Spatial instability of boundary layer along impedance wall

A numerical analysis is made of the hydro-acoustical spatial instability, apparently occurring in a mean flow with thin boundary layer along a locally reacting lined duct wall. This problem is of particular interest because unstable behaviour of liner and mean flow has been observed only very rarely. It is found that this instability quickly disappears for increasing boundary layer thickness. Specifically, for boundary-layer-thickness based Helmholtz numbers !??/c0 of the order of 0.1 the growth rate vanishes and the instability disappears. This corresponds to very thin boundary layers for practical values of frequencies that occur in aero-engine applications, which is in turn in good agreement with the fact that in industrial practice no instabilities are observed. For low duct-radius based Helmholtz numbers (?? 1), the instability exists for rather large values of ?? as an almost neutrally stable wave. This is qualitatively in good agreement with the experimental observations of Ronneberger and Auregan. It is shown by a Rayleigh-type stability criterion that impedance related hydrodynamic instabilities of temporal type do not occur for mean flows with strictly negative 2nd derivative (the usual situation).