An extended residual-based variational multiscale method for two-phase flow including surface tension

In this study, an extended residual-based variational multiscale method is proposed for two-phase flow including surface tension. The extended residual-based variational multiscale method combines a residual-based form of the variational multiscale method and the extended finite element method (XFEM). By extending the solution spaces, it is possible to reproduce discontinuities of the solution fields inside elements intersected by the interface. In particular, we propose a quasi-static enrichment to reproduce time-dependent discontinuities. Kink enrichments of both velocity and pressure as well as kink enrichments of velocity combined with jump enrichments of pressure are considered here. To capture the interface between the phases on a fixed grid, a level-set approach is used. A residual-based variational multiscale method is employed for computing both flow and interface motion. The presented method is tested for various two-phase flow examples exhibiting small and large density and viscosity ratios, with and without surface tension: a two-phase Couette flow, a Rayleigh–Taylor instability, a sloshing tank and a three-dimensional rising bubble. To the best of our knowledge, these are the first simulation results for representative time-dependent three-dimensional two-phase flow problems using an extended finite element method. Stable and accurate results are obtained for all test examples.