A hybrid level set/front tracking approach for finite element simulations of two-phase flows

In this paper we give details on the numerical realization of a new finite element method for the simulation of two‐phase flows which was recently introduced in Basting and Weismann (2013). The main ingredient is a hybrid representation of the interface between the fluid phases: An implicit description of the interface is given by a level set function and an explicit representation is obtained from aligning edges of the computational mesh to the implicitly described interface. This step is done by a black‐box optimization based mesh smoothing approach which does not change the topology of the mesh while guaranteeing optimal mesh quality. Furthermore, we make use of quadratic isoparametric elements to increase the approximation quality of the discrete interface. Due to the alignment, discontinuities of the solution variables (pressure) can be captured accurately, while a variational treatment of the curvature allows for a precise approximation of surface tension. We present our time discretization scheme for the coupled Navier–Stokes/level set equations, and discuss our space discretization based on the so called subspace projection method (SPM) to account for discontinuities across the interface. We present two numerical examples for which reference solutions exist. We consider the oscillation of a single droplet and provide our results for an established two‐phase flow benchmark problem.