## 4 Conclusions  
This work performs a numerical investigation on the evolution behaviors of separated vortices and near-wall flow in a centrifugal impeller operated in off-designed flow rate condition. The flow characteristics are simulated using the URANS approach, which is well validated, and are discussed and analyzed in terms of temporal variations and three-dimensionality characteristics of flow in the stalled and unstalled passages of the impeller. The following conclusions are obtained:  
1. At the axial cross section \(Z/b_{2}=0.15\), the APG of the pressure surface of the stalled passage at the beginning of the blade generally shows a sharp increase, while the magnitude of APG on the suction surface shows a slight increase within a vortex formation period. At \(S/S_{0}=0.4\), the wall shear stress of the blade pressure surface also increases sharply, and the peak value reaches about 2.0. The wall shear stress of the suction surface of the blade slowly increases, and the peak value reaches about 7.5 at the impeller outlet section. The large-scale vortex is seen almost blocking the flow channel which decays, merges, sheds, and grows in time. 2. At the axial cross section \(Z/b_{2}=0.5\), the AFG of the pressure surface and suction surface of the stalled passage are similar to the variation at \(Z/b_{2}=0.15\). However, the wall shear stress of the pressure surface increases rapidly at first and then slowly increases near \(S/S_{0}=0.4\). The peak value  
Figure 14: Blade-to-blade distribution of instantaneous radial velocity for the \(Z/b_{2}=0.5\) cross section: (**a**) \(t=1/12T\), (**b**) \(t=3/12T\), (**c**) \(t=5/12T\), (**d**) \(t=7/12T\), (**e**) \(t=9/12T\), and (**f**) \(t=11/12T\). of wall shear stress reaches about 4.0. Although the variation trend of the wall shear stress of the suction surface also exhibits peaks and valleys, the peak value reaches about 0.8. Comparing the wall shear stress of the same pressure surface at different axial positions, it is found that the wall shear stress increases the most at the beginning of \(S/S_{0}=0.6\). 3. At the axial cross section \(Z/b_{2}=0.15\), the APG of the pressure surface of the unstalled passage is at around the inlet of the impeller (\(S/S_{0}=0.4\)), in which FPG generally shows a decreasing trend, while the FPG of the suction surface shows a slow decrease. The wall shear stress of the pressure surface shows an increasing trend near \(S/S_{0}=0.5\), and its peak value reaches about 7.5. A large-scale vortex can be seen at the impeller outlet. At the entrance of the impeller, the wall shear stress of the suction surface varies from 1.5 to 1.7. The velocity gradient is hardly disturbed; thus, the near-wall flow of the suction surface is relative stable. 4. At the axial cross-section \(Z/b_{2}=0.5\), the FPG of the pressure and suction surfaces of the unstalled passage show similar distributions. The wall shear stress of the pressure surface is approximately the same as that of \(Z/b_{2}=0.15\). At the entrance of the impeller, the wall shear stress of the suction surface varies from 1.5 to 2.0, which is slightly higher. Although the peak value has increased by about 0.2, the flow in the vicinity of the suction surface is relatively stable. 5. From the inlet to outlet of the stalled passage, the peak positive value of radial velocity gradually approaches the suction side, and the reversed flow zone gradually expands, indicating that a large area of the vortex appears in the outlet area and deteriorates the through-flow capacity of the passage. The findings obtained by this work reflect the essential three-dimensional characteristics of the internal flow in a centrifugal impeller. Referring to the conclusions, the engineering design and optimization of such a type of impeller should take into consideration the different flow patterns from the shroud to hub sections to avoid possible flow separation that deteriorates the performances.