Title Details

Preface x

1 What is CFD? 1

1.1 Introduction 1

1.2 Brief History of CFD 4

1.3 Outline of the Book 5

References and suggested reading 7

I Fundamentals 9

2 Governing Equations 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 in Integral Form 21

2.5 Conservation Form 24

2.6 Vector Form 26

2.7 Boundary Conditions 26

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 31

2.8.2 Equilibrium and Marching Problems 32

References and Suggested Reading 33

Problems 34

3 Partial Differential Equations 37

3.1 Formulation of a PDE problem 38

3.1.1 Model Equations 38

3.1.2 Domain, Boundary and Initial Conditions, Well-posed PDE Problem 40

3.1.3 Examples 42

3.2 Mathematical Classification 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

References and suggested reading 59

Problems 60

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, 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 72

4.2.6 Mixed Derivatives 73

4.2.7 Finite Difference Approximation on Non-uniform Grids 75

4.3 Development of Finite Difference Schemes 77

4.3.1 Taylor Series Expansions 77

4.3.2 Polynomial Fitting 80

4.3.3 Development on Non-uniform Grids 80

4.4 Approximation of Partial Differential Equations 82

4.4.1 Approach and Examples 82

4.4.2 Boundary and Initial Conditions 85

4.4.3 Difference Molecule, 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

References and suggested reading 97

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 116

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 123

References and suggested reading 124

Problems 124

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 130

6.2 Stability Analysis 132

6.2.1 Neumann Method 132

6.2.2 Matrix Method 139

6.3 Implicit versus Explicit Schemes 141

References and suggested reading 143

Problems 143

7 Application to Model Equations 145

7.1 Linear Convection Equation 145

7.1.1 Simple Explicit Schemes 146

7.1.2 Simple Implicit Scheme 150

7.1.3 Leapfrog Scheme 150

7.1.4 Lax-Wendro Scheme 152

7.1.5 MacCormack Scheme 152

7.2 One-dimensional Heat Equation 152

7.2.1 Simple Explicit Scheme 153

7.2.2 Simple Implicit Scheme 154

7.2.3 Crank-Nicolson Scheme 155

7.3 Burgers and Generic Transport Equations 156

7.4 Method of Lines 158

7.4.1 Adams Methods 159

7.4.2 Runge-Kutta Methods 159

7.5 Solution of Tridiagonal Systems by Thomas Algorithm 160

References and suggested reading 164

Problems 164

II Methods 167

8 Steady-state Problems 169

8.1 Problems Reducible to Matrix Equations 169

8.1.1 Elliptic PDE 169

8.1.2 Marching Problems Solved by Implicit Schemes 174

8.1.3 Structure of Matrices 175

8.2 Direct Methods 176

8.2.1 Cyclic Reduction Algorithm 177

8.2.2 Thomas Algorithm for Block-tridiagonal Matrices 180

8.2.3 LU Decomposition 181

8.3 Iterative Methods 182

8.3.1 General Methodology 183

8.3.2 Jacobi Iterations 184

8.3.3 Gauss-Seidel Algorithm 184

8.3.4 Successive Over- and Underrelaxation 186

8.3.5 Convergence of Iterative Procedures 187

8.3.6 Multigrid Methods 189

8.3.7 Pseudo-transient Approach 192

8.4 Systems of Nonlinear Equations 193

8.4.1 Newton's Algorithm 194

8.4.2 Iteration Methods Using Linearization 195

8.4.3 Sequential Solution 196

8.5 Computational Performance 197

References and suggested reading 199

Problems 199

9 Unsteady Flows and Heat Transfer 203

9.1 Introduction 203

9.2 Compressible Flows 204

9.2.1 Equations, Mathematical Classification, and General Comments 204

9.2.2 MacCormack Scheme 208

9.2.3 Beam-Warming Scheme 210

9.2.4 Upwinding 213

9.2.5 Methods for Purely Hyperbolic Systems; TVD Schemes 216

9.3 Unsteady Conduction Heat Transfer 218

9.3.1 Overview 218

9.3.2 Simple Methods for Multidimensional Heat Conduction 219

9.3.3 Approximate Factorization 220

9.3.4 ADI Method 221

References and suggested reading 223

Problems 224

10 Incompressible Flows 227

10.1 General Considerations 227

10.1.1 Introduction 227

10.1.2 Role of Pressure 228

10.2 Discretization Approach 230

10.2.1 Conditions for Conservation of Mass by Numerical Solution 230

10.2.2 Colocated and Staggered Grids 231

10.3 Projection Method for Unsteady Flows 237

10.3.1 Explicit Schemes 238

10.3.2 Implicit Schemes 241

10.4 Projection Methods for Steady-State Flows 244

10.4.1 SIMPLE 246

10.4.2 SIMPLEC and SIMPLER 248

10.4.3 PISO 250

10.5 Other Methods 251

10.5.1 Vorticity-Stream function Formulation for Two-dimensional Flows 251

10.5.2 Artificial Compressibility 255

References and suggested reading 255

Problems 256

III Art of CFD 259

11 Turbulence 261

11.1 Introduction 261

11.1.1 A Few Words About Turbulence 261

11.1.2 Why is the Computation of Turbulent Flows Difficult? 265

11.1.3 Overview of Numerical Approaches 267

11.2 DNS 269

11.2.1 Homogeneous Turbulence 269

11.2.2 Inhomogeneous Turbulence 272

11.3 RANS 273

11.3.1 Mean Flow and Fluctuations 274

11.3.2 Reynolds-Averaged Equations 275

11.3.3 Reynolds Stresses and Turbulent Kinetic Energy 276

11.3.4 Eddy Viscosity Hypothesis 277

11.3.5 Closure Models 279

11.3.6 Algebraic Models 280

11.3.7 One-equation Models 281

11.3.8 Two-equation Models 283

11.3.9 RANS and URANS 285

11.3.10Models of Turbulent Scalar Transport 286

11.3.11 Numerical Implementation of RANS Models 287

11.4 LES 291

11.4.1 Filtered Equations 291

11.4.2 Closure Models 295

11.4.3 Implementation of LES in CFD Analysis: Numerical Resolution and Near-Wall Treatment 297

References and suggested reading 301

Problems 303

12 Computational Grids 307

12.1 Need for Irregular and Unstructured Grids 307

12.2 Irregular Structured Grids 311

12.2.1 Generation by Coordinate Transformation 311

12.2.2 Examples 313

12.2.3 Grid Quality 315

12.3 Unstructured Grids 316

12.3.1 Grid Generation 319

12.3.2 Cell Topology 320

12.3.3 Grid Quality 320

12.4 Adaptive Grids 324

References and suggested reading 326

Problems 326

13 Conducting CFD Analysis 329

13.1 Setting and Solving a CFD Problem 329

13.2 Errors and Uncertainty 332

13.2.1 Errors in CFD Analysis 333

13.2.2 Verification and Validation 339

References and suggested reading 343

Problems 344