Stability analysis of perturbed plane Couette flow

Plane Couette flow perturbed by a spanwise oriented ribbon, similar to a co nfiguration investigated experimentally at the Centre d'Etudes de Saclay, i s investigated numerically using a spectral-element code. Two-dimensional ( 2-D) steady states are computed for the perturbed configuration; these diff er from the unperturbed flows mainly by a region of counter-circulation sur rounding the ribbon. The 2-D steady flow loses stability to three-dimension al (3-D) eigenmodes at Re-c = 230, beta(c) = 1.3 for rho = 0.086 and Re(c)a pproximate to 550, beta(c)approximate to 1.5 for rho = 0.043, where beta is the spanwise wave number and 2 rho is the height of the ribbon. For rho = 0.086, the bifurcation is determined to be subcritical by calculating the c ubic term in the normal form equation from the time series of a single nonl inear simulation; steady 3-D flows are found for Re as low as 200. The crit ical eigenmode and nonlinear 3-D states contain streamwise vortices localiz ed near the ribbon, whose streamwise extent increases with Re. All of these results agree well with experimental observations. (C) 1999 American Insti tute of Physics. [S1070-6631(99)00305-0].