### Abstract

An explicit Wiener–Hopf solution is derived to describe the scattering of sound at a hard–soft wall impedance transition at x = 0, say, in a circular duct with uniform mean flow of Mach number M. A mode, incident from the upstream hard section, scatters at x = 0 into a series of reflected modes and a series of transmitted modes. Of particular interest is the role of a possible instability along the lined wall in combination with the edge singularity. If one of the "upstream" running modes is to be interpreted as a downstream-running instability, an extra degree of freedom in the Wiener–Hopf analysis occurs that can be resolved by application of some form of Kutta condition at x = 0, for example a more stringent edge condition where wall streamline deflection at the downstream side. In general, the effect of this Kutta condition is significant, but it is particularly large for the plane wave at low frequencies and should therefore be easily measurable. For small Helmholtz numbers, the reflection coefficient modulus |R 001| tends to (1 + M)/(1-M) without and to 1 with Kutta condition, while the end correction tends to 8 without and to a finite value with Kutta condition. This is exactly the same behaviour as found for reflection at a pipe exit with flow, irrespective if this is uniform or jet flow. Although the presence of the instability in the model is hardly a question anymore since it has been confirmed numerically, a proper mathematical causality analysis is still not totally watertight. Therefore, the limit of a vortex sheet, separating zero flow from mean flow, approaching the wall has been explored. Indeed, this confirms that the Helmholtz unstable mode of the free vortex sheet transforms into the suspected mode and remains unstable. As the lined-wall vortex-sheet model predicts unstable behaviour for which experimental evidence is at best rare and indirect, the question may be raised if this model is indeed a consistent simplification of reality, doing justice to the double limit of small perturbations and a thin boundary layer. Numerical time-domain methods suffer from this instability and it is very important to decide whether the instability is at least physically genuine. Experiments based on the present problem may provide a handle to resolve this stubborn question.

Original language | English |
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Pages (from-to) | 451-475 |

Journal | Journal of Engineering Mathematics |

Volume | 59 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2007 |