First, we consider Kolmogorov flow (a shear flow with a sinusoidal velocity profile) for 2D Navier-Stokes equation on a torus. Such flows, also called bar states, have been numerically observed as one type of metastable states in the study of 2D turbulence. For both rectangular and square tori, we prove that the non-shear part of perturbations near Kolmogorov flow decays in a time scale much shorter than the viscous time scale. The results are obtained for both the linearized NS equations with any initial vorticity in L^2, and the nonlinear NS equation with initial L^2 norm of vorticity of the size of viscosity. In the proof, we use the Hamiltonian structure of the linearized Euler equation and RAGE theorem to control the low frequency part of the perturbation. Second, we consider two classes of shear flows for which a sharp stability criterion is known. We show the inviscid damping in a time average sense for non-shear perturbations with initial vorticity in L^2. For the unstable case, the inviscid damping is proved on the center space. Our proof again uses the Hamiltonian structure of the linearized Euler equation and an instability index theory recently developed by Lin and Zeng for Hamiltonian PDEs.
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