Detailed Course Outline

• Introduction to Fluid Dynamics: Derivation of the Naiver-Stokes equation; Boundary and initial conditions; The incompressible Navier-Stokes equation as the zero-Mach number limit of the compressible Navier-Stokes equation; Different formulations of the Navier-Stokes equation: primitive variable formulation, vorticity formulation, impulse density formulation; Examples of one-dimensional incompressible fluid equations: the one-dimensional Stokes flow, the stagnation point flow; Lagrangian vs. Eulerian description; Other applications: examples from geophysical fluid dynamics, MHD, heat transfer, polymeric fluids, zero-Mach number combustion.
• Basic Numerical Methods for Differential Equations: ODE solvers; Advection-diffusion equation (include compact schemes); Poisson solvers; Far field boundary conditions; Finite difference; Finite volume methods
• Primitive Variable Formulation : Mac scheme and the staggered Grid; The Projection method; The Gauge method; Pressure Poisson formulation
• Vorticity Formulation: Basic issues: vorticity boundary condition, time stepping, equivalence with the MAC scheme; Compact schemes.
• Numerical Methods for compressible fluid flows:
  • First-order Monotone methods: Lax-Friedriches scheme, upwind methods and Godunov’s method, kinetic scheme and flux splitting, numerical flux functions
  • Second-order and high-resolution methods: Lax-Wendroff scheme and MacCormack scheme, flux limiters and slope limiters, PPM and ENO scheme, total variation diminishing (TVD) methods and FCT scheme
  • Boundary conditions: periodic, non-reflecting, solid walls
  • Far field boundary conditions
  • Multi-dimensional problems: dimensional splitting and unsplit methods
• Applications: