Leaky waves and the production of sound by turbulent flow from an elastic nozzle

Abstract
An analysis is made of the sound generated by axisymmetric multipole source distributions interacting with the open end of a coaxial nozzle modeled by a semi-infinite circular elastic duct. The influence of surface compliance is important at frequencies below the coincidence frequency ωc of bending waves on a plate of the same thickness as the duct wall. At frequencies between the ring frequency of the duct, ωR, and ωc, the intensity of the sound scattered from the open end is reduced relative to that produced by the same source when the duct is rigid. At lower frequencies, scattering is dominated by sound launched by leaky extensional waves of the duct, such that the intensity of the radiation exceeds that from a rigid duct by a factor of order 1(κ0a)2 ≫1, where κ0 and a are, respectively, the acoustic wavenumber and duct radius. The leaky waves propagate supersonically relative to the fluid and cause the radiation directivity to be sharply peaked in an upstream direction determined by the ratio of the sound speed in the fluid and the leaky wave phase velocity. Application of the theory is made to determine the axisymmetric component of the sound produced by low Mach number turbulent flow from the nozzle. Structural compliance would normally be expected to reduce the direct radiation produced by an adjacent turbulent flow, and this is confirmed in the present case at source frequencies between ωR and ωc. At lower frequencies, however, the effect is offset by the greater efficiency of leaky wave generation. The net result is that the overall acoustic spectral levels are similar to those for a rigid nozzle, but the directivity is significantly different. Subsonically propagating flexural waves are also generated at the nozzle with an efficiency which, in the case of a steel nozzle in water, exceeds that of sound production via the leaky waves by 30-40 dB at low frequencies. Their influence in the fluid decays rapidly with distance from the nozzle axis, but they may, in practice, make a significant contribution to the flow-generated sound if they are scattered at structural discontinuities upstream of the nozzle exit. The results are illustrated by numerical predictions for a steel nozzle in water. An appendix contains a derivation of a new formula for the sound power radiated by a leaky wave.

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