Stability of inviscid shear flow over a flexible boundary

Abstract

The stability of an inviscid flow that comprises a thin shear layer and a uniform outer flow over a flexible boundary is investigated. It is shown that the flow is temporally unstable for all wavenumbers. This instability is either Kelvin–Helmholtz-like or induced by the phase shift across the critical layer. The threshold of absolute instability is determined in the form F = F∗(1 + Cεⁿ) for ε [Lt ] 1, where F (a Froude number) and ε are, respectively, dimensionless measures of the flow speed and the shear-layer thickness, F∗ is the limiting value of F for a uniform flow, C < 0 and n = 1 in the absence (as for a broken-line velocity profile) of a phase shift across the critical layer, and C > 0 and n = 2/3 in the presence of such a phase shift. Explicit results are determined for an elastic plate (and, in an Appendix, for a membrane) with a broken-line, parabolic, or Blasius boundary-layer profile. The predicted threshold for the broken-line profile agrees with Lingwood & Peake's (1999) result for ε [Lt ] 1, but that for the Blasius profile contradicts their conclusion that the threshold for ε ↓ 0 is a ‘singular and unattainable limit’.