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Nicolas R. Gauger; Alexander Linke; Philipp W. Schroeder
On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond
SMAI-Journal of computational mathematics, 5 (2019), p. 89-129, doi: 10.5802/smai-jcm.44
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Class. Math.: 65M12, 65M15, 65M60, 76D05, 76D10, 76D17
Keywords: incompressible Navier–Stokes, pressure-robust methods, Helmholtz–Hodge projector, Discontinuous Galerkin method, divergence-free $H$(div) finite elements, structure-preserving algorithms, high-order methods, (generalised) Beltrami flows, high Reynolds number flows, material derivative

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 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.

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