Variationally derived interface stabilization for discrete multiphase flows and relation with the ghost-penalty method

This paper presents an interface stabilized method for moving and deforming interface problems in immiscible, discrete, multi-phase flows. The evolution of the interface is represented in the momentum balance equations written in an Eulerian frame via the dependence of the local value of density and viscosity of the fluids on the spatial location of the interphase boundary. The motion of the interface is tracked via a hyperbolic equation that is driven by the velocity field furnished by the Navier–Stokes equations. Zero contour of the level set field marks the boundary which is permitted to traverse through the elements. The jump in viscosity and density of the fluids across the arbitrarily deforming phase boundaries can trigger numerical instabilities in the solution. A main contribution in this work is the mathematical analysis of interface stabilization terms that appear naturally when the Variational Multiscale (VMS) method is applied to the coupled system of partial differential equations (PDEs). Two specific forms are developed: full gradient form in R^n_sd over subdomain Ω_I around the embedded interface, and a reduced form in R^n_sd−1 for face stabilization at the embedded interface Γ_I. Benchmark problems in 2D and 3D are presented to highlight the salient features of the proposed interface stabilization method and to show its range of application.