An eXtended Discontinuous Galerkin method for three-dimensional two-phase flows: Application to large amplitude oscillations of viscous drops*

We are going to present a high-order eXtended Discontinuous Galerkin (XDG) method for transient two-phase flows. The XDG method adapts the local ansatz functions to conform with the position of the interface and provides separate degrees of freedom for each phase [1,2]. This allows a sub-cell accurate approximation of the incompressible Navier-Stokes equations in their sharp-interface formulation. The interface is defined as the zero-set of a signed-distance level-set function. 

The numerical investigations focus on nonlinear axisymmetric shape oscillations of a drop in a dynamically nearly inert ambient phase. The initial deformation is given by a Legendre polynomial. We compare the numerical results with the analytical results of the weakly nonlinear theory [3]. The properties to be compared include the droplet aspect ratio over time and mode decomposition of the droplet shape. Further, we present the kinetic and surface energy over time for the numerical simulations.