Low-frequency sound radiation and generation due to the interaction of unsteady flow with a jet pipe

Abstract

In this paper we examine the low-frequency sound radiated when various types of unsteady flow interact with a jet pipe. In each case we solve the problem exactly by the Wiener-Hopf technique, producing results valid for arbitrary internal and ex- ternal Mach numbers and temperatures, discuss the importance of a Kutta condition at the duct exit, and provide an interpretation, in elementary terms, of the radiated sound field using the Lighthill acoustic analogy. A central feature is that the solutions are always obtained subject to a causality requirement, regardless of whether or not a Kutta condition is imposed at the pipe lip.When low-frequency sound propagates down the jet pipe, little of it reaches the far field, and the major disturbance outside the pipe is that associated with the jet instability waves. At subsonic jet speeds and low-enough Strouhal number these waves transport kinetic energy at a rate precisely balancing the loss of acoustic energy from the pipe, resulting in a net attenuation of the sound power. For supersonic jet condi- tions a further wave motion, the unsteady-flow counterpart of the steady wave struc- ture of an imperfectly expanded jet, is present in addition to the instability wave. We use the Lighthill acoustic analogy to show that, for high-enough jet Mach number and temperature, the sound radiation is caused largely by quadrupole sources arising from the jet instability waves. An alternative interpretation uses the acoustic analogy incorporating a mean flow due to Dowling, Ffowcs Williams and Goldstein, and expresses the far-field sound as the sum of contributions from monopoles and dipoles distributed over the duct exit. The directivity and power of the calculated far-field sound are in good agreement with experiments.We also calculate the sound scattered by the jet pipe when there is an incident external sound field, and show a previously published result to be in error. In general, the flbw phenomena produced by internal and external incident sound fields are similar. Finally, we discuss the effects of nozzle contraction. We find that the radiated sound field is little changed in character, but that the reflection properties of the nozzle may be drastically altered.