Universal law for identifying the singing vortex

Abstract

The discrete tone radiated from a cavitating tip vortex, named as ‘vortex singing’, has remained a mystery for over thirty years. In this study, based on the dispersion relation of cavity interfacial waves, the vortex singing is proved to be generated by breathing mode waves propagating from downstream to the hydrofoil tip, solely determined by the mean cavity radius (r_c), the cavitation number (σ) and the desinent cavitation number (σ_d). Then we have identified three types of vortex singing, the universal law, such as the dimensionless singing frequency for each type (ῶ = 2πfr_c/U_∞ = 0.312, 0.037 and 0.926, f is frequency and U_∞ denotes the incoming velocity) and the wavenumber for one typical type (κ = 2πr_c /λ = 0.361, λ represents the wavelength) have been first derived and validated. Furthermore, the minimum cavitation number and desinent cavitation number required for detecting each type of vortex singing are given theoretically. Importantly, we have illustrated a long-standing perplexity: why such a whistler can appear only within a narrow range of frequency, wavelength as well as the cavitation number.