## 4 Numerical Results
### Bubble Dynamics  
The tip vortex cavitation of the underwater propeller is predicted by solving the equations of bubble dynamics and trajectory for each nucleus in combination with the information of the background hydrodynamic flow field in which the regenerated tip vortex flow is embedded. Figure 20 shows the predicted cavitation inception of nuclei at the three-time instants for two propellers. The population number of nuclei for a single generation is 991 ea, three generations are used for the simulation, and thus the total number of nuclei is 2973 ea. Because the cavitation bubbles are very small when compared to the propeller's dimension, the bubbles shown in Figure 20 are enlarged to twenty times the actual size. The cavitation bubbles from the Model-A propeller appear to be larger than those from the Model-B propeller. This fact is ensured in the results shown in Figure 21.  
Figure 19: The distribution of each nuclei size. Figure 18: Domain for nuclei distribution in the flow field: (a) iso-metric view; (b) front-viewHowever, in both cases, the production patterns of TVC predicted using the current numerical method are different from those observed in the experiment, which forms the same spiral lines as the tip vortex core lines, shown in Figures 7 and 11. The difference is due to the assumed spherical symmetry of the bubble dynamics model. When an actual TVC occurs, the cavitation bubbles are known to be stretched along the vortex core trajectory, and merge with the surrounding bubbles and grow, not only in a radial variation [26]. A more complex bubble dynamics model may be needed to reflect the asymmetric deformation of and merging between the bubbles. However, the main goal of the current study is to develop an efficient and reliable numerical methodology for the prediction of the tip vortex cavitation inception and noise. The cavitation inception and the magnitude of the cavitation  
Figure 21: Time histories of the bubble radius of each nucleus according to its size for (**a**) Model-A and (**b**) Model-B. Figure 20: Predicted tip vortex cavitation flow field for (**a**) Model-A and (**b**) Model-B. noise source, which are monopolar characteristics, are not affected by the cavitation's detailed shape. This fact justifies the current approach, which will be confirmed by the following results. The time trace of bubble radius according to its initial size is compared between the Model-A and Model-B propellers in Figure 21. The number of nuclei developing into cavitation bubbles is most for the initial nuclei size of 30 and 40 \(\upmu\)m in both cases. Because the tip vortex of the Model-B is delayed and of lower strength, as shown in Figures 11-13, the maximum radius of the cavitation bubble induced by the Model-B propeller is smaller than that of the Model-A propeller. However, the total number of the cavitation bubble is almost the same for both cases, as shown in Table 6.  
The acoustic pressure due to tip vortex cavitation is computed using the information of cavitation bubbles predicted for the Model-A and Model-B propellers. Each cavitation bubble is assumed to be a point monopole source. The observer location is set to be the same as that of the corresponding experiment shown in Figure 5. The predicted acoustic pressures in the time and frequency domains are shown in Figure 22. Furthermore, the power spectral densities of acoustic pressures are compared with the measured data in the LCT of KRISO. The sampling frequency and the frequency resolution of the predicted results are 2000 kHz and 40 Hz, respectively. The Hanning window is applied to reduce leakage and aliasing. Because the noise due to TVC contributes to the spectral components in a broad frequency range over 1 kHz, the acoustic pressure is compared in the frequency range of 1 to 100 kHz. Although the predicted spectral levels are slightly less than the measured ones, there are fair agreements between the two results in terms of spectral slope over 5 kHz.