An energy stable initial boundary value problem for incompressible multi-phase flow

Abstract

We present an energy bounded initial boundary value problem (IBVP) for incompressible multi-phase flows that separately conserves mass and kinetic energy. The IBVP combine perfectly with Summation-By-Parts (SBP) operators and weak boundary conditions on Simultaneous Approximation Term (SAT) form and lead to discrete energy stability. We extend the baseline IBVP with a general interface regularization on continuous form that preserves the original energy bound. This extension is valid for all interface-capturing models with the phase advection equation in conservative form, including volume-of-fluid and diffuse interface methods. We demonstrate high-order accuracy, energy stability and conservation of mass and kinetic energy for the non-regularized scheme applied to inviscid flow. After adding interface regularization, we show that the scheme retains these properties for finite-width interfaces. We exemplify the performance of the scheme on an inviscid flow problem with a spinning drop in a closed box and an inviscid inflow-outflow problem where a drop is advected in and out of a domain.