## 4 Numerical Results
### Bubble Dynamics and Noise
Since a point monopole source is an elementary source, a specific combination of point monopole sources produces any vortex cavity deformation such as a monopole, dipole, and quadrupole sources, as schematically described in Figure 27. In this regard, there is no fundamental difference in predicting the cavitation noise by using the point source and the line source. It is noted that point sources can form a line source, but the reverse is impossible. However, the goal of the present study is to predict the CIS of underwater propellers. At the beginning of the tip vortex, intermittent bubble  
Figure 25: The number concentration density for each nuclei size. activation occurs. The result shown in Figure 26 reveals the transition from intermittent cavitation to attached stable cavitation, as the number density of nuclei increases. To highlight the ability of the present numerical method for predicting the CIS of the wing, additional computations are carried out by varying the cavitation number. The variation of the SPL spectral density spectrum predicted according to the cavitation numbers are shown and compared with  
Figure 27: Combinations of point monopole sources equivalent to corresponding line sources: (a) monopole, (b) dipole, and (c) quadrupole sources. Figure 26: Comparison of wing-tip vortex cavitation shape and PSD between experiment and bubble dynamics simulations at t = 0.0999 s; (a) N = 1721 ea; (b) N = 2869 ea; (c) N = 4303 ea; (d) N = 8606 ea; (e) N = 17,212 ea. with the acoustic analogy. First, the single-phase flow field around the wing was predicted using the RANS solver. The effects of grid resolutions and turbulence models on the numerical results are analyzed in terms of location and strength of the tip-vortex core along the mean flow direction. It was observed that the predicted trajectory of the tip-vortex core center is not sensitive to the grid resolutions and the choice of turbulence models except for the standard k-\(\epsilon\) model. However, the predicted strength of tip-vortex was sensitive to the grid-resolution and turbulence model used. The numerical solutions used more than 27 grids over the vortex core and showed convergent results in terms of minimum pressure and tangential velocity distribution across the vortex core. The numerical solution using the Reynolds stress model showed the sharpest distributions of pressure and tangential velocity across the vortex core. Furthermore, the strength of tip-vortex predicted using the Reynolds stress model maintained its value during its convection in the downstream direction. Secondly, the vortex model was applied to deal with the decay of the vortex structure along the downstream direction. This is inevitable when using the RANS solver with a moderate numerical cost. The \(\lambda_{2}\) criteria were used to trace the vortex core line from the RANS results. It was shown that the trajectory of the vortex core center detected using the \(\lambda_{2}\) criteria was consistent irrespective of grid resolutions and the turbulence models. Subsequently, the tip-vortex flow was regenerated by applying the Scully vortex model to the predicted background flow. The physical decay of tip-vortex, due to viscosity along with its convection in the downstream direction, was computed using the theoretical vortex dissipation model. Finally, the tip-vortex cavitation was predicted by applying the bubble dynamics equations to the regenerated tip-vortex flow field. Initial nuclei were distributed upstream of the wing. Its number density versus nuclei size was determined to match the experimentally measured property of the freshwater. The wing-tip vortex cavitation was simulated by computing the bubble radius variation and bubble position obtained from the bubble dynamics equations. Although the predicted tip-vortex cavitation shape does not exactly match the line-shape observed in the experiment, the bubbles growing from the nuclei appeared in the tip-vortex line and showed reasonable agreement with the camera-recorded image.