Local instabilities of flow in a flexible channel: Asymmetric flutter driven by a weak critical layer

Abstract

We examine the linear stability of two-dimensional Poiseuille flow in a long channel confined by a rigid wall and a massless damped-tensioned membrane. We seek solutions that are periodic in the streamwise spatial direction and time, solving the homogeneous eigenvalue problem using a Chebyshev spectral method and asymptotic analysis. Several modes of instability are identified, including Tollmien–Schlichting (TS) waves and traveling-wave flutter (TWF). The eigenmode for neutrally stable downstream-propagating TWF in the absence of wall damping is shown to have a novel asymptotic structure at high Reynolds numbers, not reported in symmetric flexible-walled channels, involving a weak but destabilizing critical layer at the channel centerline where the wave speed is marginally greater than the maximum Poiseuille flow speed. We also show that TS instabilities along the lower branch of the neutral curve are modified remarkably little by wall compliance, but can be either stabilized or destabilized by wall damping. We discuss the energy budget underlying TWF and briefly describe the structure of other flow-induced surface instabilities.