Unified adaptive Variational MultiScale method for two phase compressible-incompressible flows

In this paper we present a new stabilized finite element method to solve the two phase compressible-incompressible fluid flow problems using the level set method. An anisotropic mesh adaptation with a regularized interface is adopted to deal with the high discontinuity in the material properties. The coupling between the pressure and the flow velocity is ensured by introducing mass conservation terms in the momentum equation. Therefore, the same set of primitive unknowns and equations is described for both phases. The unified system is solved using a new derived Variational MultiScale stabilized finite element method. It is tested on three time-dependent liquid-gas interface problems. The numerical results show good stability and accuracy properties, permitting also to deal with important compressibility effect, high density ratio, and extremely stretched anisotropic elements.