Dynamics Near the Subcritical Transition of the 3D Couette Flow II: Above Threshold Case

This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number Re . In this work, we show that there is constant 0 > c 0 ≪ 1 0 > c_0 1 , independent of R e Re , such that sufficiently regular disturbances of size ϵ ≲ R e − 2 / 3 − δ ε Re-2/3-δ for any δ > 0 δ > 0 exist at least until t = c 0 ϵ − 1 t = c_0ε ⁻¹ and in general evolve to be O ( c 0 ) O(c_0) due to the lift-up effect. Further, after times t ≳ R e 1 / 3 t Re1/3 , the streamwise dependence of the solution is rapidly diminished by a mixing-enhanced dissipation effect and the solution is attracted back to the class of “2.5 dimensional” streamwise-independent solutions (sometimes referred to as “streaks”). The largest of these streaks are expected to eventually undergo a secondary instability at t ≈ ϵ − 1