Propagation of sound in a lined circular duct with sheared mean flow

This presentation describes an analytical investigation of the propagation of sound in a lined circular duct with sheared mean ambient flow. The main assumptions are that the ambient flow is turbulent and unaffected by the sound, the Mach number for the flow is small compared to unity, the thickness of the viscous boundary layer is small in comparison with the radius of the duct, and the acoustical lining is locally reactive. The mean flow profile away from the wall is assumed to be uniform, but the profile within the boundary layer can take one of several analytic forms. Outside the boundary layer, the acoustic mode structure is described by Bessell functions. The solution within the boundary layer is expressed in terms of Kummer functions, and the dispersion relation is obtained by matching the inner and outer solutions, taking the wall impedance into account. The dispersion relation is solved numerically for the attenuation and phase speed of the sound as a function of mode number, complex wall impedance, boundary layer thickness, and flow profile within the boundary layer. Both upstream and downstream propagation are considered. Comparisons are made with published numerical results.