×

Multiscale analysis of spectral broadening of acoustic waves by a turbulent shear layer. (English) Zbl 1434.76113

Summary: We consider the scattering of acoustic waves emitted by an active source above a plane turbulent shear layer. The layer is modeled by a moving random medium with small spatial and temporal fluctuations of its mean velocity, and constant density and speed of sound. We develop a multiscale perturbative analysis for the acoustic pressure field transmitted by the layer and derive its power spectral density when the correlation function of the velocity fluctuations is known. Our aim is to compare the proposed analytical model with some experimental results obtained for jet flows in open wind tunnels. We start with the Euler equations for an ideal fluid flow and linearize them about an ambient, unsteady inhomogeneous flow. We study the transmitted pressure field without fluctuations of the ambient flow velocity to obtain the Green’s function of the unperturbed medium with constant characteristics. Then we use a Lippmann-Schwinger equation to derive an analytical expression of the transmitted pressure field, as a function of the velocity fluctuations within the layer. Its power spectral density is subsequently computed invoking a stationary-phase argument, assuming in addition that the source is time-harmonic and the layer is thin. We finally study the influence of the source tone frequency and ambient flow velocity on the power spectral density of the transmitted pressure field and compare our results with other analytical models and experimental data.

MSC:

76Q05 Hydro- and aero-acoustics
76F40 Turbulent boundary layers
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] D. G. Alfaro Vigo, J.-P. Fouque, J. Garnier, and A. Nachbin, Robustness of time reversal for waves in time-dependent random media, Stochastic Process. Appl., 113 (2004), pp. 289-313. · Zbl 1073.35031
[2] M. Asch, W. Kohler, G. Papanicolaou, M. Postel, and B. White, Frequency content of randomly scattered signals, SIAM Rev., 33 (1991), pp. 519-625. · Zbl 0736.60055
[3] G. K. Batchelor, The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge, UK, 1953. · Zbl 0053.14404
[4] I. Bennaceur, D. C. Mincu, I. Mary, M. Terracol, L. Larchevêque, and P. Dupont, Numerical simulation of acoustic scattering by a plane turbulent shear layer: Spectral broadening study, Comput. Fluids, 138 (2016), pp. 83-98. · Zbl 1390.76131
[5] L. Borcea, G. Papanicolaou, and C. Tsogka, Theory and applications of time reversal and interferometric imaging, Inverse Problems, 19 (2003), pp. S139-S164. · Zbl 1045.94500
[6] L. Borcea, J. Garnier, G. Papanicolaou, and C. Tsogka, Enhanced statistical stability in coherent interferometric imaging, Inverse Problems, 27 (2011), 085004. · Zbl 1230.62129
[7] R. Burridge, G. Papanicolaou, P. Sheng, and B. White, Probing a random medium with a pulse, SIAM J. Appl. Math., 49 (1989), pp. 582-607. · Zbl 0686.73027
[8] L. M. B. C. Campos, The spectral broadening of sound by turbulent shear layers. Part 1. The transmission of sound through turbulent shear layers, J. Fluid Mech., 89 (1978), pp. 723-749. · Zbl 0418.76049
[9] L. M. B. C. Campos, The spectral broadening of sound by turbulent shear layers. Part 2. The spectral broadening of sound and aircraft noise, J. Fluid Mech., 89 (1978), pp. 751-783. · Zbl 0418.76050
[10] S. Candel, A. Guédel, and A. Julienne, Refraction and scattering in an open wind tunnel flow, in Proceedings of the 6th International Congress on Instrumentation in Aerospace Simulation Facilities, Ottawa, ON, 1975, pp. 288-300.
[11] S. Candel, A. Guédel, and A. Julienne, Radiation, refraction and scattering of acoustic waves in a free shear flow, in Proceedings of the 3rd AIAA Aeroacoustics Conference, Palo Alto, CA, 1976, AIAA paper 1976-544.
[12] S. Candel, A. Guédel, and A. Julienne, Résultats préliminaires sur la diffusion d’une onde acoustique par écoulement turbulent, J. Phys. Colloques, 37 (1976), pp. C1-153-C1-160.
[13] V. Clair and G. Gabard, Numerical investigation on the spectral broadening of acoustic waves by a turbulent layer, in Proceedings of the 22nd AIAA/CEAS Aeroacoustics Conference, Lyon, France, 2016, AIAA paper #2016-2701. · Zbl 1419.76544
[14] J.-F. Clouet and J.-P. Fouque, Spreading of a pulse traveling in random media, Ann. Appl. Probab., 4 (1994), pp. 1083-1097. · Zbl 0814.73018
[15] R. Ewert, O. Kornow, B. J. Tester, C. J. Powles, J. W. Delfs, and M. Rose, Spectral broadening of jet engine turbine tones, in Proceedings of the 14th AIAA/CEAS Aeroacoustics Conference, Vancouver BC, 2008, AIAA paper 2008-2940.
[16] A. Fannjiang and L. V. Ryzhik, Radiative transfer of sound waves in a random flow: Turbulent scattering, straining, and mode-coupling, SIAM J. Appl. Math., 61 (2001), pp. 1545-1577. · Zbl 0994.76091
[17] M. Fink, Time reversed acoustics, Scientific American, 281 (1999), pp. 91-97.
[18] J.-P. Fouque, J. Garnier, A. Nachbin, and K. Sølna, Time reversal refocusing for point source in randomly layered media, Wave Motion, 42 (2005), pp. 238-260. · Zbl 1189.76459
[19] J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer-Verlag, New York, 2007. · Zbl 1386.74001
[20] J. Garnier, Imaging in randomly layered media by cross-correlating noisy signals, Multiscale Model. Simul., 4 (2005), pp. 610-640. · Zbl 1089.76054
[21] J. Garnier and G. Papanicolaou, Passive Imaging with Ambient Noise, Cambridge University Press, Cambridge, UK, 2016. · Zbl 1352.86001
[22] G. H. Goedecke, R. C. Wood, H. J. Auvermann, V. E. Ostashev, D. I. Havelock, and C. Ting, Spectral broadening of sound scattered by advecting atmospheric turbulence, J. Acoust. Soc. Am., 109 (2001), pp. 1923-1934.
[23] M. E. Goldstein, Aeroacoustics, McGraw-Hill, New York, 1976.
[24] A. Guédel, Scattering of an acoustic field by a free jet shear layer, J. Sound Vib., 100 (1985), pp. 285-304.
[25] A. Ishimaru, Wave Propagation and Scattering in Random Media, Volume 2: Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing, Academic Press, San Diego, 1978. · Zbl 0873.65115
[26] Th. von Karman and L. Howarth, On the statistical theory of isotropic turbulence, Proc. A, 1938, 164 (1938), pp. 192-215. · Zbl 0018.15805
[27] S. Kröber, M. Hellmold, and L. Koop, Experimental investigation of spectral broadening of sound waves by wind tunnel shear layers, in Proceedings of the 19th AIAA/CEAS Aeroacoustics Conference, Berlin, 2013, AIAA paper 2013-2255.
[28] M. J. Lighthill, On sound generated aerodynamically: (I) General theory, Proc. A, 211 (1952), pp. 564-587. · Zbl 0049.25905
[29] G. M. Lilley, Generation of sound in a mixing region, in The Generation and Radiation of Supersonic Jet Noise, Vol. IV: Theory of Turbulence Generated Jet Noise, Noise Radiation from Upstream Sources, and Combustion Noise, Technical Report AFAPL-TR-72-53, Air Force Aero Propulsion Laboratory, Wright-Patterson Air Force Base, OH, 1972, pp. 2-69.
[30] B. A. Lippmann and J. Schwinger Variational principles for scattering processes I, Phys. Rev., 79 (1950), pp. 469-480. · Zbl 0039.42406
[31] A. McAlpine, C. J. Powles, and B. J. Tester, A weak-scattering model for tone haystacking, in Proceedings of the 15th AIAA/CEAS Aeroacoustics Conference, Miami, FL, 2009, AIAA paper 2009-3216.
[32] A. McAlpine, C. J. Powles, and B. J. Tester, A weak-scattering model for turbine-tone haystacking, J. Sound Vib., 332 (2013), pp. 3806-3831.
[33] A. McAlpine and B. J. Tester, A weak-scattering model for tone haystacking caused by sound propagation through an axisymmetric turbulent shear layer, in Proceedings of the 22nd AIAA Aeroacoustics Conference, Lyon, France, 2016, AIAA paper 2016-2702.
[34] R. F. O’Doherty and N. A. Anstey, Reflections on amplitudes, Geophys. Prospect., 19 (1971), pp. 430-458.
[35] A. D. Pierce, Wave equation for sound in fluids with unsteady inhomogeneous flow, J. Acoust. Soc. Am., 87 (1990), pp. 2292-2299.
[36] C. J. Powles, B. J. Tester, and A. McAlpine, A weak-scattering model for turbine-tone haystacking outside the cone of silence, Int. J. Aeroacoustics, 10 (2011), pp. 17-50.
[37] H. P. Robertson, The invariant theory of isotropic turbulence, Math. Proc. Cambridge Philos. Soc., 36 (1940), pp. 209-223. · Zbl 0023.42604
[38] L. V. Ryzhik, G. Papanicolaou, and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), pp. 327-370. . · Zbl 0954.74533
[39] P. Sijtsma, S. Oerlemans, T. Tibbe, T. Berkefeld, and C. Spehr, Spectral broadening by shear layers of open jet wind tunnels, in Proceedings of the 20th AIAA/CEAS Aeroacoustics Conference, Atlanta GA, 2014, AIAA paper 2014-3178.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.