## 4 Numerical Results Analysis
### Unsteady Flow Characteristic Analysis  
In order to study the pressure pulsation of the model pump, 16 virtual monitoring points are set in the middle section of the pump from the inlet pipe to the outlet of the solute during the simulation process, and the dimensionless pressure coefficient \(C_{\text{P}}\) is introduced to characterize the pressure pulsation intensity at each monitoring point [29]. \(C_{\text{P}}\) is defined by Equation (15), with \(p\) denoting the static pressure and \(\overline{p}\) standing for the averaged static pressure.  
\[C_{P}=\frac{(p-\overline{p})}{0.5\rho u_{2}{}^{2}} \tag{15}\]  
After applying the FFT (fast Fourier transform) to \(C_{\text{P}}\) in the time domain, the frequency domain amplitude coefficients \(C_{\text{P}}{}^{*}\)of the pressure pulsation at every monitoring point are obtained. The trusted sampling frequency is 6756 Hz based on DES unsteady calculation, which could cover the analysis frequency range of flow-induced noise from the experiment information. The location and spectrum of the monitoring points are shown in Figure 15. As seen from it, the spectrum curves present discrete characteristics; the main frequency of mostly monitoring points is blade passing frequency (\(6f_{0}=675\) Hz) and its multiple due to the rotor-stator interaction between the impeller and the volute. The characteristic frequency is concentrated in the low-frequency region; the pulsation peak decreases after increasing above the threshold of \(30f_{0}\). \(C_{\text{P}}{}^{*}\) appears largest at the impeller exit area, especially around the volute tongue (P6, P7, P8). Affected by unstable flow phenomena such as flow separation and stall in the impeller, pulsations also obviously exist below \(6f_{0}\). The main frequency of P11 located at the inlet of the impeller blade is \(2f_{0}\), which may be due to the stall phenomenon in the flow channel. The main frequencies of P9 and P10 are also \(6f_{0}\), which indicates that the pump's outlet pressure wave propagation is characterized by rotor-stator interaction as dipole characteristics.  
Figure 14: Vortex distribution of model pump: (**a**) 0.5 \(Q_{\text{r}}\), (**b**) 0.7 \(Q_{\text{r}}\), (**c**) 1.0 \(Q_{\text{r}}\), (**d**) 1.2 \(Q_{\text{r}}\).  
In order to understand the flow characteristics of the pump more clearly, the velocity vortex distribution located on the middle span of the pump in one blade passing time is shown in Figure 16. The symbol \(t_{0}\) presents the moment that one blade passes the volute tongue. \(T\) means 1/(6\(f_{0}\)). The shape and size of the flow vortex in each of the six flow-paths are different and there is relative motion with the direction of impeller rotation. The flow field does not change much from the volute tongue to the pump outlet during a blade passing time. Many flow pulsations occur with a period higher than one blade passage time, which is reflected by the low-frequency characteristic of the pressure pulsation for each monitoring point. Low-speed vortices areas are generated, developed, and disappear in the impeller flow path and, mostly, at the impeller outlet. The velocity varies greatly at the impeller blade inlet, which corresponds to the peak value of pressure pulsation amplitude at P11. The main reason for the change of the flow field is still the unsteady flow of the impeller and rotor-stator interaction between the blade and the volute tongue.  
Figure 15: Spectrum of pressure fluctuation in the cooling pump at \(Q_{t}\): (**a**) monitoring points location; (**b**) pressure spectrum of P1–P4; (**c**) pressure spectrum of P5–P8; (**d**) pressure spectrum of P9–P11.