## 2 Numerical Simulation of Submerged Cavitating Jet
### Analysis of Simulation Results  
Similar to the submerged free jet which was identified to have 3 distinct regions of flow: the undisturbed region of flow, the potential core region and the fully developed region [45; 46; 47], the cavitation jet also has 3 regions: the growing region, the shedding region, and the collapsing and diffusing region [48]. The damage to the target is mainly caused by the last region. The contours of pressure (mixture), volume fraction (vapor), and velocity for each nozzle, obtained through simulation results, are presented in Figure 3. The cavitating jet's cleaning effect primarily manifests in two forms: erosion due to water flow impact and denutation resulting from bubbles collapse. The velocity contours and pressure (mixture) contours in the simulation results indicate the extent of erosion caused by water flow impact, while the volume fraction (vapor) contours reveal the degree of denutation induced by bubbles collapse.  
Nozzle No.2, which has a larger orifice diameter than nozzle No.1, demonstrated a larger flow velocity coverage and a larger area after the jet reached the wall. This suggests that increasing the orifice diameter can enlarge the cleaning area. Nozzle No.3 featured an expanding cylinder segment at the outlet in comparison to nozzle No.2. The simulation results showed that nozzle No.3 had a superior large pressure holding distance and a larger area of the water vapor phase than nozzle No.2. Therefore, configuring the expanding cylinder segment can mitigate the decrease in vapor volume fraction and extend the outlet pressure holding distance. The pre-shrinking segment was removed in nozzle No.4 compared with nozzle No.3, and the simulation results indicated that although the pressure holding distance of nozzle No.4 decreased, the continuity of the vapor volume fraction increased, as well as the coverage area. The removal of the pre-shrinking segment can improve the cavitation effect of the nozzle and increase the bubble content of the jet. The outlet in nozzle No.5 was changed from conical in nozzle No.4 to spherical. Through  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Grid Element Size [mm]** & **Pressure Drop [Pa]** & **Error [\%]** \\ \hline
0.1 & 30,000,010.409 & 0 \\
0.07 & 29,985,590.485 & \(-\)0.0481 \\
0.05 & 29,998,177.102 & \(-\)0.0061 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The error in pressure drop for different grid element sizes.  
Figure 2: The numerical model of the submerged cavitating jet.  
the simulation results, it was found that the continuity of the vapor volume fraction was further improved, and the vapor showed a noticeable aggregation tendency. Consequently, it can be inferred that converting the conical outlet into a spherical one can facilitate the cavitation bubbles gathering together, thereby allowing more bubbles to reach the wall surface and enhance the bubbles denudation effect. When compared to the shear nozzle, the self-resonance pulsed Helmholtz nozzle exhibited a longer outlet velocity retention distance, but a lower vapor volume fraction. The organ pipe nozzle did not perform as effectively as the Helmholtz nozzle in terms of flow velocity and vapor volume distribution.  
Figure 3: Simulation results.