  This paper considers the discretization of the time-dependent Navier-Stokes
equations with the family of inf-sup stabilized Scott-Vogelius pairs recently
introduced in [John/Li/Merdon/Rui, arXiv:2206.01242, 2022] for the Stokes
problem. Therein, the velocity space is obtained by enriching the
H^1-conforming Lagrange element space with some H(div)-conforming
Raviart-Thomas functions, such that the divergence constraint is satisfied
exactly. In these methods arbitrary shape-regular simplicial grids can be used.
  In the present paper two alternatives for discretizing the convective terms
are considered. One variant leads to a scheme that still only involves volume
integrals, and the other variant employs upwinding known from DG schemes. Both
variants ensure the conservation of linear momentum and angular momentum in
some suitable sense. In addition, a pressure-robust and convection-robust
velocity error estimate is derived, i.e., the velocity error bound does not
depend on the pressure and the constant in the error bound for the kinetic
energy does not blow up for small viscosity. After condensation of the
enrichment unknowns and all non-constant pressure unknowns, the method can be
reduced to a $P_k-P_0$-like system for arbitrary velocity polynomial degree
$k$. Numerical studies verify the theoretical findings.
