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Cones of silence, complex rays and catastrophes: high-frequency flow–acoustic interaction effects

Published online by Cambridge University Press:  16 August 2018

J. T. Stone Open the ORCID record for J. T. Stone [Opens in a new window] ,
R. H. Self  and
C. J. Howls
J. T. Stone*
Affiliation:
Institute of Sound and Vibration Research, University of Southampton, Highfield, Southampton SO17 1BJ, UK Mathematical Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UK
R. H. Self
Affiliation:
Institute of Sound and Vibration Research, University of Southampton, Highfield, Southampton SO17 1BJ, UK
C. J. Howls*
Affiliation:
Mathematical Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UK
*
†Email addresses for correspondence: jtstone0@gmail.com, c.j.howls@soton.ac.uk
†Email addresses for correspondence: jtstone0@gmail.com, c.j.howls@soton.ac.uk
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Abstract

In this paper we develop a novel ray solver for the time-harmonic linearized Euler equations used to predict high-frequency flow–acoustic interaction effects from point sources in subsonic mean jet flows. The solver incorporates solutions to three generic ray problems found in free-space flows: the multiplicity of rays at a receiver point, propagation of complex rays and unphysical divergences at caustics. We show that these respective problems can be overcome by an appropriate boundary value reformulation of the nonlinear ray equations, a bifurcation-theory-inspired complex continuation, and an appeal to the uniform functions of catastrophe theory. The effectiveness of the solver is demonstrated for sources embedded in isothermal parallel and spreading jets, with the fields generated containing a wide variety of caustic structures. Solutions are presented across a large range of receiver angles in the far field, both downstream, where evanescent complex rays generate the cone of silence, and upstream, where multiple real rays are organized about a newly observed cusp caustic. The stability of the caustics is verified for both jets by their persistence under parametric changes of the flow and source. We show the continuation of these caustics as surfaces into the near field is complicated due to a dense caustic network, featuring a chain of locally hyperbolic umbilic caustics, generated by the tangency of rays as they are channelled upstream within the jet.

JFM classification

Acoustics: Acoustics Acoustics: Aeroacoustics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 853 , 25 October 2018 , pp. 37 - 71
Copyright
© 2018 Cambridge University Press 

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