## 4 Results and Discussion
### Analysis of Unstable Pressure Pulsations
As shown in Table 2, the OBS30 impeller reduces the average main frequency amplitude of circumferential pressure pulsation of the centrifugal pump, and the reduction rate is 13.556%. The OBS15 impeller increases the average main frequency amplitude of pressure pulsation, with a reduction rate of \(-6.83\%\), and its deterioration rate is slightly smaller than that under part-load condition. The reduction effect of the OBS30 impeller on the main frequency amplitude is unaffected by flow rate on the basis of the reduction rate of the main frequency amplitude under off-design conditions obtained in Tables 1 and 2. The OBS15 impeller is immensely affected by flow rate and has the most obvious effect on pressure pulsation reduction among the three impellers under part-load condition. No pressure fluctuation component of instantaneous fluctuation is obtained because the pressure pulsation component obtained through transient simulation is only a phase average value. To obtain the relationship between pressure pulsation intensity and flow rate at different positions in the circumferential direction of the volute in a rotation period, the dimensionless coefficient of time-dependent pressure pulsation intensity is defined as \(C_{\text{psd}}\) and is expressed is [27]:  
\[C_{\text{psd}}=\frac{\sqrt{\frac{1}{N}\sum\limits_{i=0}^{N-1}\left(p(x,y,z,t_ {i})-\frac{1}{N}\sum\limits_{i=1}^{N}p(x,y,z,t_{i})\right)^{2}}}{\frac{1}{2} \rho u_{2}^{2}}=\frac{\sqrt{\frac{1}{N}\sum\limits_{i=0}^{N-1}\left(p-\overline {p}\right)^{2}}}{\frac{1}{2}\rho u_{2}^{2}} \tag{4}\]  
where \(N\) is the number of time steps in a stable rotation period (360 in this article), \(p(x,\,y,\,z,\,t_{i})\) is the static pressure value of node (\(x,\,y,\,z\)) at the \(i\)th time step. Figure 11 shows the distribution of pressure pulsation intensity coefficient \(C_{\text{psd}}\) in the circumferential direction of the centrifugal pump with different impellers at 0.2Q\({}_{\text{d}}\) and 1.4Q\({}_{\text{d}}\). As shown in Figure 11, the \(C_{\text{psd}}\) value of the centrifugal pump with the OBS30 impeller is smaller than that of the OTE impeller under part-load condition 0.2Q\({}_{\text{d}}\) and overload condition 1.4Q\({}_{\text{d}}\). Under off-design conditions, pressure pulsation intensity \(C_{\text{psd}}\) in the circumferential direction of the volute has a significant reduction effect when the cutting angle of blade trailing edge is 30\({}^{\circ}\). The pressure pulsation intensity of the region (22.5\({}^{\circ}\) to 112.5\({}^{\circ}\)) is larger than that of the region (112.5\({}^{\circ}\) to 360\({}^{\circ}\)) under different schemes. The closer it is to the tongue, the greater the pressure pulsation intensity \(C_{\text{psd}}\) will be. The pressure pulsation intensity in the circumferential \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Blade Trailing Edge Profile** & **Frequency (Hz)** & \(\overline{C}_{\text{p}}\) & **Reduction (\%)** \\ \hline OTE & 386.87 & 0.008737 & 0 \\ OBS15 & 387.58 & 0.009334 & \(-6.83\) \\ OBS30 & 386.51 & 0.007552 & 13.56 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Reduction rate of average dominant frequency amplitude of different impellers (1.4Q\({}_{\text{d}}\)). Figure 10: The amplitude of main frequency of pressure pulsation at the different impellers under over-load condition (1.4Q\({}_{\text{d}}\)). direction of the centrifugal pump with different impellers decreases with the increase in flow rate, and the decreasing trend of the tongue area (22.5\({}^{\circ}\) to 112.5\({}^{\circ}\)) is obvious. In the tongue area, the OBS15 impeller is immensely affected by flow rate. The \(C_{\text{psd}}\) value of the OBS15 impeller in the tongue region is the largest among the impellers under part-load condition. This finding indicates that the internal flow of the OBS15 impeller is unstable under part-load condition.