Boundary layer thickness effects of the hydrodynamic instability along an impedance wall

Abstract: The Ingard-Myers condition, modelling the effect of an impedance wall under a mean fl ow by assuming a vanishingly thin boundary layer, is known to lead to an ill-posed problem in time-domain. By analysing the stability of a linear-then-constant mean flow over a mass-spring-damper liner in a 2D incompressible limit, we show that the fl ow is absolutely unstable for h smaller than a critical hc and convectively unstable or stable otherwise. This critical hc is by nature independent of wave length or frequency and is a property of liner and mean flow only. An analytical approximation of hc is given, which is complemented by a contourplot covering all parameter values. For an aeronautically relevant example, hc is shown to be extremely small, which explains why this instability has never been observed in industrial practice. A systematically regularised boundary condition, to replace the Ingard-Myers condition, is proposed that retains the effects of a finite h, such that the stability of the approximate problem correctly follows the stability of the real problem. Keywords: Aeroacoustics, Boundary layer stability, Impedance wall.