# FLOW: Studies of transition in Couette flow

Plane Couette flow, the flow between two parallel walls moving in opposite directions, is the simplest canonical example of the effect of shear on a viscous fluid. The only non-dimensional parameter ruling the flow is the Reynolds number, here defined as Re =Uh/ν, where ±U is the velocity of the two walls, h is the half-gap between them and ν is the kinematic viscosity of the fluid.

We are interested in the way sustained turbulence appears in this system around the onset of laminar-turbulent transition. So-called subcritical transition to turbulence occurs in a variety of wall-bounded shear flows when the base flow is linearly stable: On one hand, all experiments and our own simulations have reported that transition occurs for Re about 320,provided the amplitude of the initial disturbance exceeds a critical threshold. The transitional flow is then characterised by the co-existence of laminar and turbulent regions (laminar-turbulent patterns), delimited by sharp fronts travelling at a fixed velocity.These spatially localized structures are called turbulent spots and grow in time. This can be reproduced also by numerical time-integration of the incompressible Navier–Stokes equations, provided the size L of the periodic domain is large enough,typically L on the order of 100h. On the other hand, the local dynamics of the flow is starting to be better understood, based on numerical simulations in small periodic computational domains, with L of order 5h. Major progress has been the discovery of exact coherent states, either steady states,travelling wavesor periodic orbits.These states are all unstable, and they result from the nonlinear balance between streamwise rolls and streamwise streaks with an axial modulation,which are also local features of the turbulent flow.Some of these states sit on the separatrix between the basins of attraction of the laminar and the turbulent state,in the associated phase-space. This separatrix is referred to as the laminar-turbulent boundary S, and is an invariant manifold of the phase-space. A trajectory on Sis called an edge trajectory and it is believed to reach a relative attractor.It was demonstrated that various exact coherent states are approached or visited transiently by edge trajectories, and hence by transitional trajectories just above the critical threshold.However, the scenario for transition in such geometrically constrained domains is not sufficient to explain the spatiotemporal behaviour of spots observed experimentally.Here we extend the aforementioned dynamical system picture of transition in simple shear flows to larger domains, allowing for spatial localisation of incipient turbulent spots. These scenarios are expected be relevant for other shear flows where transition to turbulence is both subcritical and spatially intermittent, such as circular pipe flow or plane Poiseuille flow.

Y. Duguet, P. Schlatter, and D. S. Henningson. Formation of turbulent patterns near the onset of transition in plane Couette flow. J. Fluid Mech., 650:119–129, 2010.

Y. Duguet, P. Schlatter, and D. S. Henningson. Localized edge states in plane Couette flow. Phys. Fluids, 21(111701):1–4, 2009.