Contributions to the theory of sound production by vortex-airfoil interaction, with application to vortices with finite axial velocity defect

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Abstract
An analysis is made of the sound produced when a field of vorticity is cut by an airfoil in low-Mach-number flow. A general formula is given for the acoustic pressure when the airfoil is rigid and the chord is acoustically compact. This expresses the radiation in terms of an integral over the region occupied by the vorticity; the integrand contains factors describing the influence of the thickness, twist and camber of the airfoil. Explicit analytical results are derived for a rectilinear vortex, having small core diameter and finite axial velocity defect, which is ‘chopped’ by a non-lifting airfoil of large aspect ratio. The acoustic signature generally comprises two components, which are associated with the axial and azimuthal vorticity, the latter being determined by the velocity defect distribution within the core. Sound is generated predominantly when the core is in the neighbourhoods of the leading and trailing edges. The contribution from the trailing edge is usually small, however, because of destructive interference between sound produced by edge-diffraction of near-field energy of the vortex and that produced by vorticity shed into the wake of the airfoil to satisfy the unsteady Kutta condition that the pressure and velocity should be bounded at the edge. When the shed vorticity is assumed to convect at the same mean stream velocity as the impinging vortex, the interference is predicted to be complete, and no trailing edge sound is generated. If the shed vorticity is taken to convect at a reduced, ‘near-wake’ velocity, which might be appropriate for small-scale structures comparable in size to the diameter of the vortex core, a small but non-negligible pressure pulse is radiated from the trailing edge. A tentative comparison with experiment appears to confirm the presence of this trailing-edge pulse.