Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics

Introduction: An Initial Guide to CFD and to this Volume 1Part I The Mathematical Models for Fluid Flow Simulations at Various Levels of Approximation 1 The Basic Equations of Fluid Dynamics 1.1 General form of a conservation law 1.2 The mass conservation equation1.3 The momentum conservation law or equation of motion1.4 The energy conservation equationA1.5 Rotating frame of referenceA1.6 Advanced applications of control volume formulations Conclusions and main topics to remember 2 The Dynamical Levels of Approximation 2.1 The Navier–Stokes equations2.2 Approximations of turbulent flows 2.3 Thin shear layer approximation (TSL) 2.4 Parabolized Navier–Stokes equations2.5 Boundary layer approximation2.6 The distributed loss model2.7 Inviscid flow model: Euler equations 2.8 Potential flow model2.9 Summary3 The Mathematical Nature of the Flow Equations and Their Boundary Conditions3.1 Simplified models of a convection–diffusion equation 3.2 Definition of the mathematical properties of a system of PDEs3.3 Hyperbolic and parabolic equations: characteristic surfaces anddomain of dependence 3.4 Time-dependent and conservation form of the PDEs 3.5 Initial and boundary conditionsA.3.6 Alternative definition: compatibility relationsConclusions and main topics to remember Part II Basic Discretization Techniques4 The Finite Difference Method for Structured Grids 4.1 The basics of finite difference methods4.2 Multidimensional finite difference formulas 4.3 Finite difference formulas on non-uniform gridsA4.4 General method for finite difference formulas A4.5 Implicit finite difference formulas Conclusions and main topics to remember5 Finite Volume Method and Conservative Discretization with an Introduction to Finite Element Method 5.1 The conservative discretization 5.2 The basis of the finite volume method 5.3 Practical implementation of finite volume method A.5.4 The finite element methodA5.4.1 Finite Element Definition of Interpolation Functions A5.4.2 Finite Element Definition of the Equation Discretization: Integral Formulation A5.4.3 The Method of Weighted Residuals or Weak Formulation A5.4.4 The Galerkin MethodA5.4.5 Finite Element Galerkin Method for a Conservation Law A5.4.6 Subdomain Collocation: Finite Volume MethodConclusions and main topics to remember 6 Structured and Unstructured Grid Properties6.1 Structured Grids 6.2 Unstructured grids6.3 Surface and volume estimations6.4 Grid quality and best practice guidelinesConclusions and main topics to rememberPart III The Analysis of Numerical Schemes7 Consistency, Stability and Error Analysis of Numerical Schemes 7.1 Basic concepts and definitions 7.2 The Von Neumann method for stability analysis 7.3 New schemes for the linear convection equation 7.4 The spectral analysis of numerical errors Conclusions and main topics to remember 8 General Properties and High-Resolution Numerical Schemes 8.1 General formulation of numerical schemes A8.1.5 An Addition to the Stability AnalysisA8.1.6 An Advanced Addition to the Accuracy Barrier8.2 The generation of new schemes with prescribed order of accuracy 8.3 Monotonicity of numerical schemes 8.4 Finite volume formulation of schemes and limiters Conclusions and main topics to remember Part IV The Resolution of Numerical Schemes9 Time Integration Methods for Space-discretized Equations 9.1 Analysis of the space-discretized systems9.2 Analysis of time integration schemes9.3 A selection of time integration methods A9.4 Implicit schemes for multidimensional problems: approximate factorization methods A9.4.1 Two-Dimensional Diffusion EquationA9.4.2 ADI Method for the Convection EquationConclusions and main topics to remember10 Iterative Methods for the Resolution of Algebraic Systems10.1 Basic iterative methods10.2 Overrelaxation methods 10.3 Preconditioning techniques 10.4 Nonlinear problems 10.5 The multigrid method Conclusions and main topics to remember Appendix A: Thomas Algorithm for Tridiagonal Systems A.1. Scalar Tridiagonal Systems A.2. Periodic Tridiagonal Systems PartV Applications to Inviscid andViscous Flows 11 Numerical Simulation of Inviscid Flows 11.1 The inviscid Euler equations 11.2 The potential flow model 11.3 Numerical solutions for the potential equation 11.4 Finite volume discretization of the Euler equations 11.5 Numerical solutions for the Euler equations Conclusions and main topics to remember 12 Numerical Solutions of Viscous Laminar Flows 12.1 Navier–Stokes equations for laminar flows 12.2 Density-based methods for viscous flows 12.3 Numerical solutions with the density-based method 12.4 Pressure correction method12.5 Numerical solutions with the pressure correction method 12.6 Best practice advice Conclusions and main topics to remember