## 3 Results and Discussion
### Tip Leakage Flow Pattern and Vortex Trajectory  
Figure 6 shows the tip leakage flow pattern of the front impeller. Due to the clearance between the tip and shroud, leakage flow is generated at the clearance. The flow pattern of the leakage flow varies significantly under different flow rates. At a large flow rate (1.2 \(Q_{\text{BEP}}\)), a long helical leakage vortex is formed due to the mixing of the tip leakage flow and the mainstream in the impeller. The vortex is initially generated near the suction surface close to the leading edge, and then it develops in the impeller and gradually approaches the pressure surface and trailing edge of the next stage blade. At the optimal flow rate (1.0 \(Q_{\text{BEP}}\)), the shape of the leakage vortex is similar to that under the 1.2 \(Q_{\text{BEP}}\) flow rate, but the leakage vortex mixes more mainstream fluid, and the vortex has a greater influence on the mainstream flow field. In addition, near the outlet of the impeller passage, the distance between the vortex and the pressure surface of the next stage blade is closer. As the flow rate further decreases (0.8 \(Q_{\text{BEP}}\)), the leakage flow pattern produces significant differences compared with the large flow rate, and the leakage flow pattern gradually expands to the blade suction surface and the pressure surface of the next stage  
Figure 5: Comparison of numerical simulation and experimental results of fan efficiency. Figure 6: Tip leakage flow pattern of the front impeller: (**a**) 1.2 \(Q_{\text{BEP}}\); (**b**) 1.0 \(Q_{\text{BEP}}\); (**c**) 0.8 \(Q_{\text{BEP}}\). blade. It is worth noting that the leakage vortex does not flow out from the impeller outlet. Part of the leakage flow will return to the upstream region from the impeller inlet, and part of the leakage flow will enter the next flow passage from the tip clearance of the next stage blade and become the leakage flow of the next stage blade. Another part of the leakage flow will overflow from the leading edge of the next stage blade into the next flow passage and mix with the tip leakage flow of the next stage blade to form a stronger vortex, which further increases the influence range. It can be seen that the tip leakage flow and the overflow of the leading edge are the main causes of the backflow in the inlet region of the front impeller. Figure 7 shows the tip leakage flow pattern of the rear impeller. Under the 1.2 \(Q_{\text{BEP}}\) flow rate, similar to the leakage flow of the front impeller, a helical leakage vortex is formed due to the mixing of the leakage flow and the mainstream and flows out from the impeller outlet. As the flow rate decreases, the leakage flow pattern of the rear impeller is significantly different from that of the front impeller. When the flow rate is 1.0 \(Q_{\text{BEP}}\), the influence range of the helical leakage vortex caused by the tip leakage flow on the impeller is significantly increased. Part of the leakage flow even flows into the next flow passage from the tip clearance of the next stage blade and forms a helical vortex, which increases the intensity and influence range of the vortex in the flow passage. As the flow further decreases (0.8 \(Q_{\text{BEP}}\)), the area affected by the tip leakage flow becomes larger, blocking the impeller inlet. A portion of the leakage flow will return to the front impeller region from the rear impeller inlet, another portion will enter the next flow passage through the tip clearance of the next stage blade, and yet another part will spill over from the leading edge of the next stage blade to the next flow passage. Additionally, part of the fluid at the impeller inlet does not flow out from the impeller outlet, but overflows from the leading edge of the next stage blade to the next flow passage. Therefore, vortices are formed under the mixing of leakage flow, leading edge overflow, and the main flow. Figure 8 shows the leakage vortex structure and vortex core trajectory distribution of the front impeller.