## 4 Conclusions  
1. Due to the clearance between the blade tip and the shroud, leakage flow occurs within this gap. When the flow rate surpasses 1.0 \(Q_{\text{BEP}}\), the complex flow field near the blade tip is mainly caused by the tip leakage flow. The interaction between the tip leakage flow and the mainstream in the impeller flow passage forms a long helical leakage vortex. Under small flow rates, the tip leakage flow and the leading edge overflow are the main causes of the flow field disturbance within the impeller flow passage. 2. The flow rate has a significant impact on the structure and trajectory of the leakage vortex. For the front impeller, under large flow rates, the initial position of the vortex core is close to the middle of the blade tip. As the flow rate decreases, the initial position of the vortex core gradually moves towards the leading edge of the blade. During the propagation of the vortex from its initial position downstream, it gradually approaches the pressure side of the next stage blade, and the vortex structure evolves from an initially circular shape into a flattened elliptical shape. As the flow rate decreases, the distance between the vortex trajectory line and the blade pressure surface becomes increasingly smaller. Under the flow rates ranging from 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 similar to those of the front impeller. However, at 0.9 \(Q_{\text{BEP}}\), for the front impeller, the direction of the vortex core trajectory line changes near the leading edge of the next stage blade and ultimately extends downstream along the pressure surface 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 surface of the next stage blade. 3. The tip leakage flow and leading edge overflow form backflow within the impeller. When the flow rate exceeds 1.0 \(Q_{\text{BEP}}\), backflow primarily occurs in the area near the leakage vortex. As the flow rate decreases, the region of tip backflow significantly increases. For the front impeller, the backflow mainly appears in the region from the inlet of the impeller to the blade trailing edge, which is entirely covered by backflow. Additionally, the backflow velocity in the region near the impeller outlet is generally high. For the rear impeller, the region from the inlet of the impeller to the blade trailing edge is almost entirely covered by backflow, and this backflow will enter the flow passage of the front impeller, thereby causing backflow in the outlet region of the front impeller. 4. Under large flow rates, the spiral vortex caused by the tip leakage flow is the main cause of energy loss. However, under low flow rates, the combined action of backflow, tip leakage flow, and leading edge overflow are the key factors for the increased energy loss. Moreover, the backflow of fluid from the rear impeller inlet to the front impeller outlet will further intensify the generation of energy loss. Conceptualization, Y.C. and S.C.; methodology, Y.C., R.L. and S.C.; software, Y.C. ; validation, Y.C. and S.C.; formal analysis, Y.C. ; investigation, Y.C. ; data curation, Y.C. ; writing--original draft preparation, Y.C. ; writing--review and editing, S.C.; visualization, Y.C. ; supervision, R.L. and W.P. ; project administration, S.C.; funding acquisition, W.P. and S.C. All authors have read and agreed to the published version of the manuscript. This research was funded by the Natural Science Foundation of Hunan Province, China (2023 JJ50232) and the National Natural Science Foundation of China (5186401). Not applicable. Data Availability Statement: The original contributions presented in this study are included in the article, and further inquiries can be directed to the corresponding author. We would like to thank the technical support of Ping An Electric Co., Ltd. Conflicts of Interest: The authors declare no conflicts of interest.