# Numerical Study on Tip Vortex Cavitation Inception on a Foil
## 3 Numerical Results and Discussion
### Validation of Cavitating Flows  
This section presents the results of the numerical analysis of the tip vortex cavitation of the target elliptic planform hydrofoil. The cavitating flows around the foil were computed at \(R_{e}\) = 5.3 \(\times\) 10\({}^{5}\) for the cavitation numbers, \(\sigma\) = 0.58 and 1.15, in which the foil has an effective angle of attack \(\alpha_{e}\) = 9.5\({}^{\circ}\)[7]. Figure 12 shows a comparison of tip vortex cavitation obtained from the experimental observation and the numerical analysis using the RSM turbulence model on the grid resolution G5. The iso-surfaces of the cavitation were plotted as the vapor volume fraction of 0.5. The observed cavitation for the cavitation number of 1.15 in the experiment, attached to the tip, was highly stable both spatially and temporally and extended well into the far downstream of the water tunnel [7]. As discussed by [PERSON] et al. [7], the tip vortex cavity for the cavitation number of 0.58 showed an instability giving the appearance of a  
Figure 11: Pressure coefficient distribution on the suction and pressure surfaces of the foil: (**a**) suction side, RKE; (**b**) suction side, SKW; (**c**) suction side, RSM; (**d**) pressure side, RKE; (**e**) pressure side, SKW; (**f**) pressure side, RSM. twisted ribbon. Lastly, the current RSM results for both cavitation numbers showed a good agreement with the cavitation observations of the experiment. In particular, the numerical result for the cavitation number of 0.58 reproduced a similar the stationary wave shape of the cavity, its wavelength, and volume change downstream of the foil. On the other hand, in the current numerical analysis result, it is seen that sheet cavitation covers the surface of the foil. The same trend was found in the numerical results of [PERSON] et al. [23]. This might be due to the surface condition of the foil in the experiment and the nuclei condition in the cavitation tunnel which were not considered in both numerical analyses. The presence of cavitation changed the flow feature in the tip vortex significantly. Figure 13 presents the pressure, axial and vertical velocities and vorticity magnitude at the cross section of the tip vortex at x/c = 1.0 for fully wetted flow and cavitating flow conditions. In Figure 13b,c, the tip vortex cavitation profiles are shown by solid lines. It is seen that the pressure coefficients inside the vortex cavities for each cavitating flow condition are close to the given cavitation numbers. This is due to the low density of the vapor inside the cavity resulting in the reduction in the axial pressure gradient. Furthermore, the axial velocity in the tip vortex region generally decreased due to the decrease in the axial pressure gradient at each cavitation condition, but in the current study, it increased slightly in the vortex cavitation core for the cavitation number of 1.15. The vertical velocity gradient near the cavitation interface was distorted due to the growth of the vapor surrounded by the water as shown in Figure 13b,c. A comparison of vorticity magnitude revealed that the rotation was suppressed inside the vortex cavitation where there were changes in the pressure gradient and velocities due to the density difference between the water and vapor. The influence of cavitation on tip vortex flow is illustrated in Figure 14 where the axial velocity, vertical velocity and pressure coefficient profiles are shown along the horizontal line across the vortex core. As discussed in Figure 13, at lower cavitation number, the axial and vertical velocities greatly varied in the vortex region due to the cavitation growth and reduction in the pressure gradients. Figure 12: Development of tip vortex cavitation at different cavitation numbers: (**a**) EXP, \(\sigma=1.15\); (**b**) EXP, \(\sigma=0.58\); (**c**) CFD, \(\sigma=1.15\); (**d**) CFD, \(\sigma=0.58\).