In developing this book, we decided to emphasize applications and to provide methods for solving problems. As a result, we limited the mathematical devel­ opments and we tried as far as possible to get insight into the behavior of numerical methods by considering simple mathematical models. The text contains three sections. The first is intended to give the fundamen­ tals of most types of numerical approaches employed to solve fluid-mechanics problems. The topics of finite differences, finite elements, and spectral meth­ ods are included, as well as a number of special techniques. The second section is devoted to the solution of incompressible flows by the various numerical approaches. We have included solutions of laminar and turbulent-flow prob­ lems using finite difference, finite element, and spectral methods. The third section of the book is concerned with compressible flows. We divided this last section into inviscid and viscous flows and attempted to outline the methods for each area and give examples.

Inhalt

I: Numerical Approaches.- 1 Introduction and General Equations.- 1.1 The General Navier-Stokes Equations.- 1.2 Various Forms of the Navier-Stokes Equations.- 1.2.1 Dimensionless form.- 1.2.2 Orthogonal curvilinear coordinates.- 1.2.3 Plane flows.- 1.3 The Navier-Stokes Equations for Incompressible Flow.- 1.3.1 The primitive-variable formulation.- 1.3.2 The stream-function vorticity formulation.- 2 Finite-Difference Methods.- 2.1 Discrete Approximations.- 2.2 Solution of an Ordinary Differential Equation.- 2.2.1 The method of factorization.- 2.2.2 Iterative methods.- 2.2.3 Analogy between iterative procedures and equations of evolution.- 2.3 Analytical Solution of the Finite-Difference Problem.- 2.4 Upwind Corrected Schemes.- 2.5 Higher-Order Methods.- 2.5.1 Hermitian method.- 2.5.2 Mehrstellen and OCI methods.- 2.6 Solution of a One-Dimensional Linear Parabolic Equation.- 2.6.1 Effect of instability.- 2.6.2 Noncentered schemes.- 2.6.3 Leapfrog DuFort-Frankel scheme.- 2.7 Solution of One-Dimensional Nonlinear Parabolic and Hyperbolic Equations.- 2.7.1 Inviscid methods.- 2.7.2 Viscous methods.- 2.7.3 Boundary conditions.- 2.7.4 Implicit methods.- 2.8 Multidimensional Equation.- 2.8.1 Explicit schemes for the advection-diffusion equation.- 2.8.2 The A.D.I. method.- 2.8.3 Explicit schemes for a nonlinear equation in conservative form.- 2.8.4 Explicit splitting methods.- 2.8.5 Generalized A.D.I. methods.- 3 Integral and Spectral Methods.- 3.1 Finite-Element and Spectral-Type Methods.- 3.2 Steady-State Finite-Element Examples.- 3.3 Steady-State Spectral Method Examples.- 3.4 Time-Dependent Finite-Element Examples.- 3.5 Time-Dependent Spectral Method Examples.- 3.6 Pseudospectral Methods.- 3.7 Finite-Volume or Cell Method.- 3.7.1 Godunov method.- 3.7.2 Glimm method.- 4 Relationship Between Numerical Approaches.- 4.1 Finite-Difference Equivalent of Finite-Element Scheme.- 4.2 Finite-Difference Equivalent of Spectral Scheme.- 4.3 Finite-Difference Equivalent of Godunov Method.- 5 Specialized Methods.- 5.1 Potential Flow Solution Technique.- 5.2 Green's Functions and Stream-Function Vorticity Formulation.- 5.3 The Discrete Vortex Method.- 5.4 The Cloud-in-Cell Method.- 5.5 The Method of Characteristics.- II: Incompressible Flows.- 6 Finite-Difference Solutions of the Navier-Stokes Equations.- 6.1 The Navier-Stokes Equations in Primitive Variables.- 6.2 Steady Navier-Stokes Equations: The Artificial Compressibility Method.- 6.2.1 Description of the method.- 6.2.2 Discretization.- 6.2.3 Convergence toward a steady state.- 6.2.4 Treatment of boundary conditions.- 6.2.5 The Poisson equation for pressure.- 6.2.6 Other schemes for the artificial compressibility method.- 6.3 The Unsteady Navier-Stokes Equations.- 6.3.1 The projection and MAC methods.- 6.3.2 An iterative method.- 6.3.3 Relationship between the various methods.- 6.3.4 A perturbation (penalization) method.- 6.4 Example Solutions for Primitive Variable Formulation.- 6.4.1 Steady flow over a step.- 6.4.2 Unsteady horizontal jet in a stratified fluid.- 6.5 The Stream-Function Vorticity Formulation and Solution Approaches.- 6.5.1 The steady equations.- 6.5.2 The pseudo-unsteady methods.- 6.5.3 Boundary conditions.- 6.5.4 The iterative method.- 6.5.5 The problem of high Reynolds numbers.- 6.5.6 Approximation of the vorticity equation in conservative form.- 6.5.7 The unsteady equations.- 6.6 Example Solutions for Stream-Function Vorticity Formulation.- 6.6.1 Steady flow in a square cavity.- 6.6.2 Unsteady flow around a circular cylinder.- 7 Finite-Element Methods Applied to Incompressible Flows.- 7.1 The Galerkin Approach.- 7.2 The Least-Squares Approach.- 8 Spectral Method Solutions for Incompressible Flows.- 8.1 Inviscid Flows.- 8.2 Viscous Flows-Laminar and Transition.- 9 Turbulent-Flow Models and Calculations.- 9.1 Turbulence Closure Equations.- 9.2 Large Eddy Simulation Model.- 9.3 Turbulent-Flow Calculations with Closure Model.- 9.4 Direct Simulations of Turbulence.- III: Compressible Flows.- 10 Inviscid Compressible Flows.- 10.1 Application of Unsteady Methods.- 10.1.1 Finite-difference solutions.- 10.1.2 Cell and finite-volume solutions.- 10.2 Steady-Flow Methods Using Finite-Difference Approaches.- 10.2.1 Examples for M? 1.- 11 Viscous Compressible Flows.- 11.1 Introduction to Methods.- 11.2 Boundary Conditions.- 11.3 Finite-Difference Schemes in Uniform Cartesian Mesh.- 11.3.1 Explicit schemes.- 11.3.2 Implicit schemes.- 11.3.3 Artificial viscosity.- 11.4 Finite-Difference Schemes in Non-Cartesian Configurations.- 11.4.1 Discretization in transformed space.- 11.4.2 Discretization in the physical space.- Concluding Remarks.- Appendix A: Stability.- Appendix B: Multiple-Grid Method.- Appendix C: Conjugate-Gradient Method.