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An improved understanding of the divergence-free constraint for theincompressible Navier--Stokes equations leads to the observation that asemi-norm and corresponding equivalence classes of forces are fundamental fortheir nonlinear dynamics. The recent concept of {\em pressure-robustness}allows to distinguish between space discretisations that discretise theseequivalence classes appropriately or not. This contribution compares theaccuracy of pressure-robust and non-pressure-robust space discretisations fortransient high Reynolds number flows, starting from the observation that ingeneralised Beltrami flows the nonlinear convection term is balanced by astrong pressure gradient. Then, pressure-robust methods are shown to outperformcomparable non-pressure-robust space discretisations. Indeed, pressure-robustmethods of formal order $k$ are comparably accurate than non-pressure-robustmethods of formal order $2k$ on coarse meshes. Investigating the materialderivative of incompressible Euler flows, it is conjectured that strongpressure 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 incompressibleflow solvers at high Reynolds numbers. The arguments are supported by numericalanalysis and numerical experiments.

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