An Unfitted Higher-Order Discontinuous Galerkin Method for Incompressible Two-Phase Flow with Moving Contact Lines

Abstract

The author presents an unfitted discontinuous Galerkin method for incompressible two-phase flow applicable to dynamic regimes with significant surface tension. The method is suitable for simulations in complex geometries and a recursive algorithm is proposed, which allows the generation of piecewise linear sub-triangulations re- solving both the domain boundaries and the interface between the two immiscible phases. Hence, discontinuous finite element spaces can be employed to capture the irregularities in the solution along the interface, i.e. the jump in the pressure field and in the velocity derivatives. While the sub-triangulation is based on a linear Cartesian cut-cell approach, its resolution is decoupled from the resolution of the finite element mesh thus enabling the application of higher-order finite element spaces. The time development of the two subdomains is realized by level set methods and an unfitted discretization for the solution of the corresponding equations is described. Multiple approaches for the numerical treatment of surface tension in the context of unfitted discretizations are discussed and compared. Furthermore, these methods are extended to allow simulations with contact lines taking into account the occurrence of microscopic deformations of the contact angle. All proposed methods are verified by numerical test simulations in two and three dimensions.