From temporal to spatiotemporal dynamics in transitional plane Couette flow.

Laboratory experiments point out the existence of patterns made of alternately laminar and turbulent oblique bands in plane Couette flow (PCF) on its way to or 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}~70-80 h (where 2 h is the gap between the plates driving the flow) beyond which the transition to or 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 the 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.