# Numerical Investigation on the Water Entry of Several Different Bow-Flared Sections
## 4 Results and Discussions
### Free Surface Evolution  
Figures 12-15 show the free surface evolution during the water entry of different bow-flared sections. For the water entry of the original section S1, the free surface was separated from the section at the bulbous bow location. Then, the separated fluid impacted on the section again. In the process of secondary slamming, a small amount of air was captured, and a cavity was formed in the flow field. For the water entry of the simplified bow-flared sections, the free surface was gradually raised along the section, and the secondary slamming did not appear, as shown in Figures 13-15.  
As shown in Figure 3, the original section and the simplified sections had the same bow-flared profile when y was greater than 0.4, while the simplified section S3 had a larger width when y was smaller than 0.4. In order to analyze the influence of different simplified options on water surface elevation, wetted length and pile-up coefficient were introduced and comparatively studied. Figure 16 illustrates the wetted length r* and the reference wetted length r corresponding to the penetration depth \(\xi\). The pile-up coefficient \(\phi\)r is defined as \(\phi\)r = r*/r. It should be noted that the penetration depth of different sections is defined as the depth of the origin of the coordinate system XOY entering into the still water surface. Therefore, the lowest point of section S4 just touched the still water surface when the penetration depth \(\xi\), was 0.248 m.  
Figure 12: Free surface evolution during the water entry of the modified section “S1”.  
Figure 14: Free surface evolution during the water entry of the modified section “S3”.  
Figure 13: Free surface evolution during the water entry of the original section “S2”.  
Figure 17 shows the wetted length as a function of the penetration depth for different sections. With the increase of penetration depth, the water surface gradually rose along the section and the wetted length of different sections gradually increased. For a certain penetration depth, the wetted lengths of sections S3 and S4 were significantly larger and smaller than the wetted length of the original section S1, respectively. This was because the modification of section S3 and section S4 significantly increased and decreased the area of the lower part of the original section, respectively. The change  
Figure 16: The wetted length \(r^{*}\) and the reference wetted length \(r\) corresponding to the penetration depth \(\xi\).  
Figure 15: Free surface evolution during the water entry of the modified section “S4”.  
of the displacement at the lower part of the section further affected the increase of free surface in the bow-flared region during the water entry. Figure 18 shows the pile-up coefficient as a function of the penetration depth for different sections. It can be seen that the pile-up coefficient of section S2 was closest to that of the original section S1. The value of the pile-up coefficient of sections S1, S2, and S3 was obviously larger than that of \(\pi\)/2 predicted by [PERSON] (1986) for wedges, which was due to the obvious curvature change in sections S1-S3. Because the shape of section S4 was approximately wedge-shaped, the pile-up coefficient of section S4 was close to \(\pi\)/2.