Effects on ̂

Although sometimes ignored, in near-wall flows pressure reflec-tions from the surface have a significant effect on the pressure-straincorrelation. It isn't just rigid walls where surface effects on <j>ij and(f>ig are important. In free-surface flows the resultant pressure "re-flections" from the free surface are significant^ - indeed, it is theireffect that causes the maximum velocity in a river usually to be lo-cated below the free surface. The main effect of the wall on faj is todampen the level of u\ (the normal stress perpendicular to the wall)to about half the level found in a free shear flow. It is mainly thisreduction that makes the wall jet in stagnant surroundings spread atonly two thirds of the rate of the free jet.

Do these wall effects automatically get accounted for as one ap-proaches the wall, effectively through the boundary conditions? Agreat deal of research at present is aimed at developing models of fyjthat do automatically respond to the wall's proximity. However, the'basic' model presented above requires rather substantial correctionclose to a wall to return accurate relative levels of the stress tensor.This is usually achieved by introducing the unit vector normal to thewall (n/t) and applying a correction proportional to the turbulentlength scale, fc3/2/e, divided by the distance from the wall. The rec-ommended correction of this type combines proposals of Shir (1973)and Craft and Launder (1992) and may be written

where

^Strictly, the free surface is at a uniform pressure. What actually happens isthat agitations cause the free surface to be slightly "crinkled", i.e. the surfacetopography is not quite plane and is continually changing. However, if we imaginethe free surface to be replaced by an undeformable but frictionless lid, as wehabitually do in free-surface studies, then pressure fluctuations will exist at thelid surface.

270 B. E. Launder

In the above / = fc3/2/€ and y denotes the wall distance. Figure 2.2,

Figure 2.2: Behaviour of two near-wall flows a) Turbulence intensitiesin flat plate boundary layer b) Component of turbulence normal towall on axis of axisyrnmetric impinging jet (from Craft et a/., 1993a)Symbols: Experiment or DNS results; — Eq (2.18) wall reflection;

Gibson-Launder (1978) wall reflection.

from Craft and Launder (1992) shows the application of the basic


model using the wall correction of eq (2.18) to compute the flat plateboundary layer and the impinging turbulent jet. The latter flow is acritical turbulence modelling test case: numerous models developedby reference to flows parallel to walls give seriously incorrect pre-diction of the radial wall jet development as the flow develops awayfrom the vicinity of the stagnation point (Craft et a/., 1993a). Thefigure indicates that, for the impinging jet, the form recommendedfor 4>™j (solid line) achieves much better agreement with experimentthan does the earlier (and more widely used) proposal of Gibson andLaunder (1978) shown by a broken line.

It would, however, seriously misrepresent the situation to leavethe impression that modelling of near-wall influences was in a satis-factory state. Equation (2.18) and similar approaches may be ade-quate if one is dealing with a single plane or mildly curved surface.In most engineering applications, however, one needs to predict flowswithin an enclosure or around bodies with several distinct faces. Inthese cases the approach indicated by eq (2.18) is, at best, a schemethat requires ad hoc, case-specific interpretation while in others it issimply unworkable. In flow through a square duct, for example, thereare four vector directions normal to a wall and, correspondingly, fourwall-normal distances. Analogous problems arise in handling flowthrough tube banks, within internal combustion engines or in turbo-machinery flows. It is this geometrical complexity that has spurredefforts to eliminate from the closure the very parameters on whichtraditional 'wall-proximity' corrections depend, namely wall distanceand unit vectors.

What has come in their place? Firstly, as noted earlier, a benefitof using the non-linear models for <$>{j\ and <^,-j2 is that there is amuch diminished wall-proximity correction. Secondly, there is therecognition that, within the immediate wall vicinity, the turbulenceis varying so rapidly in the direction normal to the wall that someexplicit correction for inhomogeneity should be made to the pressure-strain process, Bradshaw et al. (1987). For example in replacing theexact expression for faji in eq (2.4a), there is the assumption thatdU'k/dx'j (that is, the mean velocity gradient evaluated at a distance rfrom the point where (f>ij is to be determined) can be replaced by thatat the point itself. Launder and Tselepidakis (1991) first proposedimproving this assumption by adopting, instead, an effective meanvelocity gradient

272 B. E. Launder

In current work at UMIST the quantity I is taken as IA1/2. Thereason for introducing A1/2 is pragmatic: the conventional lengthscale / = k3' 2/e exhibits a point of inflexion within the buffer layerwhich causes dl/dxm to undergo excessive variations in magnitudethus making it unsuitable. The smooth increase of A as one movesaway from the wall removes this problem. The quantity c/ has anoptimum magnitude of about 0.15.

Figure 2.3: Prediction of fully developed flow in a duct of squarecross section, Launder and Li (1994) a) Mean velocity contours b)Wall shear stress distribution. Symbols: Experiment, Gessner andEmery (1981) ; Cubic Model without wall correction;Basic Model.

Figure 2.3 shows an application of a closure employing eq (2.7d)(c2 = 0.55; c'2 = 0.6) together with an earlier form of eq (2.19)to the problem of flow in a duct of square cross section. The an-isotropy of the Reynolds stress field in the cross sectional plane ofthe duct induces weak secondary motions which are responsible forcausing the mean velocity contours to bulge towards the corners.The predicted flow is extremely sensitive to suppositions about walldamping and it is encouraging that agreement is reasonably good.If one applies the Basic Model with wall-reflection terms omitted no


secondary flows are predicted. However, to include wall reflection,by way of an equation like '(2.18), requires arbitrary decisions abouthow to handle the abutting walls. In the calculations shown in Fig2.3 it has been supposed that wall reflections associated with the fourwalls can be linearly superimposed treating each alone as though itwas a plane surface of infinite extent. These assumptions cannot beapplied for ducts of arbitrary cross-section and even here, the qualityof agreement is much inferior to that using the non-linear pressure-strain models with an inhomogeneity correction and no wall reflectionterm. This latter approach is particularly successful in predicting thedistribution of shear stress around the perimeter.

The use of non-linear models, together with an inhomogeneitycorrection to <pij2, is not fully satisfactory for handling flows im-pinging orthogonally (or thereabouts) on a wall. In these cases themean velocity decreases virtually linearly to zero along the stagna-tion streamline so eq (2.19) returns values negligibly different fromthe local velocity gradient. At UMIST we are presently seeking al-ternative approaches to achieving the desired damping including theadoption of an inhomogeneity correction to (f>ij\, the turbulent partof the pressure-strain process.

2.4 Diffusive Transport d,-j, dig

At least as much may be written about approximating diffusivetransport as about the non-dispersive action of pressure fluctuationsdiscussed above. Only a few brief remarks will be made, however,partly because the processes are, in most engineering circumstances,relatively uninfluential on the mean flow development, partly becausethis article is intended to provide an introduction to turbulence mod-elling, and partly because the quite elaborate models to be found inthe literature have by no means been extensively tested.

The most popular basis for representing diffusive transport insecond-moment closures is the generalized gradient diffusion hypoth-esis, GGDH, which may be written:

Thus, if <p is the instantaneous Reynolds stress u,uy:

274 B. E. Launder

or, for the transport of scalar flux

One evident weakness of these forms is that while the indices i and kon the left side of the equation can be interchanged without alteringthe resultant product, such a rearrangement on the right intrinsi-cally alters the form. No ambiguity arises because in the uûj andUfd transport equations a d/dx^ operation is applied to the triplemoments; nevertheless, the difference in character between the exactand the modelled form would appear to be a significant shortcoming.

In fact, forms very similar to (2.21) and (2.22) can be obtained bymaking sweeping closure simplifications to the transport equationsfor the triple moments (Hanjalic and Launder, 1972; Launder, 1976).In that case, the model for uÛjUk consists of three terms identical tothe right side of (2.21) but with a permutation of the indices i, j andk; likewise, that for uû^Q also consists of three terms in which w/t, it,-,0 successively occupy the position of u^ in (2.22). These somewhatmore elaborate and superficially more correct models do not, in prac-tice, seem to bring better agreement when used in numerical solvers.This could be due, at least partly, to the fact that those workers whohave adopted (2.21) and (2.22) for the triple moments have generallynot included any model for the pressure diffusion terms:

and

which appear in the uÛj and u$ transport equations. In these pres-sure terms the indices i and k are not interchangeable. Thus, perhapswe should really regard (2.21) and (2.22) as models for the completeturbulent diffusion of TZT/tZJ and w,-0 .

More rigorous and comprehensive models of triple moments havebeen put forward by Lumley (1978), Andre el al. (1979), Reynolds(1984) and Dekeyser and Launder (1984). While most approacheshave started from the exact triple-moment equations, Lumley's ap-proach is based on the orthogonal decomposition theorem in whichapproximations for the triple moments are developed from thosemade for the second moments.


2.5 Determining the Energy Dissipation Rate

The scalar 6 remains as an unknown in the models so far dis-cussed; it is to be obtained by solving a transport equation. Beforediscussing proposed forms for that equation; however, it may be help-ful to address the following frequently recurring question: how, inthe absence of reliable measurements of e, can one tell whether poorReynolds stress and mean velocity predictions arise from errors in de-termining the level of the energy dissipation rate or from weaknessesin modelling the unknown processes in the ufuj equations, especially</>{,•? While difficulties in discrimination do sometimes occur, it isoften possible to distinguish the source of the problem. Errors in ewill tend to give too high or too low energy levels; errors in fyj willtend to give the wrong relative levels of individual Reynolds stresses.Thus, Launder and Morse (1979) traced the failure to predict theswirling free jet to the pressure-strain process, fajz, because the cor-relation between streamwise and azimuthal velocity fluctuations wasof the wrong sign. For the (non-swirling) axisymmetric jet, however,the relative stress levels are reasonably correct but the predicted levelof kinetic energy is too large - a result that points to the dissipationrate equation as the main source of error.

Chou (1945) proposed the first equation for a variable propor-tional to e but current concepts in modelling the energy dissipationrate really spring from the work of Davidov (1961). Although an ex-act transport equation for 6 may be derived from the Navier-Stokesequations, this is by no means as useful as the corresponding equationfor UfUj. The reason for this is that the major terms in the equationconsist of fine-scale correlations describing the detailed mechanics ofthe dissipation process. However - from the high-Reynolds-numberhypothesis — the fine-scale motions, so far as the Reynolds stress fieldis concerned, are passive; they adjust in size as required to dissipateenergy at the rate dictated by a system of substantially larger eddieswhose structure is largely independent of viscosity. The conclusionto be drawn from all this is that, in devising a transport equationto mimic the spatial variation of e, one is essentially resorting to di-mensional analysis and intuition. All proposals can be written in theform :

where Tt denotes the net transport of c and EST stands for "extra

276 B. E. Launder

strain terms".Originally, both coefficients cei and c£2 were taken as constant

with "standard" values of about 1.44 and 1.92 respectively. Lum-ley (1975), however, argued that ct\ should be taken as zero andce2 should become a function of the second invariant A^. The subse-quent studies on buoyant diffusion undertaken by Zeman and Lumley(1979), however, recommend the retention of both types of process.Specifically they proposed

There are sound physical reasons for preferring to eliminate theterm containing c^ from the (. transport equation but Lumley's ex-periences and those of Morse (1980) show that this is not possible.The latter found that, in predicting free shear flows, taking cei aszero and c£2 a function linearly proportional to AI led to poor profileshapes. Worse, the loose coupling between the HiUj and e fields thatresulted produced strongly oscillatory rates of spread. Recent workat UMIST has concluded that there are overall benefits to reducingthe value of cei from its "standard" value and compensating by al-lowing ce2 to depend on the invariants A2 and A. The form presentlyrecommended is

The presence of the terms marked EST is an indication that,like the Reynolds stresses, the energy dissipation rate (and thus ef-fective time and length scales) appear to be rather sensitive to smallsecondary strains. Unfortunately while, in the u^Uj equations, theproduction tensor P,-j shows clearly the origin and the approximatesensitivity to particular strains, no equivalent help is available in thecase of the € equation. So, as remarked earlier, one proceeds by intu-ition. Pope (1978), in a well argued proposal, suggests the inclusionof an additional source term in the e equation proportional to

This term is zero in plane two-dimensional flows but not in an ax-isymmetric flow. The size of the coefficient was thus tuned to give the


correct rate of spread of the round jet in stagnant surroundings (us-ing the k - e EVM). Huang (1986) has made further tests on Pope'scorrection, re-optimizing the coefficient to suit the second-momentclosure he was using. When, however, he shifted from the round jetin stagnant surroundings to data of coaxial jets, he found that theseflows were predicted better with the extra term deleted.

Hanjalic and Launder (1980) recommended the addition of a termthat is superficially similar in appearance to Pope's:

The action of the term is quite different, however. The coefficient ofproportionality is negative and, in a straight two-dimensional shearflow, the main contributor to the term is —k(dUi/dx<2) . The termthus acts to reduce the sensitivity of the e source to shear strain rela-tive to normal strain. The additional term has been found to be help-ful in boundary layers in strongly adverse pressure gradients as wellas in the round jet for it allows the term — 2(i^ — u^dUi/dxi (in P^)to make a larger contribution (in comparison with —2uiU2 dU\/ dx-^)to the generation rate of e. Since (dUi/dxi) is predominantly nega-tive in these flows and u\ > u^ the result is that e becomes relativelylarger (length scales smaller), reducing the Reynolds stress levels.The originators tested their proposals over several cases but unfor-tunately not in curved flows. When that comparison was made themodification that the new term brought to the e equation was sub-stantial and its effect was to worsen agreement with experiment.

Leschziner and Rodi (1980) adopted a modification of Hanjalicand Launder's proposal in which at every node the orientation ofthe mean velocity was determined; this direction was designated asXi with x-2 orthogonal to it. With that choice made, they couldunambiguously discriminate between "normal" strains and "shear"strains. The contributions of the former to P^k in eq (2.21) weresimply multiplied by a coefficient approximately three times as largeas c€i (which was retained as the coefficient of the shear strain).With this re- interpretation, Leschziner and Rodi (1980) effectivelychanged the sign of the sensitivity of the equation to streamline cur-vature, though the effects the term produced in a second-momentclosure, were too great.

The above examples illustrate the difficulty with the f. equation.The standard version is naively simple in form and has well known

278 B. E. Launder

failures. Yet, so far, all attempts at improving it have producedmodifications which, when subjected to wider testing, have producedworse agreement than the original for some other flow. This appliesequally to the much publicized RNG form of the equation (Orszag,1993).

It is now becoming popular to compute the scalar dissipation rate€g from its own transport equation. It is a rapidly developing fieldof research that lies outside the scope of this discussion. Interestedreaders may refer to Craft and Launder (1989).

2.6 Simplifications to Second-Moment Closures

Although, at research level, full second-moment closures are nowbeing used in fairly complex recirculating flows, they place sufficientdemands on computer resource to make it desirable to devise simplerschemes, at least for "production runs". This subsection thereforedescribes briefly the main steps to simplification followed by differentworkers.

A popular step in simplification is the so-called algebraic second-moment (ASM) closure. In these schemes the transport (i.e. convec-tion and diffusion) of the Reynolds stresses is approximated in termsof the transport of their contraction, or rather the turbulence energy,k. Only the transport terms contain stress gradients, so this step re-duces the differential equations for u^Uj to a set of algebraic ones(hence the name). The most widely used stress-transport hypothesisis due to Rodi (1976):

where T^ means "net transport (convection minus diffusion) of </>" .Note that in this notation the turbulence energy equation is simply

Insertion of eq (2.26) into the stress transport equation, closedwith the help of equations (2.2), (2.8) and (2.14), produces the fol-lowing constitutive equation linking the turbulent stress and meanstrain fields:

so finally we obtain


where P = 1/2P^. The equation is in fact identical to that givenby the WET model, eq (1.24), save that now the coefficient on theright hand side is explicitly a function of P/e. Note also that thetime scale Tt emerges here as k/e.

Experience at UMIST suggests that, in wall-bounded flows wheretransport effects are of secondary importance, the ASM approachgives results very similar to that of a full-second-moment closurefor about 60% of the computational effort. The behavour is lesssatisfactory in free flows; however, particularly axisymmetric flows(Fu et al., 1987). Transport terms are then much more importantand the weakness of eq (2.26) has more serious consequences than inwall flows. The situation becomes even worse in flows with strongswirl. For this reason the user needs to be cautious when applyingthe ASM approach to free flows.

Brief remarks will now be made about some other approaches toavoid solving the full set of differential Reynolds stress equations.Specifically for application to thin, simply-sheared flows, Bradshawet al (1967) and Hanjalic and Launder (1972,1976) have made other,less general simplifications than the ASM route that allowed the num-ber of stress transport equations to be reduced. Hanjalic and Laun-der retain an equation for uiu? but express all the normal stresseseither as constant fractions of k or make use of the near constancy

1 /2of the shear-stress correlation coefficient UiU2/(u\ ufy in a two-dimensional boundary layer. Bradshaw et al. (1967) assume a simpleprescribable connection between the shear stress and the turbulenceenergy and solve a transport equation only for the latter. For mostof the shear flows considered, a direct proportionality between theshear stress and the turbulence energy was assumed:

the quantity c^ being generally taken as 0.09.One can place most confidence in eq (2.28) when turbulence is in

local equilibrium. That is, when energy generation and dissipationrates are in balance. Now, in a thin shear flow, the energy generationrate is effectively equal to —uiu^dUi/dx^- Thus, in local equilibrium,the kinetic energy equation reduces to

280 B. E. Launder

or, with the help of (2.28)

and finally,

This is the constitutive relation that is employed in the so-calledk — € (eddy viscosity) model (EVM). For the purpose of applying theidea to curved and recirculating flows, eq (2.30) is generalized in anobvious way to:

Now, eq (2.31) is a much simpler stress-strain connection than eq(2.27) - but it is also a less general one. It does not mimic the subtleresponses that the turbulent stress field makes to small perturbationsin the strain field. Nevertheless, it often allows a predicted behaviourin sufficiently close agreement with experiment for engineering pur-poses.

In dealing with thin shear flows and, especially, boundary layers,it is often the case that, instead of solving a transport equation for thedissipation rate, e can be obtained as accurately from the formula:

where / is an algebraically prescribed function of position in theboundary layer.

Thus, if one is working within the framework of an eddy viscositymodel, the expression for the turbulent viscosity becomes

The first such 1-equation EVM was proposed by Prandtl (1945) andsubsequently reinvented by Emmons (1954). It is not just in eddyviscosity treatments that a prescribed length scale has been adopted.Much of Donaldson's group's work with second-moment closures ( e . g .Donaldson et a/., 1972) has used a prescribed profile of /, as has theASM study of turbulence driven secondary flows by Launder and


Ying (1973) and Bradshaw!s "boundary-layer method" (Bradshawet al, 1967).

For historical completeness, one final stage of simplification mustbe mentioned. The introduction of eq (2.30) into (2.29) and thesubsequent elimination of k and t with the help of equations (2.32)and (2.33) leads to

or

Equation (2.34) is known as Prandtl's mixing-length hypothesisin which the mixing length, lm, is equal to cj I . Prandtl (1925) con-ceived the model (by arguments akin to those employed in the kinetictheory of gases) to deal with simple shearing wherein dlli/dx^ wasthe only significant strain. However, the local-equilibrium argumentpresented in these notes enables a more general expression to beobtained; for, in a multi-component strain field, eq (2.29) becomes:

or

This more elaborate form of the mixing length hypothesis has beenused with a surprising degree of success in the computation of flowsnear spinning discs and cones where the velocity vector is stronglyskewed.

2.7 Non-Linear Eddy Viscosity Models

Eddy viscosity schemes do not, on the whole, cope well withstrong streamline curvature, whether this arises from flow over curvedsurfaces or imparted swirling. The question arises whether, by addingnon-linear strain elements to the basic constitutive equation, the sen-sitivity to streamline curvature, that is missing from linear EVM's,can be captured. Models of this type have been proposed since the

282 B. E. Launder

early 1970 's but only relatively recently, with the prospect of applyingCFD to very complex flows, has an impetus developed to devise mod-els with a width of applicability approaching that of second-momentclosure for a computational cost similar to a linear EVM. Such non-linear EVMs have many similarities with ASM's but, for complexflows, they require much less computational effort (typically, onehalf to one quarter) due to their improved stability characteristics.Recent contributors to models of this type include Speziale (1987),Nisizima and Yoshizawa (1987), Rubinstein and Barton (1990), My-ong and Kasagi (1990), Shin et al. (1993) and Taulbee et al. (1993).Our experience at UMIST is that most of the weaknesses of the lin-ear EVM cannot be rectified by introducing just quadratic terms tothe stress-strain relation. Only by proceeding to cubic level does onefind sufficient variety in the stress-strain couplings to achieve thedesired effects. The UMIST work thus adopts the following stressstrain relation:

where

Ci C2 C3 C4 Cs CQ C'j

-0.1 0.1 0.26 -1.0 0 -0.1 0.1


Figures 2.4 and 2.5 show two recent applications of the abovemodel (Craft et a/., 1993b) to flows that defeat conventional lin-ear models, namely fully developed swirling flow in a pipe and theimpinging jet. In the former the roughly parabolic variation of thecircumferential velocity with radius is correctly reproduced while anylinear eddy viscosity model gives rise to solid body rotation. Equally,the dynamic field of the impinging jet is captured just as well with thenon-linear model as with any of the tested second-moment closures.

Figure 2.4: Non-linear Eddy Viscosity model applied to the predic-tion of swirling flow in a pipe, Craft et al. (1993b): o Experiment,Cheah et al. (1993) ; non-linear EVM- linear EVM.

Again it should be said that this is an area where rapid changesare occurring and where further testing is needed to establish rigo-rously the width of applicability of the proposed form.

284 B. E. Launder

Figure 2.5: Non-linear Eddy Viscosity model applied to the turbulentimpinging jet, Craft et al. (1993b): a) Mean velocity b) Shearstress c) rms turbulence intensity normal to wall: o Experiment,Cooper et al. (1993); non-linear EVM; linear EVM.

3 LOW REYNOLDS NUMBER TURBULENCE NEARWALLS

3.1 Introduction

The closure proposals made so far imply negligible effect of vis-cosity on the energy containing motions and no influence of themean-strain field on the dissipative ones. While we may expect theseassumptions to be reasonably valid throughout most of a turbulentflow, the no-slip condition at a rigid boundary ensures that over someregion of a turbulent wall layer, however thin, viscous effects on thetransport processes must be large. The present chapter provides abrief account of turbulent transport in this viscosity-modified sub-layer. A more extensive treatment, albeit less up to date, has beenprovided by the writer elsewhere, Launder (1986).

In round terms, molecular influences may be expected to be in-


fluential over a region extending from the surface to where the local"eddy" Reynolds number, based on a typical eddy dimension normalto the wall and the intensity of velocity fluctuation in that direction,is of order 102. Turbulence models for this region may reasonablycontain elements that introduce appropriately these viscous influ-ences. The region in question is thin (even in low-speed laboratorystudies rarely extending over more than 2mm), the processes arehighly complex and the acquisition of accurate experimental data isgreatly complicated not just by the thinness of this sublayer but bywall-proximity influences of various kinds on the instruments them-selves. It is thus no wonder that , despite the important contributionsbeing made by direct numerical simulations, our knowledge of thisimportant region of flow is still incomplete.

Turbulence models applied in this low-Reynolds-number regionhave usually been developed from some high-Reynolds-number clo-sure by introducing various viscosity-dependent terms and/or bymaking the coefficients of existing terms functions of a turbulenceReynolds number. These adaptations, while largely empirical, arechosen to ensure that certain general kinematic constraints are satis-fied. Models of this type, which we refer to as "low-Reynolds-number-treatments", are discussed in Section 3.3

Although the extreme thinness of the viscosity-affected sublayer,in some respects, complicates the task of model development, in an-other it simplifies the problem; for, streamwise convective transportwithin this sublayer is often sufficiently small compared with diffu-sive or (in the case of properties of the turbulent field) source orsink processes that it may be neglected. In cases where surfacetranspiration is absent and where the influence of force fields (in-cluding pressure gradients) on the region is negligible, the flow-fieldproperties, suitably normalized, are then functions of only a normal-distance Reynolds number. For the case of the mean velocity, theresultant distribution is known as the 'Law of the Wall'. This connec-tion between velocity, friction velocity, \/TW/p , kinematic viscosityand normal distance can be used in place of the no-slip boundarycondition to avoid the need to extend what may be a costly two-or three-dimensional numerical solution to the wall itself. This ap-proach is especially advantageous if the matching is applied outsideof the viscosity dependent region (though close enough to the wallfor streamwise transport to be negligible); for then the turbulencemodel used for the numerical computation does not have to include

286 B. E. Launder

viscous effects and, moreover, one escapes the need for the especiallyfine mesh that is inevitably required to resolve the viscous regionbecause there the curvature in the profiles of both the mean velocityand turbulence properties is so high.

However, as CFD gains in maturity and computational powerper dollar continues to double every 18 months or so, an increas-ingly large proportion of problems being tackled are ones where it isnot safe to treat the near-wall sublayer as though it is in its quasi-equilibrium, 'universal' state. That is why, in the present article, adetailed modelling of the sublayer is the only route considered.

3.2 Limiting Forms of Turbulence Correlationsin the Viscous Sublayer

We consider flow in the immediate vicinity of a wall lying in theplane x% — 0. It follows from continuity that 8u2/dx2 = 0 atx2 = 0 so, if we expand the fluctuating velocities in a Taylor seriesabout the wall, there follows:

where the a's, 6's and c's are functions of time whose mean value mustbe zero, since ul = 0. The linear variation of u\ over a significantsublayer region is well confirmed by experiment and, more recently,by direct numerical simulation.

It follows from (3.1) that

This cubic variation of turbulent stress helps explain why the viscouslayer exhibits a distinct, if very thin, region where turbulent stressis negligible compared with viscous stress which gives way, ratherrapidly, for small increase in X2, to a zone where molecular transportis of only minor importance in comparison with that due to turbulentexchange processes.

Likewise, the turbulent kinetic energy variation is given by:

Alternatively, casting the expression in dimensionless form:


where

Patel, Rod! and Scheuerer (1985) concluded from the availabledata that a+ lay in the range 0.025 < a+ < 0.05. The newerdirect-simulation data indicate a value of approximately 0.035.

From (3.1) we can also obtain the following expression for

e = v(du{ldxjf

or, in dimensionless form,

Thus, (. which, at the wall, equals the turbulence energy dissipa-tion rate6, is non-zero there. The direct numerical simulations ofwall turbulence give values of e as the wall is approached consistentwith those indicated recently from turbulence energy measurements.These simulations suggest unequivocally that the maximum level ofe occurs at the wall itself which is very much different from that pre-dicted by any pre-1990 turbulence model (these typically produceda peak level of e at x J w 12 with the wall value being less than halfthe maximum value).

By comparing (3.3) and (3.4) we may readily deduce that theapproximation

is correct up to terms linear in x%. This result has been used indifferent ways in modelling the limiting behaviour of e at the wall. Apopular strategy is to adopt e rather than e as the dissipative variable(Jones and Launder, 1972) where:

a choice which evidently permits the dependent variable to be set tozero at the wall.

eThe term v ( d m / d x ] ) ( d u ] / d x t ) , by which t differs from the true dissipationrate, vanishes at the wall

288 B. E. Launder

It may be wondered how, in the absence of turbulence energygeneration (P, like u\u?, is proportional to cc^), a non-zero dissipa-tion rate can be sustained at the wall. The answer is that viscousaction diffuses energy towards the wall (vd2k/dx^} at a rate whichjust balances the dissipation rate there.

The corresponding component terms, €{j, defined as

may be treated in the same way as e. It is of interest to compare theratio (e;j/e) with u^Uj/k. In isotropic turbulence the terms are bothequal to 2/3 Sij. In the limiting case of wall turbulence, however,the relative magnitudes of the stress and dissipation rate ratios dif-fer from component to component. The question of modelling thislimiting behaviour is taken up in the following section.

3.3 Low Reynolds Number Modelling

Preamble

The present notes are written at a time when second-momentclosure strategy for low-Reynolds-number regions is in great ferment.Models and ideas developed in the 70's and 80's are being cast asideand new activity, aimed at replacing these outmoded approaches bysomething more modern, is underway at several institutions. So faras the writer's group is concerned, the change of emphasis has arisenfrom finally accepting two important facts (which, in truth, have beenplain to see for several years) allied to an immensely important storeof new information about near-wall turbulence. The two essentialfacts are:

• The variation of Reynolds stresses through the sub-layer regionand the associated damping of the effective transport coeffi-cients seem to be affected rather little by viscosity.

• As the wall is approached, the stress field approaches a "two-dimensional" state since the fluctuations normal to the walldie out faster than those parallel to the surface. It is thisfeature that is mainly responsible for the damping of near-walltransport coefficients. It follows, therefore, that turbulence


modelling in the sublayer should be consistent with the two-dimensional limit especially as regards the pressure-strain anddissipation processes.

Substantial support for these assertions_is offered by Fig 3.1 whichexamines the experimental variation of u\jk across the near-wallregion. (In the figure u2 is denoted by v and x2 by y). According tothe high-Reynolds-number free flow model of Gibson and Launder(1978), in local equilibrium the connection between shear stress andmean velocity gradient in a simple shear is simply

Figure 3.1: Variation oiu\/k across sublayer (Launder, 1986).

(This is also the form given by the generalized gradient diffusionhypothesis GGDH). Now, low-Reynolds-number k — e models presenttheir constitutive stress-strain relation in the following form:

where CM normally takes the constant value 0.09 and /^ is the "vis-cous" damping function employed in models of that type. On com-paring these two equations it is evident that they would be identical if

290 B. E. Launder

u\lk were equal to 0.342 /M. Figure 3.1, from Launder (1986), showsthe variation of the quantity across the sublayer region as deduced byPatel, Rodi and Scheuerer, from admittedly rather imprecise experi-mental data (the uncertainty band attaching to /^ is probably largerthan that shown for u\/k}. The figure does show quite convincingly,however, that the two parameters vary in essentially the same wayacross the sublayer. The more recent direct simulation data leadto essentially the same distribution of f ^ . Thus, provided, as thewall is approached, the proper diminution of u\ can be modelled,there would be little need for other "viscous" damping. The pro-cesses that are most influential in determining the level of u\ are thepressure-strain and dissipative correlations. There are good reasonsfor supposing that the former is dependent on viscosity only withinthe near wall sublayer while u^/k increases with distance from thewall over a more extensive region. It implies, therefore, that therequisite damping of u\/k should, to a large extent, be provided bynon-viscous parameters.

The "store of new information about near-wall turbulence" notedabove refers to the results of full simulations of turbulent shear flowsthat have become available over the last five years. The first of these,the flow between plane and slightly curved parallel surfaces by Kimet al. (1987) and Moser and Moin (1984), present detailed budgets ofeach Reynolds-stress component right up to the wall itself, includingthe contributions made by the pressure-strain correlation. Thus, forthe first time, the modeller can make direct appeal to data of theprocesses to be approximated — data generated not by experimentbut by computer simulation.

Component dissipation

The process in which it appears most important to include vis-cous effects is the dissipative correlation. As remarked earlier, at highReynolds numbers, it is usually assumed that the fine scale motionis isotropic and thus:

As the Reynolds number approaches zero the energy-containing anddissipating range of motions are no longer distinct and the dissipa-tion rate is then commonly approximated (for example Rotta, 1951,Haiijalic and Launder, 1976) as:


If one accepts eq (3.6) and (3.7) as asymptotic forms valid in thelimit of very high and very low Reynolds number, it is natural togeneralize the result as:

The function fa is usually assumed to be a function of the turbu-lent Reynolds number (Rt = k2/z/e), its value going from unity tozero as Rt varies from zero to infinity. The recommended form forfs to have emerged from the computations of Hanjalic and Launder(1976) was:

so that, by the start of the fully turbulent region (Rt « 150), thedissipation is very nearly isotropic.

Equation (3.7), while attractively simple, is not an exact limitingform. It may readily be deduced (Launder and Reynolds, 1983) fromthe polynomial expansions for the velocity components that, whileit is correct if neither i nor j takes the value 2, if one of the indicesrefers to the direction normal to the wall:

and, for the u\ component,

An invariant way of expressing these results is through the unit vectornormal to the wall Hk (Launder and Reynolds, 1983, Kebede et al.,1985):

It is readily deduced that, if eq (3.12) is to satisfy equations (3.10)and (3.11), the coefficients a and /? must be unityA A model forthe dissipative correlation that satisfies both this wall-adjacent limitand the high Reynolds number, locally isotropic form can clearly be

yFor the case i = 1 , j = 1 arid i = 3 , j = 3, eq(3.12) gives

292 B. E. Launder

obtained by replacing euiUj/k in eq (3.8) by the right hand side ofeq (3.12). This has been the route followed in several closure studiesfrom the mid-eighties to the present. However, eq (3.12) cannot besaid to be a convenient equation for general application as it bringsin the unit vector normal to the wall (c.f. the discussion in Sec.2 on the desirability of removing unit vectors and wall distance).Recently the writer has discovered that the desired limiting valuesmay be obtained in a form that is free of unit vectors, namely:

The form of eq (3.13) was suggested by eq (3.5) which expresses thelimiting dissipation rate at the wall in terms of gradients of -\fk. Theexpression for t*j only departs significantly from eu^Uj/k within theviscous sublayer where ^-gradients are steep, Fig 3.2a.

The quantity c*- can only represent e^j in the immediate vicin-ity of a wall. To obtain an expression for e^ of greater width ofapplicability a composite expression of type (3.8) would need to beadopted

The question arises as to what parameter fs should depend on. Asnoted above, the turbulent Reynolds number is the usual choice butFig 3.3, which shows the DNS-computed variation of different quan-tities (including Rt~) across the near-wall region of turbulent channelflow, brings out a problem: for this case Rt reaches its maximumvalue at x% ~ 20 whereas the dissipation ratios shift towards theirisotropic levels well beyond that. It is for this reason that presentstudies at UMIST continue the practice of Launder and Tselepidakis(1991) in employing one of the stress-invariant parameters as an ar-

where 'a' takes the value 1 or 3. In the immediate vicinity of the wall u\ isnegligible compared with u\ or v% giving the desired limiting behaviour.


Figure 3.2: Variation of fij/e in near-wall turbulent flow a) e*,-/eversusus normalized wall distance b) resultant profile of e,-j/e usingeq (3.13-3.15).

gument of fs. The form currently adopted is (J R Cho, personalcommunication):

The quantity e*- can only represent c,-j in the immediate vicinityof a

which produces, with eq (3.14), the component dissipation profilesshown in Fig 3.2b.

Near-wall effects on pressure-strain process

As discussed in Section 2.3, to achieve a widely applicable for-mulation it is important to adopt a model of <foj that keeps wall-reflection effects to a minimum (and, if possible, eliminates thementirely). This is because there seems no way at present to modeltheir impact in flows with highly irregular boundary surfaces. Forthis reason we here discuss only schemes employing cubic versionsof faff because this formulation seems naturally to give rise to thedecrease of uiu^/k with increasing strain rate, such as occurs as one

294 B. E. Launder

Figure 3.3: Distribution of Rt, A<2, A$ and A across viscous sublayerof low Reynolds number channel flow of Kim et al. (1987)(fromLaunder and Tselipidakis, 1991): Rt] — A2; A3;A.

enters the viscosity-dependent sublayer. At present such models havebeen applied only in fairly simple flows so it remains to be seen howwell they extrapolate to recirculating and impinging flow regions.Simpler linear schemes have been extensively reviewed by Hanjalic(1994). _

The principal task is to make u\/k fall to zero in the correctmanner as the wall is approached. Making the coefficient Cj of ̂ i(viz eq (2.15)) vanish with A would appear to ensure that ^221 fell tozero in the 2-component limit that applies at the wall, thus causingu\jk to fall to zero. Figure 3.4, from early work of Launder andTselepidakis (1990) shows this is not the case, however. While eq(2.15) is consistent with the 2-component limit it does not enforceit.Thus u\/k remains virtually constant across the near-wall sublayer,A does not vanish at the wall and neither does cj. Replacing € in theequation for 4>iji by f. at least ensures that faji vanishes at the wall;however, it still fails to reduce u\/k except in the immediate wallvicinity. Apparently, the only way to ensure a proper decrease ofthis stress ratio is to introduce damping via the turbulent Reynoldsnumber Rt as was done for the lowest curve in Fig 3.4. Of course,agreement with the DNS results is still far from complete with that


early model. The more recent results to emerge from the UMISTgroup (to be shown below) have been obtained with the followingchoice of the coefficient of <^,-ji:

Figure 3.4: Variation of u\/k across low-Reynolds-number channelflow: A Kim at al (1987) - DNS data; Lines: Second-moment pre-dictions of Launder and Tselipidakis (1990) without wall reflection:

: ci a function of A; : as before but e replaced by e; :ci damped by turbulent Reynolds number.

The high-Reynolds-number, free-flow form of <^,-j2 is adapted intwo ways as one approaches the wall. The term with coefficient c2

gives rise to the following sink term in the shear-stress equation:

Both AW and (an - 022) increase rapidly as the wall is approachedleading to the annihilation of Uiu2. To limit the strength of this sink,the coefficient c2 is amended to c2 = 7mn(c2co, A), where C2oo is thevalue c2 would take in a free shear flow (that value depends slightlyon whether c2 takes a non-zero value; see the discussion in Section2.3). The other amendment, discussed already in connection with eq

296 B. E. Launder

(2.19), is the need to correct eq (2.6) for the rapid spatial variation ofthe mean velocity gradient. In a channel flow the correction, given byeq (2.19) or similar forms, is of substantial importance in the bufferregion but becomes negligible further from the wall. This behaviouris consistent with the conclusions reached by Bradshaw et al. (1987)from a processing of the DNS data.

Diffusion

Serious study of stress diffusion in the near-wall sublayer is onlyjust beginning. Most computations have retained the GGDH repre-sentation with the same value for the coefficient cs (0.22) as in fullyturbulent regions. It isn't that the formula is regarded as satisfac-tory, but rather that, in the near-wall fully turbulent region, trans-port is often of relatively little importance while, in the immediatevicinity of the surface, where diffusive transport balances the lossby dissipation, molecular transport outweighs diffusion by velocityfluctuations, UiUjUk- A minor improvement to the standard GGDHscheme would be to introduce e in place of e so that the turbulenttime scale remains finite as the wall is approached.

In fact, because non-zero pressure fluctuations occur on the wall,pressure diffusion dies out less rapidly than the velocity diffusion asthe wall is approached. This may be seen by considering the stressbudget in the viscous sublayer. For example, with u\u<i = a^bix^(c.f. eq (3.2)) the net viscous diffusion of UiH? is Qa^b^x^ while itsdissipation rate is ^aib^vx-^. Since the form of faji ensures a varia-tion of this process to a power of x^ greater than unity, the imbalancebetween viscous diffusion and dissipation can only be compensatedby a non-zero pressure diffusion amounting to — laib^vx^. Likewise,in the u\ equation it is easily deduced that a pressure diffusion ofmagnitude -\b\vx\ is required to offset the excess viscous diffusion(\1b\vx\) over dissipation. By contrast, in the u\ and u\ budgets,where no pressure-diffusion terms appear, the leading viscous diffu-sion and dissipation terms do balance.

Tselepidakis (1992) devised a model of pressure-diffusion amongcomponents thereby satisfying the budget in all components. Theformula adopted, however, employed the wall normal vector and forthis reason it is not reproduced here. A less restrictive form is


This formula embodies the' same underlying idea but does not employthe unit vector. However, while the imbalance in the u\ equation isentirely eliminated with this form, that in the ~u\u^ equation is onlyreduced by one half. Nevertheless, since Tselepidakis (1992) foundhis formulation to have only a secondary effect on the stress profiles,the above replacement — which removes about 75% of the (small)problem - is probably satisfactory.

Low Reynolds number effects on e

In Section 2.5 no detailed consideration was given to the exact 6equation because, at high Reynolds numbers, the scales of the con-trolling eddies and the dissipative motions are quite distinct. Thatdistinction does not apply, however, in the viscous layer: there theexact dissipation equation shows the mean strain to act as a sourceof e principally through the term:

Having regard for the fact that, in a thin shear flow, the meanvelocity is significant only in directions parallel to the wall (so index'i' denotes 1 or 3), the second term is evidently small compared withthe first since, within the sublayer, rates of change of the instanta-neous turbulence field will be much larger in direction x% than in x\or £3. The first term may, in fact, be expressed in terms of eq (3.12)or (3.13). If direction x\ is aligned with the near-wall mean velocityboth lead to the result:

Equation (3.17) suggests therefore that at x^ = 0 the value of c£ishould be 2.0 rather than the value 1.44 commonly used at high Rey-nolds numbers9. The changeover from the high-Reynolds-numberlimit to equation (3.17) should consistently adopt the function fs.Earlier work by the writer's group (Hanjalic and Launder, 1976) hasadopted a value for cf\ that was entirely independent of Reynolds

9Or the value 1.0 adopted in recent UMIST work when cC2 is a function of AIand A.

298 B. E. Launder

number. A viscosity dependent influence of mean strain was intro-duced, however, via the source term

which may be regarded as the outcome of using the generalizedgradient-diffusion hypothesis to close a term containing d2Ui/dxjdxkin the exact e equation. In this case the transportable fluctuatingquantity </> is d u { / d x j . The value 1.0 adopted for c'fl by Hanjalicand Launder (1976) was, however, 3 - 4 times as large as is usu-ally adopted; so, arguably, that choice was compensating for the factthat c£i had been held fixed. In more recent studies c'el has usuallybeen set to considerably smaller values. For example, Launder andTselepidakis (1993) take c£l = 0.43.

We now shift attention to the decay term in the dissipationrate equation which, at high-Reynolds-numbers, has been taken asc&^/k- For high-Reynolds-numbers the coefficient c& was deter-mined by reference to grid turbulence decay and it is natural, there-fore, that one should turn to low-Reynolds-number grid turbulenceto decide if modifications are needed. The most complete turbulencedecay data are those of Batchelor and Townsend (1948); these sug-gest that below a value of Rt of about 10 the decay exponent n in therelation k oc t~n increases from the high-Reynolds-number limit ofabout 1.2 to an asymptotic value of 2.5. If we assume the changeoverto be describable in terms of the local value of Rt, ce2 must be mul-tiplied by some function of Reynolds-number, /£, to give the desiredlimiting values of the decay exponents. Hanjalic and Launder (1976)took

There is no reason to suppose much similarity between the low-Reynolds-number sublayer near a wall and the viscous decay of weakgrid turbulence. However, the exponential term in eq (3.18) fallsto zero so rapidly that /E is essentially unity over the whole of theregion near the wall where turbulence is important. There is, how-ever, a more important modification to this sink term needed in wallturbulence since, as noted in Section 3.1, e is non-zero at the wallwhile k varies as x\. Even with the introduction of /£, therefore,c£2/ee2/fc tends to infinity as x2 goes to zero. Accordingly, Hanjalicand Launder (1976) replaced e2 in the sink term by ec where


As noted in Section 3.2, e varies as x\ near the wall and thus theterm c^f^l/k tends to a constant value as the wall is approached.Virtually all proposals for extending the e equation to the wall adoptthis form.

Finally, there remains the matter of viscous effects on the diffu-sion of e to be accounted for. _The writer's group has retained thehigh-Reynolds-number form, c^fc/e, as the appropriate diffusion co-efficient in the low-Reynolds-number region while Prud'homme andElghobashi (1983) multiply the coefficient ce by a viscous dampingfunction, /^. All proposals have included the exact viscous trans-port vdt/dxj as an addition to the high-Reynolds-number diffusionmodel. There is also the question of pressure diffusion to be consid-ered. In high-Reynolds-number turbulence no explicit accounting ofthis process was attempted. An order-of-magnitude estimate of thisterm in the exact e equation suggests, however, that as the wall isapproached the process becomes significant (Launder, 1986). Thissuggests that a separate approximation ought to be incorporated. Apossible model for the additional term that is significant only in theviscous region has been proposed in that article:

Tselepidakis and Launder (1993) have suggested a value of c£4 of0.92.

3.4 Applications

Low-Reynolds-number models designed to match the wall-limitingbehaviour have so far been applied only in simple shear flows. This isa little ironic because models of this type are really designed to han-dle complex mean strain patterns and far-from-equilibrium states.Nevertheless, one has to start with simple flows to calibrate any freecoefficients in the model. Figures 3.5 and 3.6 show applications fromtwo recent Ph.D studies at UMIST.

The first relates to plane channel flow and the figure compares thecomputed variation of rms velocity fluctuations across the near-wall

300 B. E. Launder

Figure 3.5: Distribution of turbulent intensities and shear stressacross near-wall region of low-Reynolds-number channel flow; Sym-bols: DNS results, Kim at al (1987) —: closure computations, Laun-der and Li (1994) a) turbulence intensities normalized by frictionvelocity b) shear stress.

sublayer with the DNS results of Kim et al. (1987). The adoptedmodel of <pij2 was the complete cubic form (c2 = 0.55; c'2 = 0.6 in eq(2.7(d)) with an inhomogeneity correction (eq (2.19)) but no "wall-reflection" term. The agreement of the closure computations withthe DNS results is reasonably satisfactory.

The second application, from Launder and Tselepidakis (1994),also relates to channel flow but here the channel rotates about the#3 axis, i.e. in orthogonal mode rotation. The resultant asymme-try in the turbulence intensity profiles arises from the direct effectof the Coriolis forces on the turbulent stresses in the x\ — x^ plane,augmenting fluctuations normal to the wall on the pressure (left)side of the channel which, in turn, increases the shear stress u\u-iand thus of streamwise fluctuations too. The model in this case wassimilar to that adopted in the previous example except that, becausethe computations were carried out rather earlier, the simplified cubicmodel of </>,-j2 was adopted (c2 = 0.6 ; c'2 = 0) and, in consequence asmall 'wall-reflection' contribution was added. Generally the closure


Figure 3.6: Application to turbulent flow in plane channel rotatingabout x3 axis (orthogonal mode rotation) a) Mean velocity profile;b) turbulent shear stress profile 8 ; c) rms turbulent intensities Sym-bols: DNS results, Kristoffersen and Andersson (1990) : Launderand Tselipidakis (1994); : Launder and Shima (1989) (fromKristoffersen and Andersson, 1990).

computations (continuous line) are quite successful in mimicking theasymmetries found in the DNS data generated by Kristoffersen and

hNote: UTo is friction velocity for non-rotating duct at same bulk Reynoldsnumber

302 B. E. Launder

Andersson (1990). For comparison, predictions obtained by ProfessorAndersson's group using the older 2nd-moment closure of Launderand Shima (1989) are also included; this model only partially com-plied with asymptotic wall-limiting behaviour. While it too capturesbroadly the asymmetry of the flow, it is also clearly not as successfulas the cubic form.

ACKNOWLEDGEMENTS

The recent developments described are based largely on the re-search discoveries of former and present students and research assis-tants in the UMIST CFD group. Particular contributions have beenmade by Drs. J R Cho, T J Craft, S Fu, N Z Ince, S-P Li, Mr. KSuga and Dr. D P Tselepidakis. The text of this article has beenprapared in TATgX format by Erdinc, Ozkan.

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Index

algebraic dissipation rate model,202

algebraic second-moment mod-els, see algebraic stressmodel

algebraic stress model, 197applications, 204-210

anisotropic eddy viscosity model,191,208

autocorrelation coefficient, 6

backscatter term, 119backstep flow, 210buoyancy effects, 250

closure problem, 189coherent structure capturing, 147coherent structures, 29, 111, see

very large eddy simu-lation

large eddy simulation, 146—148

modeling, 148compressed (expanded) isotro-

pic turbulence, 229compressible mixing layer

convective Mach number,103, 233

compressible turbulenceequations of motion, 220homogeneous shear, 229isotropic, 227

Reynolds stress equation,223

Reynolds-average equations,221

Reynolds-averages, see Favre-averages

two-equation models, 226Coriolis forces, 264

deformation work, 23diffusive transport, 248, 273direct numerical simulation (DNS),

79,115compressible, 100

homogeneous, 102incompressible

homogeneous, 85inhomogeneous, 86

resolution requirements, 81solution methods

fast vortex, 97remeshing, 96spectral element, 98vector expansion, 88viscous vortex, 91

dissipationdilatational, 26, 224

dissipation anisotropy, 194dissipation rate, 249, 257, 275

component, 290-293low Reynolds number, 297-

299

311

312 Index

dissipation wavemimber, 156

eddy damped quasi-normal Marko-vian (EDQNM), 119,135

eddy shocklets, 28eddy viscosity model, 280eigenvectors, 25elliptic relaxation model, 216ensemble average, 188equilibrium flows

dissipation, 10energy cascade, 6

equilibrium, turbulence in, 194,267

ergodic hypothesis, 188extra strain terms, 275

Favre-aver ages, 221, 244force fields, 264friction factor, 81

generalized gradient diffusion hy-pothesis, 273

heat flux, 223, 250Helmholtz decomposition, 102homogeneous turbulence, 187

integral length scale, 6integrity bases, 196isotropic turbulence, 9, 103,121,

169, 200,247,288isotropization of production model,

263

Kolmogorov, 9hypothesis of local isotropy,

195scales, 9, 80, 201universal spectrum, 156

Kurarnoto-Sivashinsky, 68

large eddy simulation (LES),109

a priori testing, 123cross term, 118cutoff filter, 127extra strain effects, 135-137filtering, 116numerical methods, 141-143subgrid-scale models, 118-

119differential, 137-138dynamic, 127-132scale similarity, 125-127shifted model, 140Smagorinsky, 119-123spectral, 132-135

law of the wall, 285Leonard term, 118low Reynolds number turbulence,

284modeling, 288

mass flux, 223mass-weighted average, see Favre-

averagesmaterial frame-indifference, 218mean scalar transport, 245mean-field closure, 247mean-field scalar transport, 247

Navier-Stokes equations, 187near-wall asymptotic behavior,

see viscous sublayernon-equilibrium flows

energy cascadedelay in crossing, 14fading memory, 20negative production, 19

spectral cascade, 13nonlinear eddy viscosity model,

281

Index 313

nonlocal effects, 210

one-equation models, 191outscatter term, see Leonard

term

power-law divergences, 162Prandtl mixing length, 190pressure Poisson equation, 258pressure reflections, 269pressure-dilatation, 225pressure-strain correlation, 210,

248near-wall, 293-296turbulence-turbulence part,

265principle of receding influence,

255proper orthogonal decomposi-

tion (POD), 33eigenfunctions, 52-66models using, 37-42wall region, 42-52

quasi-isotropic model, 259

realizability, 218regularization, 199relaxation effects, 210renormalization group (RNG),

155e—expansion, 163applications

flapping-hydrofoil, 177-181

time-dependent turboma-chinery, 174-177

turbomachinery heat trans-fer, 171-174

resummation of divergentseries, 162-168

statistical perturbation the-ory, 159-161

thermal noise, 166two-equation model, 169

Reynolds stress anisotropy, 194Reynolds-averaged equation, 243Reynolds-averaged Navier Stokes

(RANS), 112Reynolds-averages, 187rotating channel flow, 205, 300rotating homogeneous shear, 205rotational strains, 204

second-moment closure, see Rey-nolds stress

second-order closure, see Rey-nolds stress

single-point closure, 246solenoidal dissipation rate equa-

tion, 225spectral scaling, 5square duct, flow in, 208statistically steady turbulence,

187strain rate tensor, 195stress coupling, 249subgrid-scale Reynolds stress,

118supersonic mixing layer, 232

third-order diffusion, 212turbulent eddies, 7turbulent kinetic energy, 11turbulent Mach number, 224turbulent pipe flow, 81turbulent scalar flux, 246turbulent transport equations

dissipation rate, 193kinetic energy, 192Reynolds stress, 192

314 Index

near-wall models, 213-220

van Driest damping, 122very large-eddy simulation (VLES),

147,175viscous sublayer, 286Von Karman constant, 190vorticity rate tensor, 195

wall effects, 269diffusion, 296

wall modelslarge eddy simulation, 138-

141wall shear velocity, 81wavelets, 67WET concept, 253

zero-equation models, 190Baldwin-Lomax model, 190Smagorinsky model, 190

simulation and modelimg of turbulent flows_ lumley j.l

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