The dynamics of transition to turbulence in plane Couette flow

In plane Couette flow, the incompressible fluid between two plane parallel walls is driven by the motion of those walls. The laminar solution, in which the streamwise velocity varies linearly in the wall-normal direction, is known to be linearly stable at all Reynolds numbers ($Re$). Yet, in both experiments and computations, turbulence is observed for $Re \gtrsim 360$. In this article, we show that for certain {\it threshold} perturbations of the laminar flow, the flow approaches either steady or traveling wave solutions. These solutions exhibit some aspects of turbulence but are not fully turbulent even at $Re=4000$. However, these solutions are linearly unstable and flows that evolve along their unstable directions become fully turbulent. The solution approached by a threshold perturbation could depend upon the nature of the perturbation. Surprisingly, the positive eigenvalue that corresponds to one family of solutions decreases in magnitude with increasing $Re$, with the rate of decrease given by $Re^{\alpha}$ with $\alpha \approx -0.46$.