1．Basic Concepts of Fluid Flow 1
1.1 Introduction 1
1.2 Conservation Principles 3
1.3 Mass Conservation 4
1.4 Momentum Conservation 5
1.5 Conservation of Scalar Quantities 9
1.6 Dimensionless Form of Equations 11
1.7 Simplified Mathematical Models 12
1.7.1 Incompressible Flow 12
1.7.2 Inviscid(Euler)Flow 13
1.7.3 Potential Flow 13
1.7.4 Creeping(Stokes)Flow 14
1.7.5 Boussinesq Approximation 14
1.7.6 Boundary Layer Approximation 15
1.7.7 Modeling of Complex Flow Phenomena 16
1.8 Mathematical Classification of Flows 16
1.8.1 Hyperbolic Flows 17
1.8.2 Parabolic Flows 17
1.8.3 Elliptic Flows 17
1.8.4 Mixed Flow Types 18
1.9 Plan of This Book 18
2．Introduction to Numerical Methods 21
2.1 Approaches to Fluid Dynamical Problems 21
2.2 What is CFD? 23
2.3 Possibilities and Limitations of Numerical Methods 23
2.4 Components of a Numerical Solution Method 25
2.4.1 Mathematical Model 25
2.4.2 Discretization Method 25
2.4.3 Coordinate and Basis Vector Systems 26
2.4.4 Numerical Grid 26
2.4.5 Finite Approximations 30
2.4.6 Solution Method 30
2.4.7 Convergence Criteria 31
2.5 Properties of Numerical Solution Methods 31
2.5.1 Consistency 31
2.5.2 Stability 32
2.5.3 Convergence 32
2.5.4 Conservation 33
2.5.5 Boundedness 33
2.5.6 Realizability 33
2.5.7 Accuracy 34
2.6 Discretization Approaches 35
2.6.1 Finite Difference Method 35
2.6.2 Finite Volume Method 36
2.6.3 Finite Element Method 36
3．Finite Difference Methods 39
3.1 Introduction 39
3.2 Basic Concept 39
3.3 Approximation of the First Derivative 42
3.3.1 Taylor Series Expansion 42
3.3.2 Polynomial Fitting 44
3.3.3 Compact Schemes 45
3.3.4 Non-Uniform Grids 47
3.4 Approximation of the Second Derivative 49
3.5 Approximation of Mixed Derivatives 52
3.6 Approximation of Other Terms 53
3.7 Implementation of Boundary Conditions 53
3.8 The Algebraic Equation System 55
3.9 Discretization Errors 58
3.10 An Introduction to Spectral Methods 60
3.10.1 Basic Concept 60
3.10.2 Another View of Discretization Error 62
3.11 Example 63
4．Finite Volume Methods 71
4.1 Introduction 71
4.2 Approximation of Surface Integrals 72
4.3 Approximation of Volume Integrals 75
4.4 Interpolation and Differentiation Practices 76
4.4.1 Upwind Interpolation(UDS) 76
4.4.2 Linear Interpolation(CDS) 77
4.4.3 Quadratic Upwind Interpolation(QUICK) 78
4.4.4 Higher-Order Schemes 79
4.4.5 Other Schemes 81
4.5 Implementation of Boundary Conditions 81
4.6 The Algebraic Equation System 82
4.7 Examples 82
5．Solution of Linear Equation Systems 91
5.1 Introduction 91
5.2 Direct Methods 92
5.2.1 Gauss Elimination 92
5.2.2 LU Decomposition 94
5.2.3 Tridiagonal Systems 95
5.2.4 Cyclic Reduction 96
5.3 Iterative Methods 97
5.3.1 Basic Concept 97
5.3.2 Convergence 98
5.3.3 Some Basic Methods 100
5.3.4 Incomplete LU Decomposition:Stone's Method 101
5.3.5 ADI and Other Splitting Methods 105
5.3.6 Conjugate Gradient Methods 107
5.3.7 Biconjugate Gradients and CGSTAB 110
5.3.8 Multigrid Methods 112
5.3.9 Other Iterative Solvers 116
5.4 Coupled Equations and Their Solution 116
5.4.1 Simultaneous Solution 117
5.4.2 Sequential Solution 117
5.4.3 Under-Relaxation 118
5.5 Non-Linear Equations and their Solution 119
5.5.1 Newton-like Techniques 119
5.5.2 Other Techniques 121
5.6 Deferred-Correction Approaches 122
5.7 Convergence Criteria and Iteration Errors 124
5.8 Examples 129
6．Methods for Unsteady Problems 135
6.1 Introduction 135
6.2 Methods for Initial Value Problems in ODEs 135
6.2.1 Two-Level Methods 135
6.2.2 Predictor-Corrector and Multipoint Methods 138
6.2.3 Runge-Kutta Methods 140
6.2.4 Other Methods 142
6.3 Application to the Generic Transport Equation 142
6.3.1 Explicit Methods 143
6.3.2 Implicit Methods 148
6.3.3 Other Methods 151
6.4 Examples 152
7．Solution of the Navier-Stokes Equations 157
7.1 Special Features of the Navier-Stokes Equations 157
7.1.1 Discretization of Convective and Viscous Terms 157
7.1.2 Discretization of Pressure Terms and Body Forces 158
7.1.3 Conservation Properties 160
7.2 Choice of Variable Arrangement on the Grid 164
7.2.1 Colocated Arrangement 165
7.2.2 Staggered Arrangements 166
7.3 Calculation of the Pressure 167
7.3.1 The Pressure Equation and its Solution 167
7.3.2 A Simple Explicit Time Advance Scheme 168
7.3.3 A Simple Implicit Time Advance Method 170
7.3.4 Implicit Pressure-Correction Methods 172
7.4 Other Methods 178
7.4.1 Fractional Step Methods 178
7.4.2 Streamfunction-Vorticity Methods 181
7.4.3 Artificial Compressibility Methods 183
7.5 Solution Methods for the Navier-Stokes Equations 188
7.5.1 Implicit Scheme Using Pressure-Correction and a Stag-gered Grid 188
7.5.2 Treatment of Pressure for Colocated Variables 196
7.5.3 SIMPLE Algorithm for a Colocated Variable Arrange-ment 200
7.6 Note on Pressure and Incompressibility 202
7.7 Boundary Conditions for the Navier-Stokes Equations 204
7.8 Examples 206
8．Complex Geometries 217
8.1 The Choice of Grid 217
8.1.1 Stepwise Approximation Using Regular Grids 217
8.1.2 Overlapping Grids 218
8.1.3 Boundary-Fitted Non-Orthogonal Grids 219
8.2 Grid Generation 219
8.3 The Choice of Velocity Components 223
8.3.1 Grid-Oriented Velocity Components 224
8.3.2 Cartesian Velocity Components 224
8.4 The Choice of Variable Arrangement 225
8.4.1 Staggered Arrangements 225
8.4.2 Colocated Arrangement 226
8.5 Finite Difference Methods 226
8.5.1 Methods Based on Coordinate Transformation 226
8.5.2 Method Based on Shape Functions 229
8.6 Finite Volume Methods 230
8.6.1 Approximation of Convective Fluxes 231
8.6.2 Approximation of Diffusive Fluxes 232
8.6.3 Approximation of Source Terms 238
8.6.4 Three-Dimensional Grids 239
8.6.5 Block-Structured Grids 241
8.6.6 Unstructured Grids 244
8.7 Control-Volume-Based Finite Element Methods 245
8.8 Pressure-Correction Equation 247
8.9 Axi-Symmetric Problems 252
8.10 Implementation of Boundary Conditions 254
8.10.1 Inlet 255
8.10.2 Outlet 255
8.10.3 Impermeable Walls 256
8.10.4 Symmetry Planes 258
8.10.5 Specified Pressure 258
8.11 Examples 259
9．Turbulent Flows 265
9.1 Introduction 265
9.2 Direct Numerical Simulation(DNS) 267
9.2.1 Example:Spatial Decay of Grid Turbulence 275
9.3 Large Eddy Simulation(LES) 277
9.3.1 Smagorinsky and Related Models 279
9.3.2 Dynamic Models 281
9.3.3 Deconvolution Models 283
9.3.4 Example:Flow Over a Wall-Mounted Cube 284
9.3.5 Example:Stratified Homogeneous Shear Flow 287
9.4 RANS Models 292
9.4.1 Reynolds-Averaged Navier-Stokes(RANS)Equations 292
9.4.2 Simple Turbulence Models and their Application 294
9.4.3 The v2f Model 301
9.4.4 Example:Flow Around an Engine Valve 302
9.5 Reynolds Stress Models 304
9.6 Very Large Eddy Simulation 306
10．Compressible Flow 309
10.1 Introduction 309
10.2 Pressure-Correction Methods for Arbitrary Mach Number 310
10.2.1 Pressure-Velocity-Density Coupling 311
10.2.2 Boundary Conditions 315
10.2.3 Examples 319
10.3 Methods Designed for Compressible Flow 324
10.3.1 An Overview of Some Specific Methods 326
11．Efficiency and Accuracy Improvement 329
11.1 Error Analysis and Estimation 329
11.1.1 Description of Errors 329
11.1.2 Estimation of Errors 332
11.1.3 Recommended Practice for CFD Uncertainty Analysis 337
11.2 Grid quality and optimization 341
11.3 Multigrid Methods for Flow Calculation 344
11.4 Adaptive Grid Methods and Local Grid Refinement 351
11.5 Parallel Computing in CFD 356
11.5.1 Iterative Schemes for Linear Equations 357
11.5.2 Domain Decomposition in Space 360
11.5.3 Domain Decomposition in Time 363
11.5.4 Efficiency of Parallel Computing 364
12．Special Topics 369
12.1 Introduction 369
12.2 Heat and Mass Transfer 370
12.3 Flows With Variable Fluid Properties 373
12.4 Moving Grids 373
12.5 Free-Surface Flows 381
12.5.1 Interface-Tracking Methods 388
12.5.2 Hybrid Methods 396
12.6 Meteorological and Oceanographic Applications 397
12.7 Multiphase flows 399
12.8 Combustion 400
A．Appendices 405
A.1 List of Computer Codes and How to Access Them 405
A.2 List of Frequently Used Abbreviations 407
References 409
Index 421