## Divergence-free $$H(\operatorname{div})$$-FEM for time-dependent incompressible flows with applications to high Reynolds number vortex dynamics.(English)Zbl 1392.35210

Summary: In this article, we consider exactly divergence-free $$H(\operatorname{div})$$-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 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, $$H(\operatorname{div})$$-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.

### MSC:

 35Q30 Navier-Stokes equations 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 76D17 Viscous vortex flows 76M10 Finite element methods applied to problems in fluid mechanics 76E06 Convection in hydrodynamic stability 76D05 Navier-Stokes equations for incompressible viscous fluids
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