Octupole radiation of localized vortex

For a weakly compressible inviscid fluid (with the Mach number M<<1), we consider the problem of obtaining the maximum possible number of terms of the asymptotic expansion of a sound field in powers of the Mach number for the aerodynamic sound generated by localized vortices. Using Crow's approach and the matching procedure of Van Dyke, quadrupole and octupole terms are obtained in the asymptotic expansion. It is shown that higher moments of the sound field cannot be obtained in such a procedure because of the divergence of the respective integrals in the inner expansion. We discuss the possibility of representing the sound field in terms of expressions for the quadrupole and octupole moments of an incompressible flow with a retarded argument. The theory of vortex ring eigen-oscillations and the formulas for quadrupole and octupole sound obtained previously by the authors on the basis of matching asymptotical expansions are used for predictions of non- axisymmetrical octupole radiation generated by eigen-oscillations of the vortex ring. The result obtained previously for axisymmetric case is elaborated for 3D problem. The calculation of octupole term has shown that for thin vortex rings this component depends on the ratio of two small parameters and may appear to be large.