Rare events in the transition to turbulence in shear flows*

Transition to turbulence in shear flows is characterized by intermittent laminar-turbulent patterns that statistically proliferate or collapse depending on the Reynolds number. In pipe, channel or plane Couette flow, turbulence either decays to an absorbing state or proliferates via a process known as splitting. Both are effectively memoryless processes associated with large mean first passage times. The lifetimes depend super-exponentially on the Reynolds number and lead to crossing Reynolds number above which proliferation is more likely than decay. We apply a rare event algorithm, adaptative multi-level splitting (AMS), to the deterministic Navier-Stokes equations to study the transition paths and estimate large time scales that are currently out of reach for direct numerical simulations. Trajectories are selected via an importance function that describes the distance of a flow state to the one-band or two-band attractor. Splitting events approach a most-probable pathway or instanton which paves the way for an out-of-equilibrium description of transition to turbulence. Pathways are intrinsically linked with nucleation processes that approach and leave an edge state in the phase space.