A Eulerian level set/vortex sheet method for two-phase interface dynamics

A Eulerian fixed grid approach to simulate the dynamics of two-phase interfaces in the presence of surface tension forces is presented. This level set/vortex sheet method consists of a simplified system of equations that contain individual source terms describing the relevant physical processes at the phase interface explicitly. Hence, this approach provides a framework that will allow for a simplified subsequent modeling of phase interface dynamics in turbulent environments. In the presented level set/vortex sheet method, the location and the motion of the phase interface are captured by a level set equation. Topological changes of the interface, like breakup or merging, are thus handled automatically. Assuming that all vorticity is concentrated at the phase interface, the phase interface itself constitutes a vortex sheet with varying vortex sheet strength. The Eulerian transport equation for the vortex sheet strength is derived by combining its Lagrangian formulation with the level set equation. The resulting differential equation then contains source terms accounting for the stretching of the interface and the influence of surface tension, thus allowing for a detailed study of each effect individually. The results of three test problems, namely the roll-up of a vortex sheet without surface tension, the growth of the Kelvin-Helmholtz instability in the linear regime, and the long-time evolution of the Kelvin-Helmholtz instability are presented.