Localized Degenerate Control of Navier-Stokes

For incompressible Navier-Stokes on $\mathbb{T}^d$, $d=2,3$, is the system approximately controllable and/or controllable in finite-dimensional projections by a localized degenerate forcing, where the control space $E$ is a finite-dimensional subspace of $\{u\in V:\operatorname{supp}u\subset\overline{\mathcal D}\}$? Construct such an $E$ independently of the viscosity $\nu$.