  We introduce a residual-based stabilized formulation for incompressible
Navier-Stokes flow that maintains discrete (and, for divergence-conforming
methods, strong) mass conservation for inf-sup stable spaces with
$H^1$-conforming pressure approximation, while providing optimal convergence in
the diffusive regime, robustness in the advective regime, and energetic
stability. The method is formally derived using the variational multiscale
(VMS) concept, but with a discrete fine-scale pressure field which is solved
for alongside the coarse-scale unknowns such that the coarse and fine scale
velocities separately satisfy discrete mass conservation. We show energetic
stability for the full Navier-Stokes problem, and we prove convergence and
robustness for a linearized model (Oseen flow), under the assumption of a
divergence-conforming discretization. Numerical results indicate that all
properties extend to the fully nonlinear case and that the proposed formulation
can serve to model unresolved turbulence.
