  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.
