Essential Computational Fluid Dynamics

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

I Fundamentals 9

2 Governing Equations of Fluid Dynamics and Heat Transfer 11
2.1. Preliminary Concepts 11
2.2. Conservation Laws 14
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
2.8. Dimensionality and Time Dependence 31

3 Partial Different Equations 37
3.1. Model Equations: Formulation of a PDE Problem 38
3.2. Mathematical Classification of PDEs of Second Order 45
3.3. Numerical Discretization: Different Kinds of CFD 53

4 Finite Difference Method 63
4.1. Computational Grid 63
4.2. Finite Difference Approximation 65
4.3. Development of Finite Difference Schemes 77
4.4. Finite Difference Approximation of Partial Differential Equations 81

5 Finite Volume Schemes 103
5.1. Introduction and General Formulation 103
5.2. Approximation of Integrals 109
5.3. Methods of Interpolation 112
5.4. Finite Volume Method on Unstructured Grids 119
5.5. Implementation of Boundary Conditions 122

6 Numerical Stability for Marching Problems 127
6.1. Introduction and Definition of Stability 127
6.2. Stability Analysis 132
6.3. Implicit Versus Explicit Schemes â" Stability and Efficiency Considerations 142

II Methods 147

7 Application to Model Equations 149
7.1. Linear Convection Equation 150
7.2. One-Dimensional Heat Equation 157
7.3. Burgers and Generic Transport Equations 161
7.4. Method of Lines 162
7.5. Solution of Tridiagonal Systems by Thomas Algorithm 165

8 Steady-State Problems 173
8.1. Problems Reducible to Matrix Equations 173
8.2. Direct Methods 180
8.3. Iterative Methods 186
8.4. Systems of Nonlinear Equations 197
8.5. Computational Performance 202

9 Unsteady Compressible Fluid Flows and Conduction Heat Transfer 207
9.1. Introduction 207
9.2. Compressible Flows 208
9.3. Unsteady Conduction Heat Transfer 223

10 Incompressible Flows 233
10.1. General Considerations 233
10.2. Discretization Approach 236
10.3. Projection Method for Unsteady Flows 243
10.4. Projection Methods for Steady-State Flows 250
10.5. Other Methods 257

III Art of CFD 265

11 Turbulence 267
11.1. Introduction 267
11.2. Direct Numerical Simulation (DNS) 275
11.3. Reynolds-Averaged Navierâ"Stokes (RANS) Models 279
11.4. Large Eddy Simulation (LES) 297

12 Computational Grids 313
12.1. Introduction: Need for Irregular and Unstructured Grids 313
12.2. Irregular Structured Grids 316
12.3. Unstructured Grids 322
12.4. Adaptive Grids 329

13 Conducting CFD Analysis 335
13.1. Overview: Setting and Solving a CFD Problem 335
13.2. Errors and Uncertainty 339

Bibliography 349
Problems 349
Index 351