Introduction: An Initial Guide to CFD and to this Volume 1

Part 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 equation

1.3 The momentum conservation law or equation of motion

1.4 The energy conservation equation

A1.5 Rotating frame of reference

A1.6 Advanced applications of control volume formulations

Conclusions and main topics to remember

2 The Dynamical Levels of Approximation

2.1 The NavierStokes equations

2.2 Approximations of turbulent flows

2.3 Thin shear layer approximation (TSL)

2.4 Parabolized NavierStokes equations

2.5 Boundary layer approximation

2.6 The distributed loss model

2.7 Inviscid flow model: Euler equations

2.8 Potential flow model

2.9 Summary

3 The Mathematical Nature of the Flow Equations and Their Boundary Conditions

3.1 Simplified models of a convectiondiffusion equation

3.2 Definition of the mathematical properties of a system of PDEs

3.3 Hyperbolic and parabolic equations: characteristic surfaces and domain of dependence

3.4 Timedependent and conservation form of the PDEs

3.5 Initial and boundary conditions

A.3.6 Alternative definition: compatibility relations

Conclusions and main topics to remember

Part II Basic Discretization Techniques

4 The Finite Difference Method for Structured Grids

4.1 The basics of finite difference methods

4.2 Multidimensional finite difference formulas

4.3 Finite difference formulas on nonuniform grids

A4.4 General method for finite difference formulas

A4.5 Implicit finite difference formulas

Conclusions and main topics to remember

5 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 method

A5.4.1 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 Method

A5.4.5 Finite Element Galerkin Method for a Conservation Law

A5.4.6 Subdomain Collocation: Finite Volume Method

Conclusions and main topics to remember

6 Structured and Unstructured Grid Properties

6.1 Structured Grids

6.2 Unstructured grids

6.3 Surface and volume estimations

6.4 Grid quality and best practice guidelines

Conclusions and main topics to remember

Part III The Analysis of Numerical Schemes

7 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 HighResolution Numerical Schemes

8.1 General formulation of numerical schemes

A8.1.5 An Addition to the Stability Analysis

A8.1.6 An Advanced Addition to the Accuracy Barrier

8.2 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 Schemes

9 Time Integration Methods for Spacediscretized Equations

9.1 Analysis of the spacediscretized systems

9.2 Analysis of time integration schemes

9.3 A selection of time integration methods

A9.4 Implicit schemes for multidimensional problems: approximate factorization methods

A9.4.1 TwoDimensional Diffusion Equation

A9.4.2 ADI Method for the Convection Equation

Conclusions and main topics to remember

10 Iterative Methods for the Resolution of Algebraic Systems

10.1 Basic iterative methods

10.2 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 NavierStokes equations for laminar flows

12.2 Densitybased methods for viscous flows

12.3 Numerical solutions with the densitybased method

12.4 Pressure correction method

12.5 Numerical solutions with the pressure correction method

12.6 Best practice advice

Conclusions and main topics to remember
