## 4 Results and Discussion
### Comparative Analysis of Internal Flow Characteristics
The stall cell develops slowly in the axial direction of the impeller, but grows more rapidly in the radial direction. From the inlet to the outlet of the impeller passage, the rotational vortex continuously expands in scale until it eventually forms a stall vortex spanning multiple impeller passages, inducing an abrupt-type rotating stall phenomenon. The characteristics of this stall phenomenon are primarily manifested in the interaction between the tip leakage flow and the rotating vortex spanning multiple impeller passages [35]. As shown in Figure 15c, flow passage B is located near the 300\({}^{\circ}\) circumferential position, where the inlet swirl angle reaches its minimum. In this region, the tip leakage flow is significantly weaker compared to flow passage A. The tip leakage flows at the leading and trailing edges are suppressed by the main flow and move downstream along the suction side. The trailing-edge leakage flow forms strip-like patterns without developing large-scale vortex structures. As shown in Figure 15, the modified impeller exhibits significantly  
Figure 14: Blade tip leakage flow in the original impeller: (**a**) streamlines at 0.9 blade height section (**b**) three-dimensional streamlines in flow passage A (**c**) three-dimensional streamlines in flow passage B.  
Figure 15: Blade tip leakage flow in the modified impeller. (**a**) streamlines at 0.9 blade height section (**b**) three-dimensional streamlines in flow passage A (**c**) three-dimensional streamlines in flow passage B.  
reduced tip leakage flow at the leading edge and mid-span regions. The trailing-edge leakage vortex is reduced to a thin strip, and no additional vortex structures appear within the impeller passage. The overall flow is stable. The streamline distribution of the blades before and after optimization is shown in \[\eta=\frac{Q\times p_{s}}{P} \tag{9}\]0 The original impeller exhibits significant low-velocity regions and local vortices near the blade surface, leading to increased fluid energy losses and reduced impeller efficiency. In contrast, the modified impeller shows a noticeable increase in flow velocity near the blades, with the suppression of flow vortices. Phenomena such as flow separation and reattachment, which contribute to flow deterioration, are reduced. This improvement enhances the consistency of the airflow direction at the blade inlet and outlet, thereby improving the aerodynamic performance of the impeller. Tip leakage flow often results in significant axial velocity deficits, commonly referred to as flow blockage. Its presence not only reduces the flow capacity near the rotor tip but also induces substantial flow losses. To quantitatively analyze the extent of flow blockage, the average blockage factor is defined, with its formula expressed as follows [36]:  
\[\eta=\frac{Q\times p_{s}}{P} \tag{9}\]1  
In the formula, \(A_{act}\) represents the area of the flow cross-section, \(\rho W_{m}\) is the meridional momentum, and \(\left(\rho W_{m}\right)_{edge}\) is the edge momentum. At the blockage boundary, \(\rho W_{m}\) is equal to \(\left(\rho W_{m}\right)_{edge}\). Figure 17 shows the distribution of the meridional average blockage factor along the flow direction in the impeller passages A and B. From the figure, it can be seen that the blockage factors in both the original and modified designs increase from the impeller inlet to the outlet across different cross-sections. Moreover, the blockage factor in impeller passage A is significantly higher than that in passage B. This is because the maximum swirl angle occurs near the 60\({}^{\circ}\) circumferential position at the impeller inlet, where the flow passage A is greatly influenced by incoming flow distortion. In contrast, flow passage B is located near the 300\({}^{\circ}\) circumferential position at the impeller inlet, close to the minimum swirl angle, and is less affected by the incoming flow distortion.