List of Figures
List of Tables
Preface
CHAPTER 1 Introduction to the Equations of Fluid Dynamics
and the Finite Element Approximation
1.1 General remarks and classification of fluid dynamics
problems discussed in this book
1.2 The governing equations of fluid dynamics
1.2.1 Velocity, strain rates, and stresses in fluids
1.2.2 Constitutive relations for fluids
1.2.3 Mass conservation
1.2.4 Momentum conservation: Dynamic equilibrium
1.2.5 Energy conservation and equation of state
1.2.6 Boundary conditions
1.2.7 Navier-Stokes and Euler equations
1.3 Inviscid, incompressible flow
1.3.1 Velocity potential solution
1.4 Incompressible (or nearly incompressible) flows
1.5 Numerical solutions: Weak forms, weighted residual, and
finite element approximation
1.5.1 Strong and weak forms
1.5.2 Weighted residual approximation
1.5.3 The Galerkin finite element method
1.5.4 A finite volume approximation
1.6 Concluding remarks
References
CHAPTER 2 Convection-Dominated Problems: Finite Element
Approximations to the Convection-Diffusion-
Reaction Equation
2.1 Introduction
2.2 The steady-state problem in one dimension
2.2.1 General remarks
2.2.2 Petrov-Galerkin methods for upwinding in one
dimension
2.2.3 Balancing diffusion in one dimension
2.2.4 A variational principle in one dimension
2.2.5 Galerkin least-squares approximation (GLS) in one
dimension
2.2.6 Subgrid scale (SGS) approximation
2.2.7 The finite increment calculus (FIC) for stabilizing
the convective-diffusion equation in one dimension
2.2.8 Higher-order approximations
2.3 The steady-state problem in two (or three) dimensions
2.3.1 General remarks
2.3.2 Streamline (upwind) Petrov-Galerkin
weighting (SUPG)
2.3.3 Galerkin least squares (GLS) and finite increment
calculus (FIC) in multidimensional problems
2.4 Steady state: Concluding remarks
9.5 Transients: Introductory remarks
2.5.1 Mathematical background
2.5.2 Possible discretization procedures
2.6 Characteristic-based methods
2.6.1 Mesh updating and interpolation methods
2.6.2 Characteristic-Galerkin procedures
2.6.3 A simple explicit characteristic-Galerkin procedure.
2.6.4 Boundary conditions: Radiation
2.7 Taylor-Galerkin procedures for scalar variables
2.8 Steady-state condition
2.9 Nonlinear waves and shocks
2.10 Treatment of pure convection
2.11 Boundary conditions for convection-diffusion
2.12 Summary and concluding remarks
References
CHAPTER 3 The Characteristic-Based Split (CBS) Algorithm:A General Procedure for Compressible and Incompressible Flow
3.1 Introduction
3.2 Nondimensional form of the governing equations
3.3 Characteristic-based split (CBS) algorithm
3.3.1 The split: General remarks
3.3.2 The split: Temporal discretization
3.3.3 Spatial discretization and solution procedure
3.3.4 Mass diagonalization (lumping)
3.4 Explicit, semi-implicit, and nearly implicit forms
3.4.1 Fully explicit form
CHAPTER 4 Incompressible Newtonian Laminar Flows.
CHAPTER 5 Incompressible Non-Newtonian Flows
CHAPTER 6 Free Surface and Buoyancy Driven Flows.
CHAPTER 7 Compressible High-Speed Gas Flow.
CHAPTER 8 Turbulent Flows.
CHAPTER 9 Generalized Flow and Heat Transfer in Porous Media
CHAPTER 10 Shallow-Water Problems.
CHAPTER 11 Long and Medium Waves.
CHAPTER 12 Short Waves
CHAPTER 13 Fluid-Structure Interaction
CHAPTER 14 Biofluid Dynamics.
CHAPTER 15 Computer Implementation of the CBS Algorithm
APPENDIX A Self-Adjoint Differential Equations
APPENDIX B Nonconservative Form of Navier-Stokes Equations
APPENDIX C Computing the Drag Force and Stream Function.
APPENDIX D Convection-Diffusion Equations: Vector-Valued Variables
APPENDIX E Integration Formulae
APPENDIX F Edge-Based Finite Element Formulation
APPENDIX G Boundary Layer-lnviscid Flow Coupling
APPENDIX H Multigrid Method
APPENDIX I Mass-Weighted Averaged Turbulence Transport Equations
Author Index
Subject Index
