Fundamentals of Computational Fluid Dynamics

This book is intended as a textbook for a first course in computational fluid dynamics and will be of interest to researchers and practitioners as well. It emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. The linear convection and diffusion equations are used to illustrate concepts throughout. The chosen approach, in which the partial differential equations are reduced to ordinary differential equations, and finally to difference equations, gives the book its distinctiveness and provides a sound basis for a deep understanding of the fundamental concepts in computational fluid dynamics.

The field of computational fluid dynamics (CFD) has already had a significant impact on the science and engineering of fluid dynamics, ranging from a role in aircraft design to enhancing our understanding of turbulent flows. It is thus not surprising that there exist several excellent books on the subject. We do not attempt to duplicate material which is thoroughly covered in these books. In particular, our book does not describe the most recent developments in algorithms, nor does it give any instruction with respect to programming. Neither turbulence modelling nor grid generation are covered. This book is intended for a reader who seeks a deep understanding of the fundamental principles which provide the foundation for the algorithms used in CFD. As a result of this focus, the book is suitable for a first course in CFD, presumably at the graduate level. The underlying philosophy is that the theory of linear algebra and the attendant eigenanalysis of linear systems provide a mathematical framework to describe and unify most numerical methods in common use for solving the partial differential equations governing the physics of fluid flow. This approach originated with the first author during his long and distinguished career as Chief of the CFD Branch at the NASA Ames Research Center.

The chosen semi-discrete approach of a reduction procedure of partial differential equations to ordinary differential equations and finally to difference equations gives the book its distinctiveness Includes supplementary material: sn.pub/extras

Contenu

1. Introduction.- 2. Conservation Laws and the Model Equations.- 3. Finite-Difference Approximations.- 4. The Semi-Discrete Approach.- 5. Finite-Volume Methods.- 6. Time-Marching Methods for ODE'S.- 7. Stability of Linear Systems.- 8. Choosing a Time-Marching Method.- 9. Relaxation Methods.- 10. Multigrid.- 11. Numerical Dissipation.- 12. Split and Factored Forms.- 13. Analysis of Split and Factored Forms.- Appendices.- A. Useful Relations from Linear Algebra.- A.1 Notation.- A.2 Definitions.- A.3 Algebra.- A.4 Eigensystems.- A.5 Vector and Matrix Norms.- B. Some Properties of Tridiagonal Matrices.- B.1 Standard Eigensystem for Simple Tridiagonal Matrices.- B.2 Generalized Eigensystem for Simple Tridiagonal Matrices.- B.3 The Inverse of a Simple Tridiagonal Matrix.- B.4 Eigensystems of Circulant Matrices.- B.4.1 Standard Tridiagonal Matrices.- B.4.2 General Circulant Systems.- B.5 Special Cases Found from Symmetries.- B.6 Special Cases Involving Boundary Conditions.- C. The Homogeneous Property of the Euler Equations.