Computational Fluid Mechanics and Heat Transfer

Description

Cover Half Title Series Page Title Page Copyright Page Table of Contents Preface to the Fourth Edition Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Authors Part I: Fundamentals 1. Introduction 1.1 General Remarks 1.2 Comparison of Experimental, Theoretical, and Computational Approaches 1.3 Historical Perspective 2. Partial Differential Equations 2.1 Introduction 2.1.1 Partial Differential Equations 2.2 Physical Classicfiation 2.2.1 Equilibrium Problems 2.2.2 Eigenvalue Problems 2.2.3 Marching Problems 2.3 Mathematical Classicfiation 2.3.1 Hyperbolic PDEs 2.3.2 Parabolic PDEs 2.3.3 Elliptic PDEs 2.4 Well-Posed Problem 2.5 Systems of Partial Differential Equations 2.6 Other PDES of Interest Problems 3. Basics of Discretization Methods 3.1 Introduction 3.2 Finite Differences 3.3 Difference Representation of Partial Differential Equations 3.3.1 Truncation Error 3.3.2 Round-Off and Discretization Errors 3.3.3 Consistency 3.3.4 Stability 3.3.5 Convergence for Marching Problems 3.3.6 Comment on Equilibrium Problems 3.3.7 Conservation Form and Conservative Property 3.4 Further Examples of Methods for Obtaining Finite-Difference Equations 3.4.1 Use of Taylor Series 3.4.2 Use of Polynomial Fitting 3.4.3 Integral Method 3.5 Finite-Volume Method 3.6 Treatment of Diffusion Terms in General Nonorthogonal Meshes 3.6.1 Calculating the Gradient Vector 3.6.2 Discretizing the Laplace Operator 3.7 Finite Element Method (FEM) 3.7.1 Approximating Functions 3.7.2 Selection of FEM Basis Functions 3.7.3 Weak Form 3.7.4 Residual 3.7.5 Projection, Matrix Expressions, and Implementation 3.7.6 Numerical Integration, Quadrature 3.7.7 Continuous and Discontinuous Galerkin Methods 3.8 Introduction to the Use of Irregular Meshes 3.8.1 Irregular Mesh Due to Shape of a Boundary 3.8.2 Irregular Mesh Not Caused by Shape of a Boundary 3.8.3 Concluding Remarks 3.9 Stability Considerations 3.9.1 Fourier or Von Neumann Analysis 3.9.2 Stability Analysis for Systems of Equations Problems 4. Application of Numerical Methods to Selected Model Equations 4.1 Wave Equation 4.1.1 Euler Explicit Methods 4.1.2 Upstream (First-Order Upwind or Windward) Differencing Method 4.1.3 Lax Method 4.1.4 Euler Implicit Method 4.1.5 Leap Frog Method 4.1.6 Lax–Wendroff Method 4.1.7 Two-Step Lax–Wendroff Method 4.1.8 Maccormack Method 4.1.9 Second-Order Upwind Method 4.1.10 Time-Centered Implicit Method (Trapezoidal Differencing Method) 4.1.11 Rusanov (Burstein-Mirin) Method 4.1.12 Warming–Kutler–Lomax Method 4.1.13 Runge–Kutta Methods 4.1.14 Discontinuous Galerkin (DG) Method 4.1.14.1 Time Advancement 4.1.14.2 Stability and Accuracy 4.1.14.3 Filters and Limiters 4.1.14.4 Basis Functions—Condition Number 4.1.14.5 Numerical Implementation 4.2 Heat Equation 4.2.1 Simple Explicit Method 4.2.2 Richardson’s Method 4.2.3 Simple Implicit (Laasonen) Method 4.2.4 Crank–Nicolson Method 4.2.5 Combined Method A 4.2.6 Combined Method B 4.2.7 Dufort–Frankel Method 4.2.8 Keller Box and Modifies Box Methods 4.2.9 Methods for the Two-Dimensional Heat Equation 4.2.10 ADI Methods 4.2.11 Splitting or Fractional-Step Methods 4.2.12 ADE Methods 4.2.13 Hopscotch Method 4.2.14 Discontinuous Galerkin Method 4.2.15 Heat Conduction at Material Interfaces 4.2.15.1 Conduction Across a Material Interface 4.2.15.2 Contact Resistance 4.2.16 Mass Transport 4.2.17 Additional Comments 4.3 Laplace’s Equation 4.3.1 Finite-Difference Representations for Laplace’s Equation 4.3.1.1 Five-Point Formula 4.3.1.2 Nine-Point Formula 4.3.1.3 Residual Form of the Difference Equations 4.3.2 Simple Example for Laplace’s Equation 4.3.3 Direct Methods for Solving Systems of Linear Algebraic Equations 4.3.3.1 Cramer’s Rule 4.3.3.2 Gaussian Elimination 4.3.3.3 Thomas Algorithm 4.3.3.4 Advanced Direct Methods 4.3.4 Iterative Methods for Solving Systems of Linear Algebraic Equations 4.3.4.1 Gauss–Seidel Iteration 4.3.4.2 Sufficient Condition for Convergence of the Gauss–Seidel Procedure 4.3.4.3 Successive Overrelaxation 4.3.4.4 Coloring Schemes 4.3.4.5 Block-Iterative Methods 4.3.4.6 SOR By Lines 4.3.4.7 ADI Methods 4.3.4.8 Strongly Implicit Methods 4.3.4.9 Krylov Subspace Methods (GMRES) 4.3.5 Multigrid Method 4.3.5.1 Example Using Multigrid 4.3.5.2 Multigrid for Nonlinear Equations 4.4 Burgers’ Equation (Inviscid) 4.4.1 Lax Method 4.4.2 Lax–Wendroff Method 4.4.3 Maccormack Method 4.4.4 Rusanov (Burstein–Mirin) Method 4.4.5 Warming–Kutler–Lomax Method 4.4.6 Tuned Third-Order Methods 4.4.7 Implicit Methods 4.4.8 Godunov Scheme 4.4.9 Roe Scheme 4.4.10 Enquist–Osher Scheme 4.4.11 Higher-Order Upwind Schemes 4.4.12 TVD Schemes 4.5 Burgers’ Equation (Viscous) 4.5.1 FTCS Method 4.5.2 Leap Frog/Dufort–Frankel Method 4.5.3 Brailovskaya Method 4.5.4 Allen–Cheng Method 4.5.5 Lax–Wendroff Method 4.5.6 Maccormack Method 4.5.7 Briley–Mcdonald Method 4.5.8 Time-Split Maccormack Method 4.5.9 ADI Methods 4.5.10 Predictor–Corrector, Multiple-Iteration Method 4.5.11 Roe Method 4.6 Concluding Remarks Problems Part II: Application of Numerical Methods to the Equations of Fluid Mechanics and Heat Transfer 5. Governing Equations of Fluid Mechanics and Heat Transfer 5.1 Fundamental Equations 5.1.1 Continuity Equation 5.1.2 Momentum Equation 5.1.3 Energy Equation 5.1.4 Equation of State, Thermodynamic and Transport Properties 5.1.5 Chemically Reacting Flows 5.1.6 Magnetohydrodynamic Flows 5.1.7 Vector Form of Equations 5.1.8 Nondimensional Form of Equations 5.1.9 Orthogonal Curvilinear Coordinates 5.2 Averaged Equations for Turbulent Flows 5.2.1 Background 5.2.2 Reynolds Averaged Navier–Stokes Equations 5.2.3 Reynolds Form of the Continuity Equation 5.2.4 Reynolds Form of the Momentum Equations 5.2.5 Reynolds Form of the Energy Equation 5.2.6 Comments on the Reynolds Equations 5.2.7 Filtered Navier–Stokes Equations for Large-Eddy Simulation 5.3 Boundary-Layer Equations 5.3.1 Background 5.3.2 Boundary-Layer Approximation for Steady Incompressible Flow 5.3.3 Boundary-Layer Equations for Compressible Flow 5.4 Introduction to Turbulence Modeling 5.4.1 Background 5.4.2 Modeling Terminology 5.4.3 Simple Algebraic or Zero-Equation Models 5.4.4 One-Half-Equation Models 5.4.5 One-Equation Models 5.4.6 One-and-One-Half- and Two-Equation Models 5.4.7 Reynolds Stress Models 5.4.8 Subgrid-Scale Models for Large-Eddy Simulation 5.4.9 Comments on the Implementation of DES 5.4.10 Closing Comment on Turbulence Modeling 5.5 Euler Equations 5.5.1 Continuity Equation 5.5.2 Inviscid Momentum Equations 5.5.3 Inviscid Energy Equations 5.5.4 Some Useful Thermodynamic Relations 5.5.5 Vector Form of Euler Equations 5.5.6 Quasi-One-Dimensional Form of the Euler Equations 5.5.6.1 Conservation of Mass 5.5.6.2 Conservation of Momentum 5.5.6.3 Conservation of Energy 5.5.7 Simplified Forms of Euler Equations 5.5.8 Shock Equations 5.6 Multi-Phase and Multi-Fluid Flows 5.6.1 Modeling of Free Surface Flows 5.6.2 Modeling Flows with Droplets 5.7 Equations of Convection Heat Transfer 5.7.1 Boussinesq Approximation in Free and Mixed Convection 5.7.2 Phase Change 5.8 Transformation of Governing Equations 5.8.1 Simple Transformations 5.8.2 Generalized Transformation 5.9 Finite-Volume Formulation 5.9.1 Two-Dimensional Finite-Volume Method 5.9.2 Three-Dimensional Finite-Volume Method Problems 6. Numerical Methods for Inviscid Flow Equations 6.1 Introduction 6.2 Method of Characteristics 6.2.1 Linear Systems of Equations 6.2.2 Nonlinear Systems of Equations 6.3 Classical Shock-Capturing Methods 6.4 Flux Splitting Schemes 6.4.1 Steger–Warming Splitting 6.4.2 Van Leer Flux Splitting 6.4.3 Other Flux Splitting Schemes 6.4.4 Application for Arbitrarily Shaped Cells 6.5 Flux-Difference Splitting Schemes 6.5.1 Roe Scheme 6.5.2 Second-Order Schemes 6.6 Multidimensional Case in a General Coordinate System 6.7 Boundary Conditions for the Euler Equations 6.8 Methods for Solving the Potential Equation 6.8.1 Treatment of the Time Derivatives 6.8.2 Spatial Derivatives 6.9 Transonic Small-Disturbance Equations 6.10 Panel Method, Laplace’s Equation Problems 7. Numerical Methods for Boundary-Layer-Type Equations 7.1 Introduction 7.2 Brief Comparison of Prediction Methods 7.3 Finite-Difference Methods for Two-Dimensional or Axisymmetric Steady External Flows 7.3.1 Generalized Form of the Equations 7.3.2 Example of a Simple Explicit Procedure 7.3.2.1 Alternative Formulation for Explicit Method 7.3.3 Crank–Nicolson and Fully Implicit Methods 7.3.3.1 Lagging the Coefficients 7.3.3.2 Simple Iterative Update of Coefficients 7.3.3.3 Use of Newton Linearization to Iteratively Update Coefficients 7.3.3.4 Newton Linearization with Coupling 7.3.3.5 Extrapolating the Coefficients 7.3.3.6 Recommendation 7.3.3.7 Warning on Stability 7.3.3.8 Closing Comment on Crank–Nicolson and Fully Implicit Methods 7.3.4 Dufort–Frankel Method 7.3.5 Box Method 7.3.6 Other Methods 7.3.7 Coordinate Transformations for Boundary Layers 7.3.7.1 Analytical Transformation Approach 7.3.7.2 Generalized Coordinate Approach 7.3.8 Special Considerations for Turbulent Flows 7.3.8.1 Use of Wall Functions 7.3.8.2 Use of Unequal Grid Spacing 7.3.8.3 Use of Coordinate Transformations 7.3.9 Example Applications 7.3.10 Closure 7.4 Inverse Methods, Separated Flows, and Viscous–Inviscid Interaction 7.4.1 Introduction 7.4.2 Comments on Computing Separated Flows Using the Boundary-Layer Equations 7.4.3 Inverse Finite-Difference Methods 7.4.3.1 Inverse Method A 7.4.3.2 Inverse Method B 7.4.4 Viscous–Inviscid Interaction 7.5 Methods for Internal Flows 7.5.1 Introduction 7.5.2 Coordinate Transformation for Internal Flows 7.5.3 Computational Strategies for Internal Flows 7.5.3.1 Variable Secant Iteration 7.5.3.2 Lagging the Pressure Adjustment 7.5.3.3 Newton’s Method 7.5.3.4 Treating the Pressure Gradient as a Dependent Variable 7.5.4 Additional Remarks 7.6 Application to Free-Shear Flows 7.7 Three-Dimensional Boundary Layers 7.7.1 Introduction 7.7.2 Equations 7.7.3 Comments on Solution Methods for Three-Dimensional Flows 7.7.3.1 Crank–Nicolson Scheme 7.7.3.2 Krause Zigzag Scheme 7.7.3.3 Some Variations 7.7.3.4 Inverse Methods and Viscous–Inviscid Interaction 7.7.4 Example Calculations 7.7.5 Additional Remarks 7.8 Unsteady Boundary Layers Problems 8. Numerical Methods for the “Parabolized” Navier–Stokes Equations 8.1 Introduction 8.2 Thin-Layer Navier–Stokes Equations 8.3 “Parabolized” Navier–Stokes Equations 8.3.1 Derivation of PNS Equations 8.3.2 Streamwise Pressure Gradient 8.3.2.1 Iterative PNS Methods 8.3.2.2 Detecting Upstream Inufluence Regions 8.3.3 Numerical Solution of PNS Equations 8.3.3.1 Early Schemes 8.3.3.2 Beam–Warming Scheme 8.3.3.3 Roe Scheme 8.3.3.4 Other Schemes 8.3.3.5 Advanced Schemes 8.3.4 Applications of PNS Equations 8.4 Parabolized and Partially Parabolized Navier–Stokes Procedures for Subsonic Flows 8.4.1 Fully Parabolic Procedures 8.4.2 Parabolic Procedures for 3-D Free-Shear and Other Flows 8.4.3 Partially Parabolized (Multiple Space-Marching) Model 8.4.3.1 Pressure-Correction PPNS Schemes 8.4.3.2 Coupled PPNS Schemes 8.5 Viscous Shock-Layer Equations 8.6 “Conical” Navier–Stokes Equations Problems 9. Numerical Methods for the Navier–Stokes Equations 9.1 Introduction 9.2 Compressible Navier–Stokes Equations 9.2.1 Explicit Maccormack Method 9.2.2 Other Explicit Methods 9.2.3 Beam–Warming Scheme 9.2.4 Other Implicit Methods 9.2.5 Upwind Methods 9.2.6 Compressible Navier–Stokes Equations at Low Speeds 9.3 Incompressible Navier–Stokes Equations 9.3.1 Vorticity–Stream Function Approach 9.3.2 Primitive-Variable Approach 9.3.2.1 General 9.3.2.2 Coupled Approach: The Method of Artificial Compressibility 9.3.2.3 Coupled Approach: Space Marching 9.3.2.4 Pressure-Correction Approach: General 9.3.2.5 Pressure-Correction Approach: Marker-and-Cell Method 9.3.2.6 Pressure-Correction Approach: Projection (Fractional-Step) Methods 9.3.2.7 Pressure-Correction Approach: SIMPLE Family of Methods 9.3.2.8 Pressure-Correction Approach: SIMPLE on Nonstaggered Grids 9.3.2.9 Pressure-Correction Approach: Pressure Implicit with Splitting of Operators (PISO) Method 9.3.3 Example Problems 9.3.3.1 Driven Cavity Flow in Two Dimensions 9.3.3.2 Channel Flow in Two Dimensions 9.3.3.3 Two-Dimensional Backward Facing Step 9.3.3.4 Two-Dimensional Square Cavity with Natural Convection 9.4 Free-Surface Flow 9.4.1 Numerical Methods for the Level-Set Equation 9.4.2 Numerical Solution of the Reinitialization Equation 9.4.3 Mass Conservation 9.4.4 Initial and Boundary Conditions 9.4.5 Time Integration 9.4.6 Example Problems 9.4.6.1 Two-Dimensional Droplet 9.4.6.2 Broken Dam Problem Problems 10. Grid Generation 10.1 Introduction 10.2 Algebraic Methods 10.3 Differential Equation Methods 10.3.1 Elliptic Schemes 10.3.2 Hyperbolic Schemes 10.3.3 Parabolic Schemes 10.3.4 Deformation Method 10.3.5 Some Considerations in the Formulation of Grid Generators 10.4 Variational Methods 10.5 Unstructured Grid Schemes 10.5.1 Connectivity Information 10.5.2 Delaunay Triangulation 10.5.3 Bowyer Algorithm 10.6 Other Approaches 10.7 Adaptive Grids 10.8 CAD Models and Surface Representations 10.9 Higher-Order Curvilinear Gridding 10.9.1 Example of Curved Mesh Generation with An Elasticity Model Problems 11. High-Performance Computing 11.1 High-Performance Computing Hardware 11.2 Trends in Programming 11.3 Performance and Prospects: Exascale Computing, Applications Driving Growth 11.4 Some Aspects of CFD Software and Technology Appendix A: Modified Strongly Implicit Procedure Nomenclature References Index