# Dynamic Instability Analysis of a Double-Blade Centrifugal Pump
## 3 Results and Discussion  
### Pressure Pulsations of Volute at Different Flow Rates  
The dimensionless unsteady pressure coefficient is calculated in Equation (1), which is a normalization of the obtained pressure values based on the dynamic pressure in the impeller outlet (\(0.5\rho u_{2}^{2}\)). The intensity of pressure pulsation \(\mu_{\rm P}\) can be calculated in Equation (3) and the frequency of pressure pulsations is decomposed into Strouhal number (St) in Equation (4). \[c_{\rm P}=\frac{p_{i}-\overline{p}}{0.5\rho u_{2}^{2}} \tag{1}\]  
\[\overline{p}=\frac{1}{N}\sum_{i=1}^{N}p_{i} \tag{2}\]  
\[\mu_{\rm P}=\sqrt{\frac{\sum\limits_{i=1}^{N}\left(p_{i}-\overline{p}\right)^ {2}}{N}}\,/\left(0.5\rho u_{2}^{2}\right) \tag{3}\]  
\[{\rm St}=\frac{f}{z\cdot f_{n}}=\frac{f}{\rm BPF} \tag{4}\]  
where \(N=120\) is the number of samples. Figure 8 presents the intensity of pressure pulsations at different monitoring points in volume under different flow rates. It is noted that pump volume is asymmetry by having a feature of a tongue. The flow passage in the tongue area is quite narrow and pressure changes dramatically under the interference between impeller and tongue. The vicinity of the tongue (P1-P3) is the area with the most serious pressure pulsation in the entire volute channel. For the P1-P3 monitoring points, the change of flow rate caused the change of the pressure pulsation amplitude. At \(0.4\mathcal{Q}_{d}\), the change of pressure near the tongue area (P2) was the most significant (shown in Figure 9a). As the relative gap between the impeller outlet and the volute wall surface increases, the intensity of rotor-stator interference weakens, and the influence of the flow rate on the pressure pulsation decreases. For the P4-P6 monitoring points, the pressure pulsation amplitude of different flow rates is getting smaller (shown in Figure 8). For the P7 monitoring point, the pressure pulsation amplitude achieves the minimum value (shown in Figure 8), and the changes in flow rate have a slight effect on pressure pulsation (shown in Figure 9b). In Figure 9, the data were captured from the last two rotation cycles  
Figure 7: Numerical calculation results validation. of impeller, and the time required for one rotation cycle (\(T\)) is about 0.02 s. There are two main peaks and valleys in one rotation cycle that is directly linked to the number of blades (\(z=2\)). At the same time, this also shows that the rotor-stator interference between the pump's dynamic part (impeller) and static part (volute) was the main source of pressure pulsation at each monitoring point. Frequency spectra of pressure pulsations at the volute under different flow rates are shown in Figure 10. It is presented that the domain frequencies of the pressure pulsation  
Figure 8: Intensity of pressure pulsation of different monitoring points in volute. Figure 9: Pressure pulsations of volume at different flow rates: (**a**) pressure pulsation at P2; (**b**) pressure pulsation at P7. under different flow rates are the blade passing frequency (St = 1) and its multipliers (St = 2, 3, 4 and etc.). At 1.2 Qd, the amplitude of pressure pulsation at the blade passing frequency is larger when the interaction occurs in the vicinity of the tongue at P2. However, at P7, the high amplitude of pressure pulsation occurs at a low flow rate (0.4 Qd). In addition, especially at P2, it is noted that the broadband pulsation occurs at 0.4 Qd, which means some unsteady flow structures could be generated at this area, e.g., vortex, backflow and flow separation. This is clearly shown in Figure 11, where the backflow phenomena are presented in the outlet of impeller, and some vortices are produced near the suction side of the blade. Figure 10: Frequency spectra of pressure pulsations of solute under different flow rates: (**a**) P2; (**b**) P7. Actually, the flow fields at \(t_{0}\) and \(t_{2}\) are quite similar, and the flow field at \(t_{1}\) is quite similar to that at \(t_{3}\), thanks to the effect of the relative position of blade to tongue of solute.