Well-Posed Boundary Condition for Acoustic Liners in Straight Ducts with Flow

The Myers boundary condition for acoustics within flow over an acoustic lining has been shown to be ill-posed, leading to numerical stability issues in the time domain and mathematical problems with stability analyses. This paper gives a modification (for flat or cylindrical straight ducts) to make the Myers boundary condition well posed, and indeed more accurate, by accounting for a thin inviscid boundary layer over the lining and correctly deriving the boundary condition to first order in the boundary-layer 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 to 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. A time domain version of the new boundary condition is proposed, although not implemented. 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 at certain frequencies for any boundary-layer thickness.