Compressive advection and multi-component methods for interface-capturing

This paper develops methods for interface-capturing in multi-phase flows. The main novelties of these methods are: (a) multi-component modelling that embeds interface structures into the continuity equation; (b) a new family of triangle/tetrahedron finite elements, in particular the P1DG − P2 (linear discontinuous between elements velocity and quadratic continuous pressure); (c) an interface-capturing scheme based on compressive control volume advection methods and high-order finite element interpolation methods; (d) a time stepping method that allows use of relatively large time step sizes; and (e) application of anisotropic mesh adaptivity to focus the numerical resolution around the interfaces and other areas of important dynamics. This modelling approach is applied to a series of pure advection problems with interfaces as well as to the simulation of the standard computational fluid dynamics benchmark test cases of a collapsing water column under gravitational forces (in two and three dimensions) and sloshing water in a tank. Two more test cases are undertaken in order to demonstrate the many-material and compressibility modelling capabilities of the approach. Numerical simulations are performed on coarse unstructured meshes to demonstrate the potential of the methods described here to capture complex dynamics in multi-phase flows.