A cut finite element method for incompressible two-phase Navier-Stokes flows

TL;DR

This paper introduces a space-time Cut Finite Element Method for simulating two-phase incompressible Navier-Stokes flows, accurately capturing interface discontinuities without re-meshing, and ensuring stability through stabilization techniques.

Contribution

It develops a novel space-time CutFEM that handles evolving interfaces in two-phase flows without re-meshing and introduces a stabilized high-order surface tension force computation.

Findings

Accurately captures pressure and velocity discontinuities across interfaces.

Maintains stability and good condition number regardless of interface position.

Effective in 2D and 3D simulations with complex interface dynamics.

Abstract

We present a space-time Cut Finite Element Method (CutFEM) for the time-dependent Navier-Stokes equations involving two immiscible incompressible fluids with different viscosities, densities, and with surface tension. The numerical method is able to accurately capture the strong discontinuity in the pressure and the weak discontinuity in the velocity field across evolving interfaces without re-meshing processes or regularization of the problem. We combine the strategy proposed in [P. Hansbo, M. G. Larson, S. Zahedi, Appl. Numer. Math. 85 (2014), 90--114] for the Stokes equations with a stationary interface and the space-time strategy presented in [P. Hansbo, M. G. Larson, S. Zahedi, Comput. Methods Appl. Mech. Engrg. 307 (2016), 96--116]. We also propose a strategy for computing high order approximations of the surface tension force by computing a stabilized mean curvature vector. The presented space-time CutFEM uses a fixed mesh but includes stabilization terms that control the condition number of the resulting system matrix independently of the position of the interface, ensure stability and a convenient implementation of the space-time method based on quadrature in time. Numerical experiments in two and three space dimensions show that the numerical method is able to accurately capture the discontinuities in the pressure and the velocity field across evolving interfaces without requiring the mesh to be conformed to the interface and with good stability properties.