A consistent finite element method for fluid flows

A number of finite element methods for fluid rely on streamline upwind Petrov-Galerkin stabilization term, τ_SUPG, that is introduced for the fine-scale approximation. The formulation for this term is conventionally dependent on the time step size, making the solution inconsistent as the time step size is changed. We propose a new definition of τ_SUPG that produces consistent results. This method that is implemented on top of SUPG/PSPG stabilized formulation involves replacement of time step size with a physical measure of flow acceleration relative to its velocity. Our numerical experiment shows that the conventional formulation can generate up to 50% change in the solution as the time step size is reduced, whereas the new formulation converges to a unique solution. In this presentation, we will demonstrate the characteristics of both the conventional method and our proposed method using three cases: pipe flow, 2D flow over a square, and a realistic cardiovascular flow model.