Stream function vorticity revisited

A three-level consistent time-split group finite element formulation is described for a stream function vorticity representation of incompressible laminar separated flow. A consistent time-splitting is achieved by explicitly extracting directional mass and difference operators. It is shown that application of linear rectangular elements to the convective terms with a non-uniform grid introduces an artificial viscosity, but that a simple modification removes this effect. The three-level fully implicit marching algorithm is shown to converge to the steady solution faster than a Crank-Nicolson scheme when combined with a banded Gauss elimination procedure along each grid-line in turn. The present method is applied to the flow past rearward- and forward-facing steps for which the vorticity solution is known to be singular at the sharp corners. To isolate the sharp corners from the computational domain a surface layer is introduced across which the computational solution is matched to an analytic solution for the corner flow. A one-dimensional integration of the energy function and transverse extrapolation is used to obtain the pressure distribution on the body. Results, for a range of step-height Reynolds numbers of up to 250, demonstrate good agreement with corresponding experimental results.