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Low-frequency acoustic reflection at a hard-soft lining transition in a cylindrical duct with uniform flow. (English) Zbl 1180.76053
Summary: 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 S. W. Rienstra [J. Eng. Math. 59, No. 4, 451–475 (2007; Zbl 1198.76136)], 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 \(Z(\omega ) \sim - ib/\omega \) for some \(b > 0\) as \(\omega \rightarrow 0\), 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 the author and N. Peake [J. Fluid Mech. 602, 403–426 (2008; Zbl 1170.76016)], 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.

MSC:
76Q05 Hydro- and aero-acoustics
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