Stabilisation of Hydrodynamic Instabilities by Critical Layers in Acoustic Lining Boundary Layers

Abstract

Acoustics are considered in a straight cylindrical lined duct with an axial mean flow that is uniform apart from a boundary layer near the wall. Within the boundary layer, which may or may not be thin, the flow profile is quadratic and satisfies no-slip at the wall. Time-harmonic modal solutions to the linearized Euler equations are found by solving the Pridmore-Brown equation using Frobenius series. The Briggs–Bers criterion is used to ascertain the spatial stability of the modes, without considering absolute instabilities. The modes usually identified as hydrodynamic instabilities are found to interact with the critical layer branch cut, also known as the continuous spectrum. By varying the boundary-layer thickness, flow speed, frequency, and wall impedance, it is found that these spatial instabilities can be stabilized behind the critical layer branch cut. In particular, spatial instabilities are only found for a boundary layer thinner than a critical boundary-layer thickness . The behaviors observed for the uniform-quadratic sheared flow considered here are further compared to a uniform-linear sheared flow, and a uniform slipping flow under the Ingard–Myers boundary condition, where this process of stabilization is not observed. It is therefore argued that modeling a sufficiently smooth mean flow boundary layer is necessary to predict the correct stability of flow over a lined wall.