Stochastic motion of a laminar/turbulent interface in a shear flow

Abstract

Plane Couette flow is a classical prototype of a shear flow where transition to turbulence is subcritical, i.e. happens despite linear stability of the base flow. In this study we are interested in the spatio-temporal competition between the (active) turbulent phase and the (absorbing) laminar. Our three-dimensional numerical simulations show that the delimiting interface, when parallel to the streamwise direction, moves in a stochastic manner which we model as a continuous-time random walk. Statistical analysis suggests a Gaussian diffusion process and allows us to determine the average speed of this interface as a function of the Reynolds number Re, as well as the threshold in Re above which turbulence contaminates the whole domain. For the lowest value of Re, this stochastic motion competes with another deterministic regime of growth of the localised perturbations. The latter, a rather unexpected regime, is shown to be linked to the recently found localised snaking solutions of the Navier-Stokes equations. An extension of this thinking to more general orientations of the interfaces will be proposed.