A Well-posed Modified Myers Boundary Condition

Abstract

The Myers boundary condition for acoustics within flow over an acoustic lining has been shown to be illposed, leading to numerical stability issues in the time domain and mathematical problems with stability analyses. This paper gives a modification to make the Myers boundary condition well-posed, by accounting for a thin inviscid boundary layer over the lining, and correctly deriving the boundary condition to first order in the boundarylayer thickness. The modification involves two integral terms over the boundary layer. The first may be written in terms of the mass, momentum, and kinetic energy thicknesses of the boundary layer, which are shown to physically correspond a modified boundary mass, modified grazing velocity, and a tension along the boundary. The second integral term is related to the critical layer within the boundary layer. The modified boundary condition is validated against high-fidelity numerical solutions of the Pridmore-Brown equation for sheared inviscid flow in a cylinder. Absolute instability boundaries are given for certain examples, though convective instabilities appear to always be present for any boundary layer thickness.