On certain aspects of three-dimensional instability of parallel flows

Abstract

The effect of small imperfections in shear flows on the development of finite-amplitude three-dimensional disturbances in the flow is examined. A model problem is studied, one in which the basic flow is plane Poiseuille flow in a channel and the small imperfection in the form of spanwise periodic variation of the basic flow is introduced from the channel boundaries. It is shown that this has a positive effect on the growth of larger Tollmien-Schlichting wave disturbances, which are in the form of standing waves in the spanwise direction. Equations for the amplitudes of these disturbances, based on Stuart-Watson-Eckhaus theory, are derived and over a range of Reynolds number, the regions in the wavenumber plane over which equilibrium solutions are possible are identified. The possibility of three-dimensional disturbances that are oscillatory in the streamwise direction but that may be growing exponentially in a spanwise direction is examined.