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 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
Classification: 65M12, 65M15, 65M60, 76D05, 76D10, 76D17
Keywords: incompressible Navier–Stokes, pressure-robust methods, Helmholtz–Hodge projector, Discontinuous Galerkin method, divergence-free (div) finite elements, structure-preserving algorithms, high-order methods, (generalised) Beltrami flows, high Reynolds number flows, material derivative
@article{SMAI-JCM_2019__5__89_0, author = {Nicolas R. Gauger and Alexander Linke and Philipp W. Schroeder}, 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}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {5}, year = {2019}, doi = {10.5802/smai-jcm.44}, zbl = {07090176}, language = {en}, url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.44/} }
TY - JOUR 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 DA - 2019/// SP - 89 EP - 129 VL - 5 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.44/ UR - https://zbmath.org/?q=an%3A07090176 UR - https://doi.org/10.5802/smai-jcm.44 DO - 10.5802/smai-jcm.44 LA - en ID - SMAI-JCM_2019__5__89_0 ER -
%0 Journal Article %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 https://doi.org/10.5802/smai-jcm.44 %R 10.5802/smai-jcm.44 %G en %F SMAI-JCM_2019__5__89_0
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. The SMAI journal of computational mathematics, Volume 5 (2019), pp. 89-129. doi : 10.5802/smai-jcm.44. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.44/
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