## 4 Results and Discussion
### Viscous Effects on Flow Field Prediction
However, the viscous lagging model results show a significantly different pattern where, around the leading edge, no cavities have developed, and, in the trailing edge region, a large cavity is observed. For the thin-layer approach, the development of leading edge cavities is under-predicted, and they develop to approximately \(x/C=0.2\). The full viscous approach exhibits cavitation structures most similar to the experimental data, although the tail of the leading-edge cavities is slightly detached. However, the thin-layer and full viscous models exhibit similar patterns in terms of overall cavity structure. Figure 10 represent the comparison of time-averaged vapor volume fraction distribution at five different stream-wise locations for three different viscous flux vector treatment methods. As shown in  
Figure 6: Time-variation of predicted liquid volume fraction in viscous lagging case. Figure 10, most of cavity structures are developed near the hydrofoil surfaces, and above the \(y/C=0.3\) there exist no cavities. When the homogeneous mixture approach is applied, the fluid existed in mixture phase. Therefore, it is hard to distinguish the boundary between liquid water and water vapor. Usually, the range of vapor volume fraction of \(0.05\leq\alpha_{v}\leq 0.95\) is considered as water vapor phase. Therefore, the locations where the vapor volume fraction, \(\alpha_{v}\), is equal to 0.05 can be as the boundary of developed cavity. Furthermore, the corresponding values of distance from the hydrofoil surface to the values of \(\alpha_{v}=0.05\) can be as the developed cavity thickness. Based on these points of view, to examine the effects of viscous flux vector treatment methods on developed cavity distributions and thickness in detail, zoomed vapor volume fraction distribution is compared as shown in Figure 11. The viscous lagging approach shows different results of vapor volume fraction distributions compared to the full viscous method and thin-layer approximation, except for the location of \(x/C=0.2\). Because complex flow phenomena such as separation did not occur at the leading edge, the effects of viscous flux vectors are relatively weak in this location. On the other hand, the full viscous method and thin-layer approximation show almost the same results for the location below the \(x/C=0.6\). For the locations of \  
Table 2 represents the predicted time-averaged cavity thickness of three viscous flux vector treatment methods at the different stream-wise locations with experimental results by [PERSON] et al. (2018). Note that, in these results, the cavity thickness is nondimensionalized by chord length. As shown in Table 2, the viscous lagging approach shows the largest differences compared to experimental results at all of five different locations. On the other hand, the full viscous method and thin-layer approximation show good agreement with experimental results, except for the location of \(x/C=0.2\). Both of these methods show a little over-predicted result at \(x/C=0.4\) and \(x/C=0.6\), whereas a little under-predicted result at the locations of \(x/C=0.8\) and \(x/C=1.0\). Figure 8: Time-variation of predicted liquid volume fraction in thin-layer case. Figure 10: Time-averaged vapor volume fraction distribution at five stream-wise locations: (**a**) viscous lagging; (**b**) full viscous; and (**c**) thin-layer cases. Figure 9: Comparison of instantaneous cavitation structures in the trailing edge region. Figure 11: Comparison of zoomed time-averaged vapor volume fraction distribution for three different viscous flux treatment methods at five stream-wise locations: (**a**) \(x/C=0.2\); (**b**) \(x/C=0.4\); (**c**) \(x/C=0.6\); (**d**) \(x/C=0.8\); and (**e**) \(x/C=1.0\). The reason why these predicted results among three different viscous flux treatment methods show quite different cavity structures can be identified from the velocity field around the hydrofoil.