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.