# Essential Computational Fluid Dynamics

2. Edition September 2019

384 Pages, Hardcover*Textbook*

**978-1-119-47462-3**

Provides a clear, concise, and self-contained introduction to Computational Fluid Dynamics (CFD)

This comprehensively updated new edition covers the fundamental concepts and main methods of modern Computational Fluid Dynamics (CFD). With expert guidance and a wealth of useful techniques, the book offers a clear, concise, and accessible account of the essentials needed to perform and interpret a CFD analysis.

The new edition adds a plethora of new information on such topics as the techniques of interpolation, finite volume discretization on unstructured grids, projection methods, and RANS turbulence modeling. The book has been thoroughly edited to improve clarity and to reflect the recent changes in the practice of CFD. It also features a large number of new end-of-chapter problems.

All the attractive features that have contributed to the success of the first edition are retained by this version. The book remains an indispensable guide, which:

* Introduces CFD to students and working professionals in the areas of practical applications, such as mechanical, civil, chemical, biomedical, or environmental engineering

* Focuses on the needs of someone who wants to apply existing CFD software and understand how it works, rather than develop new codes

* Covers all the essential topics, from the basics of discretization to turbulence modeling and uncertainty analysis

* Discusses complex issues using simple worked examples and reinforces learning with problems

* Is accompanied by a website hosting lecture presentations and a solution manual

Essential Computational Fluid Dynamics, Second Edition is an ideal textbook for senior undergraduate and graduate students taking their first course on CFD. It is also a useful reference for engineers and scientists working with CFD applications.

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

Bibliography 7

I Fundamentals 9

2 Governing Equations of Fluid Dynamics and Heat Transfer 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 of Integral Form 22

2.5. Equations in Conservation Form 25

2.6. Equations in Vector Form 26

2.7. Boundary Conditions 27

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 32

2.8.2. Equilibrium and Marching Problems 33

Bibliography 34

Problems 34

3 Partial Different Equations 37

3.1. Model Equations: Formulation of a PDE Problem 38

3.1.1. Model Equations 38

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

3.1.3. Examples 42

3.2. Mathematical Classification of PDEs of Second Order 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

Bibliography 59

Problems 59

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 au/ax 65

4.2.2. Truncation Error, Consistency, and Order of Approximation 66

4.2.3. Other Formulas for au/ax: Evaluation of the Order of Approximation 69

4.2.4. Schemes of Higher Order for First Derivative 71

4.2.5. Higher-Order Derivatives 71

4.2.6. Mixed Derivatives 73

4.2.7. Finite Difference Approximation on Nonuniform Grids 74

4.3. Development of Finite Difference Schemes 77

4.3.1. Taylor Series Expansions 77

4.3.2. Polynomial Fitting 79

4.3.3. Development on Nonuniform Grids 80

4.4. Finite Difference Approximation of Partial Differential Equations 81

4.4.1. Approach and Examples 81

4.4.2. Boundary and Initial Conditions 85

4.4.3. Difference Molecule and 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

Bibliography 98

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 115

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 122

Bibliography 123

Problems 123

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 131

6.2. Stability Analysis 132

6.2.1. Neumann Method 132

6.2.2. Matrix Method 140

6.3. Implicit Versus Explicit Schemes - Stability and Efficiency Considerations 142

Bibliography 144

Problems 144

II Methods 147

7 Application to Model Equations 149

7.1. Linear Convection Equation 150

7.1.1. Simple Explicit Schemes 151

7.1.2. Simple Implicit Scheme 154

7.1.3. Leapfrog Scheme 155

7.1.4. Lax-Wendroff Scheme 156

7.1.5. MacCormack Scheme 157

7.2. One-Dimensional Heat Equation 157

7.2.1. Simple Explicit Scheme 157

7.2.2. Simple Implicit Scheme 159

7.2.3. Crank-Nicolson Scheme 159

7.3. Burgers and Generic Transport Equations 161

7.4. Method of Lines 162

7.4.1. Adams Methods 163

7.4.2. Runge-Kutta Methods 164

7.5. Solution of Tridiagonal Systems by Thomas Algorithm 165

Bibliography 169

Problems 169

8 Steady-State Problems 173

8.1. Problems Reducible to Matrix Equations 173

8.1.1. Elliptic PDE 174

8.1.2. Marching Problems Solved by Implicit Schemes 177

8.1.3. Structure of Matrices 179

8.2. Direct Methods 180

8.2.1. Cyclic Reduction Algorithm 181

8.2.2. Thomas Algorithm for Block-Tridiagonal Matrices 184

8.2.3. LU Decomposition 185

8.3. Iterative Methods 186

8.3.1. General Methodology 187

8.3.2. Jacobi Iterations 188

8.3.3. Gauss-Seidel Algorithm 189

8.3.4. Successive Over- and Underrelaxation 190

8.3.5. Convergence of Iterative Procedures 191

8.3.6. Multigrid Methods 194

8.3.7. Pseudo-transient Approach 197

8.4. Systems of Nonlinear Equations 197

8.4.1. Newton's Algorithm 198

8.4.2. Iteration Methods Using Linearization 199

8.4.3. Sequential Solution 201

8.5. Computational Performance 202

Bibliography 203

Problems 203

9 Unsteady Compressible Fluid Flows and Conduction Heat Transfer 207

9.1. Introduction 207

9.2. Compressible Flows 208

9.2.1. Equations, Mathematical Classification, and General Comments 208

9.2.2. MacCormack Scheme 212

9.2.3. Beam-Warming Scheme 214

9.2.4. Upwinding 218

9.2.5. Methods for Purely Hyperbolic Systems: TVD Schemes 220

9.3. Unsteady Conduction Heat Transfer 223

9.3.1. Overview 223

9.3.2. Simple Methods for Multidimensional Heat Conduction 223

9.3.3. Approximate Factorization 225

9.3.4. ADI Method 227

Bibliography 228

Problems 229

10 Incompressible Flows 233

10.1. General Considerations 233

10.1.1. Introduction 233

10.1.2. Role of Pressure 234

10.2. Discretization Approach 236

10.2.1. Conditions for Conservation of Mass by Numerical Solution 237

10.2.2. Colocated and Staggered Grids 238

10.3. Projection Method for Unsteady Flows 243

10.3.1. Explicit Schemes 244

10.3.2. Implicit Schemes 247

10.4. Projection Methods for Steady-State Flows 250

10.4.1. SIMPLE 252

10.4.2. SIMPLEC and SIMPLER 254

10.4.3. PISO 256

10.5. Other Methods 257

10.5.1. Vorticity-Streamfunction Formulation for Two-Dimensional Flows 257

10.5.2. Artificial Compressibility 261

Bibliography 261

Problems 262

III Art of CFD 265

11 Turbulence 267

11.1. Introduction 267

11.1.1. A Few Words About Turbulence 268

11.1.2. Why is the Computation of Turbulent Flows Difficult? 271

11.1.3. Overview of Numerical Approaches 273

11.2. Direct Numerical Simulation (DNS) 275

11.2.1. Homogeneous Turbulence 275

11.2.2. Inhomogeneous Turbulence 278

11.3. Reynolds-Averaged Navier-Stokes (RANS) Models 279

11.3.1. Mean Flow and Fluctuations 280

11.3.2. Reynolds-Averaged Equations 281

11.3.3. Reynolds Stresses and Turbulent Kinetic Energy 282

11.3.4. Eddy Viscosity Hypothesis 284

11.3.5. Closure Models 285

11.3.6. Algebraic Models 286

11.3.7. One-Equation Models 287

11.3.8. Two-Equation Models 289

11.3.9. RANS and URANS 291

11.3.10. Models of Turbulent Scalar Transport 292

11.3.11. Numerical Implementation of RANS Models 294

11.4. Large Eddy Simulation (LES) 297

11.4.1. Filtered Equations 298

11.4.2. Closure Models 301

11.4.3. Implementation of LES in CFD Analysis: Numerical Resolution and Near-Wall Treatment 304

Bibliography 307

Problems 309

12 Computational Grids 313

12.1. Introduction: Need for Irregular and Unstructured Grids 313

12.2. Irregular Structured Grids 316

12.2.1. Generation by Coordinate Transformation 316

12.2.2. Examples 319

12.2.3. Grid Quality 321

12.3. Unstructured Grids 322

12.3.1. Grid Generation 325

12.3.2. Cell Topology 325

12.3.3. Grid Quality 326

12.4. Adaptive Grids 329

Bibliography 331

Problems 332

13 Conducting CFD Analysis 335

13.1. Overview: Setting and Solving a CFD Problem 335

13.2. Errors and Uncertainty 339

13.2.1. Errors in CFD Analysis 339

13.2.2. Verification and Validation 346

Bibliography 349

Problems 349

Index 351