The effect of the convective momentum transfer on the acoustic boundary condition of perforated liners with grazing mean flow

Abstract

The interaction of sound with sound-permeable hard walls subjected to grazing mean flow is investigated with a focus on the sound-induced exchange of streamwise momentum between the mean flow and the wall. Two generic wall types have to be distinguished, the homogeneously permeable wall and the wall with clearly separated openings, which is a more realistic model of technically feasible walls. To begin with, the focus is on the shear stress that drives the dynamics of the shearing mean flow over the homogeneous wall. This is analyzed by means of two simple mathematical models of shear stress diffusion, which come as two equivalent pairs of differential equations either for the acoustic shear stress and the wall-normal displacement, or for the streamwise and the wall-normal components of the acoustic velocity. The physical analysis is concentrated on the relation between shear stress and the wall-normal displacement of the fluid elements, which determines the effective admittance of the wall. The shear stress is represented by the momentum transfer impedance which is defined as the ratio between the acoustic wall shear stress and the in-wall velocity evaluated at the wall. It turns out that the strong increase of the acoustic wall shear stress due to transfer of mean flow momentum to the wall is the dominating mechanism which affects the effective admittance of the wall. Nevertheless, the suitability of the momentum transfer impedance as part of a complete boundary condition of the wall is questioned. The disagreement between the predicted momentum transfer impedance and some rare experimental data obtained with real inhomogeneous walls is considered as a strong indication that some further mechanisms are invoked by the inhomogeneity of real walls which are briefly discussed with regard to future studies.