Table of Contents
Preface xvii
About the Companion Website xxi
1 What is CFD? 1
1.1. Introduction 1
1.2. Brief History of CFD 4
1.3. Outline of the Book 5
Bibliography 7
I Fundamentals 9
2 Governing Equations of Fluid Dynamics and Heat Transfer 11
2.1. Preliminary Concepts 11
2.2. Conservation Laws 14
2.2.1. Conservation of Mass 15
2.2.2. Conservation of Chemical Species 15
2.2.3. Conservation of Momentum 16
2.2.4. Conservation of Energy 20
2.3. Equation of State 21
2.4. Equations of Integral Form 22
2.5. Equations in Conservation Form 25
2.6. Equations in Vector Form 26
2.7. Boundary Conditions 27
2.7.1. Rigid Wall Boundary Conditions 28
2.7.2. Inlet and Exit Boundary Conditions 29
2.7.3. Other Boundary Conditions 30
2.8. Dimensionality and Time Dependence 31
2.8.1. Two- and One-Dimensional Problems 32
2.8.2. Equilibrium and Marching Problems 33
Bibliography 34
Problems 34
3 Partial Different Equations 37
3.1. Model Equations: Formulation of a PDE Problem 38
3.1.1. Model Equations 38
3.1.2. Domain, Boundary and Initial Conditions, and Well-Posed PDE Problem 40
3.1.3. Examples 42
3.2. Mathematical Classification of PDEs of Second Order 45
3.2.1. Classification 45
3.2.2. Hyperbolic Equations 48
3.2.3. Parabolic Equations 50
3.2.4. Elliptic Equations 52
3.2.5. Classification of Full Fluid Flow and Heat Transfer Equations 52
3.3. Numerical Discretization: Different Kinds of CFD 53
3.3.1. Spectral Methods 54
3.3.2. Finite Element Methods 56
3.3.3. Finite Difference and Finite Volume Methods 56
Bibliography 59
Problems 59
4 Finite Difference Method 63
4.1. Computational Grid 63
4.1.1. Time Discretization 63
4.1.2. Space Discretization 64
4.2. Finite Difference Approximation 65
4.2.1. Approximation of 𝜕u∕𝜕x 65
4.2.2. Truncation Error, Consistency, and Order of Approximation 66
4.2.3. Other Formulas for 𝜕u∕𝜕x: Evaluation of the Order of Approximation 69
4.2.4. Schemes of Higher Order for First Derivative 71
4.2.5. Higher-Order Derivatives 71
4.2.6. Mixed Derivatives 73
4.2.7. Finite Difference Approximation on Nonuniform Grids 74
4.3. Development of Finite Difference Schemes 77
4.3.1. Taylor Series Expansions 77
4.3.2. Polynomial Fitting 79
4.3.3. Development on Nonuniform Grids 80
4.4. Finite Difference Approximation of Partial Differential Equations 81
4.4.1. Approach and Examples 81
4.4.2. Boundary and Initial Conditions 85
4.4.3. Difference Molecule and Difference Equation 87
4.4.4. System of Difference Equations 88
4.4.5. Implicit and Explicit Methods 89
4.4.6. Consistency of Numerical Approximation 91
4.4.7. Interpretation of Truncation Error: Numerical Dissipation and Dispersion 92
4.4.8. Methods of Interpolation for Finite Difference Schemes 95
Bibliography 98
Problems 98
5 Finite Volume Schemes 103
5.1. Introduction and General Formulation 103
5.1.1. Introduction 103
5.1.2. Finite Volume Grid 105
5.1.3. Consistency, Local, and Global Conservation Property 107
5.2. Approximation of Integrals 109
5.2.1. Volume Integrals 109
5.2.2. Surface Integrals 110
5.3. Methods of Interpolation 112
5.3.1. Upwind Interpolation 112
5.3.2. Linear Interpolation of Convective Fluxes 115
5.3.3. Central Difference (Linear Interpolation) Scheme for Diffusive Fluxes 115
5.3.4. Interpolation of Diffusion Coefficients 117
5.3.5. Upwind Interpolation of Higher Order 118
5.4. Finite Volume Method on Unstructured Grids 119
5.5. Implementation of Boundary Conditions 122
Bibliography 123
Problems 123
6 Numerical Stability for Marching Problems 127
6.1. Introduction and Definition of Stability 127
6.1.1. Example 127
6.1.2. Discretization and Round-Off Error 129
6.1.3. Definition 131
6.2. Stability Analysis 132
6.2.1. Neumann Method 132
6.2.2. Matrix Method 140
6.3. Implicit Versus Explicit Schemes – Stability and Efficiency Considerations 142
Bibliography 144
Problems 144
II Methods 147
7 Application to Model Equations 149
7.1. Linear Convection Equation 150
7.1.1. Simple Explicit Schemes 151
7.1.2. Simple Implicit Scheme 154
7.1.3. Leapfrog Scheme 155
7.1.4. Lax–Wendroff Scheme 156
7.1.5. MacCormack Scheme 157
7.2. One-Dimensional Heat Equation 157
7.2.1. Simple Explicit Scheme 157
7.2.2. Simple Implicit Scheme 159
7.2.3. Crank–Nicolson Scheme 159
7.3. Burgers and Generic Transport Equations 161
7.4. Method of Lines 162
7.4.1. Adams Methods 163
7.4.2. Runge–Kutta Methods 164
7.5. Solution of Tridiagonal Systems by Thomas Algorithm 165
Bibliography 169
Problems 169
8 Steady-State Problems 173
8.1. Problems Reducible to Matrix Equations 173
8.1.1. Elliptic PDE 174
8.1.2. Marching Problems Solved by Implicit Schemes 177
8.1.3. Structure of Matrices 179
8.2. Direct Methods 180
8.2.1. Cyclic Reduction Algorithm 181
8.2.2. Thomas Algorithm for Block-Tridiagonal Matrices 184
8.2.3. LU Decomposition 185
8.3. Iterative Methods 186
8.3.1. General Methodology 187
8.3.2. Jacobi Iterations 188
8.3.3. Gauss–Seidel Algorithm 189
8.3.4. Successive Over- and Underrelaxation 190
8.3.5. Convergence of Iterative Procedures 191
8.3.6. Multigrid Methods 194
8.3.7. Pseudo-transient Approach 197
8.4. Systems of Nonlinear Equations 197
8.4.1. Newton’s Algorithm 198
8.4.2. Iteration Methods Using Linearization 199
8.4.3. Sequential Solution 201
8.5. Computational Performance 202
Bibliography 203
Problems 203
9 Unsteady Compressible Fluid Flows and Conduction Heat Transfer 207
9.1. Introduction 207
9.2. Compressible Flows 208
9.2.1. Equations, Mathematical Classification, and General Comments 208
9.2.2. MacCormack Scheme 212
9.2.3. Beam–Warming Scheme 214
9.2.4. Upwinding 218
9.2.5. Methods for Purely Hyperbolic Systems: TVD Schemes 220
9.3. Unsteady Conduction Heat Transfer 223
9.3.1. Overview 223
9.3.2. Simple Methods for Multidimensional Heat Conduction 223
9.3.3. Approximate Factorization 225
9.3.4. ADI Method 227
Bibliography 228
Problems 229
10 Incompressible Flows 233
10.1. General Considerations 233
10.1.1. Introduction 233
10.1.2. Role of Pressure 234
10.2. Discretization Approach 236
10.2.1. Conditions for Conservation of Mass by Numerical Solution 237
10.2.2. Colocated and Staggered Grids 238
10.3. Projection Method for Unsteady Flows 243
10.3.1. Explicit Schemes 244
10.3.2. Implicit Schemes 247
10.4. Projection Methods for Steady-State Flows 250
10.4.1. SIMPLE 252
10.4.2. SIMPLEC and SIMPLER 254
10.4.3. PISO 256
10.5. Other Methods 257
10.5.1. Vorticity–Streamfunction Formulation for Two-Dimensional Flows 257
10.5.2. Artificial Compressibility 261
Bibliography 261
Problems 262
III Art of CFD 265
11 Turbulence 267
11.1. Introduction 267
11.1.1. A Few Words About Turbulence 268
11.1.2. Why is the Computation of Turbulent Flows Difficult? 271
11.1.3. Overview of Numerical Approaches 273
11.2. Direct Numerical Simulation (DNS) 275
11.2.1. Homogeneous Turbulence 275
11.2.2. Inhomogeneous Turbulence 278
11.3. Reynolds-Averaged Navier–Stokes (RANS) Models 279
11.3.1. Mean Flow and Fluctuations 280
11.3.2. Reynolds-Averaged Equations 281
11.3.3. Reynolds Stresses and Turbulent Kinetic Energy 282
11.3.4. Eddy Viscosity Hypothesis 284
11.3.5. Closure Models 285
11.3.6. Algebraic Models 286
11.3.7. One-Equation Models 287
11.3.8. Two-Equation Models 289
11.3.9. RANS and URANS 291
11.3.10. Models of Turbulent Scalar Transport 292
11.3.11. Numerical Implementation of RANS Models 294
11.4. Large Eddy Simulation (LES) 297
11.4.1. Filtered Equations 298
11.4.2. Closure Models 301
11.4.3. Implementation of LES in CFD Analysis: Numerical Resolution and Near-Wall Treatment 304
Bibliography 307
Problems 309
12 Computational Grids 313
12.1. Introduction: Need for Irregular and Unstructured Grids 313
12.2. Irregular Structured Grids 316
12.2.1. Generation by Coordinate Transformation 316
12.2.2. Examples 319
12.2.3. Grid Quality 321
12.3. Unstructured Grids 322
12.3.1. Grid Generation 325
12.3.2. Cell Topology 325
12.3.3. Grid Quality 326
12.4. Adaptive Grids 329
Bibliography 331
Problems 332
13 Conducting CFD Analysis 335
13.1. Overview: Setting and Solving a CFD Problem 335
13.2. Errors and Uncertainty 339
13.2.1. Errors in CFD Analysis 339
13.2.2. Verification and Validation 346
Bibliography 349
Problems 349
Index 351
