Title Details

OLEG ZIKANOV is an Associate Professor in the Department of Mechanical Engineering at the University of Michigan–Dearborn.

PREFACE xv

1 What Is CFD? 1

1.1. Introduction / 1

1.2. Brief History of CFD / 4

1.3. Outline of the Book / 6

References and Suggested Reading / 7

I Fundamentals 9

2 Governing Equations of Fluid Dynamics and Heat Transfer 11

2.1. Preliminary Concepts / 11

2.2. Mass Conservation / 14

2.3. Conservation of Chemical Species / 15

2.4. Conservation of Momentum / 16

2.5. Conservation of Energy / 19

2.6. Equation of State / 21

2.7. Equations in Integral Form / 21

2.8. Equations in Conservation Form / 24

2.9. Equations in Vector Form / 25

2.10. Boundary Conditions / 26

2.10.1. Rigid Wall Boundary Conditions / 27

2.10.2. Inlet and Exit Boundary Conditions / 29

2.10.3. Other Boundary Conditions / 29

References and Suggested Reading / 30

Problems / 30

3 Partial Differential Equations 32

3.1. Model Equations; Formulation of a PDE Problem / 33

3.1.1. Model Equations / 33

3.1.2. Domain, Boundary, and Initial Conditions / 35

3.1.3. Equilibrium and Marching Problems / 36

3.1.4. Examples / 37

3.2. Mathematical Classification of PDE of Second Order / 40

3.2.1. Classification / 40

3.2.2. Hyperbolic Equations / 42

3.2.3. Parabolic Equations / 45

3.2.4. Elliptic Equations / 46

3.3. Numerical Discretization: Different Kinds of CFD / 46

3.3.1. Spectral Methods / 47

3.3.2. Finite Element Methods / 49

3.3.3. Finite Difference and Finite Volume Methods / 49

References and Suggested Reading / 52

Problems / 52

4 Basics of Finite Difference Approximation 55

4.1. Computational Grid / 55

4.1.1. Time Discretization / 55

4.1.2. Space Discretization / 56

4.2. Finite Differences and Interpolation / 57

4.2.1. Approximation of ∂u/∂x / 57

4.2.2. Truncation Error, Consistency, Order of Approximation / 58

4.2.3. Other Formulas for ∂u/∂x: Evaluation of the Order of Approximation / 60

4.2.4. Schemes of Higher Order for First Derivative / 62

4.2.5. Higher-Order Derivatives / 63

4.2.6. Mixed Derivatives / 64

4.2.7. Truncation Error of Linear Interpolation / 66

4.3. Approximation of Partial Differential Equations / 67

4.3.1. Approach and Examples / 67

4.3.2. Interpretation of Truncation Error: Numerical Dissipation and Dispersion / 70

4.3.3. Boundary and Initial Conditions / 73

4.3.4. Consistency of Numerical Approximation / 74

4.3.5. System of Difference Equations / 75

4.3.6. Implicit and Explicit Methods / 76

4.4. Development of Finite Difference Schemes / 78

4.4.1. Taylor Series Expansions / 79

4.4.2. Polynomial Fitting / 82

References and Suggested Reading / 83

Problems / 83

5 Finite Volume Method 86

5.1. Introduction and Integral Formulation / 86

5.1.1. Finite Volume Grid / 87

5.1.2. Global Conservation Property / 89

5.2. Approximation of Integrals / 91

5.2.1. Volume Integrals / 91

5.2.2. Surface Integrals / 92

5.3. Methods of Interpolation / 94

5.3.1. Upwind Interpolation / 95

5.3.2. Linear Interpolation / 96

5.3.3. Upwind Interpolation of Higher Order / 98

5.3.4. Interpolation on Nonorthogonal Grids / 99

5.4. Boundary Conditions / 101

References and Suggested Reading / 102

Problems / 102

6 Stability of Transient Solutions 104

6.1. Introduction and Definition of Stability / 104

6.1.1. Discretization and Round-off Error / 106

6.1.2. Definition / 107

6.2. Stability Analysis / 108

6.2.1. Neumann Method / 108

6.2.2. Matrix Method / 116

6.3. Implicit versus Explicit Schemes—Stability and

Efficiency Considerations / 118

References and Suggested Reading / 120

Problems / 120

7 Application to Model Equations 121

7.1. Linear Convection Equation / 121

7.1.1. Simple Explicit Schemes / 123

7.1.2. Other Schemes / 125

7.2. One-Dimensional Heat Equation / 128

7.2.1. Simple Explicit Scheme / 129

7.2.2. Simple Implicit Scheme / 130

7.2.3. Crank-Nicolson Scheme / 131

7.3. Burgers and Generic Transport Equations / 132

7.4. Method of Lines Approach / 134

7.4.1. Adams Methods / 134

7.4.2. Runge-Kutta Methods / 135

7.5. Implicit Schemes: Solution of Tridiagonal Systems
by Thomas Algorithm / 136

References and Suggested Reading / 140

Problems / 140

II Methods 143

8 Steady-State Problems 145

8.1. Problems Reducible to Matrix Equations / 145

8.1.1. Elliptic PDE / 145

8.1.2. Implicit Integration of Nonsteady Equations / 149

8.2. Direct Methods / 150

8.2.1. Band-Diagonal and Block-Diagonal Matrices / 151

8.2.2. LU Decomposition / 153

8.3. Iterative Methods / 153

8.3.1. General Methodology / 154

8.3.2. Jacobi Iterations / 155

8.3.3. Gauss-Seidel Algorithm / 156

8.3.4. Successive Over- and Underrelaxation / 157

8.3.5. Convergence of Iterative Procedures / 158

8.3.6. Multigrid Methods / 161

8.3.7. Pseudo-transient Approach / 164

8.4. Systems of Nonlinear Equations / 164

8.4.1. Newton’s Algorithm / 165

8.4.2. Iteration Methods Using Linearization / 166

8.4.3. Sequential Solution / 168

References and Suggested Reading / 168

Problems / 169

9 Unsteady Problems of Fluid Flows and Heat Transfer 171

9.1. Introduction / 171

9.2. Compressible Flows / 172

9.2.1. Overview and General Comments / 172

9.2.2. Explicit MacCormack Method / 176

9.2.3. Beam-Warming Method / 178

9.2.4. Upwinding / 182

9.2.5. Methods for Purely Hyperbolic Systems / 185

9.3. Unsteady Conduction Heat Transfer / 187

9.3.1. Simple Methods for Multidimensional Heat Conduction / 188

9.3.2. Approximate Factorization / 189

9.3.3. ADI Method / 191

References and Suggested Reading / 192

Problems / 193

10 Incompressible Flows 196

10.1. General Considerations / 196

10.1.1. Introduction / 196

10.1.2. Role of Pressure / 197

10.2. Discretization Approach / 198

10.2.1. Colocated and Staggered Grids / 200

10.3. Projection Method for Unsteady Flows / 205

10.3.1. Explicit Schemes / 206

10.3.2. Implicit Schemes / 209

10.4. Projection Methods for Steady-State Flows / 212

10.4.1. SIMPLE / 214

10.4.2. SIMPLEC, SIMPLER, and PISO / 216

10.5. Other Methods / 218

10.5.1. Vorticity-Streamfunction Formulation for Two-Dimensional Flows / 218

10.5.2. Artificial Compressibility / 222

References and Suggested Reading / 222

Problems / 223

III Art of CFD 225

11 Turbulence 227

11.1. Introduction / 227

11.1.1. A Few Words About Turbulence / 227

11.1.2. Why Is the Computation of Turbulent Flows Difficult? / 231

11.1.3. Overview of Numerical Approaches / 232

11.2. Direct Numerical Simulation (DNS) / 234

11.2.1. Homogeneous Turbulence / 234

11.2.2. Inhomogeneous Turbulence / 237

11.3. Reynolds-Averaged Navier-Stokes (RANS) Models / 238

11.3.1. Reynolds-Averaged Equations / 240

11.3.2. Eddy Viscosity Hypothesis / 241

11.3.3. Algebraic Models / 242

11.3.4. Two-Equation Models / 243

11.3.5. Numerical Implementation of RANS Models / 246

11.4. Large-Eddy Simulation (LES) / 249

11.4.1. Filtered Equations / 250

11.4.2. Closure Models / 253

11.4.3. Implementation of LES in CFD Analysis: Numerical Resolution and Near-Wall Treatment / 255

References and Suggested Reading / 258

Problems / 259

12 Computational Grids 261

12.1. Introduction: Need for Irregular and Unstructured Grids / 261

12.2. Irregular Structured Grids / 264

12.2.1. Generation by Coordinate Transformation / 264

12.2.2. Examples / 266

12.2.3. Grid Quality / 268

12.3. Unstructured Grids / 269

12.3.1. Grid Generation / 271

12.3.2. Finite Volume Discretization on Unstructured Grids / 272

12.3.3. Cell Topology / 274

12.3.4. Grid Quality / 275

References and Suggested Reading / 278

Problems / 278

13 Conducting CFD Analysis 280

13.1. Overview: Setting and Solving a CFD Problem / 280

13.2. Errors and Uncertainty / 283

13.2.1. Errors in CFD Analysis / 283

13.2.2. Verification and Validation / 290

13.3. Adaptive Grids / 293

References and Suggested Reading / 295

INDEX 297