Abstract
Abstract The stability of plane Couette flow to travelling-wave disturbances is studied asymptotically at high Reynolds numbers Re when the lower boundary possesses a degree of flexibility modelled by a spring-backed plate. First, it is shown that a three-dimensional (3D) linear instability exists, with streamwise and spanwise wavelengths comparable with the channel width. Building on this, nonlinear effects from the self-interaction of the wave are introduced, leading to a self-sustaining interaction between a roll/streak flow and the 3D wave. Governing nonlinear vortex-wave interaction (VWI) equations are derived and a perturbation analysis is carried out to guide a numerical investigation of the equations. The co-existence of two families of finite-amplitude solutions, each with different flow structures, is found. Numerical solutions of the VWI equations in each case show that a small wave amplitude of O(Re−1(log Re)−1/2) is all that is necessary to provoke an O(1) change to the basic Couette flow.
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