Structure-preserving discretizations of two-phase Navier-Stokes flow using fitted and unfitted approaches

TL;DR

This paper develops structure-preserving finite element methods for two-phase Navier-Stokes flow, ensuring volume conservation and energy decay, with both fitted and unfitted approaches, validated by numerical experiments.

Contribution

It introduces novel fitted and unfitted finite element methods that preserve key physical properties for two-phase flow simulations.

Findings

Methods preserve volume and energy decay at discrete level.

Numerical results demonstrate high accuracy and efficiency.

Fitted and unfitted approaches are both effective for interface evolution.

Abstract

We consider the numerical approximation of a sharp-interface model for two-phase flow, which is given by the incompressible Navier-Stokes equations in the bulk domain together with the classical interface conditions on the interface. We propose structure-preserving finite element methods for the model, meaning in particular that volume preservation and energy decay are satisfied on the discrete level. For the evolving fluid interface, we employ parametric finite element approximations that introduce an implicit tangential velocity to improve the quality of the interface mesh. For the two-phase Navier-Stokes equations, we consider two different approaches: an unfitted and a fitted finite element method, respectively. In the unfitted approach, the constructed method is based on an Eulerian weak formulation, while in the fitted approach a novel arbitrary Lagrangian-Eulerian (ALE) weak formulation is introduced. Using suitable discretizations of these two formulations, we introduce two finite element methods and prove their structure-preserving properties. Numerical results are presented to show the accuracy and efficiency of the introduced methods.