# Tip Vortex Cavitation and Induced Noise Characteristics of Hydrofoils
## 3 Results and Discussion
### Cavitation Induced Noise  
Tip vortex cavitation is generally associated with the streaks of water vapour issuing from the tip of three-dimensional hydrofoil with intense noise. The evolution of the wing tip cavitation with the decrease of the cavitation number is seen in six panels shown in Figure 7 for the elliptical planform NACA 16-020, at the angle of inclination, \(\alpha=15^{\circ}\). With the decrease in the cavitation number, six different stages were observed according to the cavitation number. They are shown as six frames from the top to the bottom in Figure 7: (1) Tip Vortex Cavitation (TVC) inception, (2) fully developed tip vortex cavitation, (3) attached tip vortex cavitation to the tip, (4) tip vortex cavitation with leading edge sheet  
Figure 5: Schematic of the fluid element of mass, \(dm\) swirling about the vortex axis at the rate, \(\omega\). The fluid element is in radial equilibrium by the balance between the pressure force and centrifugal force acting on it. cavity, (5) tip vortex cavitation with cloud cavity and (6) tip vortex cavitation breakdown. The frequency response curve (sound pressure level vs frequency) obtained by performing Fast Fourier Transform (FFT) of the raw pressure signals from the unsteady pressure transducer corresponding to these six stages are in figures Figure 8. At the inception of cavitation, no peaks were observed in the frequency spectrum, when the tip vortex cavitation becomes fully developed at \(\sigma=2.44\), frequency peaks begin to appear in the power spectrum in Figure 8. Upon further decrease of the cavitation number, the frequency peak becomes strong at 1.1 KHz, with a strong Sound Pressure Level of about 150 dB. With further decrease of the cavitation number to \(\sigma=2.14,1.89\) & 1.66, the tip vortex cavitation with leading edge cavity appears, followed by tip vortex cavitation with cloud cavity and finally the break down of the tip vortex cavitation with fully developed cloud cavity in successive stages. However, in all these three stages at \(\sigma=2.14,1.89\) & 1.66, no clear noise peak is observed in the frequency spectrum. As shown in Figure 2, the cavitation inception corresponding to stage (1) is characterized by sporadic issuance of water vapor bubble packets along the vortex core. No peak is evident in the frequency spectrum at the stage (1) at \(\sigma=6.49\) in Figure 9. With the further decrease of the cavitation number, the cavity along the core becomes stronger in the stage (2) and a mild noise peak can be seen in the frequency response curve at \(\sigma=2.44\). When \(\sigma=2.29\), the cavity at the tip of the hydrofoil gets attached to the hydrofoil and a specific frequency of about 1 kHz occurs at the stage (3), with the sound pressure level being closer to 150 dB in Figure 9. Upon further decrease of the cavitation number to \(\sigma=2.29\), the noise peak is seen to be not concentrated at about the 1 kHz. However, the SPL values remain larger than the non-cavitating case with the decrease of cavitation number. At the stage (4) when \(\sigma=2.14\), a sheet cavity can be seen around the leading edge of the hydrofoil along with the attached tip vortex cavitation. Upon further decrease of the cavitation number in the stage (5), a cloud cavity is seen in the leading edge of the hydrofoil along with the tip vortex cavitation. In the stage (6), tip vortex cavitation begins to breakdown at the lowest value of the cavitation number. However, in stage (5) and stage (6), there is no clear concentrated peak frequency in the power spectrum in Figure 9.  
Figure 6: Cavitation inception scaling with the measured lift coefficient, \(C_{L}\) and Reynolds number, \(Re\). As opposed to NACA 16-020 hydrofoil, NACA 66(2)-415 hydrofoil, being an asymmetric hydrofoil, has an attached cavitation inception as shown in Figure 10 at \(\alpha=8^{\circ}\), \(\sigma=2.14\).