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

摘要

In this article, we consider exactly divergence-free $H$(div)-conformingfinite 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 firstsemi-discrete numerical analysis for time-dependent flows is presented herewhereby special emphasis is put on pressure- and Reynolds-semi-robustness. Forconvection-dominated problems, the proposed method relies on a velocity jumpupwind stabilisation which is not gradient-based. Complementing the theoreticalresults, $H$(div)-FEM are applied to the simulation of full nonlinearNavier-Stokes problems. Focussing on dynamic high Reynolds number examples withvortical structures, the proposed method proves to be capable of reliablyhandling the planar lattice flow problem, Kelvin-Helmholtz instabilities andfreely decaying two-dimensional turbulence.