On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond

TL;DR

This paper demonstrates that pressure-robust space discretisations significantly improve the accuracy of high Reynolds number flow simulations, outperforming non-pressure-robust methods, especially on coarse meshes, and establishing pressure-robustness as essential for reliable incompressible flow modeling.

Contribution

It introduces and compares pressure-robust and non-pressure-robust discretisations, showing pressure-robust methods are more accurate for high Reynolds flows and are a necessary condition for precise simulations.

Findings

Pressure-robust methods outperform non-pressure-robust methods on coarse meshes.

Pressure-robust methods of order k match the accuracy of non-pressure-robust methods of order 2k.

Strong pressure gradients are typical in high Reynolds number flows, emphasizing pressure-robustness importance.

Abstract

An improved understanding of the divergence-free constraint for the incompressible Navier--Stokes equations leads to the observation that a semi-norm and corresponding equivalence classes of forces are fundamental for their nonlinear dynamics. The recent concept of {\em pressure-robustness} allows to distinguish between space discretisations that discretise these equivalence classes appropriately or not. This contribution compares the accuracy of pressure-robust and non-pressure-robust space discretisations for transient high Reynolds number flows, starting from the observation that in generalised Beltrami flows the nonlinear convection term is balanced by a strong pressure gradient. Then, pressure-robust methods are shown to outperform comparable non-pressure-robust space discretisations. Indeed, pressure-robust methods of formal order $k$ are comparably accurate than non-pressure-robust methods of formal order $2k$ on coarse meshes. Investigating the material derivative of incompressible Euler flows, it is conjectured that strong pressure gradients are typical for non-trivial high Reynolds number flows. Connections to vortex-dominated flows are established. Thus, pressure-robustness appears to be a prerequisite for accurate incompressible flow solvers at high Reynolds numbers. The arguments are supported by numerical analysis and numerical experiments.