Hostname: page-component-7479d7b7d-rvbq7 Total loading time: 0 Render date: 2024-07-12T14:30:40.634Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral broadening of acoustic waves by convected vortices

Published online by Cambridge University Press:  19 February 2018

Vincent Clair Open the ORCID record for Vincent Clair [Opens in a new window]  and
Gwénaël Gabard Open the ORCID record for Gwénaël Gabard [Opens in a new window]
Vincent Clair*
Affiliation:
Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK Université de Lyon, École Centrale de Lyon, LMFA UMR CNRS 5509, F-69134 Écully, France
Gwénaël Gabard
Affiliation:
Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK Le Mans Université, LAUM UMR CNRS 6613, F-72085 Le Mans, France
*
†Email address for correspondence: vincent.clair@ec-lyon.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The scattering of acoustic waves by a moving vortex is studied in two dimensions to bring further insight into the physical mechanisms responsible for the spectral broadening caused by a region of turbulence. When propagating through turbulence, a monochromatic sound wave will be scattered over a range of frequencies, resulting in typical spectra with broadband sidelobes on either side of the tone. This spectral broadening, also called ‘haystacking’, is of importance for noise radiation from jet exhausts and for acoustic measurements in open-jet wind tunnels. A semianalytical model is formulated for a plane wave scattered by a vortex, including the influence of the convection of the vortex. This allows us to perform a detailed parametric study of the properties and evolution of the scattered field. A time-domain numerical model for the linearised Euler equations is also used to consider more general sound fields, such as that radiated by a point source in a uniform flow. The spectral broadening stems from the combination of the spatial scattering of sound due to the refraction of waves propagating through the vortex, and two Doppler shifts induced by the motion of the vortex relative to the source and of the observer relative to the vortex. The fact that the spectrum exhibits sidebands is directly explained by the directivity of the scattered field which is composed of several beams radiating from the vortex. The evolution of the acoustic spectra with the parameters considered in this paper is compared with the trends observed in previous experimental work on acoustic scattering by a jet shear layer.

JFM classification

Acoustics: Acoustics Acoustics: Aeroacoustics Waves/Free-surface Flows: Wave scattering
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 841 , 25 April 2018 , pp. 50 - 80
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. National Bureau of Standards.Google Scholar
Bennaceur, I., Mincu, D. C., Mary, I., Terracol, M., Larchevêque, L. & Dupont, P. 2016 Numerical simulation of acoustic scattering by a plane turbulent shear layer: spectral broadening study. Comput. Fluids 138, 83–98.Google Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Courier Dover Publications.Google Scholar
Brown, E. & Clifford, S. 1973 Spectral broadening of an acoustic pulse propagating through turbulence. J. Acoust. Soc. Am. 54 (1), 36–39.Google Scholar
Brown, E. H. 1974 Turbulent spectral broadening of backscattered acoustic pulses. J. Acoust. Soc. Am. 56 (5), 1398–1406.Google Scholar
Brown, E. H. & Clifford, S. F. 1976 On the attenuation of sound by turbulence. J. Acoust. Soc. Am. 60 (4), 788–794.CrossRefGoogle Scholar
Campos, L. M. B. C. 1978a The spectral broadening of sound by turbulent shear layers. Part 1. The transmission of sound through turbulent shear layers. J. Fluid Mech. 89 (4), 723–749.Google Scholar
Campos, L. M. B. C. 1978b The spectral broadening of sound by turbulent shear layers. Part 2. The spectral broadening of sound and aircraft noise. J. Fluid Mech. 89 (4), 751–783.CrossRefGoogle Scholar
Candel, S. M. 1979 Numerical solution of wave scattering problems in the parabolic approximation. J. Fluid Mech. 90 (3), 465–507.CrossRefGoogle Scholar
Candel, S. M., Guédel, A. & Julienne, A. 1975 Refraction and scattering of sound in an open wind tunnel flow. In 6th International Congress on Instrumentation in Aerospace Simulation Facilities, pp. 288–300. Institute of Electrical and Electronics Engineers.Google Scholar
Candel, S. M., Guédel, A. & Julienne, A. 1976a Radiation, refraction and scattering of acoustic waves in a free shear flow. In 3rd AIAA Aero-Acoustics Conference. AIAA Paper 76-544.Google Scholar
Candel, S. M., Julliand, M. & Julienne, A. 1976b Shielding and scattering by a jet flow. In 3rd AIAA Aero-Acoustics Conference. AIAA Paper 76-545.Google Scholar
Cheinet, S., Ehrhardt, L., Juvé, D. & Blanc-Benon, P. 2012 Unified modeling of turbulence effects on sound propagation. J. Acoust. Soc. Am. 132 (4), 2198–2209.Google Scholar
Clair, V. & Gabard, G. 2016 Numerical investigation on the spectral broadening of acoustic waves by a turbulent layer. In 22nd AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2016-2701.Google Scholar
Colonius, T., Lele, S. K. & Moin, P. 1994 The scattering of sound waves by a vortex: numerical simulations and analytical solutions. J. Fluid Mech. 260, 271–298.Google Scholar
Dallois, L., Blanc-Benon, P. & Juvé, D. 2001 A wide-angle parabolic equation for acoustic waves in inhomogeneous moving media: applications to atmospheric sound propagation. J. Comput. Acoust. 9 (2), 477–494.CrossRefGoogle Scholar
Davies, B. 2012 Integral Transforms and their Applications. Springer.Google Scholar
Ehrhardt, L., Cheinet, S., Juvé, D. & Blanc-Benon, P. 2013 Evaluating a linearized Euler equations model for strong turbulence effects on sound propagation. J. Acoust. Soc. Am. 133 (4), 1922–1933.CrossRefGoogle ScholarPubMed
Ewert, R., Kornow, O., Delfs, J., Yin, J., Röber, T. & Rose, M. 2009 A CAA based approach to tone haystacking. In 15th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2009-3217.Google Scholar
Ewert, R., Kornow, O., Tester, B., Powles, C., Delfs, J. & Rose, M. 2008 Spectral broadening of jet engine turbine tones. In 14th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2008-2940.Google Scholar
Ferziger, J. H. 1974 Low-frequency acoustic scattering from a trailing vortex. J. Acoust. Soc. Am. 56 (6), 1705–1707.Google Scholar
Ford, R. & Llewellyn Smith, S. 1999 Scattering of acoustic waves by a vortex. J. Fluid Mech. 386, 305–328.Google Scholar
Georges, T. M. 1972 Acoustic ray paths through a model vortex with a viscous core. J. Acoust. Soc. Am. 51 (1), 206–209.Google Scholar
Goedecke, G. H., Wood, R. C., Auvermann, H. J., Ostashev, V. E., Havelock, D. I. & Ting, C. 2001 Spectral broadening of sound scattered by advecting atmospheric turbulence. J. Acoust. Soc. Am. 109 (5), 1923–1934.Google Scholar
Goldstein, M. E. 1976 Aeroacoustics. McGraw-Hill.Google Scholar
Guédel, A. 1985 Scattering of an acoustic field by a free jet shear layer. J. Sound Vib. 100 (2), 285–304.Google Scholar
Hargreaves, J. A., Kendrick, P. & von Hünerbein, S. 2014 Simulating acoustic scattering from atmospheric temperature fluctuations using a k-space method. J. Acoust. Soc. Am. 135 (1), 83–92.Google Scholar
Hattori, Y. & Llewellyn Smith, S. G. 2002 Axisymmetric acoustic scattering by vortices. J. Fluid Mech. 473, 275–294.Google Scholar
Howe, M. S. 1973 Multiple scattering of sound by turbulence and other inhomogeneities. J. Sound Vib. 27 (4), 455–476.CrossRefGoogle Scholar
Howe, M. S. 1975 Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech. 71 (4), 625–673.Google Scholar
Karweit, M., Blanc-Benon, P., Juvé, D. & Comte-Bellot, G. 1991 Simulation of the propagation of an acoustic wave through a turbulent velocity field: a study of phase variance. J. Acoust. Soc. Am. 89 (1), 52–62.Google Scholar
Kraichnan, R. H. 1953 The scattering of sound in a turbulent medium. J. Acoust. Soc. Am. 25 (6), 1096–1104.Google Scholar
Kröber, S., Hellmold, M. & Koop, L. 2013 Experimental investigation of spectral broadening of sound waves by wind tunnel shear layers. In 19th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2013-2255.Google Scholar
Lewis, H. R. & Bellan, P. M. 1990 Physical constraints on the coefficients of Fourier expansions in cylindrical coordinates. J. Math. Phys. 31 (11), 2592–2596.CrossRefGoogle Scholar
Lighthill, M. J. 1953 On the energy scattered from the interaction of turbulence with sound or shock waves. Math. Proc. Cambridge Philos. Soc. 49, 531–551.Google Scholar
Llewellyn Smith, S. G. & Ford, R. 2001a Three-dimensional acoustic scattering by vortical flows. I. General theory. Phys. Fluids 13 (10), 2876–2889.Google Scholar
Llewellyn Smith, S. G. & Ford, R. 2001b Three-dimensional acoustic scattering by vortical flows. II. Axisymmetric scattering by Hill’s spherical vortex. Phys. Fluids 13 (10), 2890–2900.Google Scholar
McAlpine, A., Powles, C. & Tester, B. J. 2013 A weak-scattering model for turbine-tone haystacking. J. Sound Vib. 332, 3806–3831.Google Scholar
Mohseni, K. & Colonius, T. 2000 Numerical treatment of polar coordinate singularities. J. Comput. Phys. 157, 787–795.Google Scholar
Morse, P. M. & Ingard, K. U. 1968 Theoretical Acoustics. McGraw-Hill.Google Scholar
O’Shea, S. 1975 Sound scattering by a potential vortex. J. Sound Vib. 43 (1), 109–116.Google Scholar
Ostashev, V. E., Salomons, E. M., Clifford, S. F., Lataitis, R. J., Wilson, D. K., Blanc-Benon, P. & Juvé, D. 2001 Sound propagation in a turbulent atmosphere near the ground: a parabolic equation approach. J. Acoust. Soc. Am. 109 (5), 1894–1908.Google Scholar
Powles, C. J., Tester, B. J. & McAlpine, A. 2011 A weak-scattering model for turbine-tone haystacking outside the cone of silence. Intl J. Aeroacoust. 10 (1), 17–50.Google Scholar
Schlinker, R. H. & Amiet, R. K.1980 Refraction and scattering of sound by a shear layer. NASA Contractor Rep. 3371. NASA.Google Scholar
Sijtsma, P., Oerlemans, S., Tibbe, T., Berkefeld, T. & Spehr, C. 2014 Spectral broadening by shear layers of open jet wind tunnels. In 20th AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2014-3178.Google Scholar
Tam, C. K. W. 2012 Computational Aeroacoustics: A Wave Number Approach. Cambridge University Press.Google Scholar
Tam, C. K. W. & Dong, Z. 1996 Radiation and outflow boundary conditions for direct computation of acoustic and flow disturbances in a nonuniform mean flow. J. Comput. Acoust. 4 (2), 175–201.CrossRefGoogle Scholar
Thompson, K. W. 1987 Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys. 68, 1–24.Google Scholar
Wilson, D. K., Ostashev, V. E., Goedecke, G. H. & Auvermann, H. J. 2004 Quasi-wavelet calculations of sound scattering behind barriers. Appl. Acoust. 65, 605–627.Google Scholar

Clair and Gabard supplementary movie 1

Total pressure fluctuation p' (top) and scattered pressure fluctuation p's (bottom) for a plane wave with direction α=π/2 and wavenumber kL=π/2 scattered by a vortex of strength Mv=0.05 convected at a velocity Mc=0.176.

Download Clair and Gabard supplementary movie 1(Video)
Video 57.8 MB

Clair and Gabard supplementary movie 2

Total pressure fluctuation p' (top) and scattered pressure fluctuation p's (bottom) for a point mass source with a wavenumber kL=π/2 and located at (xs,ys) = (0,-20L) scattered by a vortex of strength Mv=0.05 convected at a velocity Mc=0.176.

Download Clair and Gabard supplementary movie 2(Video)
Video 66.6 MB