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

## TL;DR

This paper develops divergence-free $H$(div)-conforming finite element methods for time-dependent incompressible flows, demonstrating robustness and accuracy in high Reynolds number vortex dynamics simulations.

## Contribution

It extends divergence-free $H$(div)-conforming FEM to time-dependent flows, including nonlinear Navier-Stokes, with pressure and Reynolds robustness and a novel upwind stabilization.

## Key findings

- Successfully simulates high Reynolds number vortex phenomena
- Proves pressure- and Reynolds-semi-robustness of the method
- Handles complex vortex dynamics reliably

## 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.