Rare, noise-induced, bypass transition in plane Couette flow can bypass instantons.

Abstract

This paper presents a study of rare noise-induced transitions from stable laminar flow to transitional turbulence in plane Couette flow, which we will term buildup. We wish to study forced paths that go all the way from laminar to turbulent flow and to focus the investigation on whether these paths share the properties of noise-induced transitions in simpler systems. The forcing noise has a red spectrum without any component in the natural, large-scale, linear receptivity range of the flow. As we decreased the forcing energy injection rate, the transitions became rare. 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 rates. On the computed trajectories, the flow manages to nonlinearly redistribute energy from the small forced scales 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 manifest themselves only after velocity tubes have reached near-turbulent amplitude, displaying a two-stage process reminiscent of the "backward" path from turbulence to laminar flow. We checked that these were not time-reversed turbulence collapse paths. As the domain size is increased from a minimal flow unit (MFU) type flow at L_{x}×L_{z}=6×4 (in half gap units) to a large domain L_{x}×L_{z}=36×24, spatial localization and then extension of the generated coherent streaks and vortices in the spanwise direction is observed in the reactive paths. The paths systematically computed in MFU display many of the characteristics of instantons that often structure noise-induced transitions: such as concentration of trajectories, exponentially increasing waiting times before transition, and Gumbel distribution of trajectory durations. However, bisections started from successive states on the reactive trajectories indicate that for all sizes and energy injection rates investigated, the trajectory lacks two key ingredients of instantons. First, they do not visit the neighborhood of the nearest saddle point and do not display the natural relaxation path from that saddle to transitional wall turbulence. This discrepancy is observed for all system sizes. Second, 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. They would explain why classical instanton calculations have proved to be tremendously difficult in wall flows.