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arXiv · 1705.10176

Divergence-free $H$(div)-FEM for time-dependent incompressible flows with applications to high Reynolds number vortex dynamics

Philipp W. Schroeder·Gert Lube
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Abstract

In this article, we consider exactly divergence-free $H$(div)-conforming finite element methods for time-dependent incompressible viscous flow problems. This is an extension of previous research concerning divergence-free $H^1$-conforming methods. For the linearised Oseen case, the first semi-discrete numerical analysis for time-dependent flows is presented here whereby special emphasis is put on pressure- and Reynolds-semi-robustness. For convection-dominated problems, the proposed method relies on a velocity jump upwind stabilisation which is not gradient-based. Complementing the theoretical results, $H$(div)-FEM are applied to the simulation of full nonlinear Navier-Stokes problems. Focussing on dynamic high Reynolds number examples with vortical structures, the proposed method proves to be capable of reliably handling the planar lattice flow problem, Kelvin-Helmholtz instabilities and freely decaying two-dimensional turbulence.

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Philipp W. Schroeder, Gert Lube. 2017-09-30. Divergence-free $H$(div)-FEM for time-dependent incompressible flows with applications to high Reynolds number vortex dynamics. https://doi.org/10.1007/s10915-017-0561-1

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