## 4 Results and Discussion
### Analysis of Unstable Flow Structure  
As a common unstable vortex structure, the trailing shedding vortex at the impeller outlet affects the internal flow of the centrifugal pump and is closely related to the unsteady pressure pulsation of the centrifugal pump [26; 28]. At present, vorticity, \(Q\)-criterion, and new \(\Omega\) methods are widely used as the main vortex identification tools in hydraulic machines, such as centrifugal pumps [29; 30; 31]. The vorticity method cannot distinguish the shear and actual rotational motions of the vortex, thereby affecting the accuracy of the results. The \(Q\)-criterion method is ineffective in identifying surface vortices near the wall and in areas with high velocity gradients. Therefore, the new \(\Omega\) method is used to identify the trailing shedding vortices at the impeller outlet, which can be expressed as [32]:  
\[\Omega=\frac{\left(\
abla\times V\cdot R\right)^{2}}{\|\
abla\times V\|_{2}^{2 }\|R\|_{2}^{2}} \tag{5}\]  
where \(R\) is the vorticity, and \(\Omega\) is the ratio of the vortex amount to the total vorticity. The value of \(\Omega\) isosurface selected in this analysis is 0.52 [32]. This study analyses the transient wake vortex distribution of each centrifugal pump when the blade passes the tongue. Figure 12 shows the shedding vortex distribution at the impeller outlet with different cutting schemes under part-load condition 0.2\(Q_{\text{d}}\). As shown in Figure 12, \(\alpha\) and \(\beta\), which are two main trailing shedding vortex regions, appear in the rotor-stator interaction regions of the impeller and the volute. In the \(\alpha\) region close to the tongue, the OTE and OBS15 impellers have a large area of shedding vortex regions. The large area of the vortex structure occupies the entire area of the tongue, indicating that many severe shedding vortices and considerable tongue impacts occur in the OTE and OBS15 impellers. Compared with the large area shedding vortices of the OTE impeller in the \(\alpha\) and \(\beta\) regions, the shedding vortex area of the OBS30 impeller is significantly reduced, and no obvious vortex structure appears at the tongue. This condition indicates that the blade with a 30\({}^{\circ}\) cutting angle at the trailing edge has good suppression effect on the shedding vortex generation of the pump under part-load condition 0.2\(Q_{\text{d}}\). Figure 11: Pressure pulsation intensity distribution in the circumferential direction of volute: (**a**) part-load condition (0.2\(Q_{\text{d}}\)); (**b**) over-load condition (1.4\(Q_{\text{d}}\)). Figure 13 shows the shedding vortex distribution at the impeller outlet with different cutting schemes under over-load condition 1.4\(Q_{\text{d}}\). As shown in Figure 13, the OTE and OBS15 impellers have a large area of shedding vortex regions in the \(\alpha\) region. The shedding vortex area in the OBS30 impeller is smaller than that of the OTE impeller. As shown in Figures 12 and 13, different cutting angles of blade trailing edge have different effects on the impeller outlet shedding vortex under different flow rates. The OBS30 impeller has a significant impact on the shedding vortex distribution at the impeller outlet and improves the flow structure at the impeller outlet. In recent years, scholars have applied axial vorticity analysis to quantitatively study the influences of blade trailing edge on the flow characteristics of centrifugal pumps and proved that the intensity of axial vorticity is closely related to the flow structure [28]. Therefore, this study analyses the axial vorticity in the impeller outlet and quantitatively compares the effects of blade trailing edge on unsteady flow in a centrifugal pump. The calculation formula is expressed as [28]:  
\[\Omega_{Z}=\frac{\partial v_{y}}{\partial x}-\frac{\partial v_{x}}{\partial y} \tag{6}\]  
Figure 14 shows the comparison of average vorticity intensity at the blade trailing edge of different schemes at part-load condition 0.2\(Q_{\text{d}}\) and overload condition 1.4\(Q_{\text{d}}\).