## 4 Results and Discussion
### Viscous Effects on Flow Field Prediction  
The effects of viscous fluxes, according to their approximate levels of predicting the generation, development, and extinction of cavitation over the hydrofoil, are investigated in terms of hydro-dynamic loadings, liquid volume fraction, and stream-wise velocity. First, the cyclic behavior of the generation, development, and extinction of cavitation needs to be identified. According to the experimental observations of [PERSON] et al. [3], the current problem shows periodic behavior in terms of cavitation distribution over the hydrofoil, and a period (T) of 50 ms, i.e., with a fundamental frequency of 20 Hz. Figure 4 shows the time-histories of the predicted lift and drag coefficients, and the corresponding power spectral densities (PSD). As can be seen in the spectrums, the predicted results exhibit the same cycles in terms of unsteady loadings, irrespective of the viscous model adopted. Therefore, all subsequent comparisons between the different viscous models are performed during this cyclic period. Note that when the acoustic analogy is applied for the prediction of far-field noise under the assumption of compact sources, these unsteady loadings are used directly as the dipole source terms for flow noise. There is good correlation between the thin layer and the full viscous models, especially in terms of broadband components. Figure 4: \(Cont.\)  
Figure 3: Comparison of time-averaged x-directional velocity distribution with experimental results by [PERSON] et al. [41]: (**a**) \(x/C=0.2\); (**b**) \(x/C=0.4\); (**c**) \(x/C=0.6\); (**d**) \(x/C=0.8\); and (**e**) \(x/C=1.0\). Figure 5 represents the predicted time-variation of total cavity volume and Figures 6-8 compare the iso-contours of the predicted distributions of the liquid volume fractions of the three different viscous models, during a single cycle. All three cases exhibit similar cyclic behavior, comprising three phases: generation, development, and extinction of cavitation. At the beginning of the cycle, the cavities are generated from the leading edge, and extend rapidly toward the trailing edge. At the end of development, the fully extended cavities cover the entire suction side of the hydrofoil. At this point, the maximum cavity length is observed. After this fully developed instant, the cavities break into two at mid-chord. On the front side of the hydrofoil, the attached cavities continuously weaken toward the leading edge, while, on the other side, the cavities are rolled-up near the trailing edge and shed in a downstream direction. Although the three cases show similar cyclic behavior in terms of cavitation flow patterns, the detailed cavitation structures differ. In the case of the viscous lagging approach, it developed at \(t/T^{*}=0.0\sim 0.2\), and then split at \(t/T^{*}=0.3\). However, the cavity structures in the thin-layer and full viscous cases developed relatively slowly between \(t/T^{*}=0.0\sim 0.4\), and the fully developed cavities split at approximately \(t/T^{*}=0.5\). In addition, the thin-layer approach exhibited a quicker dissipation process of shedding cavities between \(t/T^{*}=0.5\sim 0.9\). Figure 4: Predicted time-variation and power spectral density of non-dimensional hydrodynamic forces: (**a**) lift force; (**b**) drag force. Figure 5: Time-variations of predicted cavity volume of three different viscous flux vector treatment methods. Additionally, the cavitation distribution at the moment of minimum cavitation size also differs. Figure 9 compares the instantaneous cavity structures at the minimum developing moment (\(t/T^{*}=0.9\)) in viscous lagging, full viscous and thin-layer cases. At this instant, the experimental data by [PERSON] et al. [3] shows that the cavities begin developing from the leading edge, and reach \(x/C=0.4\). In the trailing edge region, small attached cavities are observed.