Computational Methods for Fluid Flow

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.- 10.2.2 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.