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[Lt ]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.