## 3 Results and Discussion
### Backflow Velocity Distribution of Tip Clearance  
To investigate the backflow velocity distribution characteristics within the tip clearance, an analysis of the backflow velocity distribution on different SPN (SPN is the abbreviation for Span Normalize, representing the dimensionless distance from the hub to the shroud) surfaces was conducted. Figure 11 illustrates the distribution of different SPN surfaces within the impeller. In Figure 11, LE and TE denote the leading edge and trailing edge of the blade, respectively, and PS and SS represent the pressure side and suction side of the blade, respectively. In the figure, SPN = 0 indicates the hub, and SPN = 1.0 indicates the shroud. As analyzed previously, there will be varying degrees of backflow in the impeller flow passage, especially at low flow rates. The regions with backflow will block the flow of the mainstream fluid, thereby causing significant energy loss. Therefore, clarifying the distribution of backflow regions within the impeller is very important for further understanding the fluid flow characteristics inside the impeller. In the impeller flow passage, an axial velocity lower than 0 m/s indicates that the axial velocity direction is opposite to the mainstream flow direction. If the axial velocity of the fluid in the flow passage is negative, it indicates that the fluid is backflowing in the flow passage. The negative sign only represents the occurrence of backflow. In the following text, backflow velocity represents the negative axial velocity. Therefore, Figures 12 and 13 present the backflow velocity distribution of the front and rear impeller. Figure 11: Distribution of different SPN surfaces in the impeller. Figure 10: Distance from the core of the vortex to the shroud: (**a**) Front impeller; (**b**) Rear impeller. Figure 12: Backflow velocity distribution of the front impeller: (**a**) 1.2 \(Q_{\text{REP}}\); (**b**) 1.0 \(Q_{\text{REP}}\); (**c**) 0.8 \(Q_{\text{REP}}\). Figure 13: Backflow velocity distribution of the rear impeller: (**a**) 1.2 \(Q_{\text{REP}}\); (**b**) 1.0 \(Q_{\text{REP}}\); (**c**) 0.8 \(Q_{\text{REP}}\). Figure 12 shows that the range affected by the backflow region is quite limited at 1.2 \(Q_{\rm BEP}\), occurring only in the flow passage of the SPN = 0.99 surface and at a relatively low velocity. The backflow mainly appears in the region influenced by the leakage vortex. At 1.0 \(Q_{\rm BEP}\), the range of the backflow region in the flow passage of the SPN = 0.99 surface increases, and the backflow velocity is greater. However, no backflow is observed when SPN \(\leq\) 0.95. As the flow rate decreases to 0.8 \(Q_{\rm BEP}\), the backflow region can be divided into two parts. One part is the region from the impeller inlet to the blade trailing edge, which is entirely covered by backflow, and the backflow velocity at the blade leading edge is significantly greater than other regions. The other part is the region near the outlet of the impeller flow passage, where the backflow velocity is generally high. It is worth noting that backflow regions still exist in the flow passage of the SPN = 0.85 surface, indicating that the influence range of the tip leakage flow and leading edge overflow is substantial under low flow rates. From Figure 13, it can be observed that when the flow rate exceeds 1.0 \(Q_{\rm BEP}\), the distribution of the backflow region in the rear impeller resembles that of the front impeller. However, under low flow rates, there are significant differences in the backflow distributions between the front and rear impellers. At the flow condition of 0.8 \(Q_{\rm BEP}\), in the SPN = 0.99 surface, the region from the inlet of the rear impeller to the trailing edge of the blade is almost entirely covered by backflow. This backflow will enter the flow passage of the front impeller, thereby causing backflow in the outlet region of the front impeller.