Arbitrary Lagrangian-Eulerian finite element approximations for axisymmetric two-phase flow

TL;DR

This paper develops and analyzes ALE finite element methods for simulating axisymmetric two-phase flows, ensuring volume preservation and stability, with numerical tests demonstrating their accuracy and efficiency.

Contribution

It introduces new ALE finite element schemes with volume preservation and stability properties for axisymmetric two-phase flow simulations.

Findings

Methods achieve volume conservation at the discrete level

Numerical results demonstrate high accuracy in bubble and droplet simulations

Proposed schemes are stable and efficient for complex interface dynamics

Abstract

We analyze numerical approximations for axisymmetric two-phase flow in the arbitrary Lagrangian-Eulerian (ALE) framework. We consider a parametric formulation for the evolving fluid interface in terms of a one-dimensional generating curve. For the two-phase Navier-Stokes equations, we introduce both conservative and nonconservative ALE weak formulations in the 2d meridian half-plane. Piecewise linear parametric elements are employed for discretizing the moving interface, which is then coupled to a moving finite element approximation of the bulk equations. This leads to a variety of ALE methods, which enjoy either an equidistribution property or unconditional stability. Furthermore, we adapt these introduced methods with the help of suitable time-weighted discrete normals, so that the volume of the two phases is exactly preserved on the discrete level. Numerical results for rising bubbles and oscillating droplets are presented to show the efficiency and accuracy of these introduced methods.