Synopsis for Numerical Solutions of Navier-Stokes Equations

Students will study a range of solution methods for the unsteady laminar Navier-Stokes equations in two dimensions and develop their own solutions for one flow from a range of test problems.

Syllabus

1. Introduction: Navier Stokes equations in 2D, special solutions, limiting cases: Stokes equations, Euler equations. Boundary conditions, finite domain and infinite domains.
2. Incompressible, unsteady laminar 2D flows: Formulation, boundary conditions, finite difference approximation, Poisson equation solver — multigrid. Vorticity transport equation: upwind, Lax-Wendroff and higher order discretisation, implementation of boundary conditions. Model Problem I: driven cavity flow. Conservative and non-conservative formulation. Model Problem II: movement of tracer in driven cavity. Problems with inflow and outflow boundaries, channel flows. Model problems III: Coanda bifurcation. Boundary layer equations, derivation and numerical solution.
3. Compressible flow: Formulation, characteristic solutions, finite volume methods, flux approximations. Gudonov schemes, Roe's approximate Riemann solver. Model problem IV: shock tube.
4. Turbulent flow: Formulation, Reynolds stress, closure problem, eddy viscosity, mixing length and k-epsilon models. 2D channel flow. Model Problem V: k-epsilon solution for turbulent channel flow.