*5.1. Pump Performance Validation*

To validate the numerical method, the prototype pump with seven diffuser vanes was manufactured and measured in an open test bed. The detailed schematic diagram of this test bed can be found in our previous paper [22,38]. The comparison of pump performance in terms of predicted and experimental results are shown in Figure 6. Actually, the model ESP used in the performance test has seven stages, so the single-stage and power single-stage head were converted by the total power and total head, and then compared with the CFD predicted performance. As shown in Figure 6, the experimental results are slightly lower than the CFD results, but they have the same changing trend under different flow rates. The pump efficiency predicted by CFD has good agreement with the experimental efficiency. The comparison confirms the precision and accuracy of the numerical method used in this study. For the sources of error between the CFD and the test, one of the most important reasons is the one stage computational domain was used by considering the balance of computer capability and meshes number, while the tested prototype pump had seven stages. Generally speaking, the flow at the impeller inlet of the first stage is irrotational flow, but the flow at the impeller inlet of other stages is rotational flow, so in most cases the pump power and head of the first stage is higher than that of the other stages [22,38]. Another reason is that the leakage flow was not considered in this study, which means the volume losses caused by leakage flow was neglected. The real pump model with seven stages could be simulated in a powerful computer in a future study to obtain more arcuate results.

**Figure 6.** Comparison of pump performance between CFD and experimental results.

#### *5.2. Pressure Fluctuation Analysis*

In this study, the pressure coefficient is used to measure the pressure fluctuation. The pressure coefficient is defined as:

$$C\_p = \frac{p - \overline{p}}{\frac{1}{2}\rho u^2} \tag{13}$$

where *p* is the static pressure at the location of the monitoring points; *u* is the circumferential velocity of impeller outlet; and *p* is mean static pressure at the monitoring points during the rotation of the impeller. After 10 cycles of impeller rotation, the pressure fluctuations at the inlet and outlet of the impeller show the regular periodicity. In the process of the impeller rotating 360 degrees (360 time steps), the pressure fluctuation at each monitoring point was collected to analyze the unsteady flow in the flow channel.

To compare the transient pressure fluctuation characteristics in the flow channel of pump, seven monitoring points were arranged in the same position in three scenarios of space diffuser, 6P7–6P1 represents the pump flow channel of the six-vane configuration, 7P7–7P1 represents the pump flow channel of the seven-vane configuration, 8P7–8P1 represents the pump flow channel of the eight-vane configuration. The position of the seven monitoring points in the flow channel is shown in Figure 7.

**Figure 7.** Location of monitoring points.

Figure 8 presents the pressure fluctuation for each monitoring point in the pump flow channel of the six-vane configuration. Generally speaking, the pressure fluctuation shows stable periodicity at the impeller outlet and inlet. During the process of impeller rotating 360 degrees, five peaks and five troughs are seen, which is equal to the number of impeller blades. The pressure fluctuation amplitude at P1 located at the impeller inlet is the lowest one. The pressure fluctuation amplitude at P3 is the most intense one, which is located at the intersection of the diffuser vanes and the impeller blades. The static pressure changes along with the rotation of the impeller. In particular, when the impeller blade rotates from the pressure surface to the suction surface, the static pressure changes rapidly from the smallest value to the largest value. By comparing the amplitude of each monitoring points, we could find that the pressure fluctuation signal is generated at the impeller outlet, and in the process of its transfer to the pump outlet, its intensity gradually decays.

**Figure 8.** Pressure fluctuation of the six-vane configuration.

Figures 9 and 10 show time domain diagram of pressure fluctuation at each monitoring point for the configurations of 7 vanes and 8 vanes, respectively. It can be seen that pressure fluctuation at the monitoring point in the impeller inlet (P1, P2) and the impeller outlet (P3, P4) is similar to that of the six-vane configuration, exhibiting again five peaks and five troughs. It shows that at the impeller outlet and the impeller inlet, the pressure fluctuation is mainly affected by the impeller blade number, and the diffuser vane number has no effect on the pressure fluctuation at these locations. However, after entering the diffuser vane, the pressure fluctuations at P5, P6, and P7 change with the increase of the diffuser vane number and the pulse intensity increases with the increase of the vane number. Especially for the eight-vane configuration, the pressure fluctuation amplitude at the diffuser outlet is relatively large.

**Figure 9.** Pressure fluctuation of the seven-vane configuration.

**Figure 10.** Pressure fluctuation of the eight-vane configuration.

#### *5.3. Frequency Domain Analysis*

In the rotating coordinate system, the static and dynamic interference between the diffuser vanes and the impeller blades is similar to the pressure fluctuation of sinusoid and cosine waves. This transient periodic fluctuation can be decomposed into Fourier series. In this study, we used Fast Fourier Transform (FFT) to analyze the frequency domain characteristics of the pressure fluctuation between the pump stages. The fundamental frequency of the internal pressure fluctuation is generally shaft frequency (*f* s), so the frequency domain is analyzed by comparing with the shaft frequency *f* s. The frequency is divided by the fundamental frequency to get a multiple of the fundamental frequency as a horizontal coordinate. The rotating speed of the investigated pump is 2850 r/min, so the shaft frequency is 47.5 Hz.

Figure 11 presents the spectrum of pressure fluctuation at each monitoring point in the pump flow channel of the six-blade configuration. The frequency distribution at each monitoring point is basically the same, and fluctuating frequency appears at the double leaf frequency of Blade Passing Frequency (BPF, *f*/*f* <sup>s</sup> = 5) and its harmonic frequencies. It is further explained that the static and dynamic interference at the outlet of the impeller is the main cause of the pressure fluctuation, and the pressure fluctuation signal is gradually fading during the process of transferring at the inlet side of the impeller and the outlet side of the pump.

Figures 12 and 13 show the spectrum of pressure fluctuation at each monitoring point in the seven-vane and eight-vane configurations, respectively. Compared to the six-vane configuration, the frequency domain characteristics of pressure fluctuation are very similar. The closer to the impeller, the greater the pressure coefficient pulsation amplitude, and the fluctuating amplitude shows an obvious tendency of obviously increasing. As the number of the diffuser vanes increases, the amplitude of pressure fluctuation at each monitoring point in the diffuser vane increases gradually, which

indicates that the velocity of the fluid is converted to pressure energy by the diffuser vanes. With the increase of static pressure, the influence of BPF decays gradually, and the effect of the diffuser becomes gradually obvious. The lower number of diffuser vanes is beneficial to obtain a weak pressure fluctuation intensity.

**Figure 11.** Frequency domain analysis of the six-vane configuration.

**Figure 12.** Frequency domain analysis of the seven-vane configuration.

**Figure 13.** Frequency domain analysis of the eight-vane configuration.
