Optimal impedance of a lined plane surface with inviscid sheared grazing flow

The presence of sheared grazing flow changes the acoustic absorption of a lined surface. Since acoustic waves are refracted through the boundary layer between the lined surface and the mean flow, the reflection and absorption coefficients are modified by the presence of the boundary layer, and therefore so too is the optimal impedance of the surface. A common simplification is to assume a uniform flow and take into account the refraction in the impedance boundary condition. However, there is still debate in the academic community about what is the appropriate boundary condition. We investigate the optimal impedance of a lined plane surface with an inviscid sheared flow. Optimal impedances are found both using numerical solutions to the Pridmore-Brown equation, which is the exact solution for a sheared parallel flow, and using the convected Helmholtz equation, that assumes a uniform flow, together with different boundary conditions that attempt to model the boundary layer. Results suggest that, for frequencies and Mach number typically found in turbofan aero-engines, boundary layer effects are of great importance at particular angles of incidence. We find the first order asymptotic boundary condition of Brambley (2011) [1] "Well-posed boundary condition for acoustic liners in straight ducts with flow", AIAA Journal 49, 127-128, reproduces the same optimal impedance as the Pridmore-Brown equation in most cases. The impact of the velocity profile shape is also investigated, and results suggest the same optimal impedance is expected irrespective of the velocity profile provided the boundary layer displacement thickness is the same.