## 3 Results and Discussion
### Swirling Flow Field  
Due to the influence of the runner, the flow inside the draft tube presents a swirling flow field. In addition to axial flow, there is also a certain degree of swirling flow in the draft tube inlet. The swirling flow intensity can be expressed by the swirling number S, which  
Figure 8: Head loss of the draft tube. Figure 9: Head loss in the middle elbow section. is the ratio of swirling momentum flux to axial momentum flux and is a dimensionless number. The calculation method is shown in the following equation,  
\[\text{S}= \frac{1}{R}\frac{\int_{0}^{\text{R}}r^{2}V_{\text{c}}V_{\text{a}}d_{ \text{r}}}{\int_{0}^{R}rV_{\text{a}}^{2}d_{\text{r}}} \tag{10}\]  
where \(V_{\text{c}}\) is the circumferential velocity, \(V_{\text{a}}\) is the axial velocity, \(R\) is the radius of the draft tube, and \(\text{r}\) represents the radial distance. In the draft tube, FSS is the strengthening section of the swirling flow, and MES is the fast attenuation section of the swirling flow, as shown in Figure 10. The flow in the draft tube gradually changes from the dominance of swirling momentum flux to the dominance of axial momentum flux. In the first half of the MES, most of the fluid is dominated by the circumferential velocity and is thrown downstream along the draft tube wall. RSS is the continuous decrease in swirling flow to the basic disappearance section. The swirling number at the inlet of the draft tube is the largest in NLN, followed by SLN, and is the smallest in ELN. In FSS, the increase rate of swirling number for SLN is the smallest, ELN is slightly larger, and NLN is the largest. In MES, the decrease rate of the swirling number of ELN is the largest; however, it first decays to the predominance of circumferential velocity. At the exit cross-section of MES, the swirling number of NLN, SLN, and ELN decreases sequentially and remains throughout the RSS. In RSS, the swirling number for SLN decreases to a minimum and then increases slightly, but ELN has maintained a decreasing or flat trend. Therefore, it can be seen that the locking nut significantly reduces the circumferential velocity of the draft tube inlet, and the larger the nut size, the more obvious the circumferential flow reduction. To a certain extent, the attenuation speed of swirling flow for MES is also accelerated, and the larger the nut size, the faster the swirling flow attenuation speed. The swirling flow contains axial and circumferential velocity components, and the circumferential velocity component is the main reason for the hydraulic loss of the draft tube, as shown in Figure 11. The circumferential velocity of FSS and MES decreases rapidly, while the circumferential velocity of RSS decreases slowly. The circumferential velocity of the draft tube inlet is the largest in ELN, followed by NLN, and the smallest in SLN. In FSS and MES, the circumferential velocity of ELN decreases the fastest, and the circumferential velocities of SLN and NLN decrease at essentially the same rate; however, both are smaller than that of ELN. In RSS, the circumferential velocity of ELN remains almost constant, and both the SLN and NLN decrease in circumferential velocity at a lower rate and are essentially the same. The circumferential velocity of SLN decreases to the same as that of ELN at the RSS outlet cross-section. Figure 10: Swirling flow changes in the draft tube. The circumferential velocity of the draft tube increases gradually from the center to the wall, as shown in Figure 12. For NLN, there are two elliptical low circumferential velocity zones in the middle of the FSS inlet, and it develops into an eccentric elliptical low circumferential velocity zone when the MES inlet and this low-velocity zone gradually expands and maintains until the RSS inlet. For SLN, there is an eccentric elliptical low circumferential velocity zone located close to the locking nut in the FSS inlet, and this low-velocity zone gradually expands and maintains up to the RSS inlet.