# Tip Vortex Cavitation and Induced Noise Characteristics of Hydrofoils
## 3 Results and Discussion  
### Cavitation Inception  
The cavitation inception for the NACA 16-020 hydrofoil at the angle of inclination of \(15^{\circ}\) and cavitation number, \(\sigma=2.97\) is shown in Figure 2. Cavitation inception is characterized by streaks of water vapor pocket along the low pressure vortex core trailing downstream of the NACA 16-020 hydrofoil. Such sporadic streaks of varying lengths and time periods are noticeable along the vortex core, once the cavitation inception happens. Water vapor bubble pockets of varying lengths get advected along the vortex core sporadically, once the cavitation inception happens and they are encircled in Figure 2. The attached cavitation inception for the NACA 66(2)-415 hydrofoil at the angle of inclination of \(\alpha=8^{\circ},\sigma=2.14\) is shown in Figure 3 by an arrow. The attached tip vortex is discretly identifiable by a continuous line from the tip of the vortex, advecting downstream. The area of the water vapor bubble streaks advecting sporadically at irregular time intervals along the vortex core was measured by binarizing the RGB images once the cavitation inception was noticed. The cavity area obtained from the high-speed imaging with time is given in Figure 4 for NACA 16-020 hydrofoil at \(\alpha=15^{\circ},\sigma=2.97\) as red curve and for NACA 66(2)-415 hydrofoil at \(\alpha=8^{\circ},\sigma=2.14\) as a green curve. The cavity area and the time period associated with the cavity inception is sporadic in nature as observed by the the time series in Figure 4. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & & **Capacity** & **Linearity [\%]** & **Hysteresis [\%]** & **Repeatability [\%]** \\ \hline \(F_{x}\) & 20 kgf & \(0.623\) mVolt/Volt & \(-0.01\) & \(-0.03\) & \(0.03\) \\ \hline \(F_{y}\) & 50 kgf & \(1.559\) mVolt/Volt & \(0.01\) & \(-0.04\) & \(0.03\) \\ \hline \(M_{z}\) & 1 kgf.m & \(0.874\) mVolt/Volt & \(-0.01\) & \(-0.01\) & \(0.05\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance characteristics of the dynamometer. Figure 1: Schematic of the experimental set-up. Three component dynamometer, high speed cameras and the wall-mounted pressure transducers are synchronized via a LABVIEW interface. Finding the cavitation inception number for various hydrofoils under various angles of incidence and Reynolds number is an important step in the cavitation inception prediction. In order to derive an expression for the cavitation inception number, we summarize the derivation given by [PERSON] [10] here below by considering the radial equilibrium of a fluid element swirling about the vortex axis. Let us consider a fluid element of mass \(dm\), swirling at the rotation rate, \(\omega\) at a distance \(r\) from the vortex axis. The force balance on the fluid element is shown in Figure 5.  
\[(P+dp)(r+dr)d\theta-Prd\theta = dm\frac{v_{\theta}^{2}}{r}, \tag{1}\] \[\frac{\partial P}{\partial r} = \rho\frac{v_{\theta}^{2}}{r} \tag{2}\]  
The balance between the centrifugal force and the pressure gradient on a fluid element is given in Equation (1). Upon neglecting the higher order terms and considering the tangential velocity to be independent of the azimuthal position, \(\theta\), we arrive at Equation (2). Figure 4: Cavity area variation with time for NACA 16-020 at the instance of cavity inception, \(\alpha=15^{\circ},\sigma=2.97\) and for NACA 66(2)-415 at the instance of cavity inception, \(\alpha=8^{\circ},\sigma=2.14\). Figure 3: Fully developed inception of tip vortex cavitation NACA 66(2)-415, \(\alpha=8^{\circ},\sigma=2.14\). Figure 2: Inception of tip vortex cavitation NACA 16-020, \(\alpha=15^{\circ},\sigma=2.97\).