Journal of Engineering Mathematics, Vol. 10, No. 3, July 1976 Noordhoff International Publishing-Leyden Printed in The Netherlands 231 The stability of inviscid plane Couette flow in the presence of random fluctuations M. J. MANTON Department of Mathematics, Monash University, Clayton, Victoria, Australia. L. A. MYSAK* Department of Mathemi~tics, University of British Columbia, Vancouver, British Columbia, Canada. (Received June 23, 1975) SUMMARY The stability of a plane parallel flow which varies randomly across a channel is investigated. The mean velocity prone corresponds to plane Couette flow which is stable in the absence of fluctuations. The mean component of an infinitesimal disturbance in the flow is governed by an equation which is analogous to the Rayleigh equation arising in the classical stability analysis of an inviscid flow. The flow is found to be unstable when the correlation scale of the fluctuations in the basic flow is small compared with the channel width. 1. Introduction Plane Couette flow of an inviscid fluid is theoretically stable with respect to infinitesimal disturbances [1]. Similarly, analysis of a linear profile in a viscous fluid shows it to be stable [2]. On the other hand, such a flow is found experimentally to become unstable at a Reynolds number of a few thousand. But any physical realisation of a plane Couette flow is not straightforward and is liable to introduce random fluctuations into the basic flow. For example, fluctuations would be produced by any eccentricity of the walls of a large radius annulus. In the present work, we therefore consider the stability of a plane parallel basic flow which varies randomly across the channel but has a linear mean velocity. The fluid is taken to be inviscid, and so any unstable disturbance found by this analysis has a corresponding unstable disturbance in a viscous fluid at high Reynolds number. On the other hand, a viscous fluid might support further unstable disturbances at a finite Reynolds number [31. The random system is analysed by the methods developed to study random wave fields [4, 5]. A second order ordinary differential equation, analogous to the Rayleigh equation, is derived to describe the behaviour of the mean vorticity of an infinitesimal disturbance. It is found that the flow is unstable provided that the correlation scale of the fluctuations in the basic flow is small compared with the channel width. This is not surprising when it is recalled that a deterministic basic profile is unstable if it contains an inflexion point [3]. * Also, Institute of Oceanography. Journal of Engineering Math., Vol. 10 (1976) 231-241