Theory of Vortex Sound

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.