Computational Fluid Dynamics

Computational Fluid Dynamics, Spring 2005.

A graduate course in Mathematics at the Department of Mathematics, National Tsing Hua University.

Instructor: Professor Wei-Cheng Wang,

Class meeting: 綜三 734. Meeting schedule (temporary): T789.

First meeting: Mar 1st. 3:20 PM

Grading: 3-4 sets of homework assignments, including paper work and basic programming. Some reading assignments as well.

Course description: The incompressible Navier-Stokes equation is a complicated system of PDE of mixed type. The efficient computation of the fluid interacting with the physical boundary is the central issue of computational fluid dynamics in past decades. The purpose of this course is to explore various equivalent formulations of incompressible flows suited for numerical simulations, with emphasis on proper implementation of boundary conditions which results in stable and accurate numerical schemes. We will also introduce tools of error analysis for various formulations. We'll emphasize on the analysis aspect with some programming requirement. Students are required to do independent homework assignments on paper works and programming using modern languages (any one of matlab, c, c++, fortran ...). Some knowledges on numerical programming. and a preliminary undergraduate course of numerical analysis (or equivalent) is required.

Textbook and References: Book draft by Prof. Weinan E (Princeton U.) and JianGuo Liu (U. Maryland).

Topics and Contents:

Introduction to Navier Stokes Equation: Derivation of NSE, boundary conditions, incompressible flow as zero Mach number limit of compressible flow.
Equivalent Formulations of incompressible NSE: Vorticity-Stream formulation; Vorticity velocity formulation; Impulse(Gauge)-velocity formulation.
Numerical Methods for Simple Model Problems: Reynolds number, spatial scales and boundary layer; Fast and iterative Poisson solver; Numerical methods for heat equation; Passive convection diffusion equation.
Methods Based on Vorticity Formulations: Method for vorticity transport equation; Vorticity boundary condition; Higher order finite difference scheme; Proper finite element formulation; Multiply connected region.
Projection Methods: Discrete Hodge decomposition; Projection method on staggered grid; Projection method on standard grid; Normal mode analysis; Co-volume method;
Gauge Method: Gauge transformation; Selection of gauge and boundary condition.
Methods Based on Accurate Pressure Poisson Solver: Explicit treatment of pressure; local pressure boundary condition; divorcing pressure from NSE.
Variants of incompressible NSE (if time permits): Incompressible MHD equation; Incompressible Boussinesq equation; Geophysical Fluids for Oceans and Atmosphere.

Other sources of useful information:

Netlib is an excellent net resource. It contains tons of reliable numerical software that can be downloaded and links to most technical information on numerical analysis and scientific computing on the internet. The ftp mirror site in Taiwan is at the National Center for High Performance Computing.

Matlab Information

Matlab is an excellent environment for experimentation in scientific computing. It is not the most efficient environment for large-scale simulations, although with some effort, Matlab can be used in conjunction with C and Fortran routines. Matlab is a product of The Mathworks. For a tutorial, see A Free Matlab Online Tutorial or Another Tutorial or Matlab Primer
You can start from our class note on matlab before reading these materials.

A short tutorial for C programming language.

Homework assignments:

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