Slope limiting the velocity field in a discontinuous Galerkin divergence free two-phase flow solver

TL;DR

This paper introduces slope limiting techniques for divergence-free velocity fields in a discontinuous Galerkin solver to effectively handle sharp density discontinuities in two-phase flows, preventing Gibbs oscillations while conserving mass.

Contribution

It presents a divergence-free slope limiter and a simplified scalar component limiter tailored for exactly divergence-free velocity fields in two-phase flow simulations.

Findings

Both limiters outperform naive methods in stabilizing solutions.

The methods effectively handle high density ratios and Reynolds numbers.

Mass conservation and suppression of Gibbs oscillations are achieved.

Abstract

Solving the Navier-Stokes equations when the density field contains a large sharp discontinuity---such as a water/air free surface---is numerically challenging. Convective instabilities cause Gibbs oscillations which quickly destroy the solution. We investigate the use of slope limiters for the velocity field to overcome this problem in a way that does not compromise on the mass conservation properties. The equations are discretised using the interior penalty discontinuous Galerkin finite element method that is divergence free to machine precision. A slope limiter made specifically for exactly divergence free (solenoidal) fields is presented and used to illustrated the difficulties in obtaining convectively stable fields that are also exactly solenoidal. The lessons learned from this are applied in constructing a simpler method based on the use of an existing scalar slope limiter applied to each velocity component. We show by numerical examples how both presented slope limiting methods are vastly superior to the naive non-limited method. The methods can solve difficult two-phase problems with high density ratios and high Reynolds numbers---typical for marine and offshore water/air simulations---in a way that conserves mass and stops unbounded energy growth caused by the Gibbs phenomenon.