Finite element MAC scheme in general curvilinear coordinates

Abstract

A combined study of F.D. and F.E. methods for 2-D incompressible Navier-Stokes flows is undertaken. In primitive variable formulation, major difficulties are connected to spurious numerical oscillations which may arise from the enforcement of the incompressibility constraint. With regard to this problem, various F.D. schemes differ essentially according to the variable location on the mesh points, while F.E. schemes are analogously differentiated by the interpolation functions adopted for the different variables. In the present paper, we propose an F.E. analog of MAC scheme, which can be accomplished by different interpolation functions for the two velocity components. This new F.E. scheme-although based on low order approximations-eliminates all spurious oscillations. The extension to curvilinear quadrilateral elements—which is needed in order to achieve geometrical versatility—requires the problem to be formulated in general curvilinear coordinates and the contravariant velocity components to be assumed as variables. Some numerical results are presented and discussed, in order to assess the capabilities of the proposed model.