Discontinuous Galerkin methods for two-phase flows with soluble surfactants

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The general research objective is to develop and implement a high-accurate numerical solver for the computation of multiphase flows including phase interfaces and interfacial transport equations with the Discontinuous Galerkin (DG) method. Specifically, it is intended to extend the newly developed DG library BoSSS (Bounded Support Spectral Solver) for single phase flows by an interface tracking method combined with the cut-cell method and non-smooth basis functions. The cut-cell method is proposed to accurately compute surface and volume integrals for those cells which are split by the phase interface. Further the non-smooth basis functions such as the Heaviside function allow for a sharp delimitation between different fluid properties. In this connection the interface tracking method is based on level-set and volume-of-fluid, or most likely a combination of both, to ensure both local mass preservation and a highly accurate front position. Finally, the interfacial equation is intended to be solved in an analogous fashion to the level-set formulation where the interface, a two-dimensional manifold, is embedded and solved in a manifold of dimension three. Similar to the signed-distance function for the level-set method certain embedding conditions are to be developed to couple the interfacial transport equations, which live on a two-dimensional manifold, into a three-dimensional manifold. The key goal is that the combination of the latter methods may avoid serious shortcomings of classical schemes such as mass deficit at the interface or artificially induced flows due to numerical errors in the computation of surface curvature to name only a few.