*3.3. Cavitation Distribution Analysis for Di*ff*erence Pressure Di*ff*erences*

Figure 11 demonstrates the vapor distributions inside the sleeve regulating valve for different pressure difference with the valve core displacement of 60 mm. In general, when cavitation causes severe valve damages, the vapor volume fraction of each computational cell, α, is higher than 0.5. Figure 11a–c depicts three-dimensional isosurfaces of αlarger than 0.5 in valves with the valve core displacement of 60 mm for different pressure differences.

**Figure 11.** Vapor distributions inside the sleeve regulating valve for different pressure difference with the valve core displacement of 60 mm (**a**) 2 MPa; (**b**) 5 MPa; (**c**) 8 MPa.

For the pressure difference of 2 MPa, the vapor appears at the orifices inlet of the sleeve and the distribution region is very small. When the pressure difference is increased from 2 MPa to 5 MPa, the vapor is still concentrated at the orifice inlet of the sleeve but the distribution region is enlarged slightly. Furthermore, when the pressure difference is 8 MPa, the vapor begins to appear at the outlet of orifices in the sleeve and the distribution region is further enlarged. As a whole, the cavitation intensity and density are tiny when the valve core displacement is 60 mm and the pressure difference has a slight influence on the cavitation intensity and density inside the regulating valve.

When the valve core displacement is decreased from 60 mm to 30 mm, as is shown in Figure 12, the effect of the pressure difference on the cavitation intensity and density is enhanced. The vapor distribution region is still very small when the pressure difference is 2 MPa. However, when the pressure difference is increased from 2 MPa to 4 MPa, the cavitation distribution region is obviously enlarged and the vapor begins to appear on the sealing surface of the valve core. There is no doubt the increase of the pressure difference induces a more serious cavitation. Moreover, when the pressure difference reaches the maximum value of 8 MPa, the vapor almost fills the orifices in the valve and the total sealing surface is filled with vapor. Meanwhile, the vapor distribution region begins to extend downward along the surface of the valve chamber.

**Figure 12.** Vapor distributions inside the sleeve regulating valve for different pressure difference with the valve core displacement of 30 mm (**a**) 2 MPa; (**b**) 5 MPa; (**c**) 8 MPa.

To further quantify the influence of the pressure difference on cavitation characteristics of the sleeve regulating valve, the total vapor volume is also calculated using the following equation:

$$V\_v = \iiint\limits\_{\Omega} adV\tag{9}$$

where α denotes the vapor volume fraction in an element. The total vapor volume demonstrates the whole vapor caused by cavitation, so the total vapor volume can represent the intensity of the cavitation. The higher the total vapor volume is, the more intense the cavitation intensity is. The total vapor volume quantifies the cavitation intensity.

Figure 13 depicts the total vapor volume variation with the increase of the pressure difference when the valve core displacement is 60 mm and 30 mm. It can be found that the total vapor volumes for two opening state are increased with the increase of the pressure difference. However, the increase trends for two opening states are totally different. When the pressure difference is lower than 3 MPa, the total vapor volumes for two opening states are close to 0. The above phenomenon indicates that the cavitation intensity is still very low for both opening states when the pressure difference is lower than 3 MPa. The low-pressure difference induces the slight cavitation. When the pressure difference is higher than 3 MPa, the increase trend for the half opening state begins to be steep suddenly, while the increase trend for the half full opening state is still very gentle.

**Figure 13.** Total vapor volumes inside the valve for different pressure differences with the valve core displacement of 60 mm and 30 mm.

To estimate the probable occurrence of cavitation, the cavitation index σ*<sup>v</sup>* is calculated and denoted as

$$
\sigma\_v = \frac{p\_d - p\_v}{p\_u - p\_d} \tag{10}
$$

where *pu* is the inlet pressure, *p*<sup>d</sup> is the outlet pressure, and the lower the σ*<sup>v</sup>* is, the higher is the potential. In general, when the σ*<sup>v</sup>* is lower than 1.0, the cavitation will occur. When the σ*<sup>v</sup>* is lower than 0.5, the cavitation phenomenon will be stable.

In this study, the inlet and outlet pressure are determined by the inlet and outlet boundary conditions. The saturation vapor pressure is set as a constant of 1.5 MPa. Therefore, the cavitation index for the valve at specified pressure conditions is certain. To further quantify the actual influence of the cavitation index on the actual cavitation intensity in the sleeve regulating valve, the total vapor volumes variation with the change of the cavitation index is shown in Figure 14. It can be seen that for different opening states, the actual effects of the cavitation index on the total vapor volumes are really different, and the lower the cavitation index is, the higher the difference is for different opening states. As the cavitation index is higher than 0.2, for both two opening state, the variation of the cavitation index has a small influence on the total vapor volume and cavitation intensity. When the cavitation index is lower than 0.2, the total vapor volume is increased with a steep trend at the half opening state, which indicates that a small variation of cavitation index induces a more drastic variation of the total vapor volume. When the valve core displacement is 60 mm, the variation trend of total vapor volume is gentler than the trend at the half opening state when the cavitation is lower than 0.2. As a whole, the relation between actual cavitation intensity and cavitation index is not simply linear and is changed with the change of valve core displacement. When the cavitation index is higher than 0.2, the cavitation index's effects on cavitation intensity are small. As the cavitation index is lower than 0.2, the cavitation index's effects on cavitation intensity are intense.

**Figure 14.** Total vapor volumes inside the valve for different cavitation index with the valve core displacement of 60 mm and 30 mm.

#### **4. Conclusions**

The cavitation occurring in a sleeve regulating valve for different pressure differences and valve core displacements has been numerically investigated in this study, and the effects of pressure difference and valve core displacement have been revealed using the proposed numerical model.

First, the flow streamlines, pressure distribution and vapor distribution for different opening state are obtained. A high-velocity and low-pressure region appear behind the valve sleeve because of the sudden decrease of the cross-section area. According to the predicted vapor distribution, the vapor is mainly concentrated in the edge of orifice inlet at the full opening state. The decrease of the valve core displacement induces the enlargement of the vapor distribution region and the increase of the vapor density.

Second, the pressure and flow streamlines for different pressure difference are analyzed. With the increase in pressure difference, the pressure at the orifice inlet of the sleeve is increased while the pressure drop when flowing through the sleeve is increased. The region with high pressure at the center of the sleeve is enlarged with the increase of the pressure difference, while the value of pressure is increased. The inlet velocity is increased with the increase of the pressure difference. The velocity rise when flowing through the sleeve is increased while the velocity at the center of valve chamber is increased with the increase of pressure difference.

Last, the cavitation distributions inside the sleeve regulating valve for different pressure differences are analyzed. It can be seen that the increase of the pressure difference induces a more serious cavitation. The pressure difference has a slight influence on the cavitation intensity and density inside the regulating valve when the valve core displacement is 60 mm. When the valve core displacement is decreased from 60 mm to 30 mm, the effects of the pressure difference on the cavitation intensity are enhanced. The relation between actual cavitation intensity and cavitation index is not simply linear and is changed with the change of valve core displacement. When the cavitation index is higher than 0.2, the cavitation index's effects on cavitation intensity are small. As the cavitation index is lower than 0.2, the cavitation index's effects on cavitation intensity are intense.

**Author Contributions:** Conceptualization, C.Q. and C.-H.J.; methodology, C.Q. and C.-H.J.; software, C.Q.; validation, C.-H.J., H.Z. and J.-Y.W.; formal analysis, C.Q.; investigation, H.Z.; resources, C.Q.; data curation, C.-H.J.; writing—original draft preparation, C.Q.; writing—review and editing, C.-H.J.; visualization, J.-Y.W.; supervision, Z.-J.J.; project administration, Z.-J.J.; funding acquisition, Z.-J.J.

**Funding:** This research was funded by the National Natural Science Foundation of China (NSFC), grant number 51875514; the Zhejiang Key Research & Development Project, grant number 2019C01025; and the Zhejiang Quality and Technical Supervision Research Project, grant number 20180117.

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