## 3 Results and Discussion
### Vortex Field  
There is always an irregular-shaped, low-pressure zone before the RSS of the draft tube, occupying part of the draft tube section, indicating the existence of an eccentric vortex strip in the draft tube. The dominant circumferential velocity component at the draft tube inlet and the coiling of the shear layer between the central return flow and the external swirl flow form the vortex strip. The draft tube vortex pattern is shown in Figure 20. For NLN, there are vortices of uneven distribution and different sizes in FSS. There are two long conical vortex strips in the draft tube. The main vortex strip is longer and extends to the RSS. The secondary vortex band is shorter and only extends to the first half of the MES. For SLN, there are vortices of uneven distribution and different sizes in FSS, but the number and size are significantly smaller than that of NLN. There is only one vortex strip in the draft tube, extending to RSS; however, the length is slightly smaller than that of NLN. The vortex strip in MES is more deviated from the wall and closer to the axis than that of NLN. For ELN, there are vortices of uneven distribution and different sizes in FSS. The vortex strip in the draft tube does not show a long cone shape, but a cloud-like distribution that approximately covers the entire tube. The vortex strip in MBE is distributed closer to the wall and hardly extends to RSS. The surface area of the draft tube vortex strip in each cross-section along the flow direction shows a trend of first increasing and then decreasing, as shown in Figure 21. For NLN, the vortex surface area of FSS accounts for 28% of the total surface area of the draft tube vortex, that of MES accounts for 66%, and that of RSS accounts for 6%. For SLN, the vortex strip surface area of each section is significantly smaller than that of NLN, but the vortex strip surface area proportion of MES increases to 69%. It can be seen that the locking nut can significantly reduce the size of the vortex strip, but the degree of the reduction of the MES is smaller than that of other sections. For ELN, both FSS and RSS vortex strip surface areas are significantly reduced compared to that of NLN, and the vortex strip surface area in the MES is significantly increased compared to that of NLN; however, the vortex strip surface area decreases sharply in RSS, which shows that the extended locking nut contributes little to the reduction in vortex strip before RSS but can reduce the vortex strip sharply in RSS. Figure 21: The surface area of the draft tube vortex. Figure 20: Vortex pattern of draft tube: (**a**) NLN; (**b**) SLN; (**c**) ELN. Vortex intensity was characterized by velocity invariant \(Q\), the definition of \(Q\) is:  
\[Q\ =C_{Q}\Big{(}\Omega^{2}-S^{2}\Big{)} \tag{13}\]  
where \(S\) is the absolute value of the strain rate and \(\Omega\) is the absolute value of vortices. When the velocity invariant \(Q>0\), it indicates that the rotation rate of the fluid in this region is greater than the strain rate, and the flow vortex structure is dominant. The vortex intensity of the draft tube decreases sharply then fluctuates slightly and finally stabilizes, as shown in Figure 22. For NLN, the vortex strength is greatest at the FSS inlet, decreases to a minimum by the second half of the MES, and is essentially smooth by RSS. For SLN, the vortex intensity at the FSS inlet is smaller than that of NLN and decreases to the minimum value in the middle part of MES, which is larger than that of NLN. There is a peak in the second half of MES, and a second trough appears at the RSS inlet, which increases and tends to be stable. The vortex intensity after entering RSS is basically consistent with that of NLN. For ELN, the vortex intensity at the FSS inlet is smaller than that of NLN and decreases to the minimum value in the first half of MES, which is smaller than that of NLN. There is a peak in the second half of MES, and a second trough appears at the RSS inlet, which increases and tends to be stable, with a stable value smaller than that of NLN.