A higher-order finite element method with unstructured anisotropic mesh adaption for two phase flows with surface tension

TL;DR

This paper introduces a high-order finite element method with anisotropic mesh adaptation for simulating two-phase flows with surface tension, achieving higher accuracy and optimal convergence rates near interfaces.

Contribution

It develops a fully implicit, stabilized finite element framework with anisotropic mesh adaptation for improved simulation of two-phase flows with surface tension.

Findings

Achieves higher-order convergence rates in smooth interface regions.

Demonstrates improved accuracy and efficiency in 2D and 3D bubble simulations.

Maintains mesh complexity by adaptive element stretching near interfaces.

Abstract

A novel finite element framework is proposed for the numerical simulation of two phase flows with surface tension. The Level-Set (LS) method with piece-wise quadratic (P2) interpolation for the liquid-gas interface is used in order to reach higher-order convergence rates in regions with smooth interface. A balanced-force implementation of the continuum surface force model is used to take into account the surface tension and to solve static problems as accurately as possible. Given that this requires a balance between the discretization used for the LS function, and that used for the pressure field, an equal-order P2/P2/P2 scheme is proposed for the Navier-Stokes and LS advection equations, which are strongly coupled with each other. This fully implicit formulation is stabilized using the residual-based variational multiscale framework. In order to improve the accuracy and obtain optimal convergence rates with a minimum number of elements, an anisotropic mesh adaption method is proposed where the unstructured mesh is kept as fine as possible close to the zero iso-value of the P2 LS function. Elements are automatically stretched in regions with flat interface in order to keep the complexity fixed during the simulation. The accuracy and efficiency of this approach are demonstrated for two and three dimensional simulations of a rising bubble.