A new upwind finite element formulation for convection dominated transport problem is presented. The basis of formulation is the optimal weighting function concept based on variational principle, suggested by Shen.
The optimal weighting functions for triangular element and for quadrilateral element are modified and developed, respectively. Their accuracies are demonstrated on several example problems and compared with previous upwind method, particularly streamline-upwind/Petrov-Galerkin (SU/PG) method by Brooks and Hughes which is the most commonly used upwind finite element method.
The present results exhibit almost equal effects compared with SU/PG. The method for triangular element shows slightly larger crosswind diffusion effect than that for quadrilateral element. For quadrilateral element, however, spurious oscillating modes appear similar to SU/PG when the streamline is skewed to the mesh at a certain angle.
The method for quadrilateral element is applied to solving two dimensional, steady, laminar Navier-Stokes equation, using the segregated velocity-pressure solution scheme, suggested by Rice and Schnipke, that employs an equal-order velocity-pressure solution scheme, suggested by Rice and Schnipke, that employs an equal-order velocity-pressure approximation.
Comprehensive computational results for backward-facing step flow for Reynolds number of 100 through 900 and lid-driven cavity flow for Reynolds number of 400 through 7500 are presented and compared with the experimental data and computational results which use finer girds. The results for backward-facing step flow case are in good agreement with the experiments and previously predicted results. However the predictions for cavity flow are not good for high Reynolds number. It is thought that these disagreements are caused by the crosswind diffusion and spurious oscillation due to the vortices which have significant curvature.
