### Samenvatting

An explicit Wiener-Hopf solution is derived to describe the scattering of duct modes at a hard-soft wall
impedance transition in a circular duct with uniform mean flow. Specifically, we have a circular duct r =
1,-8 <x <8 with mean flow Mach number M > 0 and a hard wall along x <0 and a wall of impedance
Z along x > 0. A minimum edge condition at x = 0 requires a continuous wall streamline r = 1 + h(x, t ), no more singular than h = O(x1/2) for x ¿ 0. A mode, incident from x <0, 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, we have an extra degree of freedom in the Wiener-Hopf analysis that can be resolved by application of some form of Kutta condition at x = 0, for example a more stringent edge condition where h = O(x3/2) at the downstream side. The question of the instability requires an investigation of the modes in the complex frequency
plane and therefore depends on the chosen impedance model, since Z = Z(¿) is essentially frequency
dependent. The usual causality condition by Briggs and Bers appears to be not applicable here because it requires a temporal growth rate bounded for all real axial wave numbers. The alternative Crighton-Leppington criterion, however, is applicable and confirms that the suspected mode is usually unstable. 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 ¿ ¿ 0, the modulus tends to |R001| ¿ (1 + M)/(1 - M) without and to 1 with Kutta condition, while the end correction tends to8without 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.

Originele taal-2 | Engels |
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Titel | 11th AIAA/CEAS Aeroacoustics Conference (Monterey CA, USA, May 23-25, 2005) |

Aantal pagina's | 19 |

Status | Gepubliceerd - 2005 |

Evenement | 11th AIAA/CEAS Aeroacoustics Conference, May 23-25, 2005, Monterey, CA, USA - Monterey, CA, Verenigde Staten van Amerika Duur: 23 mei 2005 → 25 mei 2005 |

### Congres

Congres | 11th AIAA/CEAS Aeroacoustics Conference, May 23-25, 2005, Monterey, CA, USA |
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Land | Verenigde Staten van Amerika |

Stad | Monterey, CA |

Periode | 23/05/05 → 25/05/05 |

Ander | 11th AIAA/CEAS Aeroacoustics Conference |

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## Citeer dit

Rienstra, S. W., & Peake, N. (2005). Modal scattering at an impedance transition in a lined flow duct. In

*11th AIAA/CEAS Aeroacoustics Conference (Monterey CA, USA, May 23-25, 2005)*[AIAA-2005-2852]