## 3 Results and Discussion
### Tip Leakage Flow Pattern and Vortex Trajectory
The Q-criterion method is used for the visualization of the vortex structure, with a threshold of 500,000 s\({}^{-2}\). Cross-sections are set along the vortex structure to observe the vortex intensity distribution in the tip region. The figure provides the vortex intensity contour maps for some of these cross-sections. The positions with the maximum vortex intensity on each cross-section are marked as the vortex core, denoted by white dots in the figure. To obtain the vortex core trajectory, the coordinates of the vortex core on each cross-section were extracted and plotted as a curve in the blade-to-blade surface. The blue curve in the figure represents the vortex trajectory line. Figure 7: Tip leakage flow pattern of the rear impeller: (**a**) 1.2 \(Q_{\text{BEP}}\); (**b**) 1.0 \(Q_{\text{BEP}}\); (**c**) 0.8 \(Q_{\text{BEP}}\). approaches the pressure side of the next stage blade. Moreover, the vortex structure evolves from an initially circular shape into an elliptical shape. With the decrease in flow rate, the distance between the vortex trajectory line and the pressure side of the blade becomes increasingly smaller. It can also be observed from Figure 8 that under different flow rates, the vortex core trajectory line forms a certain angle with the blade. This angle increases as the flow rate decreases. It is worth noting that when the flow rate is between 1.0 \(Q_{\text{BEP}}\) and 1.2 \(Q_{\text{BEP}}\), the vortex trajectory line shows an overall linear trend in the streamline development diagram. However, when the flow rate decreases to 0.9 \(Q_{\text{BEP}}\), there is a significant change in the direction of the vortex trajectory line. Especially at the flow condition of 0.8 \(Q_{\text{BEP}}\), the vortex trajectory line resembles a curve. It should be pointed out that under low flow rates, as the vortex propagates downstream, the vortex intensity on the cross-section will significantly decrease, with a reduction greater than that observed between 1.0 \(Q_{\text{BEP}}\) and 1.2 \(Q_{\text{BEP}}\). However, the area of the vortex structure will markedly increase, indicating that the leakage flow will entrain more mainstream fluid under low flow rates, thereby expanding its influence range. At the same time, a vortex structure appears near the leading edge of the blade under low flow rates, indicating that part of the fluid overflows from the leading edge into the flow passage of the next stage blade. Figure 9 shows the leakage vortex structure and vortex core trajectory distribution of the rear impeller. As can be seen from Figure 9, under the flow rates of 1.0 \(Q_{\text{BEP}}\) to 1.2 \(Q_{\text{BEP}}\), the vortex structure and vortex core trajectory line of the rear impeller are relatively close to those of the front impeller, but the distance between the vortex trajectory line of the rear impeller and the pressure side of the blade is even closer. When the flow rate decreases to 0.9 \(Q_{\text{BEP}}\), there are significant differences in the vortex structure and vortex core trajectory line of the rear impeller compared to the front impeller, especially in the vortex core trajectory. For the front impeller, the vortex core trajectory line changes direction near the leading edge of the next stage blade and ultimately extends downstream along the pressure side of the next stage blade. In contrast, the trajectory line of the rear impeller still shows an overall linear trend and ultimately extends towards the suction side of the next stage blade, indicating that the leakage flow will overflow from the leading edge of the next stage blade to the next flow passage. As the flow rate decreases to 0.8 \(Q_{\text{BEP}}\), the endpoint of the vortex core trajectory line will appear at the impeller inlet, and the vortex will block the impeller inlet and cause a significant backflow.