Essential Computational Fluid Dynamics

Essential Computational Fluid Dynamics......Page 3
Contents......Page 9
Preface......Page 17
1.1. Introduction......Page 19
1.2. Brief History of CFD......Page 22
1.3. Outline of the Book......Page 24
References and Suggested Reading......Page 25
I Fundamentals......Page 27
2.1. Preliminary Concepts......Page 29
2.2. Mass Conservation......Page 32
2.3. Conservation of Chemical Species......Page 33
2.4. Conservation of Momentum......Page 34
2.5. Conservation of Energy......Page 37
2.7. Equations in Integral Form......Page 39
2.8. Equations in Conservation Form......Page 42
2.9. Equations in Vector Form......Page 43
2.10. Boundary Conditions......Page 44
2.10.1. Rigid Wall Boundary Conditions......Page 45
2.10.3. Other Boundary Conditions......Page 47
Problems......Page 48
3 Partial Differential Equations......Page 50
3.1.1. Model Equations......Page 51
3.1.2. Domain, Boundary, and Initial Conditions......Page 53
3.1.3. Equilibrium and Marching Problems......Page 54
3.1.4. Examples......Page 55
3.2.1. Classification......Page 58
3.2.2. Hyperbolic Equations......Page 60
3.2.3. Parabolic Equations......Page 63
3.3. Numerical Discretization: Different Kinds of CFD......Page 64
3.3.1. Spectral Methods......Page 65
3.3.3. Finite Difference and Finite Volume Methods......Page 67
Problems......Page 70
4.1.1. Time Discretization......Page 73
4.1.2. Space Discretization......Page 74
4.2.1. Approximation of ∂u/∂x......Page 75
4.2.2. Truncation Error, Consistency, Order of Approximation......Page 76
4.2.3. Other Formulas for ∂u/∂x: Evaluation of the Order of Approximation......Page 78
4.2.4. Schemes of Higher Order for First Derivative......Page 80
4.2.5. Higher-Order Derivatives......Page 81
4.2.6. Mixed Derivatives......Page 82
4.2.7. Truncation Error of Linear Interpolation......Page 84
4.3.1. Approach and Examples......Page 85
4.3.2. Interpretation of Truncation Error: Numerical Dissipation and Dispersion......Page 88
4.3.3. Boundary and Initial Conditions......Page 91
4.3.4. Consistency of Numerical Approximation......Page 92
4.3.5. System of Difference Equations......Page 93
4.3.6. Implicit and Explicit Methods......Page 94
4.4. Development of Finite Difference Schemes......Page 96
4.4.1. Taylor Series Expansions......Page 97
4.4.2. Polynomial Fitting......Page 100
Problems......Page 101
5.1. Introduction and Integral Formulation......Page 104
5.1.1. Finite Volume Grid......Page 105
5.1.2. Global Conservation Property......Page 107
5.2.1. Volume Integrals......Page 109
5.2.2. Surface Integrals......Page 110
5.3. Methods of Interpolation......Page 112
5.3.1. Upwind Interpolation......Page 113
5.3.2. Linear Interpolation......Page 114
5.3.3. Upwind Interpolation of Higher Order......Page 116
5.3.4. Interpolation on Nonorthogonal Grids......Page 117
5.4. Boundary Conditions......Page 119
Problems......Page 120
6.1. Introduction and Definition of Stability......Page 122
6.1.1. Discretization and Round-off Error......Page 124
6.1.2. Definition......Page 125
6.2.1. Neumann Method......Page 126
6.2.2. Matrix Method......Page 134
6.3. Implicit versus Explicit Schemes—Stability and Efficiency Considerations......Page 136
Problems......Page 138
7.1. Linear Convection Equation......Page 139
7.1.1. Simple Explicit Schemes......Page 141
7.1.2. Other Schemes......Page 143
7.2. One-Dimensional Heat Equation......Page 146
7.2.1. Simple Explicit Scheme......Page 147
7.2.2. Simple Implicit Scheme......Page 148
7.2.3. Crank-Nicolson Scheme......Page 149
7.3. Burgers and Generic Transport Equations......Page 150
7.4.1. Adams Methods......Page 152
7.4.2. Runge-Kutta Methods......Page 153
7.5. Implicit Schemes: Solution of Tridiagonal Systems by Thomas Algorithm......Page 154
Problems......Page 158
II Methods......Page 161
8.1.1. Elliptic PDE......Page 163
8.1.2. Implicit Integration of Nonsteady Equations......Page 167
8.2. Direct Methods......Page 168
8.2.1. Band-Diagonal and Block-Diagonal Matrices......Page 169
8.3. Iterative Methods......Page 171
8.3.1. General Methodology......Page 172
8.3.2. Jacobi Iterations......Page 173
8.3.3. Gauss-Seidel Algorithm......Page 174
8.3.4. Successive Over- and Underrelaxation......Page 175
8.3.5. Convergence of Iterative Procedures......Page 176
8.3.6. Multigrid Methods......Page 179
8.4. Systems of Nonlinear Equations......Page 182
8.4.1. Newton’s Algorithm......Page 183
8.4.2. Iteration Methods Using Linearization......Page 184
References and Suggested Reading......Page 186
Problems......Page 187
9.1. Introduction......Page 189
9.2.1. Overview and General Comments......Page 190
9.2.2. Explicit MacCormack Method......Page 194
9.2.3. Beam-Warming Method......Page 196
9.2.4. Upwinding......Page 200
9.2.5. Methods for Purely Hyperbolic Systems......Page 203
9.3. Unsteady Conduction Heat Transfer......Page 205
9.3.1. Simple Methods for Multidimensional Heat Conduction......Page 206
9.3.2. Approximate Factorization......Page 207
9.3.3. ADI Method......Page 209
References and Suggested Reading......Page 210
Problems......Page 211
10.1.1. Introduction......Page 214
10.1.2. Role of Pressure......Page 215
10.2. Discretization Approach......Page 216
10.2.1. Colocated and Staggered Grids......Page 218
10.3. Projection Method for Unsteady Flows......Page 223
10.3.1. Explicit Schemes......Page 224
10.3.2. Implicit Schemes......Page 227
10.4. Projection Methods for Steady-State Flows......Page 230
10.4.1. SIMPLE......Page 232
10.4.2. SIMPLEC, SIMPLER, and PISO......Page 234
10.5.1. Vorticity-Streamfunction Formulation for Two-Dimensional Flows......Page 236
References and Suggested Reading......Page 240
Problems......Page 241
III Art of CFD......Page 243
11.1.1. A Few Words About Turbulence......Page 245
11.1.2. Why Is the Computation of Turbulent Flows Difficult?......Page 249
11.1.3. Overview of Numerical Approaches......Page 250
11.2.1. Homogeneous Turbulence......Page 252
11.2.2. Inhomogeneous Turbulence......Page 255
11.3. Reynolds-Averaged Navier-Stokes (RANS) Models......Page 256
11.3.1. Reynolds-Averaged Equations......Page 258
11.3.2. Eddy Viscosity Hypothesis......Page 259
11.3.3. Algebraic Models......Page 260
11.3.4. Two-Equation Models......Page 261
11.3.5. Numerical Implementation of RANS Models......Page 264
11.4. Large-Eddy Simulation (LES)......Page 267
11.4.1. Filtered Equations......Page 268
11.4.2. Closure Models......Page 271
11.4.3. Implementation of LES in CFD Analysis: Numerical Resolution and Near-Wall Treatment......Page 273
References and Suggested Reading......Page 276
Problems......Page 277
12.1. Introduction: Need for Irregular and Unstructured Grids......Page 279
12.2.1. Generation by Coordinate Transformation......Page 282
12.2.2. Examples......Page 284
12.2.3. Grid Quality......Page 286
12.3. Unstructured Grids......Page 287
12.3.1. Grid Generation......Page 289
12.3.2. Finite Volume Discretization on Unstructured Grids......Page 290
12.3.3. Cell Topology......Page 292
12.3.4. Grid Quality......Page 293
Problems......Page 296
13.1. Overview: Setting and Solving a CFD Problem......Page 298
13.2.1. Errors in CFD Analysis......Page 301
13.2.2. Verification and Validation......Page 308
13.3. Adaptive Grids......Page 311
References and Suggested Reading......Page 313
INDEX......Page 315