Does rare, noise-induced, bypass transition in plane Couette flow bypass instantons ?

Abstract

We study rare noise induced paths that go all the way from stable laminar to transitional turbulent plane Couette flow and investigate whether these paths share the properties of classical noise induced transitions. The rare paths from laminar to turbulent flow are computed using Adaptive Multilevel Splitting, a rare event simulation method, and are validated against Direct Numerical Simulations at moderately small energy injection rate. The flow is forced outside its natural scales and redistribute energy to the unforced large scales so that the reactive trajectories display forced streamwise velocity tubes at the natural scale of velocity streaks. As the trajectory proceeds, these tubes gradually grow in amplitude until they cross the separatrix between laminar and turbulent flow. Streamwise vortices are visible after velocity tubes have reached near turbulent amplitude. As the domain size is increased from Lx * Lz=6 *4 to 36*24, spatial localisation then extension of the generated coherent streaks and vortices in the spanwise direction is observed. The paths computed in MFU display many of the characteristics of instantons, that often structure noise induced transitions: concentration of trajectories, exponentially increasing waiting times before transition, and Gumbel distribution of trajectory durations. However, bisections started on the reactive trajectories indicate that the trajectory lack two key ingredients of instantons. They do not visit the neighbourhood of the nearest saddle point and do not display the natural relaxation path from that saddle to transitional wall turbulence. The reactive paths do not concentrate more and more around the same trajectory as energy injection rate is decreased, but instead gradually move in phase space. They might reconnect with instantons at very small energy injection rate and exceedingly long waiting times.