A Stable Parametric Finite Element Discretization of Two-Phase Navier--Stokes Flow

TL;DR

This paper introduces a stable, parametric finite element method for simulating two-phase Navier--Stokes flow, ensuring good volume conservation, mesh quality, and stability, validated through numerical experiments in 2D and 3D.

Contribution

It presents a novel variational formulation and a coupled finite element scheme that improves interface evolution accuracy and stability in two-phase flow simulations.

Findings

Unconditionally stable finite element scheme.

Good volume conservation properties.

Mesh quality remains high over time.

Abstract

We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Navier--Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. The parametric finite element approximation of the evolving interface is then coupled to a standard finite element approximation of the two-phase Navier--Stokes equations in the bulk. Here enriching the pressure approximation space with the help of an XFEM function ensures good volume conservation properties for the two phase regions. In addition, the mesh quality of the parametric approximation of the interface in general does not deteriorate over time, and an equidistribution property can be shown for a semidiscrete continuous-in-time variant of our scheme in two space dimensions. Moreover, our finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results in two and three space dimensions.