Computational Fluid Mechanics and Heat Transfer

Cover......Page 1
COMPUTATIONAL FLUID MECHANICS AND HEAT TRANSFER......Page 3
Dedication......Page 5
CONTENTS......Page 6
PREFACE......Page 13
PREFACE TO THE FIRST EDITION......Page 16
Part I. Fundamentals......Page 19
10. GRID GENERATION......Page 0
Part II. Application of Numerical Methods to the Equations of Fluid Mechanics and Heat Transfer......Page 20
1.1 GENERAL REMARKS......Page 21
1.2 COMPARISON OF EXPERIMENTAL, THEORETICAL, AND COMPUTATIONAL APPROACHES......Page 23
1.3 HISTORICAL PERSPECTIVE......Page 28
2.2.1 Equilibrium Problems......Page 33
2.2.2 Marching Problems......Page 37
2.3 MATHEMATICAL CLASSIFICATION......Page 40
2.3.1 Hyperbolic PDEs......Page 44
2.3.2 Parabolic PDEs......Page 47
2.3.3 Elliptic PDEs......Page 50
2.4 THE WELLPOSED PROBLEM......Page 51
2.5 SYSTEMS OF EQUATIONS......Page 53
2.6 OTHER DIFFERENTIAL EQUATIONS OF INTEREST......Page 58
PROBLEMS......Page 59
3.1 INTRODUCTION......Page 63
3.2 FINITE DIFFERENCES......Page 64
3.3.1 Truncation Error......Page 70
3.3.2 Round-Off and Discretization Errors......Page 72
3.3.4 Stability......Page 73
3.3.6 A Comment on Equilibrium Problems......Page 75
3.3.7 Conservation Form and Conservative Property......Page 76
3.4 FURTHER EXAMPLES OF METHODS FOR OBTAINING FINITE-DIFFERENCE EQUATIONS......Page 78
3.4.1 Use of Taylor Series......Page 79
3.4.2 Use of Polynomial Fitting......Page 83
3.4.3 Integral Method......Page 87
3.4.4 Finite-Volume (Control-Volume) Approach......Page 89
3.5.1 Irregular Mesh Due to Shape of a Boundary......Page 94
3.5.2 Irregular Mesh Not Caused by Shape of a Boundary......Page 100
3.6 STABILITY CONSIDERATIONS......Page 101
3.6.1 Fourier or von Neumann Analysis......Page 102
3.6.2 Stability Analysis for Systems of Equations......Page 109
PROBLEMS......Page 114
APPLICATION OF NUMERICAL METHODS TO SELECTED MODEL EQUATIONS......Page 119
4.1.1 Euler Explicit Methods......Page 120
4.1.2 Upstream (First-Order Upwind or Windward) Differencing Method......Page 121
4.1.3 Lax Method......Page 130
4.1.4 Euler Implicit Method......Page 131
4.1.5 Leap Frog Method......Page 134
4.1.6 Lax-Wendroff Method......Page 135
4.1.7 Two-step Lax-Wendroff Method......Page 136
4.1.9 Second-Order Upwind Method......Page 137
4.1.10 Time-Centered Implicit Method (Trapezoidal Differencing Method)......Page 138
4.1.11 Rusanov (Burstein-Mirin) Method......Page 140
4.1.12 Warming-Kutler-Lomax Method......Page 141
4.1.13 Runge-Kutta Methods......Page 142
4.1.14 Additional Comments......Page 143
4.2.1 Simple Explicit Method......Page 144
4.2.2 Richardson\'s Method......Page 147
4.2.4 Crank-Nicolson Method......Page 148
4.2.6 Combined Method B......Page 150
4.2.7 DuFort-Frankel Method......Page 151
4.2.8 Keller Box and Modified Box Methods......Page 152
4.2.9 Methods for the Two-Dimensional Heat Equation......Page 155
4.2.10 AD1 Methods......Page 157
4.2.12 ADE Methods......Page 160
4.2.13 Hopscotch Method......Page 161
4.3 LAPLACE’S EQUATION......Page 162
4.3.1 Finite-Difference Representations for Laplace’s Equation......Page 163
4.3.2 Simple Example for Laplace\'s Equation......Page 164
4.3.3 Direct Methods for Solving Systems of Linear Algebraic Equations......Page 166
4.3.4 Iterative Methods for Solving Systems of Linear Algebraic Equations......Page 171
4.3.5 Multigrid Method......Page 183
4.4 BURGERS’ EQUATION (INVISCID)......Page 194
4.4.1 Lax Method......Page 199
4.4.2 Lax- Wendroff Method......Page 202
4.4.3 MacCormack Method......Page 205
4.4.4 Rusanov (Burstein-Mirin) Method......Page 206
4.4.5 Warming-Kutler-Lomax Method......Page 207
4.4.6 Tuned Third-Order Methods......Page 208
4.4.7 Implicit Methods......Page 210
4.4.8 Godunov Scheme......Page 213
4.4.9 Roe Scheme......Page 216
4.4.10 Enquist-Osher Scheme......Page 220
4.4.11 Higher-Order Upwind Schemes......Page 222
4.4.12 TVD Schemes......Page 225
4.5 BURGERS’ EQUATION (VISCOUS)......Page 235
4.5.1 lTCS Method......Page 238
4.5.3 Brailovskaya Method......Page 243
4.5.4 Allen-Cheng Method......Page 244
4.5.6 MacCormack Method......Page 245
4.5.7 Briley-McDonald Method......Page 247
4.5.8 Time-Split MacCormack Method......Page 248
4.5.10 Predictor-Corrector, Multiple-Iteration Method......Page 250
PROBLEMS......Page 252
5.1 FUNDAMENTAL EQUATIONS......Page 264
5.1.1 Continuity Equation......Page 265
5.1.2 Momentum Equation......Page 267
5.1.3 Energy Equation......Page 270
5.1.4 Equation of State......Page 272
5.1.5 Chemically Reacting Flows......Page 274
5.1.6 Vector Form of Equations......Page 278
5.1.7 Nondimensional Form of Equations......Page 279
5.1.8 Orthogonal Curvilinear Coordinates......Page 281
5.2.1 Background......Page 287
5.2.2 Reynolds Averaged Navier-Stokes Equations......Page 288
5.2.3 Reynolds Form of the Continuity Equation......Page 290
5.2.4 Reynolds Form of the Momentum Equations......Page 291
5.2.5 Reynolds Form of the Energy Equation......Page 293
5.2.6 Comments on the Reynolds Equations......Page 295
5.2.7 Filtered Navier-S tokes Equations for Large-Eddy Simulation......Page 298
5.3.1 Background......Page 300
5.3.2 Boundary-Layer Approximation for Steady Incompressible Flow......Page 301
5.3.3 Boundary-Layer Equations for Compressible Flow......Page 310
5.4.2 Modeling Terminology......Page 314
5.4.3 Simple Algebraic or Zero-Equation Models......Page 316
5.4.4 One-Half-Equation Models......Page 323
5.4.5 One-Equation Models......Page 325
5.4.6 One-and-One-Half- and Two-Equation Models......Page 328
5.4.7 Reynolds Stress Models......Page 332
5.4.8 Subgrid-Scale Models for Large-Eddy Simulation......Page 335
5.5 EULER EQUATIONS......Page 336
5.5.1 Continuity Equation......Page 337
5.5.2 Inviscid Momentum Equations......Page 338
5.5.3 Inviscid Energy Equations......Page 341
5.5.4 Additional Equations......Page 342
5.5.5 Vector Form of Euler Equations......Page 343
5.5.6 Simplified Forms of Euler Equations......Page 344
5.5.7 Shock Equations......Page 346
5.6.1 Simple Transformations......Page 348
5.6.2 Generalized Transformation......Page 353
5.7.1 Two-Dimensional Finite-Volume Method......Page 357
5.7.2 Three-Dimensional Finite-Volume Method......Page 362
PROBLEMS......Page 363
6.1 INTRODUCTION......Page 366
6.2 METHOD OF CHARACTERISTICS......Page 367
6.2.1 Linear Systems of Equations......Page 368
6.2.2 Nonlinear Systems of Equations......Page 376
6.3 CLASSICAL SHOCK-CAPTURING METHODS......Page 380
6.4 FLUX SPLITTING SCHEMES......Page 390
6.4.1 Steger-Warming Splitting......Page 391
6.4.2 Van Leer Flux Splitting......Page 396
6.4.3 Other Flux Splitting Schemes......Page 398
6.4.4 Application for Arbitrarily Shaped Cells......Page 400
6.5 FLUX-DIFFERENCE SPLITTING SCHEMES......Page 401
6.5.1 Roe Scheme......Page 403
6.5.2 Second-Order Schemes......Page 410
6.6 MULTIDIMENSIONAL CASE IN A GENERAL  COORDINATE SYSTEM......Page 413
6.7 BOUNDARY CONDITIONS FOR THE EULER EQUATIONS......Page 417
6.8 METHODS FOR SOLVING THE POTENTIAL, EQUATION......Page 428
6.9 TRANSONIC SMALL-DISTURBANCE EQUATIONS......Page 443
6.10 METHODS FOR SOLVING LAPLACE’S EQUATION......Page 446
PROBLEMS......Page 452
7.1 INTRODUCTION......Page 455
7.2 BRIEF COMPARISON OF PREDICTION METHODS......Page 456
7.3.1 Generalized Form of the Equations......Page 457
7.3.2 Example of a Simple Explicit Procedure......Page 459
7.3.3 Crank-Nicolson and Fully Implicit Methods......Page 461
7.3.4 DuFort-Frankel Method......Page 473
7.3.5 Box Method......Page 476
7.3.6 Other Methods......Page 479
7.3.7 Coordinate Transformations for Boundary Layers......Page 480
7.3.8 Special Considerations for Turbulent Flows......Page 484
7.3.9 Example Applications......Page 487
7.3.10 Closure......Page 490
7.4.1 Introduction......Page 492
7.4.2 Comments on Computing Separated Flows Using the Boundary-Layer Equations......Page 493
7.4.3 Inverse Finite-Difference Methods......Page 496
7.4.4 Viscous-Inviscid Interaction......Page 503
7.5.1 Introduction......Page 510
7.5.3 Computational Strategies for Internal Flows......Page 512
7.6 APPLICATION TO FREE-SHEAR FLOWS......Page 522
7.7.1 Introduction......Page 526
7.7.2 The Equations......Page 527
7.7.3 Comments on Solution Methods for Three-Dimensional Flows......Page 533
7.7.4 Example Calculations......Page 542
7.8 UNSTEADY BOUNDARY LAYERS......Page 544
PROBLEMS......Page 546
8.1 INTRODUCTION......Page 550
8.2 THIN-LAYER NAVIER-STOKES EQUATIONS......Page 554
8.3 “PARABOLIZED” NAVIER-STOKES EQUATIONS......Page 558
8.3.1 Derivation of PNS Equations......Page 559
8.3.2 STREAMWISE PRESSURE GRADIENT......Page 568
8.3.3 Numerical Solution of PNS Equations......Page 575
8.3.4 Applications of PNS Equations......Page 595
8.4.1 Fully Parabolic Procedures......Page 598
8.4.2 Parabolic Procedures for 3-D Free-Shear and Other Flows......Page 605
8.4.3 Partially Parabolized (Multiple Space-Marching) Model......Page 606
8.5 VISCOUS SHOCK-LAYER EQUATIONS......Page 622
8.6 “CONICAL” NAVIER-STOKES EQUATIONS......Page 627
PROBLEMS......Page 630
9.1 INTRODUCTION......Page 634
9.2 COMPRESSIBLE NAVIER-STOKES EQUATIONS......Page 635
9.2.1 Explicit MacCormack Method......Page 638
9.2.2 Other Explicit Methods......Page 645
9.2.3 Beam-Warming Scheme......Page 646
9.2.4 Other Implicit Methods......Page 653
9.2.5 Upwind Methods......Page 654
9.2.6 Compressible Navier-Stokes Equations at Low Speeds......Page 655
9.3 INCOMPRESSIBLE NAVIER-STOKES EQUATIONS......Page 662
9.3.1 Vorticity-Stream Function Approach......Page 663
9.3.2 Primitive-Variable Approach......Page 672
PROBLEMS......Page 690
10.1 INTRODUCTION......Page 692
10.2 ALGEBRAIC METHODS......Page 694
10.3.1 Elliptic Schemes......Page 701
10.3.2 Hyperbolic Schemes......Page 707
10.3.3 Parabolic Schemes......Page 710
10.4 VARIATIONAL METHODS......Page 711
10.5 UNSTRUCTURED GRID SCHEMES......Page 713
10.5.1 Connectivity Information......Page 715
10.5.2 Delaunay Triangulation......Page 716
10.5.3 Bowyer Algorithm......Page 718
10.6 OTHER APPROACHES......Page 721
10.7 ADAPTIVE GRIDS......Page 723
PROBLEMS......Page 725
NOMENCLATURE......Page 727
REFERENCES......Page 734
INDEX......Page 772
APPENDIX A. SUBROUTINE FOR SOLVING  A TRIDIAGONAL SYSTEM OF EQUATIONS......Page 782
APPENDIX B. SUBROUTINES FOR SOLVING BLOCK TRIDIAGONAL SYSTEMS OF EQUATIONS......Page 784
APPENDIX C. THE MODIFIED STRONGLY IMPLICIT PROCEDURE......Page 792
APPENDIX D. FINITE-VOLUME DISCRETIZATION FOR GENERAL CONTROL VOLUMES......Page 798
TrUe LiAr......Page 803