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

Gauger, Nicolas R.; Linke, Alexander; Schroeder, Philipp W.

doi:10.5802/smai-jcm.44

Gauger, Nicolas R.¹ ; Linke, Alexander ² ; Schroeder, Philipp W.³

¹ Chair for Scientific Computing, TU Kaiserslautern, 67663 Kaiserslautern, Germany
² Weierstrass Institute, 10117 Berlin, Germany
³ Institute for Numerical and Applied Mathematics, Georg-August-Universität Göttingen, 37083 Göttingen, Germany

The SMAI Journal of computational mathematics, Tome 5 (2019), pp. 89-129.

Résumé

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 $2 k$ 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.

Publié le : 2019-09-09

Zbl | 2 citations dans Numdam

DOI : 10.5802/smai-jcm.44

Classification : 65M12, 65M15, 65M60, 76D05, 76D10, 76D17
Mots clés : 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

Affiliations des auteurs :

Gauger, Nicolas R. ¹ ; Linke, Alexander ² ; Schroeder, Philipp W. ³

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     journal = {The SMAI Journal of computational mathematics},
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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, Tome 5 (2019), pp. 89-129. doi : 10.5802/smai-jcm.44. http://archive.numdam.org/articles/10.5802/smai-jcm.44/

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