Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-14T03:29:30.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Sound scattering by a vortex: case of exponentially decaying velocity

Published online by Cambridge University Press:  19 May 2021

Dmitry A. Gadzhiev Open the ORCID record for Dmitry A. Gadzhiev [Opens in a new window]  and
Alexander M. Gaifullin
Dmitry A. Gadzhiev*
Affiliation:
Phystech School of Aerospace Technology, Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region141701, Russian Federation Central Aerohydrodynamic Institute, 1 Zhukovsky Street, Zhukovsky, Moscow Region140180, Russian Federation
Alexander M. Gaifullin
Affiliation:
Phystech School of Aerospace Technology, Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region141701, Russian Federation Central Aerohydrodynamic Institute, 1 Zhukovsky Street, Zhukovsky, Moscow Region140180, Russian Federation
*
†Email address for correspondence: gadzhiev@phystech.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The well-known two-dimensional problem of a plane acoustic wave scattering by a point vortex is ill-posed because the vortex velocity field decays as $r^{-1}$ at infinity. We show that for problem to be well-posed, the velocity field must decay more rapidly than as $r^{-3/2}$. We propose a reformulation where the point vortex is screened by a vortical mantle with the opposite total circulation such that the velocity field is proportional to $r^{-1} \exp (-r^2 / L^2)$. The vortex effective radius $L$ is assumed long compared with the acoustic wavelength $\lambda$. In the Born approximation, the scattered field is the solution to the inhomogeneous Helmholtz equation with the Sommerfeld radiation condition that can be represented as convolution of the source term with Green's function. The asymptotic evaluation as $\lambda / L \to 0$ shows the analogy to plane-wave diffraction by a slit of width $L$. In the Fraunhofer region $r \gg L^2 / \lambda$, the solution is an outgoing cylindrical wave that peaks at small scattering angles $\theta = O(\lambda / L)$ and matches the expression with a singularity on $\theta = 0$ known from the case of the point vortex (Pitaevskii, Sov. Phys. JETP, vol. 8, 1959, pp. 888–890) as $\theta \gg \lambda / L$. In a part of the region of geometrical acoustics $r \ll L^2 / \lambda$ including the region $r \ll L$ where the vortex velocity field is proportional to $r^{-1}$, the solution matches the second known expression (Sakov, Acoust. Phys., vol. 39, 1993, pp. 280–282) that does not decay at infinity. The results are confirmed by numerical integration.

JFM classification

Acoustics: Aeroacoustics Waves/Free-surface Flows: Wave scattering
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 918 , 10 July 2021 , A46
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Aharonov, Y. & Bohm, D. 1959 Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491.CrossRefGoogle Scholar
Balmforth, N.J., Smith, S.G.L. & Young, W.R. 1998 Enhanced dispersion of near-inertial waves in an idealized geostrophic flow. J. Mar. Res. 56, 1–40.CrossRefGoogle Scholar
Belyaev, I.V. & Kopiev, V.F. 2008 On the statement of the problem of sound scattering by a cylindrical vortex. Acoust. Phys. 54, 603–614.CrossRefGoogle Scholar
Berry, M.V., Chambers, R.G., Large, M.D., Upstill, C. & Walmsley, J.C. 1980 Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue. Eur. J. Phys. 1, 154–162.CrossRefGoogle Scholar
Berthet, R. & Coste, C. 2003 Using a partial-wave method for sound–mean-flow scattering problems. Phys. Rev. E 67, 036604.CrossRefGoogle ScholarPubMed
Berthet, R., Fauve, S. & Labbé, R. 2003 Study of the sound-vortex interaction: direct numerical simulations and experimental results. Eur. Phys. J. B 32, 237–242.Google Scholar
Berthet, R. & Lund, F. 1995 The forward scattering of sound by vorticity. Phys. Fluids 7, 2522–2524.CrossRefGoogle Scholar
Blokhintsev, D.I. 1946 Acoustics of a nonhomogeneous moving medium. NACA Tech. Memo. 1399.Google Scholar
Borovikov, V.A. 1994 Uniform Stationary Phase Method. Institution of Electrical Engineers.Google Scholar
Brychkov, Y.A. & Savischenko, N.V. 2016 Some properties of the owen t-function. Integr. Transf. Spec. F. 27, 163–180.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M.E. 2003 Remote recoil: a new wave-mean interaction effect. J. Fluid Mech. 492, 207–230.CrossRefGoogle Scholar
Candel, S.M. 1979 Numerical solution of wave scattering problems in the parabolic approximation. J. Fluid Mech. 90, 465–507.CrossRefGoogle Scholar
Clair, V. & Gabard, G. 2015 Numerical assessment of the scattering of acoustic waves by turbulent structures. In 21st AIAA/CEAS Aeroacoustics Conference, p. 2680. AIAA.CrossRefGoogle 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.CrossRefGoogle Scholar
Coste, C., Lund, F. & Umeki, M. 1999 Scattering of dislocated wave fronts by vertical vorticity and the Aharonov-Bohm effect. I. Shallow water. Phys. Rev. E 60, 4908–4916.CrossRefGoogle ScholarPubMed
Doronina, O.A. & Zhdanova, N.S. 2013 Numerical simulation of acoustic waves scattering by isolated vortex structures. Math. Models Comput. Simul. 25, 85–94.Google Scholar
Fabrikant, A.L. 1982 Sound scattering by a vortex. Sov. Phys. Acoust. 28, 694–695.Google Scholar
Fabrikant, A.L. 1983 Sound scattering by vortex flows. Sov. Phys. Acoust. 29, 152–155.Google Scholar
Fedoryuk, M.V. 1963 The stationary phase method for multidimensional integrals. Zh. Vych. Mat. 2, 145–150.Google Scholar
Ferziger, J.H. 1974 Low-frequency acoustic scattering from a trailing vortex. J. Acoust. Soc. Am. 56, 1705–1707.CrossRefGoogle Scholar
Fetter, A.L. 1964 Scattering of sound by a classical vortex. Phys. Rev. 136, 1488–1493.CrossRefGoogle Scholar
Ford, R. 1994 Gravity wave radiation from vortex trains in rotating shallow water. J. Fluid Mech. 281, 81–118.CrossRefGoogle Scholar
Ford, R. & Smith, S.G.L. 1999 Scattering of acoustic waves by a vortex. J. Fluid Mech. 386, 305–328.CrossRefGoogle Scholar
Gadzhiev, D.A., Gaifullin, A.M. & Zubtsov, A.V. 2020 On vortex generation by a rotating cylinder. Fluid Dyn. 55, 965–981.CrossRefGoogle Scholar
Gaifullin, A.M. 2015 Vortex Flows (in Russian). Nauka.Google Scholar
Golemshtok, G.M. & Fabricant, A.L. 1980 Scattering and amplification of sound waves by cylindrical vortex. Sov. Phys. Acoust. 26, 383–390.Google Scholar
Gromov, P.R., Ezerskii, A.B. & Fabrikant, A.L. 1982 Sound scattering by a vortex wake behind a cylinder. Acoust. Phys. 28, 452–455.Google Scholar
Horne, W. 1983 Measurements of the scattering of sound by a line vortex. AIAA Paper 1983-0676.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, 625–673.CrossRefGoogle Scholar
Howe, M.S. 1983 On the scattering of sound by a vortex ring. J. Sound Vib. 87, 567–571.CrossRefGoogle Scholar
Howe, M.S. 1999 On the scattering of sound by a rectilinear vortex. J. Sound Vib. 227, 1003–1017.CrossRefGoogle Scholar
Iwatsu, R. & Tsuru, H. 2013 Numerical simulation of acoustic scattering from a circular vortex. Theor. Appl. Mech. Japan 61, 95–104.Google Scholar
Kambe, T. & Oo, U.M. 1981 Scattering of sound by a vortex ring. J. Phys. Soc. Japan 50, 3507–3516.CrossRefGoogle Scholar
Karabasov, S.A., Kopiev, V.F. & Goloviznin, V.M. 2009 On a classical problem of acoustic wave scattering by a free vortex: numerical modelling. In 15th AIAA/CEAS Aeroacoustics Conference (30th AIAA Aeroacoustics Conference), p. 3234. AIAA.CrossRefGoogle Scholar
Klimov, V.V. & Prozorovskii, V.L. 1987 Scattering of acoustic waves by a three-dimensional vortex. Acoust. Phys. 33, 79–81.Google Scholar
Kop'ev, V.F. & Chernyshev, S.A. 2000 Vortex ring oscillations, the development of turbulence in vortex rings and generation of sound. Phys.-Usp. 43, 663–690.CrossRefGoogle Scholar
Kopiev, V.F. & Belyaev, I.V. 2010 On long-wave sound scattering by a rankine vortex: non-resonant and resonant cases. J. Sound Vib. 329, 1409–1421.CrossRefGoogle Scholar
Kopiev, V.F. & Leontiev, E.A. 1983 Acoustic instability of an axial vortex. Sov. Phys. Acoust. 29, 192–198.Google Scholar
Kopiev, V.F. & Leontiev, E.A. 1987 Radiation and scattering of sound from a vortex ring. Fluid Dyn. 22, 83–95.Google Scholar
Krylov, V.V. 1989 Basic Principles of Sound Radiation and Scattering (in Russian). Moscow University Press.Google Scholar
Labbe, R. & Pinton, J.F. 1998 Propagation of sound through a turbulent vortex. Phys. Rev. Lett. 81, 1413–1416.CrossRefGoogle Scholar
Landau, L.D. & Lifshitz, E.M. 1987 Fluid Mechanics. Pergamon.Google Scholar
Lighthill, M.J. 1953 On the energy scattered from the interaction of turbulence with sound or shock waves. Math. Proc. Cambridge 49, 531–551.CrossRefGoogle Scholar
Lindsay, R.B. 1948 Compressional wave front propagation through a simple vortex. J. Acoust. Soc. Am. 20, 89–94.CrossRefGoogle Scholar
Lund, F. & Rojas, C. 1989 Ultrasound as a probe of turbulence. Physica D 37, 508–514.CrossRefGoogle Scholar
Manneville, S., Roux, P., Tanter, M., Maurel, A., Fink, M., Bottausci, F. & Petitjeans, P. 2001 Scattering of sound by a vorticity filament: an experimental and numerical investigation. Phys. Rev. E 63, 036607.CrossRefGoogle ScholarPubMed
McIntyre, M.E. 2009 Spontaneous imbalance and hybrid vortex–gravity structures. J. Atmos. Sci. 66, 1315–1326.CrossRefGoogle Scholar
McIntyre, M.E. 2019 Wave–vortex interactions, remote recoil, the Aharonov–Bohm effect and the Craik–Leibovich equation. J. Fluid Mech. 881, 182–217.CrossRefGoogle Scholar
Monin, A.S. & Yaglom, A.M. 2013 Statistical Fluid Mechanics, Volume II: Mechanics of Turbulence. Courier Corporation.Google Scholar
Naugolnykh, K. 2013 Sound scattering by a vortex dipole. J. Acoust. Soc. Am. 133, 1882–1884.CrossRefGoogle ScholarPubMed
Nazarenko, S.V. 1994 Absorption of sound by vortex filaments. Phys. Rev. Lett. 73, 1793–1796.CrossRefGoogle ScholarPubMed
Nazarenko, S.V., Zabusky, N.J. & Scheidegger, T. 1995 Nonlinear sound–vortex interactions in an inviscid isentropic fluid: a two-fluid model. Phys. Fluids 7, 2407–2419.CrossRefGoogle Scholar
Oseen, C.W. 1912 Uber die wirbelbewegung in einer reibenden flussigkeit. Ark. Mat. Astron. Fys. 7.Google Scholar
O'Shea, S. 1975 Sound scattering by a potential vortex. J. Sound Vib. 43, 109–116.CrossRefGoogle Scholar
Owen, D.B. 1956 Tables for computing bivariate normal probabilities. Ann. Math. Statist. 27, 1075–1090.CrossRefGoogle Scholar
Pitaevskii, L.P. 1959 Calculation of the phonon part of the mutual friction force in superfluid helium. Sov. Phys. JETP 8, 888–890.Google Scholar
Plougonven, R. & Zeitlin, V. 2002 Internal gravity wave emission from a pancake vortex: an example of wave–vortex interaction in strongly stratified flows. Phys. Fluids 14, 1259–1268.CrossRefGoogle Scholar
Reinschke, J., Möhring, W. & Obermeier, F. 1997 Scattering of sound waves by a cylindrical vortex: a semi-analytical theory. J. Fluid Mech. 333, 273–299.CrossRefGoogle Scholar
Rellich, F. 1943 Über das asymptotische verhalten der lösungen von$\ldots u+\ldots u= 0$ in unendlichen gebieten. Jahresber. Dtsch. Math. Ver. 53, 57–65.Google Scholar
Sakov, P.V. 1993 Sound scattering by a vortex filament. Acoust. Phys. 39, 280–282.Google Scholar
Salant, R.F. 1969 Acoustic rays in two-dimensional rotating flows. J. Acoust. Soc. Am. 46, 1153–1157.CrossRefGoogle Scholar
Schot, S.H. 1992 Eighty years of Sommerfeld's radiation condition. Hist. Math. 19, 385–401.CrossRefGoogle Scholar
Smith, S.G.L. 1999 Near-inertial oscillations of a barotropic vortex: trapped modes and time evolution. J. Phys. Oceanogr. 29, 747–761.2.0.CO;2>CrossRefGoogle Scholar
Smith, S.G.L. 2002 Scattering of acoustic waves by a superfluid vortex. J. Phys. A 35, 3597–3607.CrossRefGoogle Scholar
Smith, S.G.L. & Ford, R. 2001 Three-dimensional acoustic scattering by vortical flows. I. General theory. Phys. Fluids 13, 2876–2889.CrossRefGoogle Scholar
Sozou, C. 1990 Resonant interaction of a sound wave with a cylindrical vortex. J. Acoust. Soc. Am. 87, 2342–2348.CrossRefGoogle Scholar
Sveshnikov, A.G., Bogolyubov, A.N. & Kravtsov, V.V. 2004 Lectures on Mathematical Physics (in Russian). MSU.Google Scholar
Tanaka, K. & Ishii, S. 1982 Scattering of a plane sound wave by a vortex pair. J. Phys. Soc. Japan 51, 1992–1999.CrossRefGoogle Scholar
Tatarski, V.I. 2016 Wave Propagation in a Turbulent Medium. Courier Dover Publications.Google Scholar
Thomas, J. 2017 New model for acoustic waves propagating through a vortical flow. J. Fluid Mech. 823, 658–674.CrossRefGoogle Scholar
Thomas, J., Smith, K.S. & Bühler, O. 2017 Near-inertial wave dispersion by geostrophic flows. J. Fluid Mech. 817, 406–438.CrossRefGoogle Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic Press.Google Scholar
Vivanco, F., Melo, F., Coste, C. & Lund, F. 1999 Surface wave scattering by a vertical vortex and the symmetry of the Aharonov-Bohm wave function. Phys. Rev. Lett. 83, 1966–1969.CrossRefGoogle Scholar
Wei, J.-Y., Liu, J.-Y., Mahmood, W. & Zhao, Q. 2017 Scattering of electromagnetic wave by vortex flow. Phys. Lett. A 381, 1463–1469.CrossRefGoogle Scholar