Computational Methods for Fluid Dynamics

In its third revised and extended edition the book offers an overview of the techniques used to solve problems in fluid mechanics on computers. The authors describe in detail the most often used techniques. Included are advanced techniques in computational fluid dynamics, such as direct and large-eddy simulation of turbulence. Moreover, a new section deals with grid quality and an extended description of discretization methods has also been included. Common roots and basic principles for many apparently different methods are explained. The book also contains a great deal of practical advice for code developers and users.

Table of Contents:
1. Basic Concepts of Fluid Flow.- 1.1 Introduction.- 1.2 Conservation Principles.- 1.3 Mass Conservation.- 1.4 Momentum Conservation.- 1.5 Conservation of Scalar Quantities.- 1.6 Dimensionless Form of Equations.- 1.7 Simplified Mathematical Models.- 1.7.1 Incompressible Flow.- 1.7.2 Inviscid (Euler) Flow.- 1.7.3 Potential Flow.- 1.7.4 Creeping (Stokes) Flow.- 1.7.5 Boussinesq Approximation.- 1.7.6 Boundary Layer Approximation.- 1.7.7 Modeling of Complex Flow Phenomena.- 1.8 Mathematical Classification of Flows.- 1.8.1 Hyperbolic Flows.- 1.8.2 Parabolic Flows.- 1.8.3 Elliptic Flows.- 1.8.4 Mixed Flow Types.- 1.9 Plan of This Book.- 2. Introduction to Numerical Methods.- 2.1 Approaches to Fluid Dynamical Problems.- 2.2 What is CFD?.- 2.3 Possibilities and Limitations of Numerical Methods.- 2.4 Components of a Numerical Solution Method.- 2.4.1 Mathematical Model.- 2.4.2 Discretization Method.- 2.4.3 Coordinate and Basis Vector Systems.- 2.4.4 Numerical Grid.- 2.4.5 Finite Approximations.- 2.4.6 Solution Method.- 2.4.7 Convergence Criteria.- 2.5 Properties of Numerical Solution Methods.- 2.5.1 Consistency.- 2.5.2 Stability.- 2.5.3 Convergence.- 2.5.4 Conservation.- 2.5.5 Boundedness.- 2.5.6 Realizability.- 2.5.7 Accuracy.- 2.6 Discretization Approaches.- 2.6.1 Finite Difference Method.- 2.6.2 Finite Volume Method.- 2.6.3 Finite Element Method.- 3. Finite Difference Methods.- 3.1 Introduction.- 3.2 Basic Concept.- 3.3 Approximation of the First Derivative.- 3.3.1 Taylor Series Expansion.- 3.3.2 Polynomial Fitting.- 3.3.3 Compact Schemes.- 3.3.4 Non-Uniform Grids.- 3.4 Approximation of the Second Derivative.- 3.5 Approximation of Mixed Derivatives.- 3.6 Approximation of Other Terms.- 3.7 Implementation of Boundary Conditions.- 3.8 The Algebraic Equation System.- 3.9 Discretization Errors.- 3.10 An Introduction to Spectral Methods.- 3.10.1 Basic Concept.- 3.10.2 Another View of Discretization Error.- 3.11 Example.- 4. Finite Volume Methods.- 4.1 Introduction.- 4.2 Approximation of Surface Integrals.- 4.3 Approximation of Volume Integrals.- 4.4 Interpolation and Differentiation Practices.- 4.4.1 Upwind Interpolation (UDS).- 4.4.2 Linear Interpolation (CDS).- 4.4.3 Quadratic Upwind Interpolation (QUICK).- 4.4.4 Higher-Order Schemes.- 4.4.5 Other Schemes.- 4.5 Implementation of Boundary Conditions.- 4.6 The Algebraic Equation System.- 4.7 Examples.- 5. Solution of Linear Equation Systems.- 5.1 Introduction.- 5.2 Direct Methods.- 5.2.1 Gauss Elimination.- 5.2.2 LU Decomposition.- 5.2.3 Tridiagonal Systems.- 5.2.4 Cyclic Reduction.- 5.3 Iterative Methods.- 5.3.1 Basic Concept.- 5.3.2 Convergence.- 5.3.3 Some Basic Methods.- 5.3.4 Incomplete LU Decomposition: Stone’s Method.- 5.3.5 ADI and Other Splitting Methods.- 5.3.6 Conjugate Gradient Methods.- 5.3.7 Biconjugate Gradients and CGSTAB.- 5.3.8 Multigrid Methods.- 5.3.9 Other Iterative Solvers.- 5.4 Coupled Equations and Their Solution.- 5.4.1 Simultaneous Solution.- 5.4.2 Sequential Solution.- 5.4.3 Under-Relaxation.- 5.5 Non-Linear Equations and their Solution.- 5.5.1 Newton-like Techniques.- 5.5.2 Other Techniques.- 5.6 Deferred-Correction Approaches.- 5.7 Convergence Criteria and Iteration Errors.- 5.8 Examples.- 6. Methods for Unsteady Problems.- 6.1 Introduction.- 6.2 Methods for Initial Value Problems in ODEs.- 6.2.1 Two-Level Methods.- 6.2.2 Predictor-Corrector and Multipoint Methods.- 6.2.3 Runge-Kutta Methods.- 6.2.4 Other Methods.- 6.3 Application to the Generic Transport Equation.- 6.3.1 Explicit Methods.- 6.3.2 Implicit Methods.- 6.3.3 Other Methods.- 6.4 Examples.- 7. Solution of the Navier-Stokes Equations.- 7.1 Special Features of the Navier-Stokes Equations.- 7.1.1 Discretization of Convective and Viscous Terms.- 7.1.2 Discretization of Pressure Terms and Body Forces.- 7.1.3 Conservation Properties.- 7.2 Choice of Variable Arrangement on the Grid.- 7.2.1 Colocated Arrangement.- 7.2.2 Staggered Arrangements.- 7.3 Calculation of the Pressure.- 7.3.1 The Pressure Equation and its Solution.- 7.3.2 A Simple Explicit Time Advance Scheme.- 7.3.3 A Simple Implicit Time Advance Method.- 7.3.4 Implicit Pressure-Correction Methods.- 7.4 Other Methods.- 7.4.1 Fractional Step Methods.- 7.4.2 Streamfunction-Vorticity Methods.- 7.4.3 Artificial Compressibility Methods.- 7.5 Solution Methods for the Navier-Stokes Equations.- 7.5.1 Implicit Scheme Using Pressure-Correction and a Staggered GridGri.- 7.5.2 Treatment of Pressure for Colocated Variables.- 7.5.3 SIMPLE Algorithm for a Colocated Variable Arrangement.- 7.6 Note on Pressure and Incompressibility.- 7.7 Boundary Conditions for the Navier-Stokes Equations.- 7.8 Examples.- 8. Complex Geometries.- 8.1 The Choice of Grid.- 8.1.1 Stepwise Approximation Using Regular Grids.- 8.1.2 Overlapping Grids.- 8.1.3 Boundary-Fitted Non-Orthogonal Grids.- 8.2 Grid Generation.- 8.3 The Choice of Velocity Components.- 8.3.1 Grid-Oriented Velocity Components.- 8.3.2 Cartesian Velocity Components.- 8.4 The Choice of Variable Arrangement.- 8.4.1 Staggered Arrangements.- 8.4.2 Colocated Arrangement.- 8.5 Finite Difference Methods.- 8.5.1 Methods Based on Coordinate Transform ation.- 8.5.2 Method Based on Shape Functions.- 8.6 Finite Volume Methods.- 8.6.1 Approximation of Convective Fluxes.- 8.6.2 Approximation of Diffusive Fluxes.- 8.6.3 Approximation of Source Terms.- 8.6.4 Three-Dimensional Grids.- 8.6.5 Block-Structured Grids.- 8.6.6 Unstructured Grids.- 8.7 Control-Volume-Based Finite Element Methods.- 8.8 Pressure-Correction Equation.- 8.9 Axi-Symmetric Problems.- 8.10 Implementation of Boundary Conditions.- 8.10.1 Inlet.- 8.10.2 Outlet.- 8.10.3 Impermeable Walls.- 8.10.4 Symmetry Planes.- 8.10.5 Specified Pressure.- 8.11 Examples.- 9. Turbulent Flows.- 9.1 Introduction.- 9.2 Direct Numerical Simulation (DNS).- 9.2.1 Example: Spatial Decay of Grid Turbulence.- 9.3 Large Eddy Simulation (LES).- 9.3.1 Smagorinsky and Related Models.- 9.3.2 Dynamic Models.- 9.3.3 Deconvolution Models.- 9.3.4 Example: Flow Over a Wall-Mounted Cube.- 9.3.5 Example: Stratified Homogeneous Shear Flow.- 9.4 RANS Models.- 9.4.1 Reynolds-Averaged Navier-Stokes (RANS) Equations.- 9.4.2 Simple Turbulence Models and their Application.- 9.4.3 The v2f Model.- 9.4.4 Example: Flow Around an Engine Valve.- 9.5 Reynolds Stress Models.- 9.6 Very Large Eddy Simulation.- 10. Compressible Flow.- 10.1 Introduction.- 10.2 Pressure-Correction Methods for Arbitrary Mach Number.- 10.2.1 Pressure-Velocity-Density Coupling.- 10.2.2 Boundary Conditions.- 10.2.3 Examples.- 10.3 Methods Designed for Compressible Flow.- 10.3.1 An Overview of Some Specific Methods.- 11. Efficiency and Accuracy Improvement.- 11.1 Error Analysis and Estimation.- 11.1.1 Description of Errors.- 11.1.2 Estimation of Errors.- 11.1.3 Recommended Practice for CFD Uncertainty Analysis.- 11.2 Grid quality and optimization.- 11.3 Multigrid Methods for Flow Calculation.- 11.4 Adaptive Grid Methods and Local Grid Refinement.- 11.5 Parallel Computing in CFD.- 11.5.1 Iterative Schemes for Linear Equations.- 11.5.2 Domain Decomposition in Space.- 11.5.3 Domain Decomposition in Time.- 11.5.4 Efficiency of Parallel Comput ing.- 12. Special Topics.- 12.1 Introduction.- 12.2 Heat and Mass Transfer.- 12.3 Flows With Variable Fluid Properties.- 12.4 Moving Grids.- 12.5 Free-Surface Flows.- 12.5.1 Interface-Tracking Methods.- 12.5.2 Hybrid Methods.- 12.6 Meteorological and Oceanographic Applications.- 12.7 Multiphase flows.- 12.8 Combustion.- A. Appendices.- A.l List of Computer Codes and How to Access Them.- A.2 List of Frequently Used Abbreviations.- References.