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# Essential Computational Fluid Dynamics

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# Essential Computational Fluid Dynamics

ISBN: 978-1-118-15113-6 August 2011 320 Pages

## Description

This book serves as a complete and self-contained introduction to the principles of Computational Fluid Dynamic (CFD) analysis. It is deliberately short (at approximately 300 pages) and can be used as a text for the first part of the course of applied CFD followed by a software tutorial. The main objectives of this non-traditional format are: 1) To introduce and explain, using simple examples where possible, the principles and methods of CFD analysis and to demystify the `black box’ of a CFD software tool, and 2) To provide a basic understanding of how CFD problems are set and which factors affect the success and failure of the analysis. Included in the text are the mathematical and physical foundations of CFD, formulation of CFD problems, basic principles of numerical approximation (grids, consistency, convergence, stability, and order of approximation, etc), methods of discretization with focus on finite difference and finite volume techniques, methods of solution of transient and steady state problems, commonly used numerical methods for heat transfer and fluid flows, plus a brief introduction into turbulence modeling.

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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