__

- - -.

-_ ~_ti_ ._-

.---N

Turbulence Modelingfor CFD

by

David C . Wilcox

DCW Industries, Inc.La Canada, California

Dedicated to my Wife

BARBARA

my Children

KINLEY and BOB

and my Dad

Turbulence Modeling for CFD

Copyright © 1993, 1994 by DCW Industries, Inc . All rights reserved .

First Printing :

July, 1993Second Printing :

November, 1994 (with corrections)

No part of this book may be reproduced or transmitted in any form or byany means, electronic or mechanical, including photocopying, recording, orany information storage and retrieval system, without permission in writingfrom DCW Industries, Inc .

DCW Industries, Inc .5354 Palm Drive, La Canada, California 91011818/790-3844 (FAX) 818/952-1272

This book was prepared with LATEX, and was printed in the United Statesof America by Griffin Printing, Glendale, California .

Library of Congress Cataloging in Publication Data

Wilcox, David C.Turbulence Modeling for CFD / David C . Wilcox-1st ed .Includes bibliography, index and 3 z inch floppy disk .1 . Turbulence-Mathematical Models .2 . Fluid Dynamics-Mathematical Models .

TA357.5.T87 W542

1993

93-224752

ISBN 0-9636051-0-0

About the Author

Dr. David C . Wilcox, was born in Wilmington, Delaware . He did hisundergraduate studies from 1963 to 1966 at the Massachusetts Institute ofTechnology, graduating with a Bachelor of Science degree in Aeronauticsand Astronautics . From 1966 to 1967, he was an Engineer Scientist Spe-cialist at McDonnell Douglas Aircraft Division in Long Beach, California,working for A. M. O . Smith . His experience with McDonnell Douglas wasprimarily in subsonic and transonic flow calculations . From 1967 to 1970, heattended the California Institute of Technology, graduating with a Ph.D . inAeronautics . In 1970 he joined TRW Systems, Inc . in Redondo Beach, Cal-ifornia, as a Member of the Technical Staff. He performed studies of bothhigh- and low-speed fluid-mechanical and heat-transfer problems, such asturbulent hypersonic flow and thermal radiation from a flame . From 1972to 1973, he was a staff scientist for Applied Theory, Inc ., in Los Angeles,California, responsible for scientific-project management . He participateddirectly in many research efforts involving numerical computation and anal-ysis of a wide range of fluid flows such as separated turbulent flow, tran-sitional flow and hypersonic plume-body interaction . In 1973, he foundedDCW Industries, Inc ., a La Canada, California firm engaged in engineeringresearch, software development and publishing, for which he is currently thePresident . He has taught several fluid mechanics and applied mathematicscourses at the University of Southern California and at the University ofCalifornia, Los Angeles .

Dr . Wilcox has published many papers and reports on turbulence mod-eling, computational fluid dynamics, boundary-layer separation, boundary-layer transition, thermal radiation, and rapidly rotating fluids . He is anAssociate Fellow of the American Institute of Aeronautics and Astronau-tics (AIAA) and has served as an Associate Editor for the AIAA Journal .

Contents

Notation

xi

Preface

xvii

1 Introduction1 .1

Definition of an Ideal Turbulence Model

. . . . . . . . . . .1 .2

How Complex Does a Turbulence Model Have to Be? . . . .1 .3

Comments on the Physics of Turbulence

.

. . .

. . . .

. . .1 .4

A Brief History of Turbulence Modeling

.

.

.

.

.

.

.

.

.

.

.

2 The Closure Problem

3

11125

2.12.22.32.4

Reynolds Averaging . . . . . . . . . . . . . . . . . . . . . .Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . .Reynolds-Averaged Equations . . . . . . . . . . . . . . . . .The Reynolds-Stress Equation . . . . . . . . . . . . . . . . .

11151517

Algebraic Models 233.1 Molecular Transport of Momentum . . . . . . . . . . . . . . 243.2 The Mixing-Length Hypothesis . . . . . . . . . . . . . . . . 273.3 Application to Free Shear Flows . . . . . . . . . . . . . . . 30

3.3 .1 The Far Wake . . . . . . . . . . . . . . . . . . . . . 323.3 .2 The Mixing Layer . . . . . . . . . . . . . . . . . . . 383.3.3 The Jet . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Modern Variants of the Mixing-Length Model . . . . . . . . 443.4 .1 Cebeci-Smith Model . . . . . . . . . . . . . . . . . . 503.4 .2 Baldwin-Lomax Model . . . . . . . . . . . . . . . . . 52

3 .5 Application to Wall-Bounded Flows . . . . . . . . . . . . . 533.5 .1 Channel and Pipe Flow . . . . . . . . . . . . . . . . 533.5 .2 Boundary Layers . . . . . . . . . . . . . . . . . . . . 59

3 .6 Separated Flows . . . . . . . . . . . . . . . . . . . . . 61

vi

CONTENTS

3 .7

The 1/2-Equation Model . . . . . . . . . . . . . . . . . . . .

6567

737477838487

909295104105110122126126128131131132133138138146160163

171171172174180183189195203

3 .8 Range of Applicability . . . . . . . . . . . . . . . . . . . . .

4 Turbulence Energy Equation Models4 .1 The Turbulence Energy Equation . . . . . . . . . . . . . . .4 .2 One-Equation Models . . . . . . . . . . . . . . . . . . . . .4 .3 Two-Equation Models . . . . . . . . . . . . . . . . . . . . .

4.3 .1 The k-w Model . . . . . . . . . . . . . . . . . . . . .4.3 .2 The k-c Model . . . . . . . . . . . . . . . . . . . . .4.3 .3 Other Two-Equation Models . . . . . . . . . . . . .

4.4 Closure Coefficients . . . . . . . . . . . . . . . . . . . . . . .4.5 Application to Free Shear Flows . . . . . . . . . . . . . . .4.6 Perturbation Analysis of the Boundary Layer . . . . . . . .

4.6 .1 The Log Layer . . . . . . . . . . . . . . . . . . . . .4.6 .2 The Defect Layer . . . . . . . . . . . . . . . . . . . .4.6 .3 The Viscous Sublayer . . . . . . . . . . . . . . . . .

4.7 Surface Boundary Conditions . . . . . . . . . . . . . . . . .4 .7 .1 Wall Functions . . . . . . . . . . . . . . . . . . . . .4 .7 .2 Surface Roughness . . . . . . . . . . . . . . . . . . .4.7 .3 Surface Mass Injection . . . . . . . . . . . . . . . . .

4 .8 Application to Wall-Bounded Flows . . . . . . . . . . . . .4.8 .1 Channel and Pipe Flow . . . . . . . . . . . . . . . .4.8 .2 Boundary Layers . . . . . . . . . . . . . . . . . . . .

4.9 Low-Reynolds-Number Effects . . . . . . . . . . . . . . . . .4.9 .1 Asymptotic Consistency . . . . . . . . . . . . . . . .4.9 .2 Transition . . . . . . . . . . . . . . . . . . . . . . . .

4.10 Separated Flows . . . . . . . . . . . . . . . . . . . . . . . .4.11 Range of Applicability . . . . . . . . . . . . . . . . . . . . .

5 Effects of Compressibility5.1 Physical Considerations . . . . . . . . . . . . . . . . . . . .5 .2 Favre Averaging . . . . . . . . . . . . . . . . . . . . . . . .5 .3 Favre-Averaged Equations . . . . . . . . . . . . . . . . . . .5 .4 Compressible-Flow Closure Approximations . . . . . . . . .5 .5 Dilatation Dissipation . . . . . . . . . . . . . . . . . . . . .5 .6 Compressible Law of the Wall . . . . . . . . . . . . . . . . .5 .7 Compressible Boundary Layers . . . . . . . . . . . . . . . .5 .8 Shock-Induced Boundary-Layer Separation . . . . . . . . .

6.7 Application to Separated Flows . . . . . . . .6.8 Range of Applicability . . . . . . . . . . . . .

7 Numerical Considerations7.1 Multiple Time Scales and Stiffness7.2 Numerical Accuracy Near Boundaries . . . .

. . .

. . .

. . . . . . . . . . . . . .. . .

. .

. .

. .

. .

. .

. .

. 261

. 266

273273

. 2757.2 .1 Solid Surfaces . . . . . . . . . . . . . . . . . . . . . . 2757.2 .2 Turbulent/Nonturbulent Interfaces . . . . . . . . . . 279

7.3 Parabolic Marching Methods . . . . . . . . . . . . . . . . . 2877.4 Elementary Time-Marching Methods . . . . . . . . . . . . . 2927.5 Block-Implicit Methods . . . . . . . . . . . . . . . . . . . . 2977.6 Solution Convergence and Grid Sensitivity . . . . . . . . . . 303

8 New Horizons 3138.1 Background Information . . . . . . . . . . . . . . . . . . . . 3138.2 Direct Numerical Simulation . . . . . . . . . . . . . . . . . 3168.3 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . 3228.4 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

A Cartesian Tensor Analysis 331

B Rudiments of Perturbation Methods 337

C Companion Software 349C.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

C.1 .1 Program Structure . . . . . . . . . . . . . . . . . . . 350C.1 .2 Program Input . . . . . . . . . . . . . . . . . . . . . 351C.1 .3 Program Output . . . . . . . . . . . . . . . . . . . . 352

C.2 Free Shear Flows . . . . . . . . . . . . . . . . . . . . . . . . 353

CONTENTS

6 Beyond the Boussinesq Approximation

vii

2136.1 Boussinesq-Approximation Deficiencies . . . . . . . . . . . . 2136 .2 Nonlinear Constitutive Relations . . . . . . . . . . . . . . . 2186 .3 Second-Order Closure Models . . . . . . . . . . . . . . . . . 223

6 .3 .1 Closure Approximations . . . . . . . . . . . . . . . . 2246.3 .2 Launder-Reece-Rodi Model . . . . . . . . . . . . . . 2316.3 .3 Wilcox Multiscale Model . . . . . . . . . . . . . . . 232

6.4 Application to Homogeneous Turbulent Flows . . . . . . . . 2356.5 Application to Free Shear Flows . . . . . . . . . . . . . . . 2426.6 Application to Wall-Bounded Flows . . . . . . . . . . . . . 243

6.6 .1 Surface Boundary Conditions . . . . . . . . . . . . . 2446.6 .2 Channel and Pipe Flow . . . . . . . . . . . . . . . . 2486.6 .3 Boundary Layers . . . . . . . . . . . . . . . . . . . . 253

vin

C.3 Channel and Pipe Flow

. . . . . . . . . . . . . . . . . . . .

364C.3 .1

Program PIPE : Channel and Pipe Flow . . . . . . .

365C.3 .2

Program PLOTP: Plotting Utility . . . . . . . . . .

367C.4 Boundary-Layer Perturbation Analysis . . . . . . . . . . . .

371C.4 .1

Program SUBLAY: Viscous Sublayer . . . . . . . . .

373C.4 .2

Program DEFECT: Defect Layer . . . . . . . . . . .

375C.4 .3

Program PLOTS: Sublayer Plotting Utility . . . . .

376C.4 .4 Program PLOTD: Defect-Layer Plotting Utility . . .

379C.5 Miscellaneous Routines . . . . . . . . . . . . . . . . . . . . .

382C.5.1

Function ERF: Error Function . . . . . . . . . . . .

383C.5.2 Subroutine NAMSYS: Fortran Portability . . . . . .

384C.5.3

Subroutine RKGS : Runge-Kutta Integration

. . . .

386C.5.4 Subroutine RTNI : Newton's Iterations . . . . . . . .

388C.5.5

Subroutine TRI: Tridiagonal Matrix Inversion . . . .

389C.6 Diskette Contents . . . . . . . . . . . . . . . . . . . . . . . .

390

D Program EDDYBL

391D.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

D.1 .1 Acknowledgments . . . . . . . . . . . . . . . . . . . 391D.1 .2 Required Hardware and Software . . . . . . . . . . .

392D.2 Getting Started Quickly . . . . . . . . . . . . . . . . . . . .

392D.3 Installing SETEBL . . . . . . . . . . . . . . . . . . . . . . .

397D.3.1

Boot-Console Installation . . . . . . . . . . . . . . .

397D.3.2 Remote-Terminal Installation . . . . . . . . . . . . .

398D.4 Installing EDDYBL

. . . . . . . . . . . . . . . . . . . . . .

398D.5

Running a General Case . . . . . . . . . . . . . . . . . . . .

399D.5 .1 Preliminary Operations . . . . . . . . . . . . . . . .

399D.5 .2 Units Selection . . . . . . . . . . . . . . . . . . . . . 400D.5 .3 Main Parameters . . . . . . . . . . . . . . . . . . . . 400D.5 .4 Taking a Lunch Break . . . . . . . . . . . . . . . . .

402D.5 .5 Edge/Wall Conditions . . . . . . . . . . . . . . . . .

403D.5 .6

Preparing Edge/Wall Condition Data Files

. . . . .

404D.5.7 Generating Edge/Wall Conditions . . . . . . . . . .

406D.5.8

Initial Profiles . . . . . . . . . . . . . . . . . . . . . .

406D.5.9

Selecting a Turbulence Model . . . . . . . . . . . . .

408D.5.10 Logical Unit Numbers and Plotting Files

. . . . . .

408D.5.11 Running the Boundary-Layer Program . . . . . . . .

410D.5.12 Restart Run . . . . . . . . . . . . . . . . . . . . . . .

410

CONTENTS

C.2.1 Program WAKE: Far Wake . . . . . . . . . . . . . . 355C.2.2 Program MIXER: Mixing Layer . . . . . . . . . . . 357C.2 .3 Program JET: Plane, Round and Radial Jet . . . . . 359C.2 .4 Program PLOTF: Plotting Utility . . . . . . . . . . 361

D.5 .13 Gas Properties and Profile Printing

. . . . . . . . .

411D.5 .14 Selecting Laminar, Transitional or Turbulent Flow .

412412413415419420423423424426429432

E Plotting Program Details

433E.1 Font Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433E.2 Video Devices . . . . . . . . . . . . . . . . . . . . . . . . . .

433E.3 Plotting Colors . . . . . . . . . . . . . . . . . . . . . . . . .

435E.4 Hardcopy Devices . . . . . . . . . . . . . . . . . . . . . . . . 435

Bibliography

437

D.6 Applicability and Limitations . . . . . . . . . . . . . . . . .D.7 EDDYBL Output Parameters . . . . . . . . . . . . . . . . .D .8 Program PLOTEB : Plotting Utility . . . . . . . . . . . . .D .9 Adapting to Other Compilers/Systems . . . . . . . . . . . .D .10 Compile and Link Commands . . . . . . . . . . . . . . . . .D .11 Additional Technical Information . . . . . . . . . . . . . . .

D .11.1 Mean-Flow Equations . . . . . . . . . . . . . . . . .D .11.2 k-w and Multiscale Model Equations . . . . . . . . .D .11.3 k-e Model Equations . . . . . . . . . . . . . . . . . .D.11.4 Transformed Equations . . . . . . . . . . . . . . . .

D.12 Software Package Modules . . . . . . . . . . . . . . . . . . .

Notation

This section includes the most commonly used notation in this book . Inorder to avoid departing too much from conventions normally used in liter-ature on turbulence modeling and general fluid mechanics, a few symbolsdenote more than one quantity.

English Symbols

Symbol Definitiona

Speed of soundaijkl

Rapid pressure-strain tensorA� , B, C,a ,D� Coefficients in tridiagonal matrix equationAo

Van Driest damping constantAid

Slow pressure-strain tensorbid

Dimensionless Reynolds-stress anisotropy tensorB

Additive constant in the law of the wallcb l , cb2

Closure coefficientscf

Skin friction based on edge velocity, r�,/(2pU')cf�

Skin friction based on freestream velocity, rw/(2PU.)cwi, cwt, cw3

Closure coefficientsCl, C2

Closure coefficientsCP , Cw k

Closure coefficientsCD , CE

Closure coefficientsCdif, CK1,b

Closure coefficientsCK

Kolmogorov constantCM, CL2

Closure coefficientsCP

Specific heat at constant pressure ; pressure coefficientC� CE

Closure coefficientsCS

Smagorinsky constantCv

Specific heat at constant volumeC

Shear-layer spreading rate

xii

NOTATION

CE1, CE2, CE3

Closure coefficientsCTl, CT2

Closure coefficientsC,,

Closure coefficientCij

LES cross-term stress tensorCijk

Turbulent transport tensorD

Drag per unit body widthDij

Production tensor, rjm8Um/0xj +TjmBUm/8xie

Specific internal energy ; small-eddy energyE

Total energy ; viscous damping functionE(K)

Energy spectral densityE(77)

Dimensionless self-similar dissipation rateEh

Discretization errorfl, fi, f2, fs

Viscous damping functionsf, f�

Turbulence flux vectorsF(77)

Dimensionless self-similar streamfunctionFKleb(y; b)

Klebanoff intermittency functionF, F�

Mean-flow flux vectorsG

Amplitude factor in von Neumann stability analysisG(x -

)

LES filterh

Specific enthalpyH

Total enthalpy ; channel height ; shape factor, b"/0H(x)

Heaviside step functioni, j, k

Unit vectors in x, y, x directionsUnit (identity) matrix

II, III

Stress tensor invariantsj

Two-dimensional (j - 0), axisymmetric (j - 1) indexJ

Specific momentum flux (flux per unit mass)k

Kinetic energy of turbulent fluctuations per unit massk 9

Geometric progression ratiokR

Surface roughness heightK

Distortion parameterK(r7)

Dimensionless self-similar turbulence kinetic energyKE , K,

Effective Karman constant for compressible flowsKn

Knudsen numberTurbulence length scale; characteristic eddy size

P�,fPMeanfree pathtmix

Mixing lengthL

Characteristic length scaleLij

Leonard stress tensorM

Mach numberMijki

Rapid pressure-strain tensorM~

Convective Mach number

NOTATION x111

Mt Turbulence Mach number, 2k/aAll Closure coefficientN(q) Dimensionless self-similar eddy viscosityNCFL CFL numberNW Constant in near-wall solution for wN(ui) Navier-Stokes operatorp Instantaneous static pressurepij Instantaneous momentum-flux tensorP Mean static pressurepij Production tensor, Ti�,BUj/8x�, +Tjr�8Ui/8x�,Pk, Pu� P, Net production per unit dissipation of k, w, E

PrL, PrT Laminar, turbulent Prandtl numberqj Heat-flux vectorqw Surface heat fluxqLj , qTj Laminar, turbulent mean heat-flux vectorQij LES stress tensor, Cij + RijQ Dependent variable vectorr, 8, x Cylindrical polar coordinatesR Pipe radius ; channel half height ; perfect gas constantRij SGS Reynolds stress tensorRij (x,t ;r) Two-point velocity correlation tensorR Radius of curvatureRij (x, t; t') Autocorrelation tensorR+ Sublayer scaled radius or half height, u,R/vRp, Rk, Ru, Closure coefficients in viscous damping functionsReL Reynolds number based on length LReT Turbulence Reynolds number, k'l't/vRe, Sublayer scaled radius or half height, R+RiT Turbulence Richardson numberRy Near-wall turbulence Reynolds number, k 1 / 2y/vsij Instantaneous strain-rate tensors, S Source-term vectorsS Source term - production minus dissipationSij Mean strain-rate tensorSij Oldroyd derivative of SijSe , Sk, S� , Sw Source terms in a similarity solutionSB Dimensionless surface mass injection functionSR Dimensionless surface roughness functiont Timetij Instantaneous viscous stress tensorT Temperature; characteristic time scale

Greek Symbols

Symbol Definitiona, a*

Closure coefficients&, ,13, y

Closure coefficientsao , ao

Closure coefficients in viscous damping functions

xiv NOTATION

T' Freestream turbulence intensityu, v, w Instantaneous velocity components in x, y, z directionsUi Instantaneous velocity in tensor notationu Instantaneous velocity in vector notationu', v', w' Fluctuating velocity components in x, y, z directionsu; Fluctuating velocity in tensor notationu` Fluctuating velocity in vector notationii, v"", tv- Favre-averaged velocity components in x, y, z directionsiii Favre-averaged velocity in tensor notationu Favre-averaged velocity in vector notationu", v", w" Favre fluctuating velocity components in x, y, z directionsu;' Favre fluctuating velocity in tensor notationu" Favre fluctuating velocity ; fluctuating molecular velocityUrms , vrms RMS fluctuating velocity components in x, y directionsu;u? Temporal average of fluctuating velocitiesU, Friction velocity, T-w/pwu Velocity perturbation vectorU, V, W Mean velocity components in x, y, z directionsUi Mean velocity in tensor notationU Mean velocity in vector notationU+ Dimensionless, sublayer-scaled, velocity, U/u,U�, Maximum or centerline velocityU(rl) Dimensionless self-similar streamwise velocityvmix Mixing velocitywith Thermal velocityv�, Surface injection velocityV(,q) Dimensionless self-similar normal velocityW(77) Dimensionless self-similar specific dissipation ratex, y, x Rectangular Cartesian coordinatesxi Position vector in tensor notationx Position vector in vector notationy+ Dimensionless, sublayer-scaled, distance, uT y/vy2 y+ at first grid point above surfacey�, Inner/outer layer matching point

NOTATION

aT , vT, WT

Defect-layer similarity parametersa, A*

Closure coefficients,QT

Equilibrium parameter, (b*/r,,)dP/dxy

Specific heat ratio, CPI,b

Boundary layer or shear layer thickness6*

Displacement thickness, fo (1 - PU) dy

6*

Velocity thickness, fo (1 - v) dybx

Finite-difference matrix operator6;jKronecker delta0

LES filter width0(x)

Clauser thickness, U,6*/u,OQ, Ox, Ay

Incremental change in Q, x, yAt Timestepe

Dissipation per unit massEd

Dilatation dissipatione,

Solenoidal dissipatione ;j

Dissipation tensorEtjk

Permutation tensorSecond viscosity coefficient

71

Kolmogorov length scale ; similarity variableB

Momentum thickness, L PU (1 - U) dyrc

Karman constant ; thermal conductivity ; wavenumberkti

Effective Karman constant for flows with mass injectiona

Taylor microscaleAmax

Largest eigenvaluep

Molecular viscosity/-IT

Eddy viscosityPT;

Inner-layer eddy viscosityPT.

Outer-layer eddy viscosityv

Kinematic molecular viscosity, p/pVT

Kinematic eddy viscosity, FAT/p

Dimensionless streamwise distance*,

Closure coefficientsColes' wake-strength parameter

n+a

Pressure-strain correlation tensorp

Mass densitya, tT*

Closure coefficientsok, Q E

Closure coefficients011, O"L2

Closure coefficients0"T, Q ,rl, oT2

Closure coefficients

xv

xvi

NOTATION

rr(x)

Nonequilibrium parameter

0'ij

Instantaneous total stress tensorT

Kolmogorov time scale; turbulence dissipation timeTij

Reynolds stress tensorTturnover

Eddy turnover timeTXy

Reynolds shear stressTxx , Tyy , Tzz

Normal Reynolds stressesTw

Surface shear stressv

Kolmogorov velocity scale; closure coefficientDimensionless parameter, (vw/pu3)dP/dx

X

Free shear layer closure coefficientStreamfunction

Ok, 0e, Ow

Parabolic marching scheme coefficientsw

Specific dissipation rate ; vorticity vector magnitude

Other

Symbol Definition&f/(9q

Turbulence flux-Jacobian matrixOF/OQ

Mean-flow flux-Jacobian matrixOs/Oq

Source-Jacobian matrix

Subscripts

Symbol DefinitionDNS

Direct Numerical Simulatione

Boundary-layer-edge valueeq

Equilibrium valueLES

Large Eddy Simulation0

Centerline valuev Viscousw

Wall (surface) value00

Freestream value

Superscripts

Symbol Definition+

Sublayer-scaled value

Preface

This book has been developed from the author's lecture notes used in pre-senting a post-graduate course on turbulence modeling at the University ofSouthern California . While several computational fluid dynamics (CFD)texts include some information about turbulence modeling, very few textsdealing exclusively with turbulence modeling have been written . As a con-sequence, turbulence modeling is regarded by many CFD researchers as"black magic," lacking in rigor and physical foundation . This book hasbeen written to show that turbulence modeling can be done in a systematicand physically sound manner . This is not to say all turbulence modelinghas been done in such a manner, for indeed many ill-conceived and ill-fatedturbulence models have appeared in engineering journals . Even this au-thor, early in his career, devised a turbulence model that violated Galileaninvariance of the time-averaged Navier-Stokes equations! However, withjudicious use of relatively simple mathematical tools, systematic construc-tion of a well-founded turbulence model is not only possible but can be anexciting and challenging research project .

Thus, the primary goal of this book is to provide a systematic approachto developing a set of constitutive equations suitable for computation ofturbulent flows . The engineer who feels no existing turbulence model issuitable for his or her needs and wishes to modify an existing model or todevise a new model will benefit from this feature of the text . A methodologyis presented in Chapters 3 and 4 for devising and testing such equations .The methodology is illustrated in great detail for two-equation turbulencemodels . However, it is by no means limited to such models and is usedagain in Chapter 6 for a full Reynolds-stress model, but with less detail .A secondary goal of this book is to provide a rational way for deciding

how complex a model is needed for a given problem . The engineer whowishes to select an existing model that is sufficient for his or her needswill benefit most from this feature of the text . Chapter 3 begins with thesimplest turbulence models and subsequent chapters chart a course leadingto some of the most complex models that have been applied to a nontrivial

xvii

xviii

PREFACE

turbulent flow problem . Two things are done at each level of complexity.First, the range of applicability of the model is estimated . Second, many ofthe applications are repeated for all of the models to illustrate how accuracychanges with complexity .

The methodology makes extensive use of tensor analysis, similarity so-lutions, singular perturbation methods, and numerical procedures . Thetext assumes the user has limited prior knowledge of these mathemati-cal concepts and provides what is needed both in the main text and inthe Appendices . For example, Appendix A introduces rudiments of tensoranalysis to facilitate manipulation of the Navier-Stokes equation, which isdone extensively in Chapter 2 . Chapter 3 shows, in detail, the way a sim-ilarity solution is generated . Similarity solutions are then obtained for theturbulent mixing layer, jet and far wake . Appendix B presents elementsof singular perturbation theory. Chapters 4, 5 and 6 use the methods todissect model-predicted features of the turbulent boundary layer .

No book on turbulence-model equations is complete without a discus-sion of numerical solution methods . Anyone who has ever tried to obtain anumerical solution to a set of turbulence transport equations can attest tothis . Often, standard numerical procedures just won't work and alternativemethods must be found to obtain accurate converged solutions . Chapter 7focuses on numerical methods and elucidates some of the commonly encoun-tered problems such as stiffness, sharp turbulent-nonturbulent interfaces,and difficulties attending turbulence related time scales .

The concluding chapter presents a brief overview of new horizons in-cluding direct numerical simulation (DNS), large-eddy simulation (LES)and the interesting mathematical theory of chaos .

Because turbulence modeling is a key ingredient in CFD work, the textwould be incomplete without companion software implementing numericalsolutions to standard turbulence model equations . Appendices C and Ddescribe several computer programs that are included on the floppy diskaccompanying the book . The programs all have a similar structure and canbe easily modified to include new turbulence models .

The material presented in this book is appropriate for a one-semester,first or second year graduate course, or as a reference text for a CFD course .Successful study of this material requires an understanding of viscous-flowand boundary-layer theory. Some degree of proficiency in solving partialdifferential equations is also needed . A knowledge of computer program-ming, preferably in FORTRAN, will help the reader gain maximum benefitfrom the companion software described in the Appendices .

I extend my thanks to Dr . L . G . Redekopp of USC for encouraging andsupporting development of the course for which this book is intended . Afriend of many years, Dr . P . Bradshaw, reviewed the entire manuscript as I

wrote it, and taught me a lot through numerous discussions, comments andsuggestions that greatly improved the final draft. Another long time friend,Dr . D. D . Knight, helped me understand why I had to write this book,reviewed the manuscript from cover to cover and offered a great deal ofphysical and computational insight in the process . My favorite mathematicsteacher, Dr . D. S . Cohen, made sure I omitted the dot over every t andcrossed every z in Appendix B. Drs. F. R. Menter and C. C. Horstman werekind enough to provide results of several of their computations in digitalform . Thanks are also due for the support and help of several friends andcolleagues, most notably Drs. P. J . Roache, C . G. Speziale and R. M. C.So .

I thank the nine students who were the first to take the course that thisbook was written for . Their patience was especially noteworthy, partic-ularly in regard to typographical errors in the homework problems! Thatoutstanding group of young engineers is D. Foley, R. T . Holbrook, N . Kale,T.-S . Leu, H . Lin, T. Magee, S. Tadepalli, P. Taniguchi and D. Wallace.

Finally, I owe a lifelong debt to my loving wife Barbara for toleratingthe hectic pace first in college and then in the business world. Without her,this book would not have been possible .

David C. Wilcoz