A new formulation of Discontinuous Galerkin method for interface capturing approaches

In computational fluid dynamics, the numerical simulation of free-surface flows using interface capturing approaches (level set, pseudo-concentration) requires accurate resolution of a transport equation. In the present work, this equation is solved using a new formulation of the Lesaint–Raviart discontinuous Galerkin method, which is particularly suitable for problems without inflow boundaries. The motion is described by the incompressible Navier–Stokes equations with surface tension, discretized using the standard $P2-P1$ Taylor–Hood finite element method in space. Surface tension effects are represented using the so-called Continuum Surface Force model. Finally, the proposed approach is implemented and tested on several benchmarks, including the single rising bubble and the Rayleigh–Taylor instability.