  Discretization of Navier-Stokes' equations using pressure-robust finite
element methods is considered for the high Reynolds number regime. To counter
oscillations due to dominating convection we add a stabilization based on a
bulk term in the form of a residual-based least squares stabilization of the
vorticity equation supplemented by a penalty term on (certain components of)
the gradient jump over the elements faces. Since the stabilization is based on
the vorticity equation, it is independent of the pressure gradients, which
makes it pressure-robust. Thus, we prove pressure-independent error estimates
in the linearized case, known as Oseen's problem. In fact, we prove an
$O(h^{k+\frac12})$ error estimate in the $L^2$-norm that is known to be the
best that can be expected for this type of problem. Numerical examples are
provided that, in addition to confirming the theoretical results, show that the
present method compares favorably to the classical residual-based SUPG
stabilization.
