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ZblAn 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 are comparably accurate than non-pressure-robust methods of formal order 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.
DOI: 10.5802/smai-jcm.44
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
Gauger, Nicolas R. 1 ; Linke, Alexander 2 ; Schroeder, Philipp W. 3
Gauger, Nicolas R.; Linke, Alexander; Schroeder, Philipp W. On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond. The SMAI Journal of computational mathematics, Volume 5 (2019), pp. 89-129. doi: 10.5802/smai-jcm.44
@article{SMAI-JCM_2019__5__89_0,
author = {Gauger, Nicolas R. and Linke, Alexander and Schroeder, Philipp W.},
title = {On high-order pressure-robust space discretisations, their advantages for incompressible high {Reynolds} number generalised {Beltrami} flows and beyond},
journal = {The SMAI Journal of computational mathematics},
pages = {89--129},
year = {2019},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {5},
doi = {10.5802/smai-jcm.44},
zbl = {07090176},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.44/}
}
TY - JOUR AU - Gauger, Nicolas R. AU - Linke, Alexander AU - Schroeder, Philipp W. TI - On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond JO - The SMAI Journal of computational mathematics PY - 2019 SP - 89 EP - 129 VL - 5 PB - Société de Mathématiques Appliquées et Industrielles UR - http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.44/ DO - 10.5802/smai-jcm.44 LA - en ID - SMAI-JCM_2019__5__89_0 ER -
%0 Journal Article %A Gauger, Nicolas R. %A Linke, Alexander %A Schroeder, Philipp W. %T On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond %J The SMAI Journal of computational mathematics %D 2019 %P 89-129 %V 5 %I Société de Mathématiques Appliquées et Industrielles %U http://geodesic.mathdoc.fr/articles/10.5802/smai-jcm.44/ %R 10.5802/smai-jcm.44 %G en %F SMAI-JCM_2019__5__89_0
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