## 3 Results and Discussion
### Vortex Structure  
The TLV structure can be characterized by the proposed novel omega vortex identification method. The equations are as follows [9]:  
\[\Omega=\frac{\left\|\overset{\rightarrow}{\Omega}\right\|^{2}}{\left\|\overset{ \rightarrow}{S}\right\|^{2}+\left\|\overset{\rightarrow}{\Omega}\right\|^{2}+ \varepsilon}, \tag{5}\]  
\[\varepsilon=0.001\left(\left\|\overset{\rightarrow}{\Omega}\right\|^{2}- \left\|\overset{\rightarrow}{S}\right\|^{2}\right)_{\text{max}}, \tag{6}\]  
where \(\overset{\rightarrow}{\Omega}\) and \(\overset{\rightarrow}{S}\) are the vortex vector and strain rate tensor, respectively; \(\varepsilon\) is set to ensure that the denominator does not equal 0.  
\(H_{n}\) quantitatively identifies the vortex structure [28; 29]. The equation of standard helicity is expressed as:  
\[H_{n}=\frac{\overset{\rightarrow}{\Omega}\cdot\overset{\rightarrow}{W}}{ \left|\overset{\rightarrow}{\Omega}\right|\left|\overset{\rightarrow}{W} \right|}, \tag{7}\]  
where \(\overset{\rightarrow}{W}\) is the relative velocity vector.  
\(H_{n}\) can represent the vortex core when it equals to 1 or \(-1\). The vortex structure in the impeller colored by \(H_{n}\) is shown in Figure 7. Figure 8 depicts streamline released from the middle line of the tip clearance. The value \"M\" denotes the meridional distance of the blade; the values \"0.0 M\" and \"1.0 M\" indicate blade inlet and outlet, respectively. Under the high-flow condition, the streamline released at 0.0-0.1 M flows along the tip clearance. Then, the streamline flows to the suction surface, which follows the leakage flow released at 0.1-0.3 M. The streamline released at 0.3-0.6 M forms the main TLV (TLV 1). \(H_{n}\) of TLV 1 is approximately equal to one. The direction of the main TLV vector is the same as the direction of the relative velocity. TLV 1 flows to the downstream and moves to the pressure side of the adjacent blade. As a result, a passage vortex (PV) occurs. The vast majority of the streamline released at 0.6-1.0 M forms the secondary TLV or even a tertiary leakage vortex. They merge and develop with the main TLV, resulting in an increase in the leakage vortex strength.  
Figure 7: \(Cont.\)Under the BEP condition, the initial position of the TLV located in the blade inlet, as well as the circumferential motion trend, increases. The tip leakage flow includes three parts. First, TLV 3 is formed by the streamline released at 0.0-0.1 M, which flows to the downstream along the suction surface with a relatively stable flow state. Moreover, TLV 4 is formed by the streamline released at 0.1-0.6 M. TLV 4 moves to the adjacent blade in the opposite direction of the impeller rotation and breaks after collision with the wall surface. The vortex breakdown results in the appearance of PV. Finally, TLV 5, which extends to the impeller outlet, is formed by the streamline released at 0.6-1.0 M.  
The vortex structure is complex under the low-discharge condition. The vortex structure comprises the corner vortex (CV) and reverse-flow vortex (RFV), except for TLV and PV. The CV caused by the positive attack angle appears in the connection of the hub and the blade. The TLV 6 generated at the blade inlet moves toward the upstream of the impeller and breaks when it is close to the adjacent blade. Part of the leakage vortex induces PV,  
Figure 8: Tip streamline distribution along the streamwise direction: (**a**) 1.6 \(Q_{d}\); (**b**) 1.0 \(Q_{d}\); (**c**) 0.7 \(Q_{d}\).  
Figure 7: Vortex structure defined by the proposed omega vortex identification method: (**a**) 1.6 \(Q_{d}\); (**b**) 1.0 \(Q_{d}\); (**c**) 0.7 \(Q_{d}\).  
whereas the rest forms the RFV. The streamline released at 0.2-0.6 M flows through the adjacent blade. Thus, leading-edge overflow occurs.  
In addition to the above vortices, the wake vortex and tip separation vortex (TSV) are also formed under all conditions. The area of the TSV is the smallest under the BEP condition. The intensity of TSV increases when the operation deviates from the BEP condition.