On vortical flow and sound generation

The acoustic field induced by an unsteady vortical flow is analyzed. In particular, the limit where the reference length scale I of the vortical flow is much smaller than the reference length scale of the acoustic field is considered. At low Mach number M the leading-order solution of the vortical flow is an incompressible viscous flow. The far field behavior of the flow is matched to the five quadrupoles of the acoustic field and their strengths are related to linear combinations of second moments of vorticity. The compressibility effect of the vortical flow, of O(M²), induces an acoustic source which is of the same order as the quadrupoles. We also relate the strength of the acoustic source to the total dissipation energy of the incompressible vortical flow. Hence, if viscous effects are neglected, the acoustic source term will be absent. These results are applied to turbulent flows and their implications to the mean flow are discussed. The results are also employed to study the sound generated by the motion and decay of slender vortex filament(s), with effective core size δ 〈 I. It is shown that the monopole is of the same order as the quadrupoles if (I/δ)²R_e^-1 = O(1) where R_e is the Reynolds number based upon length I.