Low-frequency acoustic reflection at a hard–soft lining transition in a cylindrical duct with uniform flow

Abstract

The low-frequency limit of the reflection coefficient for downstream-propagating sound in a cylindrical duct with uniform mean flow at a sudden hard–soft wall impedance transition is considered. The scattering at such a transition for arbitrary frequency was analysed by Rienstra (2007, J Eng Maths 59:451–475), who, having derived an exact analytic solution, also considered the plane-wave reflection coefficient, R ₀₁₁, in the low-frequency limit, and it is this result that is reconsidered here. This reflection coefficient was shown to be significantly different with or without the application of a Kutta-like condition and the corresponding inclusion or exclusion of an instability wave over the impedance wall, assuming an impedance independent of frequency. This analysis is here rederived for a frequency-dependent locally-reacting impedance, and a dramatic difference is seen. In particular, the Kutta condition is shown to have no effect on R ₀₁₁ in the low-frequency limit for impedances with Z(ω) ~ −ib/ω for some b > 0 as ω → 0, which includes the mass–spring–damper and Helmholtz resonator impedances, although, interestingly, not the enhanced Helmholtz resonator model. This casts doubt on the usefulness of the low-frequency plane-wave reflection coefficient as an experimental test for the presence of instability waves over the surface of impedance linings. The plane-wave reflection coefficient is also derived in the low-frequency limit for a thin shell boundary, based on the scattering analysis of Brambley and Peake (2008, J Fluid Mech 602:403–426), who suggested the model as a well-posed regularization of the mass–spring–damper impedance. The result might be interpretable as evidence for the nonexistence of an instability over an acoustic lining.