A high-order discontinuous Galerkin pressure robust splitting scheme for incompressible flows

TL;DR

This paper introduces a high-order discontinuous Galerkin splitting scheme that is pressure robust and divergence-free, improving the accuracy and stability of simulations for high Reynolds number incompressible flows.

Contribution

It presents a novel Helmholtz flux postprocessing technique in projection methods that achieves pointwise divergence-free velocity fields and pressure robustness in high-order DG schemes.

Findings

The new scheme is pressure robust and divergence-free.

Numerical experiments confirm high-order accuracy.

The method effectively simulates underresolved turbulent flows.

Abstract

The accurate numerical simulation of high Reynolds number incompressible flows is a challenging topic in computational fluid dynamics. Classical inf-sup stable methods like the Taylor-Hood element or only $L^2$-conforming discontinuous Galerkin (DG) methods relax the divergence constraint in the variational formulation. However, unlike divergence-free methods, this relaxation leads to a pressure-dependent contribution in the velocity error which is proportional to the inverse of the viscosity, thus resulting in methods that lack pressure robustness and have difficulties in preserving structures at high Reynolds numbers. The present paper addresses the discretization of the incompressible Navier-Stokes equations with high-order DG methods in the framework of projection methods. The major focus in this article is threefold: i) We present a novel postprocessing technique in the projection step of the splitting scheme that reconstructs the Helmholtz flux in $H(\text{div})$. In contrast to the previously introduced $H(\text{div})$ postprocessing technique, the resulting velocity field is pointwise divergence-free in addition to satisfying the discrete continuity equation. ii) Based on this Helmholtz flux $H(\text{div})$ reconstruction, we obtain a high order in space, pressure robust splitting scheme as numerical experiments in this paper demonstrate. iii) With this pressure robust splitting scheme, we demonstrate that a robust DG method for underresolved turbulent incompressible flows can be realized.