Theory of vortex sound with special reference to vortex rings

From the fundamental conservation equations of mass, momentum and en-ergy for an ideal fluid of uniform entropy, one can derive a wave equation of aerodynamic sound, which can be reduced to a compact form, called the wave equation of vortex sound with a source term of the form ρ0 div(ω × v) for the acoustic pressure p′, where ρ0 is the undisturbed density. We can consider generation of acoustic waves by vortex-vortex interactions on the basis of this equation. As an axisymmetric problem, we can deduce an expression of the wave profile generated by head-on collision of two coaxial vortex rings, which is characterized as qudrupolar emission. On the other hand, based on the matched asymptotic expansion equivalent to the multipole expansions, one can derive a formula of wave pressure excited by time-dependent vorticity field localized in space. It can be applied to var-ious problems of vortex sound such as an interaction of a vortex-ring with a solid body, and an oblique collision of two vortex rings as well as the head-on collision problem where the same formula as before can be derived. In the case of vortex-body interaction, it is found that the acoustic wave profile can be related to the time derivative of volume flux (through the vortex ring) of an imaginary potential flow around the body (as if like the Faraday’s law), which is characterized as dipolar emission. The case of interaction of a vortex and a sharp edge is studied in terms of the original equation of aerodynamic sound becasue an edge plate is usually a non-compact body. All the wave emissions were detected in the experiments of the author’s group.