A probabilistic protocol for the assessment of transition and control

Transition to turbulence dramatically alters the properties of fluid flows. In most canonical shear flows, the laminar flow is linearly stable and a finite-amplitude perturbation is necessary to trigger transition. Controlling transition to turbulence is achieved via the broadening or narrowing of the basin of attraction of the laminar flow. In this paper, a novel methodology to assess the robustness of the laminar flow and the efficiency of control strategies is introduced. It relies on the statistical sampling of the phase-space neighbourhood around the laminar flow in order to assess the transition probability of perturbations as a function of their energy. This approach is applied to a canonical flow (plane Couette flow) and provides invaluable insight: in the presence of the chosen control, transition is significantly suppressed whereas plausible scalar indicators of the nonlinear stability of the flow, such as the edge-state energy, do not provide conclusive predictions. The methodology presented here in the context of transition to turbulence is applicable to any nonlinear system displaying finite-amplitude instability.