P1: FYX/FYX CB416-FM P2: FYX/FYX CB416-Chung QC: FCH/UKS November 1, 2001 T1: FCH 19:59 Char Count= COMPUTATIONAL FLUID DYNAMICS T J CHUNG University of Alabama in Huntsville iii P1: FYX/FYX CB416-FM P2: FYX/FYX CB416-Chung QC: FCH/UKS November 1, 2001 T1: FCH 19:59 Char Count= PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 47 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon ´ 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org C Cambridge University Press 2002 This book is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2002 Printed in the United Kingdom at the University Press, Cambridge Typefaces Times Ten 10/12.5 pt and Helvetica Neue Condensed A catalog record for this book is available from the British Library Library of Congress Cataloging in Publication data Chung, T J., 1929– Computational fluid dynamics / T J Chung p cm Includes bibliographical references and index ISBN 0-521-59416-2 Fluid dynamics – Data processing QA911 C476 2001 532 05 0285 – dc21 I Title 00-054671 ISBN 521 59416 hardback iv System LATEX 2ε [TB] P1: FYX/FYX CB416-FM P2: FYX/FYX CB416-Chung QC: FCH/UKS November 1, 2001 T1: FCH 19:59 Char Count= Contents Preface page xxi PART ONE PRELIMINARIES Introduction 1.1 General 1.1.1 Historical Background 1.1.2 Organization of Text 1.2 One-Dimensional Computations by Finite Difference Methods 1.3 One-Dimensional Computations by Finite Element Methods 1.4 One-Dimensional Computations by Finite Volume Methods 1.4.1 FVM via FDM 1.4.2 FVM via FEM 1.5 Neumann Boundary Conditions 1.5.1 FDM 1.5.2 FEM 1.5.3 FVM via FDM 1.5.4 FVM via FEM 1.6 Example Problems 1.6.1 Dirichlet Boundary Conditions 1.6.2 Neumann Boundary Conditions 1.7 Summary References Governing Equations 2.1 Classification of Partial Differential Equations 2.2 Navier-Stokes System of Equations 2.3 Boundary Conditions 2.4 Summary References 3 11 11 13 13 14 15 15 16 17 17 20 24 26 29 29 33 38 41 42 PART TWO FINITE DIFFERENCE METHODS Derivation of Finite Difference Equations 3.1 Simple Methods 3.2 General Methods 3.3 Higher Order Derivatives 45 45 46 50 vii P1: FYX/FYX CB416-FM P2: FYX/FYX CB416-Chung QC: FCH/UKS November 1, 2001 T1: FCH 19:59 Char Count= viii CONTENTS 3.4 Multidimensional Finite Difference Formulas 3.5 Mixed Derivatives 3.6 Nonuniform Mesh 3.7 Higher Order Accuracy Schemes 3.8 Accuracy of Finite Difference Solutions 3.9 Summary References Solution Methods of Finite Difference Equations 4.1 Elliptic Equations 4.1.1 Finite Difference Formulations 4.1.2 Iterative Solution Methods 4.1.3 Direct Method with Gaussian Elimination 4.2 Parabolic Equations 4.2.1 Explicit Schemes and von Neumann Stability Analysis 4.2.2 Implicit Schemes 4.2.3 Alternating Direction Implicit (ADI) Schemes 4.2.4 Approximate Factorization 4.2.5 Fractional Step Methods 4.2.6 Three Dimensions 4.2.7 Direct Method with Tridiagonal Matrix Algorithm 4.3 Hyperbolic Equations 4.3.1 Explicit Schemes and Von Neumann Stability Analysis 4.3.2 Implicit Schemes 4.3.3 Multistep (Splitting, Predictor-Corrector) Methods 4.3.4 Nonlinear Problems 4.3.5 Second Order One-Dimensional Wave Equations 4.4 Burgers’ Equation 4.4.1 Explicit and Implicit Schemes 4.4.2 Runge-Kutta Method 4.5 Algebraic Equation Solvers and Sources of Errors 4.5.1 Solution Methods 4.5.2 Evaluation of Sources of Errors 4.6 Coordinate Transformation for Arbitrary Geometries 4.6.1 Determination of Jacobians and Transformed Equations 4.6.2 Application of Neumann Boundary Conditions 4.6.3 Solution by MacCormack Method 4.7 Example Problems 4.7.1 Elliptic Equation (Heat Conduction) 4.7.2 Parabolic Equation (Couette Flow) 4.7.3 Hyperbolic Equation (First Order Wave Equation) 4.7.4 Hyperbolic Equation (Second Order Wave Equation) 4.7.5 Nonlinear Wave Equation 4.8 Summary References Incompressible Viscous Flows via Finite Difference Methods 5.1 General 5.2 Artificial Compressibility Method 53 57 59 60 61 62 62 63 63 63 65 67 67 68 71 72 73 75 75 76 77 77 81 81 83 87 87 88 90 91 91 91 94 94 97 98 98 98 100 101 103 104 105 105 106 106 107 P1: FYX/FYX CB416-FM P2: FYX/FYX CB416-Chung QC: FCH/UKS November 1, 2001 T1: FCH 19:59 Char Count= CONTENTS 5.3 Pressure Correction Methods 5.3.1 Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) 5.3.2 Pressure Implicit with Splitting of Operators 5.3.3 Marker-and-Cell (MAC) Method 5.4 Vortex Methods 5.5 Summary References Compressible Flows via Finite Difference Methods 6.1 Potential Equation 6.1.1 Governing Equations 6.1.2 Subsonic Potential Flows 6.1.3 Transonic Potential Flows 6.2 Euler Equations 6.2.1 Mathematical Properties of Euler Equations 6.2.1.1 Quasilinearization of Euler Equations 6.2.1.2 Eigenvalues and Compatibility Relations 6.2.1.3 Characteristic Variables 6.2.2 Central Schemes with Combined Space-Time Discretization 6.2.2.1 Lax-Friedrichs First Order Scheme 6.2.2.2 Lax-Wendroff Second Order Scheme 6.2.2.3 Lax-Wendroff Method with Artificial Viscosity 6.2.2.4 Explicit MacCormack Method 6.2.3 Central Schemes with Independent Space-Time Discretization 6.2.4 First Order Upwind Schemes 6.2.4.1 Flux Vector Splitting Method 6.2.4.2 Godunov Methods 6.2.5 Second Order Upwind Schemes with Low Resolution 6.2.6 Second Order Upwind Schemes with High Resolution (TVD Schemes) 6.2.7 Essentially Nonoscillatory Scheme 6.2.8 Flux-Corrected Transport Schemes 6.3 Navier-Stokes System of Equations 6.3.1 Explicit Schemes 6.3.2 Implicit Schemes 6.3.3 PISO Scheme for Compressible Flows 6.4 Preconditioning Process for Compressible and Incompressible Flows 6.4.1 General 6.4.2 Preconditioning Matrix 6.5 Flowfield-Dependent Variation Methods 6.5.1 Basic Theory 6.5.2 Flowfield-Dependent Variation Parameters 6.5.3 FDV Equations 6.5.4 Interpretation of Flowfield-Dependent Variation Parameters 6.5.5 Shock-Capturing Mechanism 6.5.6 Transitions and Interactions between Compressible and Incompressible Flows ix 108 108 112 115 115 118 119 120 121 121 123 123 129 130 130 132 134 136 138 138 139 140 141 142 142 145 148 150 163 165 166 167 169 175 178 178 179 180 180 183 185 187 188 191 P1: FYX/FYX CB416-FM P2: FYX/FYX CB416-Chung QC: FCH/UKS November 1, 2001 T1: FCH 19:59 Char Count= x CONTENTS 6.5.7 Transitions and Interactions between Laminar and Turbulent Flows 6.6 Other Methods 6.6.1 Artificial Viscosity Flux Limiters 6.6.2 Fully Implicit High Order Accurate Schemes 6.6.3 Point Implicit Methods 6.7 Boundary Conditions 6.7.1 Euler Equations 6.7.1.1 One-Dimensional Boundary Conditions 6.7.1.2 Multi-Dimensional Boundary Conditions 6.7.1.3 Nonreflecting Boundary Conditions 6.7.2 Navier-Stokes System of Equations 6.8 Example Problems 6.8.1 Solution of Euler Equations 6.8.2 Triple Shock Wave Boundary Layer Interactions Using FDV Theory 6.9 Summary References Finite Volume Methods via Finite Difference Methods 7.1 General 7.2 Two-Dimensional Problems 7.2.1 Node-Centered Control Volume 7.2.2 Cell-Centered Control Volume 7.2.3 Cell-Centered Average Scheme 7.3 Three-Dimensional Problems 7.3.1 3-D Geometry Data Structure 7.3.2 Three-Dimensional FVM Equations 7.4 FVM-FDV Formulation 7.5 Example Problems 7.6 Summary References 193 195 195 196 197 197 197 197 204 204 205 207 207 208 213 214 218 218 219 219 223 225 227 227 232 234 239 239 239 PART THREE FINITE ELEMENT METHODS Introduction to Finite Element Methods 8.1 General 8.2 Finite Element Formulations 8.3 Definitions of Errors 8.4 Summary References Finite Element Interpolation Functions 9.1 General 9.2 One-Dimensional Elements 9.2.1 Conventional Elements 9.2.2 Lagrange Polynomial Elements 9.2.3 Hermite Polynomial Elements 9.3 Two-Dimensional Elements 9.3.1 Triangular Elements 243 243 245 254 259 260 262 262 264 264 269 271 273 273 P1: FYX/FYX CB416-FM P2: FYX/FYX CB416-Chung QC: FCH/UKS November 1, 2001 T1: FCH 19:59 Char Count= CONTENTS 9.3.2 Rectangular Elements 9.3.3 Quadrilateral Isoparametric Elements 9.4 Three-Dimensional Elements 9.4.1 Tetrahedral Elements 9.4.2 Triangular Prism Elements 9.4.3 Hexahedral Isoparametric Elements 9.5 Axisymmetric Ring Elements 9.6 Lagrange and Hermite Families and Convergence Criteria 9.7 Summary References 10 Linear Problems 10.1 Steady-State Problems – Standard Galerkin Methods 10.1.1 Two-Dimensional Elliptic Equations 10.1.2 Boundary Conditions in Two Dimensions 10.1.3 Solution Procedure 10.1.4 Stokes Flow Problems 10.2 Transient Problems – Generalized Galerkin Methods 10.2.1 Parabolic Equations 10.2.2 Hyperbolic Equations 10.2.3 Multivariable Problems 10.2.4 Axisymmetric Transient Heat Conduction 10.3 Solutions of Finite Element Equations 10.3.1 Conjugate Gradient Methods (CGM) 10.3.2 Element-by-Element (EBE) Solutions of FEM Equations 10.4 Example Problems 10.4.1 Solution of Poisson Equation with Isoparametric Elements 10.4.2 Parabolic Partial Differential Equation in Two Dimensions 10.5 Summary References 11 Nonlinear Problems/Convection-Dominated Flows 11.1 Boundary and Initial Conditions 11.1.1 Incompressible Flows 11.1.2 Compressible Flows 11.2 Generalized Galerkin Methods and Taylor-Galerkin Methods 11.2.1 Linearized Burgers’ Equations 11.2.2 Two-Step Explicit Scheme 11.2.3 Relationship between FEM and FDM 11.2.4 Conversion of Implicit Scheme into Explicit Scheme 11.2.5 Taylor-Galerkin Methods for Nonlinear Burgers’ Equations 11.3 Numerical Diffusion Test Functions 11.3.1 Derivation of Numerical Diffusion Test Functions 11.3.2 Stability and Accuracy of Numerical Diffusion Test Functions 11.3.3 Discontinuity-Capturing Scheme 11.4 Generalized Petrov-Galerkin (GPG) Methods 11.4.1 Generalized Petrov-Galerkin Methods for Unsteady Problems 11.4.2 Space-Time Galerkin/Least Squares Methods xi 284 286 298 298 302 303 305 306 308 308 309 309 309 315 320 324 327 327 332 334 335 337 337 340 342 342 343 346 346 347 347 348 353 355 355 358 362 365 366 367 368 369 376 377 377 378 P1: FYX/FYX CB416-FM P2: FYX/FYX CB416-Chung QC: FCH/UKS November 1, 2001 T1: FCH 19:59 Char Count= xii CONTENTS 11.5 Solutions of Nonlinear and Time-Dependent Equations and Element-by-Element Approach 11.5.1 Newton-Raphson Methods 11.5.2 Element-by-Element Solution Scheme for Nonlinear Time Dependent FEM Equations 11.5.3 Generalized Minimal Residual Algorithm 11.6 Example Problems 11.6.1 Nonlinear Wave Equation (Convection Equation) 11.6.2 Pure Convection in Two Dimensions 11.6.3 Solution of 2-D Burgers’ Equation 11.7 Summary References 380 380 381 384 391 391 391 394 396 396 12 Incompressible Viscous Flows via Finite Element Methods 12.1 Primitive Variable Methods 12.1.1 Mixed Methods 12.1.2 Penalty Methods 12.1.3 Pressure Correction Methods 12.1.4 Generalized Petrov-Galerkin Methods 12.1.5 Operator Splitting Methods 12.1.6 Semi-Implicit Pressure Correction 12.2 Vortex Methods 12.2.1 Three-Dimensional Analysis 12.2.2 Two-Dimensional Analysis 12.2.3 Physical Instability in Two-Dimensional Incompressible Flows 12.3 Example Problems 12.4 Summary References 399 399 399 400 401 402 403 405 406 407 410 13 Compressible Flows via Finite Element Methods 13.1 Governing Equations 13.2 Taylor-Galerkin Methods and Generalized Galerkin Methods 13.2.1 Taylor-Galerkin Methods 13.2.2 Taylor-Galerkin Methods with Operator Splitting 13.2.3 Generalized Galerkin Methods 13.3 Generalized Petrov-Galerkin Methods 13.3.1 Navier-Stokes System of Equations in Various Variable Forms 13.3.2 The GPG with Conservation Variables 13.3.3 The GPG with Entropy Variables 13.3.4 The GPG with Primitive Variables 13.4 Characteristic Galerkin Methods 13.5 Discontinuous Galerkin Methods or Combined FEM/FDM/FVM Methods 13.6 Flowfield-Dependent Variation Methods 13.6.1 Basic Formulation 13.6.2 Interpretation of FDV Parameters Associated with Jacobians 13.6.3 Numerical Diffusion 418 418 422 422 425 427 428 428 431 433 434 435 411 413 416 416 438 440 440 443 445 P1: FYX/FYX CB416-FM P2: FYX/FYX CB416-Chung QC: FCH/UKS November 1, 2001 T1: FCH 19:59 Char Count= CONTENTS 13.6.4 Transitions and Interactions between Compressible and Incompressible Flows and between Laminar and Turbulent Flows 13.6.5 Finite Element Formulation of FDV Equations 13.6.6 Boundary Conditions 13.7 Example Problems 13.8 Summary References 14 Miscellaneous Weighted Residual Methods 14.1 Spectral Element Methods 14.1.1 Spectral Functions 14.1.2 Spectral Element Formulations by Legendre Polynomials 14.1.3 Two-Dimensional Problems 14.1.4 Three-Dimensional Problems 14.2 Least Squares Methods 14.2.1 LSM Formulation for the Navier-Stokes System of Equations 14.2.2 FDV-LSM Formulation 14.2.3 Optimal Control Method 14.3 Finite Point Method (FPM) 14.4 Example Problems 14.4.1 Sharp Fin Induced Shock Wave Boundary Layer Interactions 14.4.2 Asymmetric Double Fin Induced Shock Wave Boundary Layer Interaction 14.5 Summary References 15 Finite Volume Methods via Finite Element Methods 15.1 General 15.2 Formulations of Finite Volume Equations 15.2.1 Burgers’ Equations 15.2.2 Incompressible and Compressible Flows 15.2.3 Three-Dimensional Problems 15.3 Example Problems 15.4 Summary References 16 Relationships between Finite Differences and Finite Elements and Other Methods 16.1 Simple Comparisons between FDM and FEM 16.2 Relationships between FDM and FDV 16.3 Relationships between FEM and FDV 16.4 Other Methods 16.4.1 Boundary Element Methods 16.4.2 Coupled Eulerian-Lagrangian Methods 16.4.3 Particle-in-Cell (PIC) Method 16.4.4 Monte Carlo Methods (MCM) 16.5 Summary References xiii 446 447 449 452 459 460 462 462 463 467 471 475 478 478 480 480 481 483 483 486 489 489 491 491 492 492 500 502 503 507 508 509 510 514 518 522 522 525 528 528 530 530 ... in Publication data Chung, T J., 1929– Computational fluid dynamics / T J Chung p cm Includes bibliographical references and index ISBN 0-521-59416-2 Fluid dynamics – Data processing QA911 C476... covers the basic concepts, procedures, and applications of computational methods in fluids and heat transfer, known as computational fluid dynamics (CFD) Specifically, the fundamentals of finite... General 27.2 Governing Equations in Relativistic Fluid Dynamics 27.2.1 Relativistic Hydrodynamics Equations in Ideal Flows 27.2.2 Relativistic Hydrodynamics Equations in Nonideal Flows 27.2.3 Pseudo-Newtonian