Nonlinear Neutral Modes in Zero-Mass-Flux Plane Poiseuille-Couette Flow

Abstract

Summary Plane Poiseuille–Couette flow with zero mean advection velocity, a flow well known to be linearly stable, is analysed at asymptotically large Reynolds number R. A nonlinear neutral mode structure is uncovered which is self-sustained due to the interaction of the viscous wall layers with a predominantly inviscid nonlinear internal critical layer. The O(1) phase speed and wavenumber of the disturbance are computed as functions of the O(R−1/3) wave amplitude, with the formula for the amplitude determined analytically. Two distinct coherent structures are found: one involving a single critical layer and a second with two critical layers. The single critical layer case has a long wavelength/small amplitude limit for which an analytical solution exists, while the dual critical layer structure possesses two solution branches and exists above a threshold amplitude. Both states distort the mean flow by an O(R−1/6) amount, and in the long wavelength limit the distortion becomes comparable in size with the original basic flow. The relevance of this work to the experimental observation of self-sustained states in this flow is discussed.