## 4 Numerical Results  
### Natural Cavitation  
Studies on natural cavitation have experimental data cited by many researchers [1], which have been verified in various ways [10; 11]. In this paper, computational analysis of natural cavitation around a hemispherical head cylinder shape was performed using code developed in-house.  
In this study, a grid sensitivity analysis was performed. It consists of a total of 5 steps of the grid and is as follows. Grid-1 (48 \(\times\) 20 nodes), Grid-2 (98 \(\times\) 40 nodes), Grid-3 (148 \(\times\) 80 nodes), Grid-4 (222 \(\times\) 120 nodes), and Grid-5 (296 \(\times\) 200 nodes). Here, Grid-1 and Grid-2 are considered as the coarse grid, Grid-3 as the medium grid, and Grid-4 and Grid-5 as the fine grid. These grids use corresponding minimum grid sizes of 0.023, 0.009, 0.004, 0.0025 and 0.0015 in the normal direction near the wall. Figure 1 presents the time-averaged surface pressure coefficient distribution compared to that of the experiment. As can be seen from the results, grid resolution contributes to the prediction of cavitation dynamics. Current analysis results do not give good results on a coarse grid. The length of cavitation is predicted to be too short. The results of the medium grid show the quasi-stable state, and the results obtained with the fine grid show the unsteady behavior. As mentioned in [PERSON] [20], improvements in the grid tend to improve cavitation dynamics and pressure solutions. In this paper, computational analysis was performed by applying a grid system that can define a fine grid based on the size near the wall.  
Figure 1: Sensitivity of grid resolution to surface pressure prediction.  
Figure 2 compares the surface pressure coefficients at various cavitation numbers in experimental results [1] and analysis. The number s/d is dimensioned by the length and diameter of the hemispherical cylinder. The surface pressure coefficient (C\({}_{\rm P}\)) is a dimensionless difference between the pressure on the surface and the ambient flow, and is generally multiplied by \(-1\) to the result of Equation (18). When the cavitation number determined by Equation (16) is 0.2, C\({}_{\rm P}\) cannot be lower than \(-0.2\), and when the value is \(-0.2\), it can be determined that a complete cavity is formed. When the number of cavities is 0.2 and 0.3, it is shown that the process of generating and developing cavities is in good agreement when comparing the experimental results with the analysis. There are some differences at the end of the cavitation because of the increased instability caused by the re-entrant jet occurring at the end of the cavitation. This is a physical phenomenon common to experiments and computational analysis. Figure 3 shows the results of the computational analysis of natural cavitation over time. Over time, the cavitation repeats the process of creation, destruction, and regeneration. The size of the cavitation shape is larger as the cavitation number is smaller. Also, since the natural cavitation is a phase change process caused by local pressure drop, it can be confirmed that the cavitation disappears as the pressure is restored. This is a very important physical phenomenon that is distinguished from ventilated cavitation through artificial injection.  
Figure 2: Surface pressure coefficient on the hemispherical head cylinder at various cavitation numbers.