## 4 Results Analysis
### Pressure Pulsation Characteristics  
Fluctuation of hydraulic radial force on the impeller is closely related with pressure fluctuation of the internal unsteady flow field. On the mid-section of discharge chamber, five monitoring nodes (P1-P5) close to the volute wall were chosen to investigate the pressure fluctuation laws. The distance between each node and the volute wall is 0.02 m. As seen in Figure 15, P1 locates near the volute tongue and the rest four points are on the X-axis and Y-axis, respectively. Flow rate \(Q_{des}\) was taken as an example, under which the pressure fluctuations at five monitoring points in one impeller revolution were shown in Figure 16. Figure 16 shows the dominate pressure fluctuation period is 1/6 of the rotating period of the impeller, which agrees with the experimental regularity obtained by [PERSON] et al. (2012). However, pressure values and their amplitude variations are different for each monitoring point. Similar pressure fluctuation features were observed under other flow rates.  
Figure 14: Hydraulic radial force versus time under different flow conditions: (**a**) X direction; (**b**) Y direction.  
Figure 15: Position setting of the pressure monitoring nodes.  
To further compare pressure fluctuation characteristics under different flow rates, the time-varying curves \(P(node,t)\) of pressure at different monitoring nodes are decomposed into the fixed component \(\overline{P}(node)\) and the fluctuating component \(\overline{P}(node,t)\), as shown in Equation (11).  
\[P\left(node,t\right)=\overline{P}(node)+\overline{P}(node,t) \tag{11}\]  
where the numerical value of \(\overline{P}(node)\) is defined the arithmetic mean value of pressure in one revolution  
\[\overline{P}\left(node\right)=\frac{1}{N}\sum_{i=0}^{N-1}P(node,t_{0}+i \Delta t) \tag{12}\]  
where \(N\) is the number of time steps, \(t_{0}\) is the initial moment and \(\Delta t\) is the time step length.  
Thus the value of \(\overline{P}(node,t)\) is the difference between instantaneous pressure value \(P(node,t)\) and \(\overline{P}(node)\)  
\[\overline{P}(node,t)=P(node,t)-\overline{P}(node) \tag{13}\]  
\(\overline{P}(node,t)\) is worth in-depth study as it indicates the instability of the inner flow of centrifugal pump. Hence, the dimensionless quantity namely the pressure fluctuation intensity coefficient \(C_{p}^{\prime}\) is defined as the root-mean-square of \(\overline{P}(node,t)\) divided by reference pressure \(P_{\textit{ref}}\)  
\[C_{p}^{\prime}=\frac{\overline{P}_{\textit{rms}}(node)}{P_{\textit{ref}}} \tag{14}\]  
where \(\overline{P}_{\textit{rms}}(node)\) and \(P_{\textit{ref}}\) are shown in Equations (15) and (16), respectively.  
\[\overline{P}_{\textit{rms}}(node)=\sqrt{\frac{1}{N}\sum_{i=0}^{N-1}\overline{ P}(node,t_{0}+i\
abla t)} \tag{15}\]  
\[P_{\textit{ref}}=\frac{1}{2}\rho u_{2}^{2} \tag{16}\]  
where \(u_{2}\) is the circumferential speed at the impeller outlet.  
Figure 17 shows \(C_{p}^{\prime}\) of the five monitoring nodes under different flow conditions. \(C_{p}^{\prime}\) at P1 is higher than that at other nodes, and it presents a V-shaped variation trend with the increase of flow rate,  
Figure 16: Pressure fluctuation of different monitoring points under \(Q_{\text{des}}\).  
where it reaches the maximum value (about 0.02) under \(Q_{0.6}\). \(C_{p}^{\prime}\) decreases to the minimum (slightly higher than 0.01) under \(Q_{0.9}\). Subsequently, it increases continuously with the continuous increase of flow rate, but it is still less than that in part-load conditions. \(C_{p}^{\prime}\) values at P2-P5 are significantly lower than that at P1, and they are generally higher under part-load conditions.