  We study the incompressible Navier-Stokes equations in the two-dimensional
strip $\mathbb{R} \times [0,L]$, with periodic boundary conditions and no
exterior forcing. If the initial velocity is bounded, we prove that the
solution remains uniformly bounded for all times, and that the vorticity
distribution converges to zero as $t \to \infty$. We deduce that, after a
transient period, a laminar regime emerges in which the solution rapidly
converges to a shear flow governed by the one-dimensional heat equation. Our
approach is constructive and gives explicit estimates on the size of the
solution and the lifetime of the turbulent period in terms of the initial
Reynolds number.
