  We prove that some discretization schemes for the 2D Navier-Stokes equations
subject to a random perturbation converge in $L^2(\Omega)$. This refines
previous results which only established the convergence in probability of these
numerical approximations. Using exponential moment estimates of the solution of
the stochastic NS equations and convergence of a localized scheme, we can prove
strong convergence of fully implicit and semi-implicit time Euler
discretizations, and of a splitting scheme. The speed of the
$L^2(\Omega)$-convergence depends on the diffusion coefficient and on the
viscosity parameter.
