Stability analysis of perturbed plane Couette flow

Plane Couette flow perturbed by a spanwise oriented ribbon, similar to a configuration investigated experimentally at the Centre d'Etudes de Saclay, is investigated numerically using a spectral-element code. 2D steady states are computed for the perturbed configuration; these differ from the unperturbed flows mainly by a region of counter-circulation surrounding the ribbon. The 2D steady flow loses stability to 3D eigenmodes at Re = 230, beta = 1.3 for rho = 0.086 and Re = 550, beta = 1.5 for rho = 0.043, where Re is the Reynolds number, beta is the spanwise wavenumber and rho is the half-height of the ribbon. For rho = 0.086, the bifurcation is determined to be subcritical by calculating the cubic term in the normal form equation from the timeseries of a single nonlinear simulation; steady 3D flows are found for Re as low as 200. The critical eigenmode and nonlinear 3D states contain streamwise vortices localized near the ribbon, whose streamwise extent increases with Re. All of these results agree well with experimental observations

Transition from the Couette-Taylor system to the plane Couette system

We discuss the flow between concentric rotating cylinders in the limit of large radii where the system approaches plane Couette flow. We discuss how in this limit the linear instability that leads to the formation of Taylor vortices is lost and how the character of the transition approaches that of planar shear flows. In particular, a parameter regime is identified where fractal distributions of life times and spatiotemporal intermittency occur. Experiments in this regime should allow to study the characteristics of shear flow turbulence in a closed flow geometry.Comment: 5 pages, 5 figure

Finite-amplitude equilibrium states in plane Couette flow

A numerical bifurcation study in plane Couette flow is performed by computing successive finite-amplitude equilibrium states, solutions of the Navier-Stokes equations. Plane Couette flow being linearly stable for all Reynolds numbers, first twodimensional equilibrium states are computed by extending nonlinear travelling waves in plane Poiseuille flow through the Poiseuille-Couette flow family to the plane Couette flow limit. The resulting nonlinear states are stationary with a spatially localized structure ; they are subject to two-dimensional and three-dimensional secondary disturbances. Three-dimensional disturbances dominate the dynamics and three-dimensional stationary equilibrium states bifurcating at criticality on the twodimensional equilibrium surface are computed. These nonlinear states, periodic in the spanwise direction and spatially localized in the streamwise direction, are computed for Reynolds numbers (based on half the velocity difference between the walls and channel half-width) close to 1000. While a possible relationship between the computed solutions and experimentally observed coherent structures in turbulent plane Couette flow has to be assessed, the present findings reinforce the idea that subcritical transition may be related to the existence of finite-amplitude states which are (unstable) fixed points in a dynamical systems formulation of the Navier-Stokes system