## 4 Discussion and Analysis of Results  
### Verification of Calculation Model Reliability  
An open water experiment refers to an experiment conducted on a propeller model in a uniform inflow, and the hydrodynamic performance of the propeller obtained using direct and indirect measurements in an open water experiment is called propeller open water performance. The simulation accuracy of propeller open water performance reflects the rationality of the selected calculation model and grid settings. In this paper, we conduct a prediction for the open water performance of the PPTC propeller, select an advance coefficient range of 0.1334-1.0683, and compare the predicted results with the experimental results [55]. The thrust coefficient \(K_{T}\) and torque coefficient \(K_{Q}\) are respectively given by Equations (12) and (13):  
\[K_{T}=\frac{T}{\rho n^{2}D^{4}} \tag{12}\]  
\[K_{Q}=\frac{Q}{\rho n^{2}D^{5}} \tag{13}\]  
We plot in Figure 5 the curves of the calculated open water performance and official experimental data. It can be observed from Figure 5 that both the propeller thrust and torque decrease significantly with the increase in the advance coefficient, and the relative errors between the simulation results of the thrust coefficient and experimental results  
Figure 4: Diagram of the nuclei release position. are less than 5%, as are the relative errors between the simulation results of the torque coefficient and experimental results. This shows that the numerical simulation of the Eulerian flow field based on this set of grids is relatively accurate and reliable, laying a foundation for the subsequent calculations of discrete bubble models. The verification of the discrete bubble model has been detailed in previous studies [54]; therefore, we will not further elaborate here. In real-world engineering applications, the optics criterion used for observing whether tiny cavities occur at a tip vortex through model experiments is usually used to determine cavitation inception. Therefore, in order to verify the reliability of a bubble dynamics method in a propeller cavitation inception numerical simulation, it is necessary to compare simulation results with the results of the traditional Eulerian cavitation model and experiment using the optics criterion. We conduct a Eulerian flow field simulation on the operating condition of tip vortex cavitation inception, the advance coefficient of the cavitation inception operating condition is \(J=0.9983\), \(\sigma_{n}=6.39\), and the rotational speed is 1259.1 rpm, i.e., \(n=1259.1\) rpm. The inlet and outlet boundary conditions are respectively given by Equations (14) and (15):  
\[J=\frac{V}{nD} \tag{14}\]  
\[\sigma_{n}=\frac{P_{out}-P_{v}}{0.5\rho(nD)^{2}} \tag{15}\]  
The calculation results are shown in Figure 6. Figure 6a shows the photo taken during the experiment. Figure 6b shows the results obtained by adopting our proposed numerical prediction method based on the Eulerian-Lagrangian framework in this paper, and Figure 6c shows the isosurface with a vapor volume of 0.1 calculated based on the traditional ZGB cavitation model under the Eulerian framework. To facilitate observation, the bubbles in Figure 6b are magnified ten times their actual diameters. The bubble diameter is characterized by the color of the bubble, and the redder the color, the larger the bubble diameter. It can be clearly observed that the position and morphology of the incipient cavitation predicted by the numerical prediction method proposed in this paper are highly consistent with the results illustrated in the photo taken in experiments under the same operating conditions. In addition, it is worth mentioning that the morphology of the inceptive tip vortex cavitation bubble obtained with the simulation based on the methods proposed in this paper is slightly shorter compared with that obtained from experiments, which is closely related with the initial release position and initial concentration of nuclei.