/ / ....r. b2!]- 1 ·� ;: 0 .. .... !0 2 �. ........ • ,I 0 -3 _, ·"·_, Contents Notation xi xvii Preface 1 Introduction 1 1.1 Definition of an Ideal· Turbulence Model 2 1.2 How Complex Must a Turbulence Model Be? . 2 1.3 Comments on the Physics of Turbulence . . . 3 1.3.1 Importance of Turbulence in Practical Situations 3 1.3. 2 General Properties of Turbulence . . . . . . . 5 1.3.3 The Smallest Scales of Turbulence . . . . . . . 10 1.3.4 Spectral Representation and the Kolmogorov -5/3 Law ll 1.3.5 The Law of the Wall . . 15 1.3. 6 Power Laws . . . . . . . . . . 20 1.4 A Brief History of Turbulence Modeling 23 Problems . . . . . . . . . . . . . 28 2 The Closure Problem 33 2.1 Reynolds Averaging . 34 2.2 Correlations . . . 39 2.3 Reynolds-Averaged Equations 39 2.4 The Reynolds-Stress Equation 41 2.5 The Scales of Turbulence. . . 43 2.5.1 Turbulence Intensity . 44 2.5.2 Two-Point Correlation Tensors and Related Scales 45 Problems . . 50 3 Algebraic Models 53 3.1 Molecular Transport of Momentum 54 3.2 The Mixing-Length Hypothesis 57 3.3 Application to Free Shear Flows . . 60 3 .3 .1 The Far Wake . . 62 3 .3 .2 The Mixing Layer 67 3 .3.3 The Jet ... 70 3 .4 Modem Variants of the Mixing-Length Model 74 3 .4.1 Cebeci-Smith Model . . . 79 3.4.2 Baldwin-Lomax Model . .. 81 3 .5 Application to Wall-Bounded Flows 84 3 .5.1 Channel and Pipe Flow 84 3.5.2 Boundary Layers 89 3.6 Separated Flows . . . . . 94 3 .7 The 1/2-Equation Model 96 3 .8 Range of Applicability 100 Problems . . . . . . 102 4 One-Equation and Two-Equation Models 107 4 .1 The Turbulence Energy Equation 108 4.2 One-Equation Models . . 111 4.3 Two-Equation Models 122 4.3 .1 The k-w Model . 124 4.3.2 The k-c Model 128 4.3. 3 OtherT wo-Equation Models . 13 1 4.4 Closure Coefficients . . . . . . . . . 13 3 4.5 Application to Free Shear Flows . . . 13 6 4.5.1 Developing the Similarity Solution 13 7 4.5.2 Numerical Solution . . . . . . . . 14 3 4.5 .3 Sensitivity to Finite Freestream Boundary Conditions 14 7 4.5.4 Cross Dift\Jsion . . . . . . . . . . 151 4.5 .5 The Round-Jet/Plane-Jet Anomaly . . 154 4.6 Perturbation Analysis of the Boundary Layer 156 4.6.1 The Log Layer . . . . 156 4 .6.2 The Defect Layer . . 161 4.6. 3 The Viscous Sublayer 175 4.7 Surface Boundary Conditions 180 4.7.1 Wall Functions . . . . 181 4.7.2 Surface Roughness 182 4.7.3 Surface Mass Injection . 186 4.8 Application to Wall-Bounded Flows 187 4.8.1 Channel and Pipe Flow 187 4.8.2 Boundary Layers . . . . 189 4.9 Low-Reynolds-Number Effects . 192 4.9.1 Asymptotic Consistency 193 4.9.2 Transition . . . . . . . . ZOO 4.9.3 Channel and Pipe Flow . . . 210 4.9.4 Boundary-Layer Applications 212 4.10 Application to Separated Flows 218 4.1 J Range of Applicability 227 Problems .. 230 5 Effects of Compressibility 239 5.1 Physical Considerations . 239 5.2 Favre Averaging. ... . 241 5.3 Favre-Averaged Equations 243 5.4 Compressible-Flow Closure Approximations 249 5.4.1 Reynolds-Stress Tensor. . . . . . . 250 5.4.2 Turbulent Heat-Flux Vector . . . . 250 5.4.3 Molecular Diffusion and Turbulent Transport . 250 5.4.4 Dilatation Dissipation . . . . . . .. . . . 251 5.4.5 Pres�ure Diffusion and Pressure Dilatation . . 253 5.4.6 Pressure Work .... . ... . . . . . . . 254 5.4.7 k-w Model Equations for Compressible Flows 255 5.5 Mixing-Layer Compressibility Corrections . . . . . . . 257 5.5.1 The Sarkar/Zeman/Wilcox Compressibility Corrections 258 5.5.2 Applications . . . . . 259 5.6 Compressible Law of the Wall . . . . 262 5.6.1 Derivation. . . . . . . . .. . 262 5.6.2 The Effect of Cross Diffusion 268 5.7 Compressible Boundary Layers. . . . . 269 5.8 Shock-Induced Boundary-Layer Separation 275 5.8.1 The Earliest Applications . . . . . 275 5.8.2 The Use of Wall Functions for Shock-Separated Flows 277 5.8.3 The Next Two Decades of "Progress" . . . . . . . . . 279 5.8.4 Effect of the Stress Limiter on Shock-Separated Flows 280 5.8.5 Transonic Flow Over an Axisynunetric Bump . . . . 283 5.8.6 Mach 2 Flow Past a Backward-Facing Step . . . 284 5.8.7 Mach 3 Compression Corners and Reflecting Shocks 285 5.8.8 Mach II Reflecting-Shock. . . . . . . . . . . . 289 5.8.9 The Reattachment Point Heat-Transfer Anomaly 290 5.8.10 Three-Dimensional Applications. 292 5.9 Summary 295 Problems . .. . ..... . . . . .. . 297 6 Beyond the Boussinesq Approximation 303 6.1 Boussinesq-Approximation Deficiencies 303 6.2 Nonlinear Constitutive Relations . 308 6.2.1 The Earliest Formulations . . 308 6.2.2 Algebraic Stress Models . . 311 6.2.3 Relation to the Stress Limiter 317 6.2.4 Lag Model . . .... . 320 6.3 Stress-Transport Models . . . . . 322 6.3.1 Closure Approximations 323 6.3.2 Launder-Reece-Rodi Model 330 6.3.3 Wilcox Stress-w Model. . . 332 6.4 Application to Homogeneous Turbulent Flows 334 6.5 Application to Free Shear Flows . . . . . . . . 34 0 6.6 Application to Wall-Bounded Flows . . . . . . 343 6.6.1 Surface Boundary Conditions/Viscous Modifications . 34 3 6.6.2 Channel and Pipe Flow 34 8 6.6.3 Rotating Channel Flow. 351 6.6.4 Boundary Layers . . . . 352 6. 7 Application to Separated Flows 361 6.7.1 Incompressible Backward-Facing Step 361 6.7.2 Transonic Flow Over an Axisymmetric Bump 365 6.7.3 Mach 3 Compression Corners and Reflecting Shocks 366 6.7.4 Hypersonic Shock-Separated Flows 370 6.8 Range of Applicability 371 Problems . . . . . . . . . . . . . . 373 7 Numerical Considerations 381 7.1 Multiple Time Scales and Stiffness 381 7.2 Numerical Accuracy Near Boundaries 383 7. 2.1 Solid Surfaces . . . . . . 383 7.2.2 Turbulent/Nonturbulent Interfaces 387 7.2.3 Sensitivity to Freestream Boundary Conditions 395 7.2.4 Viscous-Interface Layer . . . 397 7.3 Parabolic Marching Methods . . . . . 399 7.4 Elementary Time-Marching Methods . 403 7.5 Block-Implicit Methods .. . . .. . . 4 09 7.6 Solution Convergence and Grid Sensitivity 414 7.6.1 Iteration Convergence and Grid Convergence . 414 7. 6.2 Richardson Extrapolation . . . 416 7. 6.3 Grid Convergence Index . 417 7.6.4 Near-Wall Grid-Point Spacing . 418 Problems . . . . . . . . . . . 420 8 New Horizons 427 8.1 Background Information 428 8.2 Direct Numerical Simulation . 431 8.3 Large Eddy Simulation 436 8.3.1 Filtering. . . . . . . 437 8.3.2 Subgrid-Scale (SGS) Modeling 440 8.3.3 "Off the Wall" Boundary Conditions 442 8.3.4 Applications . . .. . . 444 8.4 Detached Eddy Simulation 446 8.4.1 DES-Blending Functions . 446 8.4.2 Applications 451 8.5 Chaos . . . .. . 452 8.6 Further Reading . 455 Problems . ' . . 456 A Cartesian Tensor Analysis 459 Problems . . . . . . . . . 464 B Rudiments of Perturbation Methods 465 Problems . . . . . . . ' . . . . 475 c Companion Software 477 Bibliography 479 Index 509 , Tu rbulence Modeling for CFD Third Edftion • by • David C. Wilcox • DCW Industries Thrbulence Modeling for CFD Third edition copyright © 2006 by DCW Industries, Inc. All rights reserved. First Printing: November, 2006 No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any in­ formation storage and retrieval system, without permission in writing from DCW Industries, Inc. DCW Industries, Inc. 5354 Palm Drive, La Canada, California 91011 8181790-3844 (FAX) 818/952-1272 World Wide Web: http:// .dcwindustries.com www This book was prepared with Jb.TE(C as implemented by Personal T:EX, Inc. of Mill Valley, California. was printed and bound in the United States of America by It Birmingham Press, Inc., San Diego, California. Library of Congress Cataloging in Publication Data Wilcox, David C. Turbulence Mod�ing for CFD David C. Wilcox--3rd edition I Includes bibliography, index and Compact Disk. 1. Turbulence--Mathematical Models. 2. Fluid Dynamics-Mathematical Models. Catalog Card Number 2006908990 ISBN 978-1-928729-08-2 ( 1-928729-08-8) , Dedicated to Wife my BARBARA Children my KINLEY and BOB and Dad my -- - About t he Author Dr. David C. Wilcox, was born in Wilmington, Delaware. He did his under­ graduate studies from 1963 to 1966 at the Massachusetts Institute of Technology, and graduating with a Bachelor of Science degree in Aeronautics Astronautics. From I 966 to 1967, he was employed by the McDonnell Douglas Aircraft Di­ vision in Long Beach, California, and began his professional career under the guidance of A. M. Smith. His experience with McDonnell Douglas focused 0. on subsonic and transonic flow calculations. From 1967 to 1970, he attended the Califomia Institute of Technology, graduating with a �h.D. in Aeronautics. In 1970 he joined TRW Systems, Inc. in Redondo Beach, �California, where he perfonned studies of both high- and low-speed fluid-mechanical and heat-transfer problems, such as turbulent hypersonic flow and th l radiation from a flame. erma From 1972 to 1973, he was a staff scientist for Applied Theory, Inc. , in Los Angeles, California. He participated in many research efforts involving numer­ ical computation and analysis of fluid flows such as separated turbulent flow, transitional flow and hypersonic plume-body interaction. In 1973, he founded DCW Industries, Inc., a La Cafl.ada, California firm en­ gaged in engineering research and book publishing, for which he is currently Pres­ ident. He has taught several fluid mechanics and applied mathematics courses at the University of Southern California and at the University of California, Los Angeles. Dr.W ilcox has numerous publications on turbulence modeling, computational fluid dynamics, boundary-layer separation, boundary-layer transition, thermal ra­ diation, and rapidly�otating fluids. His book publications include texts entitled Elements of Fluid Mechanics, Basic Fluid Mechanics Perturb ation Methods and in the C omputer Age. He is Associate Fellow of the American Institute of Aeronautics and an Astronautics (AIAA) and has served as an Associate Editor for the AIAA Journal.