Development of a finite element/discontinuous Galerkin/level set approach for the simulation of incompressible two phase flow

In this paper, we present a combined finite element/ discontinuous Galerkin/ level set method to simulate the incompressible two phase flow. The level set method is employed to capture the moving interface because of its simplicity and efficiency when dealing with the significant interface deformations. Due to the hyperbolic nature of the level set equation and the level set re-initialization equation, we apply the second order Runge Kutta Discontinuous Galerkin (RKDG) method to get the stable and precise results. Moreover, in order to obtain the accurate velocity field, the hybrid continuous and discontinuous Galerkin approach is utilized to solve the incompressible Navier-Stokes equations. In our combined method, there is no need to re-initialize the level set function in every time step and the re-initialization process is implemented after suitable time steps. The desirable mass conservation property is able to be guaranteed with a mass correction technique in the combined algorithm. In addition, the stabilization terms can be avoided in the whole computational process. Several challenging problems, i.e., a rising bubble, the dam-break flow, the Rayleigh-Taylor instability, and the complex metal casting process in the engineering applications are investigated to evaluate the feasibility, validity and practicability of our approach for solving the problem of the incompressible two phase flow.