*3.3. Free Surface-Cavitation Interaction at Di*ff*erent Water Depths*

In the previous section, the effect of a cavitation near a free surface was analyzed. For high-speed vehicles, the distance between the underwater component and the free surface is constantly changing during movement, and the effect of the free surface on the cavitation flow varies at different depths. To study the effect of depth on the cavitation, we carried out the cavitating flows over the hydrofoil at five different water depths.

Figure 14 shows the cavity evolution for a typical cycle at five different depths. Similar typical cavitation characteristics can be observed in all cases: At *t*1, the sheet cavitation begins to develop downstream of the leading edge of the hydrofoil. Also, the cloud cavitation reaches its maximum. From *t*<sup>1</sup> to *t*<sup>1</sup> + 3*T*1/6, the sheet cavitation develops and reaches its maximum at *t*<sup>1</sup> + 3*T*1/6. The cloud cavitation moves downstream and becomes smaller because of collapsing.

**Figure 14.** Comparison of cavity evolution in one typical cycle compared at five different water depths.

From *t*<sup>1</sup> + 4*T*1/6 to *t*<sup>1</sup> + 5*T*1/6, sheet cavitation begins shedding with the effect of the re-entrant flow, and then the next cycle begins. These characteristics are the same in the other four cases, where *T*2, *T*3, *T*4, and *T*<sup>5</sup> represent different cycles for different cases.

The previous section described the influence on cavitation near a free surface, which shows a certain regularity at different depths. As the hydrofoil moves to greater depths, the cloud cavitation at the maximum size (*ti*) increases, and the sheet cavitation at the maximum size (*ti* + 3*Ti*/6) shows the same trend. Thus, the increase in water depth can promote the cavitation intensity at the same cavitation number.

The cavity volume fluctuations proved the above conclusion, which is shown in Figure 15. Five complete cycles have passed before the results are recorded, and the results are convergent. The peaks of periodic fluctuations under the same working conditions are different, indicating that cavity evolution is a strong unsteady phenomenon. Although it exhibits a certain periodicity, the flow characteristics of each cycle are not exactly the same. The cavity volume increases with increasing depth from the free surface, indicating that the effect of the free surface on cavitation decreases as the depth from the free surface increases. As the depth from the free surface increases, the fluctuations of the cavity volume become more obvious. It indicates that the volume of the cavitation shedding is larger, which is consistent with the conclusion obtained in Figure 14. Hence, the free surface inhibits the development of cavitation, leading to a reduction in cavitation volume.

**Figure 15.** Comparison of the cavity volume fluctuations for five different water depths.

As mentioned above, the depth from the free surface affects the shape of the cavitation, and changes in cavitation shape alter the fluctuation characteristics of the pressure. Based on the pressure fluctuations of case 2 (*h* = 0.25*C*) shown above, Figure 16 presents the pressure fluctuation on the suction surface of hydrofoil at P3 with the other cases. Since the cavitation phenomenon is highly unstable, the pressure fluctuations obtained by numerical simulation are not stable at the peak. The fluctuation of the pressure is generated when the sheet cavitation is cut off by the re-entrant flow. When the sheet cavitation breaks off, and the cavitation again covers the monitoring point, the pressure value will be smooth close to the saturated vapor pressure. Moreover, the fluctuation of the pressure curves expresses the different frequencies of the cavitation cycle. It can be seen that as the depth of the hydrofoil increases, the frequency of the cavity shedding decreases, gradually approaching the frequency of the cavity shedding when there is no free surface. This result further demonstrates that the free surface inhibits the evolution of the cavity and increases the frequency of cavitation shedding.

**Figure 16.** Comparison of the pressure fluctuation on the suction surface of hydrofoil at P3 in different water depth condition.

The corresponding PSD is shown in Figure 17. The maximum value of the curve represents the frequency of the cavity shedding. The pressure fluctuations in Figure 16 show that the duration of a cavitation period increases as the depth below the free surface increases, and thus, the frequency decreases, as verified by Figure 17. The frequency of the cavitation shedding period was extracted for each case. It can be seen that as the depth of the hydrofoil increases, the frequency of cavitation shedding decreases. This result may be attributed to the near free surface condition, the reduced cavitation length, and the reduced time required for development and shedding.

The effect of the free surface on the hydrofoil cavitation length also affects the lift resistance of the hydrofoil, which is an important factor for assessing the extent to how the free surface affects the cavitation and hydrofoil. Figure 18a shows the lift coefficient of the hydrofoil. It can be seen that the lift coefficient exhibits periodic fluctuations, consistent with the cavitation shedding period. The coefficient also shows strong instability, which is related to the unsteadiness of cavitation. To study the relationship between the different depth and the lift coefficient, the average lift coefficients for each operating condition is shown in Figure 18b. It can be seen that the average value of the lift coefficient decreases as the depth from the free surface decreases, which results that the relationship between lift coefficient and depth is linear.

**Figure 17.** Comparison of the power spectral density of pressure fluctuations in different water depths.

**Figure 18.** *Cont.*

**Figure 18.** Comparison of the lift coefficient fluctuation and the average value at different water depth. (**a**) Lift coefficient fluctuations at different water depths; (**b**) Average lift coefficient at different water depths.

Figure 19a shows the drag coefficient of the hydrofoil at different water depths. The drag coefficient exhibits a cyclical trend similar to that of the lift coefficient curve, although the drag coefficient is much smaller than the lift coefficient. The average drag coefficient for different depths is shown in Figure 19b. As observed for the lift coefficient, the average drag coefficient increases with the increasing depth. Also the average drag coefficient has a linear development as the depth increasing.

**Figure 19.** *Cont.*

**Figure 19.** Comparison of the drag coefficient fluctuation and the average value at different water depths. (**a**) Drag coefficient fluctuations at different water depths; (**b**) Average drag coefficient at different water depths.

The effect of free surface on hydrofoil cavitation was analyzed above. Similarly, the cavitation of the hydrofoil near a free surface will also affect the free surface and form a wave. Figure 20 presents a water volume fraction scalar of the cross-section in the direction of the hydrofoil span of the hydrofoil at the same cavitation cycle stage. The shape of the free surface affected by the hydrofoil is shown in the figure. As the distance from the free surface increases, the influence of the hydrofoil on the free surface gradually decreases, and the liquid surface gradually becomes flat. In addition to the wave above the hydrofoil, the free surface behind the hydrofoil exhibits fluctuations, corresponding to the rectangular portion outside the dotted circle. As shown in Figure 20, as the depth of the hydrofoil increases, the turbulence at the hydrofoil tail also decreases. At the same time, the fluctuation of the free surface behind the hydrofoil is weakened. Hence, free surface evolution due to hydrofoil cavitation occurs not only above the hydrofoil, but also behind the hydrofoil.

**Figure 20.** Volume fraction of water on the mid-section plane for different depths at the same cavitation cycle stage.

Positional information for the free surface was extracted from Figure 19 and is shown in Figure 21. The hydrofoil in the figure indicates the relative position of the hydrofoil under various working conditions and does not reflect the true depth. It can be seen that the wave height of the free surface above the hydrofoil decreases as the depth from the free surface increases. For *h* = 0.25*C*, 0.5*C*, 0.75*C* and 1*C*, the wavelength increases with increasing depth. However, for *h* = 1.25*C*, the wavelength is shorter than that for *h* = 1*C*. We speculate that there may be a transition point of the wavelength between *h* = 1*C* and *h* = 1.25*C*. At this point, the wavelength reaches a maximum above the hydrofoil. As a result, the effects of hydrofoil cavitation on the wave height are uniformly changed and there is a maximum wave length between *h* = 1.25*C* and *h* = 1*C* cases. We will carry out more detailed work in the future to study the mechanism of this situation.

**Figure 21.** Free surface change caused by hydrofoil cavitation near the free surface in different depth at the same cavitation cycle stage.

#### **4. Conclusions**

In this paper, the unsteady cavitation dynamics over a NACA66 hydrofoil near free surface is investigated using the LES method coupled with the Schnerr-Sauer cavitation model. The interaction between free surface and cavitation is evaluated by analyzing cavity evolution, pressure pulsation, hydrodynamic loads and dynamic mode. The main conclusions are as follows:


**Author Contributions:** The manuscript was written by T.S. and Q.X.; all authors discussed the original idea; conceptualization, T.S. and L.Z.; methodology, T.S.; software, Q.X.; validation, T.S., Q.X. and L.Z.; formal analysis, T.S.; investigation, H.W.; resources, C.X.; data curation, Q.X.; writing—original draft preparation, T.S.; writing—review and editing, T.S.; visualization, C.X.; supervision, L.Z.; project administration, L.Z.; funding acquisition, L.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (grant number 51709042), the China Postdoctoral Science Foundation (2019T120211, 2018M631791), the Natural Science Foundation of Liaoning Province of China (20180550619), the Liao Ning Revitalization Talents Program (XLYC1908027).

**Conflicts of Interest:** The authors declare no conflict of interest.
