## 4 Results and Discussion
### Analysis of Unstable Pressure Pulsations
The calculation formula is expressed as [26]:  
\[\overline{C}_{\text{P-n}}=\frac{\sum\limits_{i=1}^{n}C_{\text{P-n}}}{n}(n=17) \tag{2}\]  
where \(C_{\text{P-n}}\) represents the the static pressure coefficient amplitude of the 17 monitoring points at main frequency \(f_{\text{BPF}}\). The expression of the main frequency amplitude reduction rate is expressed as follows [26]:  
\[A_{\text{d}}=\frac{\overline{C_{\text{P-OTE}}}-\overline{C_{\text{P-n}}}}{ \overline{C_{\text{P-OTE}}}}\times\
icefrac{{\text{o/o}}}{{\text{o}}} \tag{3}\]  
where \(\overline{C_{\text{P-OTE}}}\) is the magnitude of the average static pressure coefficient of the original impeller (OTE) at \(f_{\text{BPF}}\), and \(\overline{C_{\text{P-n}}}\) represents the average static pressure coefficient amplitude of different impellers. The OBS30 impeller reduces the average main frequency amplitude of pressure pulsation of the centrifugal pump, and the reduction rate is 13.04%. Figure 7: Frequency domain of pressure pulsation for centrifugal pumps with different blade trailing edges under part-load condition (0.2\(Q_{\text{d}}\)): (**a**) monitoring point SP\({}_{1}\); (**b**) monitoring point SP\({}_{8}\). Figure 9 shows the frequency distributions of pressure pulsation monitoring points SP\({}_{1}\) and SP\({}_{8}\) under overload condition 1.4Q\({}_{d}\). As shown in Figure 9, the main frequencies of pressure pulsation at monitoring point SP\({}_{1}\) near the tongue area and pressure pulsation monitoring point SP\({}_{8}\) far from the tongue area of the centrifugal pump with different blade edges are near the blade passing frequency \(f_{\text{BFF}}\). The frequency domain amplitude is mainly concentrated at the blade passing frequency and its high harmonic frequency. The main frequency amplitudes of different impellers at SP\({}_{1}\) near the tongue region are larger than that at SP\({}_{8}\) far from the tongue region. At different monitoring points, the main frequency amplitude of the OBS30 impeller is smaller than that of the OTE impeller, the reduction at tongue SP\({}_{1}\) is larger, and the reduction value is 0.0234. This condition indicates that the 30\({}^{\circ}\) cutting angle of blade trailing edge improves the pressure pulsation at the near tongue and far from tongue areas under overload condition. Figure 10 show the main frequency amplitude of pressure pulsation at the different impellers under overload condition 1.4Q\({}_{d}\). !As shown in Figure 10, the near tongue region has a large main frequency amplitude ranging from 22.5\({}^{\circ}\) to 67.5\({}^{\circ}\) under overload condition, which is smaller than the region (22.5\({}^{\circ}\) to 112.5\({}^{\circ}\)) under part-load condition. This condition is because the unstable flow structure at the tongue and the unstable vortex area at the tongue gradually decrease with the increase in flow rate. The OBS30 impeller reduces the main frequency amplitude of pressure pulsation. Under overload \begin{table}
\begin{tabular}{c c c c} \hline
**Blade Trailing Edge Profile** & **Frequency (Hz)** & \(\overline{C}_{\text{p}}\) & **Reduction (\%)** \\ \hline OTE & 387.49 & 0.011172 & 0 \\ OBS15 & 386.33 & 0.012681 & \(-\)13.50 \\ OBS30 & 386.53 & 0.009715 & 13.04 \\ \hline \end{tabular}
\end{table} 
Table 1: Reduction rate of average dominant frequency amplitude of different impellers (0.2Q\({}_{d}\)). Figure 8: The amplitude of main frequency of pressure pulsation at the different impellers under part-load condition (0.2Q\({}_{d}\)). Figure 9: Frequency domain of pressure pulsation for centrifugal pumps with different blade trailing edges under over-load condition (1.4Q\({}_{d}\)): (**a**) monitoring point SP\({}_{1}\); (**b**) monitoring point SP\({}_{8}\). condition, the main frequency amplitude of the OBS15 impeller remains large. Table 2 shows the reduction rates of the blade trailing edge to the average main frequency amplitude of the centrifugal pump under overload condition 1.4Q\({}_{\text{d}}\).