  We consider the 2D incompressible Navier-Stokes equations on
$\mathbb{T}\times \mathbf{R}$, with initial vorticity that is $\delta$ close in
$H^{log}_xL^2_{y}$ to $-1$(the vorticity of the Couette flow $(y,0)$). We prove
that if $\delta\ll \nu^{1/2}$, where $\nu$ denotes the viscosity, then the
solution of the Navier-Stokes equation approaches some shear flow which is also
close to Couette flow for time $t\gg \nu^{-1/3}$ by a mixing-enhanced
dissipation effect and then converges back to Couette flow when $t\to +\infty$.
In particular, we show the nonlinear enhanced dissipation and the inviscid
damping results in the almost critical space $H^{log}_xL^2_{y}\subset
L^2_{x,y}$.
