Computational Fluid Dynamics
|Department of Mathematics
|Simon Fraser University
|Tel: (778) 782-3553
This course will introduce students to a variety of computational
approaches for solving the equations governing fluid dynamics, focusing
on finite difference and finite volume techniques. Theoretical
background material will be introduced as necessary, but the focus of
the course will be on the numerical methods, their accuracy and
stability, and their practical application in computing real fluid
flows. Students will gain experience writing their own codes, as well
as employing existing publicly-available software packages.
Applications will be drawn from problems arising in wave propagation,
incompressible fluids, compressible gas dynamics, and porous media.
Previous courses in ordinary and partial differential equations (such as
MATH 310 or MATH 314) are required, as is some experience in computer
programming (any language is fine although knowledge of MATLAB would be
particularly helpful). A course in fluid dynamics (such as MATH 462)
would be helpful, but is not required.
Flow over a cylinder.
|| Lid-driven cavity flow.
Fingering in a porous medium.
The grade for this course will be made up of homework
assignments (60%) and a project (40%). There will be no final
There is no textbook for this course. Material will be drawn from a
number of texts, some of which are held on reserve in the library:
- P. Colella and E. G. Puckett, Modern Numerical Methods
for Fluid Flow, course notes, 1994.
[distributed as PDF]
- C. Pozrikidis, Introduction to Theoretical and Computational
Fluid Dynamics (Oxford University Press), 1997.
[hardcopy on reserve]
Alternately: C. Pozrikidis, Fluid Dynamics: Theory,
Computation, and Numerical Simulation (Springer), 2009.
- R. J. LeVeque, Finite Volume Methods for Hyperbolic
Problems (Cambridge University Press), 2002.
- K. W. Morton and D. F. Mayers, Numerical Solution of
Partial Differential Equations: An Introduction first or second
ed. (Cambridge University Press), 1994 or 2005.
[hardcopy on reserve]
- D. A. Nield and A. Bejan, Convection in Porous Media, 3rd
ed. (Springer), 2006.
- 1. Background and Governing Equations (1 week):
Navier-Stokes equations; boundary conditions; simplifications and
extensions; analytical solutions.
- 2. Finite Differences for Linear Problems (1.5 weeks):
Consistency, stability and convergence; CFL condition; Lax Equivalence
Theorem; von Neuman stability analysis; common upwind and
centered schemes; finite volume approach; time-stepping.
Applications: scalar advection; heat equation; wave equation.
- 3. Incompressible Fluid Flow (3 weeks):
Stokes equations; Pressure Poisson equation; Navier-Stokes equations;
Applications: creeping flow; potential flow; driven cavity
- 4. Porous Media Flow (3 weeks):
Darcy's Law; capillarity; porous medium equation and nonlinear
diffusion; IMPES method; Brinkman-Forchheimer extension.
Applications: oil reservoir simulation; groundwater transport;
- 5. Nonlinear Wave Propagation (3 weeks):
Hyperbolic conservation laws; nonlinear systems; Riemann solvers;
Applications: gas dynamics; shallow water waves; traffic and
pedestrian flow; atmospheric transport.
- Additional Topics:
Interspersed throughout the course, I will also introduce ideas
related more generally to scientific computing, reproducible research,
and software design.