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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: [email protected]
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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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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