Acoustics in a Two-Deck Viscothermal Boundary Layer over an Impedance Surface

The acoustics of a mean flow boundary layer over an impedance surface or acoustic lining is considered. By considering a thick mean flow boundary layer (possibly due to turbulence), the boundary-layer structure is separated asymptotically into two decks, with a thin weakly viscous mean flow boundary layer and an even thinner strongly viscous acoustic sublayer, without requiring a high frequency. Using this, analytic solutions are found for the acoustic modes in a cylindrical lined duct. The mode shapes in each region compare well with numerical solutions of the linearized compressible Navier-Stokes equations, as does a uniform composite asymptotic solution. A closed-form effective impedance boundary condition is derived, which can be applied to acoustics in inviscid slipping flow to account for both shear and viscosity in the boundary layer. The importance of the boundary layer is demonstrated in the frequency domain, and the new boundary condition is found to correctly predict the attenuation of upstream-propagating cut-on modes, which are poorly predicted by existing inviscid boundary conditions. Stability is also investigated, and the new boundary condition is found to yield good results away from the critical layer. A time-domain formulation of a simplified version of the new impedance boundary condition is proposed.