From temporal to spatiotemporal dynamics in transitional plane Couette flow

Abstract: Laboratory experiments point out the existence of patterns made of alternately laminar and turbulent oblique bands in plane Couette flow in its way to/from turbulence as the Reynolds number R is varied. Many previous theoretical and numerical works on the problem have considered small aspect-ratio systems subjected to periodic boundary conditions, while experiments correspond to the opposite limit of large aspect-ratio. Here, by means of fully resolved direct numerical simulations of the Navier-Stokes equations at decreasing R, we scrutinize the transition from temporal to spatiotemporal behavior in systems of intermediate sizes. We show that there exists a streamwise crossover size of order L_x~70-80h (where 2h is the gap between the plates driving the flow) beyond which the transition to/from turbulence in PCF is undoubtedly a spatiotemporal process, with typical scenario [turbulent flow]->[riddled regime]->[oblique pattern]->[laminar flow]', whereas below that size it is more a temporal process describable in terms of finite-dimensional dynamical systems with scenario[chaotic flow]->[laminar flow]' (via chaotic transients). In the crossover region, the `[oblique pattern]' stage is skipped, which leads us to suggest that an appropriate rendering of the patterns observed in experiments needs a faithful account of streamwise correlations at scales at least of the order of that crossover size.