## 4 Numerical Results
### Bubble Dynamics and Noise  
Figure 21 compares the predicted tip-vortex cavitation with the measured tip-vortex cavitation. Figure 21a shows the snapshot of tip-vortex cavitation, which is recorded using the CCD video camera, as explained in Section 3. Figure 21b-i show the predicted time-history of bubble behavior in the flow field. The black dashed line denotes the vortex core trajectory that is extracted from the experiment, and the blue sphere denotes the bubble. The total number of nuclei considered in this study is 10,326 and the time interval for bubble simulation is 0.5 \(\upmu\)s. Among the nuclei convected through the regenerated flow field, a few grow into cavitation bubbles as they pass through the vortex core, which is one of the lower pressure regions. The bubbles move downstream along the vortex core trajectory and repeat the compression and expansion of the volume in response to changes in the surrounding pressure. Figure 22 shows the time-history of the bubble radius for the nuclei of the initial radius 30 \(\upmu\)m and 40 \(\upmu\)m which contribute dominantly to the radiated acoustic pressure. The corresponding acoustic  
Figure 21: Comparison of wing-tip vortex cavitation between the experiment and bubble dynamics simulation; (**a**) experiment; (**b–i**) numerical results. Figure 20: Location of the initial nuclei in the flow field. pressure is computed using Equation (13) for the nuclei. Given that a single bubble is assumed to be a point monopole sound source, acoustic pressure due to the acceleration in bubble volume can be predicted using the time-history of bubble radius, as expressed in Equation (13). The observer position is set at the measured point, as shown in Figure 9. The next generation of nuclei follows the same trajectory as the current generation of nuclei because nuclei are repeatedly generated upstream of the wing in the steady-state flow field. Thus, the subsequent variations in the volume and location of the bubbles are repeated generation by generation. The power spectral density (PSD) of acoustic pressure is calculated using the signal processing parameters shown in Table 4.  
Figure 23 compares the predicted PSD and the one-third octave band spectrum of sound pressure level (SPL) with the measured. The experimental setup was presented in Section 3. Given that the cavitation noise is impulsive, it contributes mainly to broadband noise in a high-frequency range. There is a good agreement between the two results in the frequency range exceeding 5 kHz. The difference between the two results in the frequency range higher than 5 kHz is approximately within 10 dB in terms of one-third octave band levels. Figure 24 shows the time-variation in radius and radial velocity of a particular nucleus of initial radius, 40 \(\upmu\)m, as it moves in the downstream direction. The lifetime of this bubble can be divided into three phases according to its dynamical characteristics. The first phase is a growing phase, the second phase is the shrinking and rebounding phase, and the final phase is a free vibration phase. [PERSON] separated these phases into two parts. One part corresponds to the stable acoustic cavitation, which consists of bubble oscillation with the production of subharmonics, and the other part corresponds to transient acoustic cavitation, which is composed of explosive cavitation growth and violent collapse [28]. Typically, the time-scale is shorter when bubbles shrink than that when they grow, and thus bubbles release louder acoustic pressure when they shrink as compared to that when they grow. As shown in Figure 24a,b, the radial velocity changes more rapidly in the decaying and \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Signal Processing Setup**} \\ \hline  & **Numerical** & **Experiment** \\ \hline Time resolution (\(\Delta\)t, s) & \(0.5\times 10^{-6}\) & \(3.90625\times 10^{-6}\) \\ \hline Frequency resolution (\(\Delta\)f, Hz) & \(11.1\) & \(1\) \\ \hline Acquisition time (T, s) & \(0.1\) & \(60\) \\ \hline \hline \end{tabular}
\end{table}