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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: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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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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References

Abrahams, I. D., Kriegsmann, G. A. & Reiss, E. L. 1989 On the development of caustics in shear flows over rigid walls. SIAM J. Appl. Maths 49 (6), 1652–1664.Google Scholar
Allgower, E. L. & Georg, K. 1990 Numerical Continuation Methods. Springer.Google Scholar
Amodei, D., Keers, H., Vasco, D. & Johnson, L. 2006 Computation of uniform wave forms using complex rays. Phys. Rev. E 73 (3), 036704.Google Scholar
Avila, G. S. S. & Keller, J. B. 1963 The high-frequency asymptotic field of a point source in an inhomogeneous medium. Commun. Pure Appl. Maths 16 (4), 363–381.Google Scholar
Balsa, T. F. 1976 The far field of high frequency convected singularities in sheared flows, with an application to jet-noise prediction. J. Fluid Mech. 74 (02), 193–208.Google Scholar
Berry, M. V. & Upstill, C. 1980 Catastrophe optics: morphologies of caustics and their diffraction patterns. Prog. Opt. 18, 257–346.Google Scholar
Bleistein, N. & Handelsman, R. A. 1986 Asymptotic Expansions of Integrals. Dover.Google Scholar
Candel, S. M. 1977 Numerical solution of conservation equations arising in linear wave theory: application to aeroacoustics. J. Fluid Mech. 83 (03), 465–493.Google Scholar
Červený, V., Popov, M. M. & Pšenčík, I. 1982 Computation of wave fields in inhomogeneous media – Gaussian beam approach. Geophys. J. Intl 70 (1), 109–128.Google Scholar
Chapman, C. J. 1999 Caustics in cylindrical ducts. Proc. R. Soc. Lond. A 455, 2529–2548.Google Scholar
Chapman, S. J., Lawry, J. M. H., Ockendon, J. R. & Tew, R. H. 1999 On the theory of complex rays. SIAM Rev. 41 (3), 417–509.Google Scholar
Connor, J. N. L. & Curtis, P. R. 1984 Differential equations for the cuspoid canonical integrals. J. Math. Phys. 25, 2895–2902.Google Scholar
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77 (02), 397–413.Google Scholar
Durbin, P. A. 1983a High frequency Green function for aerodynamic noise in moving media. Part I. General theory. J. Sound Vib. 91 (4), 519–525.Google Scholar
Durbin, P. A. 1983b High frequency Green function for aerodynamic noise in moving media. Part II. Noise from a spreading jet. J. Sound Vib. 91 (4), 527–538.Google Scholar
Freund, J. B. & Fleischman, T. G. 2002 Ray traces through unsteady turbulence. AIAA J. 1 (1), 83–96.Google Scholar
Goldstein, M. E. 1976 Aeroacoustics. McGraw-Hill.Google Scholar
Goldstein, M. E. 1982 High frequency sound emission from moving point multipole sources embedded in arbitrary transversely sheared mean flows. J. Sound Vib. 80 (4), 499–522.Google Scholar
Goldstein, M. E.1991 Noise from turbulent shear flows. In Aeroacoustics of Flight Vehicles: Theory and Practice. Volume 1. Noise Sources (ed. H. H. Hubbard), pp. 291–310. DTIC Document.Google Scholar
Goldstein, M. E. 2003 A generalized acoustic analogy. J. Fluid Mech. 488, 315–333.Google Scholar
Grikurov, V. E. 1980 Caustic overlap in a surface waveguide and a generalization of the ray method. Radiophys. Quantum Electron. 23 (9), 690–695.Google Scholar
Hanyga, A. 1996 Point-to-curve ray tracing. Pure Appl. Geophys. 148 (3–4), 387–420.Google Scholar
Hanyga, A. 1997 Canonical functions of asymptotic diffraction theory associated with symplectic singularities. In Symplectic Singularities and Geometry of Gauge Fields, vol. 36, pp. 57–71. Banach Center Publications.Google Scholar
Hayes, W. D. 1970 Kinematic wave theory. Proc. R. Soc. Lond. A 320 (1541), 209–226.Google Scholar
Ilário, C. R., Azarpeyvand, M., Rosa, V., Self, R. H. & Meneghini, J. R. 2017 Prediction of jet mixing noise with Lighthill’s acoustic analogy and geometrical acoustics. J. Acoust. Soc. Am. 141 (2), 1203–1213.Google Scholar
Jones, D. S. 1976 The mathematical theory of noise shielding. Prog. Aerosp. Sci. 17 (3), 149–229.Google Scholar
Keller, H. B. 1992 Numerical Methods for Two-point Boundary-value Problems. Dover.Google Scholar
Keller, H. B. & Perozzi, D. J. 1983 Fast seismic ray tracing. SIAM J. Appl. Maths 43 (4), 981–992.Google Scholar
Lighthill, M. J. 1952 On sound generated aerodynamically. I. General theory. Proc. R. Soc. Lond. A 211 (1107), 564–587.Google Scholar
Lighthill, M. J. 1954 On sound generated aerodynamically. II. Turbulence as a source of sound. Proc. R. Soc. Lond. A 222 (1148), 1–32.Google Scholar
Lilley, G. M. 1958 On noise from air jets. Aero. Res. Counc. 20, 376-N40-FM 2724.Google Scholar
Mani, R., Gliebe, P. R. & Balsa, T. F.1978 High velocity jet noise source location and reduction. Task 2, Fed. Aviation Admin. Rep., FAA-RD-76-II.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. McGraw-Hill.Google Scholar
Norris, A. N. 1986 Complex point-source representation of real point sources and the Gaussian beam summation method. J. Opt. Soc. Am. A 3 (12), 2005–2010.Google Scholar
Olver, F. W. J., Olde Daalhuis, A. B., Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., Miller, B. R. & Saunders, B. V.2016 NIST digital library of mathematical functions. Release 1.0.13 of 2016-09-16. Available from http://dlmf.nist.gov/.Google Scholar
Pereyra, V., Lee, W. H. K. & Keller, H. B. 1980 Solving two-point seismic-ray tracing problems in a heterogeneous medium. Part 1. A general adaptive finite difference method. Bull. Seismol. Soc. Am. 70 (1), 79–99.Google Scholar
Pierce, A. D. 1981 Acoustics: An Introduction to its Physical Principles and Applications. McGraw-Hill.Google Scholar
Poston, T. & Stewart, I. 2014 Catastrophe Theory and its Applications. Dover.Google Scholar
Ribner, H. S.1995 An extension of the Lighthill theory of jet noise to encompass refraction and shielding. NASA Tech. Mem. 110163.Google Scholar
Sambridge, M. S. & Kennett, B. L. N. 1990 Boundary value ray tracing in a heterogeneous medium: a simple and versatile algorithm. Geophys. J. Intl 101, 157–168.Google Scholar
Sneddon, I. N. 2006 Elements of Partial Differential Equations. Dover.Google Scholar
Stone, J. T., Self, R. H. & Howls, C. J. 2017 Aeroacoustic catastrophes: upstream cusp beaming in Lilley’s equation. Proc. R. Soc. Lond. A 473, 20160880.Google Scholar
Stone, J. T., Self, R. H. & Howls, C. J.2014 A complex ray-tracing tool for high-frequency mean-field flow interaction effects in jets. AIAA Paper 2014-2757.Google Scholar
Tam, C. K. W. & Pastouchenko, N. N. 2002 Noise from fine-scale turbulence of nonaxisymmetric jets. AIAA J. 40 (3), 456–464.Google Scholar
Tester, B. J. & Morfey, C. L. 1976 Developments in jet noise modelling – theoretical predictions and comparisons with measured data. J. Sound Vib. 46 (1), 79–103.Google Scholar
Thom, R. 1989 Structural Stability and Morphogenesis, Advanced Books Classic Series. Addison-Wesley.Google Scholar
Wundrow, D. W. & Khavaran, A. 2004 On the applicability of high-frequency approximations to Lilley’s equation. J. Sound. Vib. 272 (3), 793–830.Google Scholar