Vortex ring eigen-oscillations as a source of sound

TLDR: In this paper, a complete description of long-wave vortex ring oscillations in an ideal incompressible fluid, and an examination of sound radiation by these oscillations of a thin vortex ring in a weakly compressible fluid are investigated.

Abstract: Two coupled problems are investigated: a complete description of long-wave vortex ring oscillations in an ideal incompressible fluid, and an examination of sound radiation by these oscillations in a weakly compressible fluid. The first part of the paper relates to the problem of eigen-oscillations of a thin vortex ring (μ < 1) in an ideal incompressible fluid. The solution of the problem is obtained in the form of an asymptotic expansion in the small parameter μ. The complete set of three-dimensional eigen-oscillations and axisymmetric modes (two-dimensional oscillations) is obtained. It is shown that, unlike the vortex column oscillations which have the form of simple angular harmonics, the majority of eigen-oscillations of a thin vortex ring have a more complex form which is a combination of two harmonics in the leading approximation. This leads to dramatic changes in the efficiency of sound radiation produced by modes of the vortex ring in comparison with the corresponding modes of the vortex column. In the second part of the paper the solution obtained is used to investigate the process of sound radiation by vortex perturbations in a weakly compressible fluid. The vortex ring eigen-oscillations are classified according to their sound radiation efficiency. It is shown that the modes with the dimensionless frequency ω 1/2 radiate sound most efficiently. They are two isolated modes, two infinite families of Bessel modes and a set of axisymmetric modes. The frequencies of these modes are in the interval Δω = O(μ). The results obtained are compared with known experimental data on acoustic radiation of a turbulent vortex ring. Within the limits of the theory derived an explanation of the main characteristics of sound radiation is presented.