Theory of Vortex Sound

TLDR

Aerodynamic sound is generated by moving vortices; changes in vortex circulation or area produce dipole fields, but in free flow the net dipole vanishes, resulting in quadrupole radiation from moving vorticity elements. The study demonstrates that vorticity in a slightly compressible fluid can be regarded as the source of both the hydrodynamic and acoustic parts of the flow field. By treating vorticity as the common basis, a slightly compressible flow is compared to its incompressible counterpart to evaluate the sound‑radiation formula. The theory allows estimation of sound from flows described by vorticity, as shown for two rectilinear vortices and applied to free shear and jet flows, illustrating its usefulness for similarity methods.

Abstract

Physical arguments are followed by mathematical developments to show how aerodynamic sound is generated as a result of the movement of vortices, or of vorticity, in an unsteady fluid flow. Changes in circulation or area of a vortex ring give rise to a dipole sound field, the former being illustrated by oscillating flow about a fixed sphere, and the latter by a simple model for the aeolian tone attributable to the stretching of vortex rings. Because in a free flow there can be no change of the total vortex strength (circulation times area), there is no net dipole strength, but each moving element of vorticity still causes local dipolelike flow; each element of moving vorticity acts with some equal and opposite movement elsewhere in the flow so that together they form an oblique quadrupole, although the total effect must be reducible to an assembly of lateral quadrupoles. A cardinal result is that the vorticity in a slightly compressible fluid can be considered to induce the whole flow field, both the hydrodynamic part and the acoustic part. With vorticity taken as the common basis, a slightly compressible flow is compared to the corresponding incompressible one, which may be used in the evaluation of the sound-radiation formula. The theory is particularly well-structured to estimate sound from flows described in terms of vorticity: the sound field is determined for two rectilinear vortices spinning about an axis between them, and its basis for similarity methods is demonstrated in application to free shear flow and jet flow.