## 4 Results and Discussion
### Analysis of Unstable Pressure Pulsations  
Figure 6 shows the layout of the pressure pulsation monitoring points SP\({}_{1}\)-SP\({}_{17}\) in the numerical simulation. As shown in Figure 6, 17 monitoring points are set in the circumferential direction of the volute. Except for monitoring point SP\({}_{2}\), the two adjacent monitoring points differ by 22.5\({}^{\circ}\). Monitoring point SP\({}_{2}\) is set at an angle of 30\({}^{\circ}\) between the tongue and the negative direction of the \(x\) axis. Dimensionless coefficient \(C_{\rm P}\) of pressure is used to study the pressure pulsation of centrifugal pumps with different blade trailing edges. The specific formula of static pressure coefficient \(C_{\rm P}\) is expressed as follows [21]:  
\[C_{\rm P}=\frac{p-\overline{p}}{0.5\rho u_{2}^{2}} \tag{1}\]  
where \(p\) represents the static pressure value of each monitoring point, \(\overline{p}\) represents the average static pressure value of each monitoring point in one rotation period, \(\rho\) represents the density of the fluid medium, and \(u_{2}\) represents the peripheral velocity of the impeller outlet. Figure 7 shows that pressure pulsation monitoring point SP\({}_{1}\) is near the tongue area, and pressure pulsation monitoring point SP\({}_{8}\) is far from the tongue area of the centrifugal pump with different blade trailing edges under part-load condition 0.2\(Q_{d}\). As shown in Figure 7a, the frequency domain amplitudes of the centrifugal pump with different trailing edge blades are mainly concentrated in the low-frequency region of monitoring point SP\({}_{1}\) near the tongue region. The main frequency is near the blade passing frequency \(f_{\rm BPF}\). In addition to the main frequency, the other low-frequency regions have  
Figure 5: Numerical calculation results comparison of three model pumps: (**a**) head coefficient; (**b**) efficiency. Figure 6: Numerical simulation of pressure pulsation monitoring points. large amplitudes, which could be caused by the recirculation and unstable vortex structure at SP\({}_{1}\) under part-load condition 0.2\(Q_{\text{d}}\). The main frequency amplitude of the OBS30 impeller is smaller than that of the original OTE impeller, and its value is 0.0266. The OBS15 impeller's main frequency amplitude is greater than that of the OTE impeller, and its value is 0.0379. As shown in Figure 7b, the frequency domain amplitudes of the centrifugal pump with different trailing edge blades are mainly concentrated at the blade passing frequency, their high harmonic frequencies are located at monitoring point SP\({}_{8}\) far from the tongue, and the amplitude of the remaining low frequency regions is small. The main frequency amplitude of the OBS30 impeller is smaller than that of the OTE impeller, and the reduction value is 0.0021. This condition indicates that an improvement is observed on the pressure pulsation of the near tongue and far from tongue areas when the cutting angle of blade trailing edge is 30\({}^{\circ}\) under part-load condition. Figure 8 shows the main frequency amplitude of pressure pulsation at the different impellers under part-load condition 0.2\(Q_{\text{d}}\). The ordinate is the main frequency amplitude of each monitoring point, and the abscissa is the angle between each monitoring point in the volute circumferential direction that is in the negative direction of the \(x\) axis. As shown in Figure 8, the main frequency amplitude of the region near the tongue (22.5\({}^{\circ}\) to 112.5\({}^{\circ}\)) of the centrifugal pump with different blade trailing edges is larger than that of the region far from the tongue. The main frequency amplitude of each monitoring point of the OBS30 impeller is smaller than that of the OTE impeller. The main frequency amplitude of the OBS15 impeller is greater than that of the OTE impeller. Table 1 shows the reduction rate of blade trailing edge to the average main frequency amplitude of the centrifugal pump under part-load condition 0.2\(Q_{\text{d}}\).