Direct Numerical Simulation of Multi-Phase Flows using Extended Discontinuous Galerkin Methods

The scientific study of multi-phase flows is a challenging task for analytical and experimental works. Thus, sophisticated and specialized numerical methods are in need for the direct numerical simulation of such problems. In this work a high-order multi-phase flow solver on the basis of the extended Discontinuous Galerkin (extended DG/XDG) method is developed. This allows the direct numerical simulation of the transient incompressible two-phase Navier-Stokes equations in their sharp interface formulation. The approximation space of the local ansatz-functions is adapted to be conform to the position of the interface. The interface, described as a level-set function, is discretized by a standard DG method that enables a sub-cell accurate representation of sharp jumps in the pressure field and kinks in the velocity field. For the numerical treatment of the surface tension force the Laplace-Beltrami formulation without regularization is implemented. Stability issues regarding the energy conservation of the solver are addressed. The developed solver is validated against a wide range of typical two-phase surface tension driven flow phenomena including capillary waves, an oscillating droplet and a rising bubble. Allowing the simulation of dynamic contact line problems, the generalized Navier boundary condition is adapted for the XDG discretization. The results regarding the rise of liquid in a capillary build the basis of a new benchmark setup for capillarity driven flow problems. Another extension of the solver is the implementation of the coupled two-phase heat equation in context of the XDG method. Furthermore, the discretization for both the Navier-Stokes equations and the heat equation is extended to allow a mass and energy flow across the interface. This way the velocity field exhibits a sharp jump and the temperature field shows a kink at the interface. A first basic validation is provided against analytical solutions. This work presents a multi-purpose flow solver for the direct simulation of multi-phase flows involving dynamic contact lines and phase changes due to evaporation. It is based on the XDG method to allow a sub-cell accurate approximation in context of the sharp interface formulation.