## 4 Results and Discussion
### Viscous Effects on Hydro-Acoustic Field Prediction  
Hydro-acoustic fields due to the cavitation flow of the Clark-Y hydrofoil are predicted using the integral from of the FW-H equation, with an input from the results of the predicted flow fields discussed in the previous section. The predicted hydro-acoustic fields are compared in terms of noise directivity and spectrum between the three viscous models used for the flow simulation. The time variation of cavity volume changes is used to model the monopole source term that is the first term in the right-hand side of Equation (19). The volume integration is conducted over the volume containing all the developed cavity structure. For the dipole source that is the second term in the right-hand side of Equation (19), hydrodynamic forces on the hydrofoil surfaces are used directly in terms of lift and drag forces. Therefore, the surface integration is conducted over the hydrofoil surfaces. The variation in the retarded time is neglected because the Helmholtz number based on the chord length of the hydrofoil is around 0.47 at 10 kHz. For the extraction of source data, the flow field results are saved every 50-time steps (2.6 \(\times\) 10\({}^{-5}\) s.) and total three-cycle data are used. The predicted acoustic wave directivity for each case is presented in Figure 15.  
As can be seen in Figure 18, the three cases exhibit similar propagation patterns where the lift dipole is dominant. In the case of the dipole source term, the radiating pattern has two lobes with a 180\({}^{\circ}\) phase difference. The main direction of the lobes depends on the dominance of the lift or drag dipole forces. As shown in Figure 3, the lift time-variation is significantly greater than that of drag, therefore, hydro-acoustic waves due to dipole sources are affected primarily by the lift force, and thus exhibit the lift-dipolar pattern. Here, the directivity is inclined with respect to the mean flow due to the angle of attack. However, the directivity due to the monopole source exhibits an omni-directional radiation pattern, as it depends on the time derivative. The predicted sound pressure levels (SPL) of each noise source for the three viscous cases are compared in Figures 19 and 20. For the monopole source component, the viscous lagging approach has the largest SPLs, whereas the thin-layer approach presents the smallest SPLs. These results can be explained qualitatively by comparing the observed cavity sizes. With the viscous lagging approach, the largest cavity structures are predicted. This leads to the largest SPLs of the monopole sources, as shown in Figures 5-9, as the monopole source strength is typically proportional to the time variation of cavity volume changes. Note that, because a monopole has an omni-directional radiation pattern, the predicted acoustic pressure spectrum due to monopole sources at different azimuth angles has the same values. Figure 17: Comparison of required computational time per 50 time steps for three viscous cases. Figure 19: Predicted hydro-acoustic pressure due to monopole source: (**a**) directivity, spectrum at three azimuth angles of (**b**) 350\({}^{\circ}\); (**c**) 40\({}^{\circ}\); and (**d**) 80\({}^{\circ}\). Figure 18: Predicted directivity of radiated hydro-acoustic noise: (**a**) viscous lagging; (**b**) fill viscous; and (**c**) thin-layer cases. In the case of dipolar noise source components, the thin-layer and full viscous cases exhibit approximately the same SPLs, but the viscous lagging approach has the lowest SPLs, except at azimuth angles of 350\({}^{\circ}\) and 170\({}^{\circ}\). These results are related to the variation of hydro-dynamic forces. As shown in Figure 4, the viscous lagging approach exhibits lower spectral magnitudes across the entire frequency range for the lift force, and \(\ell\)\(\geq\)200 Hz for the drag force.