A simple stabilized finite element method for solving two phase compressible-incompressible interface flows

摘要

When computing interface flows between compressible (gas) and incompressible (liquid) fluids, one faces at least to the following difficulties: (1) transition from a gas density linked to the local temperature and pressure by an equation of state to a liquid density mainly constant in space, (2) proper approximation of the divergence constraint in incompressible regions and (3) wave transmission at the interface. The aim of the present paper is to design a global (i.e. the same for each phase) numerical method to address easily this coupling. To this end, the same set of primitive unknowns and equations is used everywhere in the flow, but with a dynamic parameterization that changes from compressible to incompressible regions. On one hand, the compressible Navier-Stokes equations are considered under weakly compressibility assumption so that a non-conservative formulation can be used. On the other hand, the incompressible non-isothermal model is retained. In addition, the level set transport equation is used to capture the interface position needed to identify the local characteristics of the fluid and to recover the adequate local modelling. For space approximation of Navier-Stokes equations, a Galerkin least-squares finite element method is used. Two essential elements for defining this numerical scheme are the stabilization and the computation of element integral of the approximated weak form. Since very different concerns motivate the need for stabilization in compressible and incompressible flows, the first difficulty is to design a stabilization operator suitable for both types of flows especially in mixed elements. In addition, some integral of discontinuous functions must be correctly computed to ensure interfacial wave transmission. To overcome these two difficulties, specific averages are computed especially near the interface. Finally, the level set transport equation is computed by a quadrature free Discontinuous Galerkin method. Numerical strategies are performed and validated for ID and 2D applications.