A mixed upwind/central WENO scheme for incompressible two-phase flows

Abstract

In this paper, an accurate and efficient numerical scheme is developed to solve the Navier-Stokes equations for two-phase flow systems with the nonlinear inertial term in conservation form. The major difficulty of discretizing the nonlinear inertial term in conservation form in two-phase flows is rooted in the discontinuous nature of momentum across interfaces. To overcome this difficulty, we developed the mixed Upwind/Central WENO scheme so that no grid deformation or mapping, or explicit interface reconstruction is necessary. Our mixed Upwind/Central WENO scheme also removes the dependence on the interface tracking/capturing method so it is compatible with any popular interface tracking/capturing methods. To decouple the overall equations of the two-phase system, an efficient semi-implicit projection scheme is constructed, which shares the efficiency of pressure stabilization method while enforces divergence-free condition to machine precision, to benefit interface advection, reduction consistency as well as large density ratio. The accuracy and reduction consistency of this projection scheme are validated by numerical experiments. Several benchmark cases are performed to show the capability of the overall scheme handling realistic two-phase problems with multiple physical effects, complex interface evolution and/or large density ratio of the order of (ρwater/ρair). Both quantitative and qualitative comparisons have been made and very good agreements with analytical solutions, published numerical and/or experimental results are achieved.