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.

Table of Contents:
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