Stability Analysis of Perturbed Plane Couette Flow

Plane Couette ow perturbed by a spanwise oriented ribbon, similar to a con guration investigated experimentally at CEA-Saclay, is investigated numerically using a spectral-element code. 2D steady states are computed for the perturbed con guration; these di er from the unperturbed ows mainly by a region of counter-circulation surrounding the ribbon. The 2D steady ow loses stability to 3D eigenmodes at Rec = 230; c = 1:3 for = 0:086 and Rec 550; c 1:5 for = 0:043, where is the spanwise wavenumber and 2 is the width of the ribbon. 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. The critical eigenmode and nonlinear 3D states contain streamwise vortices localized in the streamwise direction. The streamwise extent of the vortices in the nonlinear state decreases with decreasing Re. All of these results agree well with experimental observations.