Vortex ring eigen-oscillations as a source of sound

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[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.

Linearized oscillations of a vortex column: the singular eigenfunctions

“…The velocity eigenfunction is restricted to r < a, so the exterior irrotational region remains quiescent. These two-dimensional core eigenmodes find a mention in Kopiev & Chernyshev (1997) in the context of vortex ring oscillations. Note that g (r/a) may be expanded in terms of any of the standard orthogonal families, and each of these expansions will lead to a particular, denumerably infinite, representation of the core eigenmodes.…”

“…The analogue of the two-dimensional column disturbances are the axisymmetrical ring modes that do not depend on the coordinate along the ring perimeter. For the isochronous ring, where the ratio of the azimuthal vorticity to the transverse radial distance is a constant (in the limit of small-cored rings, this is only one of an infinite set of vorticity distributions that allow for a steadily propagating distribution of vorticity (Fraenkel 1970)), the axisymmetric CS modes have indeed been shown to exhibit a twin-vortex-sheet structure, the vortex sheets being in the form of hollow tori (Kopiev & Chernyshev 1997). Although the original analysis was for rings with a small cross-section, and restricted to the CS modes within the ring cores, similar conclusions would apply to the CS modes that govern the evolution of vortical disturbances in the much larger envelope of irrotational fluid that is entrained by the propagating ring.…”

Calculation of the Acoustic Spectrum of a Cylindrical Vortex in Viscous Heat-Conducting Gas Based on the Navier–Stokes Equations

“…We compared the high-frequency oscillations with the acoustic radiation of turbulent vortex rings [17].…”

“…The vortex rings investigated in the other paper [17] were generated in an anechoic chamber by means of a piston-driven vortex generator with a nozzle diameter d = 4 cm and an initial jet ejection velocity 30 ms −1 (corresponding to Reynolds number Re = 6.8 × 10 4 ). The ring noise was determined from the averaged spectrum in a series of 12 selected time samples of length 31.2 ms starting after 220 ms from the initiation of the ring (this corresponds to the part of the path at a distance from 200 to 230 cm from the nozzle orifice), manifesting itself as strong peaking of the spectrum in a narrow frequency band (∆ω = 300 Hz) with the maximum near the frequency ω 0 = 1200 Hz.…”

Vortex ring oscillations, the development of turbulence in vortex rings and generation of sound

“…¿ÕÑ ÒÑÎÇ, ÐÇÒÑÔÓÇAEÔÕÄÇÐÐÑ ÑÒËÔÞÄÂáÜÇÇ AEÇ×ÑÓÏÂÙËá ÍÂÉAEÑÌ ÄËØÓÇÄÑÌ ÐËÕË, ÄÒÇÓÄÞÇ ÃÞÎÑ ËÔÒÑÎßÊÑÄÂÐÑ AEÎâ ÑÒËÔÂÐËâ àÄÑÎáÙËË ÄÑÊÏÖÜÇÐËÌ ÊÂÄËØÓÇÐÐÞØ ÒÑÕÑÍÑÄ Ä ÓÂÃÑÕÇ [37]. ¥Îâ ËÔÔÎÇAEÑÄÂÐËâ ÏÂÎÞØ ÍÑÎÇÃÂÐËÌ ÄËØÓÇÄÑÅÑ ÍÑÎßÙ àÕÑÕ ÒÑAEØÑAE ËÔÒÑÎß-ÊÑÄÂÎÔâ Ä ÓÂÃÑÕÇ [38], ÅAEÇ ÄÒÇÓÄÞÇ ÃÞÎË ÒÓÂÄËÎßÐÑ ÑÒËÔÂÐÞ ÃÑÚÍÑÑÃÓÂÊÐÞÇ ÏÑAEÞ ÄËØÓÇÄÑÅÑ ÍÑÎßÙÂ,  ÊÂÕÇÏ Ë AEÓÖÅËÇ ÍÑÎÇÃÂÐËâ [32]. ³ËÔÕÇÏ ÖÓÂÄÐÇÐËÌ, ÑÒËÔÞÄÂáÜÂâ àÄÑÎáÙËá ÒÑÎâ ÔÏÇÜÇÐËâ, ÏÑÉÇÕ ÃÞÕß ÒÑÎÖÚÇРËÊ ÔËÔÕÇÏÞ ÎËÐÇÂÓËÊÑ-ÄÂÐÐÞØ ÖÓÂÄÐÇÐËÌ (2.6), (2.7) ÒÓâÏÑÌ ÒÑAEÔÕÂÐÑÄÍÑÌ ÔÑÑÕÐÑÛÇÐËâ (2.5).…”