Directivity of acoustic field generated by leapfrogging vortices

Directivity of the acoustic field of leapfrogging vortices is investigated in the present paper toward the goal to gain insights into sound propagation of free vortical flows and uncover the fundamental physics of aeroacoustics. To this end, frequency-domain Lighthill's acoustic analogy is derived and validated quantitatively. It is sufficient to account for the Doppler effect and investigate the acoustic directivity. Our study finds that the mean flow causes the convection of the propagation of the sound waves and variation of the magnitude of the acoustic pressure. The magnitude of the rms (root-mean-square) pressure fluctuation is equal to that of the acoustic pressure without the mean flow multiplied by the negative square of Doppler's factor. Furthermore, based on the spatial decomposition approach of acoustic sources, the noise radiated from leapfrogging vortices results from the linear superposition of the acoustic fields generated by the upper and lower vortex pairs alone. The superdirectivity of sound originates from the interference of the acoustic sources. The present findings and approaches are promising to provide a framework to study more complex flows.