*4.3. Pressure Pulsation*

Figure 21 shows the pressure pulsation time domain diagram and comparison of pressure pulsation amplitude of the monitoring probes P13–P15 in a period under condition A, B, and C. The main frequency, the secondary frequency, and the corresponding pressure amplitude of the monitoring probes P13–P15 under condition A, B, and C are listed in Table 2.

**Figure 21.** Pressure pulsation time domain diagram and pressure pulsation amplitude (P13–P15). (**a**) Condition A (best efficiency point). (**b**) Condition B (start point of hydraulic unstable zone). (**c**) Condition C (valley point of hydraulic unstable zone). (**d**) Pressure pulsation amplitude.


**Table 2.** Frequency domains of P13–P15.

Under the same condition, the pressure pulsation trend of monitoring probes P13–P15 is completely unanimous, but the amplitudes increase sequentially. This indicates that the farther away from the pump shaft, the smaller the pressure pulsation amplitude, the closer to the pump shaft, and the greater the pressure pulsation amplitude due to the stream disturbance. Under different conditions, the pressure pulsation trend on each monitoring probe is quite different, accompanied by poor periodicity, small flow rate, and pressure pulsation amplitude.

The main frequency and secondary frequency of each monitoring probe are 26.25 Hz and 40.83 Hz, corresponding to the 2.25 times and 3.5 times shaft frequency. For the same monitoring probe, great variation exists between the pressure pulsation amplitude of main frequency (MF) and secondary frequency (SF). The pressure pulsation amplitude of MF is three times that of SF. Under different monitoring probes, the pressure pulsation amplitudes of MF and SF decrease as the flow rate reduces and small variation occurs between the pressure pulsation amplitude of MF and SF.

Figure 22 shows the pressure pulsation time domain diagram and comparison of pressure pulsation amplitude of the monitoring probes P10–P12 in a period under different conditions. Table 3 lists the main frequency, the secondary frequency, and the corresponding pressure amplitude of the monitoring probes P10–P12 under different conditions.

**Figure 22.** Pressure pulsation time domain diagram and pressure pulsation amplitude (P10–P12). (**a**) Condition A (best efficiency point). (**b**) Condition B (start point of hydraulic unstable zone). (**c**) Condition C (valley point of hydraulic unstable zone). (**d**) Pressure pulsation amplitude.



The pressure pulsation amplitude increases from P10 to P12 under the same condition. The pressure pulsation amplitude at P12 is 1.04, 1.03, and 1.01 times of P10 under condition A, B, and C. The closer to the shaft, the greater pressure pulsation amplitude, meaning the pressure pulsation amplitude is affected by the shaft. Under different conditions, the pressure pulsation amplitude and flow rate of each monitoring probe shows a positive relationship.

Under condition A, B, and C the MF and SF on the outlet of prolonged inlet section is completely the same with those in the prolonged inlet section. The rotating impeller has not yet been able to exert a dominant influence on the flow on the outlet of prolonged inlet section. Such a frequency shows no relationship with the blade frequency. The pressure pulsation amplitudes of MF and SF enlarges from P10 to P12 and the pressure pulsation amplitude of MF is also three times of SF.

The pressure pulsation time domain diagram and comparison of pressure pulsation amplitude of the monitoring probes P7–P9 in a period under different conditions is shown in Figure 23. The main frequency, the secondary frequency and the corresponding pressure amplitude of the monitoring probes P7–P9 under different conditions are listed in Table 4.

In general, the effect of the impeller on the inflow starts from the stream entering into the impeller, but actually the effect of the blade on the inflow begins when the water flow does not enter the impeller, mainly manifested by the pre-spin action on the water flow. Under the same condition, the pressure pulsation amplitude on the monitoring probes P7–P9 is gradually reduced, where P7 is the monitoring probe near the shroud and P9 is the monitoring probe near the hub. Thus, the pressure pulsation amplitude gradually increases from the hub to the shroud. The pressure pulsation amplitude at the monitoring probe P7 is 1.02 times, 1.05 times, and 1.01 times of P9 under condition A, B, and C. Hence, the pressure pulsation amplitude on each monitoring probe is positively correlated with the flow rate under different working conditions.


**Table 4.** Frequency domains of P7–P9.

**Figure 23.** *Cont*.

**Figure 23.** Pressure pulsation time domain diagram and pressure pulsation amplitude (P7–P9). (**a**) Condition A (best efficiency point). (**b**) Condition B (start point of hydraulic unstable zone). (**c**) Condition C (valley point of hydraulic unstable zone). (**d**) Pressure pulsation amplitude.

The frequency domain of monitoring probes set at the inlet of the impeller is observed and analyzed. The MF and SF on the impeller inlet monitoring probe are not certain, wherein the MF is 70 Hz or 26.25 Hz, and the SF is 70 Hz, 140 Hz, or 26.25 Hz. Under condition A, the MF of P7 and P8 is 70 Hz, which is the blade frequency, but the MF of P9 is 2.25 times of the shaft frequency. Under condition B, and C, the MF on each monitoring probe is also the blade frequency, but the SF is 26.25 Hz or 140 Hz. Under all conditions, the pressure pulsation amplitude of MF for P7 and P8 is quite different from the pressure pulsation amplitude of SF. However, the pressure pulsation amplitude of MF of P9 is basically consistent with SF. The monitoring probe P7 is pre-installed near the shroud, and the MF and SF are 70 Hz and 140 Hz, respectively, which is 1 and 2 times the blade frequency. Thus, the blade frequency plays a dominant role in the pressure pulsation near the shroud under condition A, B, and C. The location of monitoring probe P8 is between the shroud and the hub, and the MF is 70 Hz, the SF is 26.25 Hz under condition A and C, but is 140 Hz under condition B. Therefore, the blade frequency is still the dominant factor, but the shaft frequency also begins to play a certain role in it. The monitoring probe P9 is pre-set near the hub. Thus, 26.25 Hz and 70 Hz occur alternately in the MF and the SF of P9, and the pressure pulsation amplitudes of the MF is matched roughly with the SF. This indicates that the pressure pulsation near the hub is affected by both the impeller and the pump shaft.

Figure 24 shows the pressure pulsation time domain diagram and comparison of pressure pulsation amplitude of the monitoring probes P4–P6 in a period under different conditions. Table 5 lists the main frequency, the secondary frequency, and the corresponding pressure amplitude of the monitoring probes P4–P6 under different conditions.

Under the same condition, the pressure pulsation amplitude of P4–P6 decreases successively. Thus, the pressure pulsation amplitude near the shroud of the impeller inlet is higher than near the hub of the impeller outlet. Under condition A, B, and C, the pressure pulsation amplitude of P4 is 1.03 times, 1.04 times, and 1.07 times of P6. Under different conditions, the pressure pulsation amplitude decreases with the reducing flow rate, which indicates that the pressure pulsation amplitude has a positive correlation with the flow rate.

**Figure 24.** Pressure pulsation time domain diagram and pressure pulsation amplitude (P4–P6). (**a**) Condition A (best efficiency point). (**b**) Condition B (start point of hydraulic unstable zone). (**c**) Condition C (valley point of hydraulic unstable zone). (**d**) Pressure pulsation amplitude.


**Table 5.** Frequency domains of P4–P6.

Under condition A, the MF and SF of each monitoring probe are 26.25 Hz and 70 Hz. The pressure pulsation amplitude corresponding to MF is slightly higher than SF. Under condition B, the MF and the SF of each monitoring probe are also 26.25 Hz and 70 Hz. For monitoring probes P4–P6, the pressure pulsation amplitude of MF is 1.98 times, 1.95 times, and 1.55 times of SF. Under condition C, the MF and SF of the monitoring probe P4 are 70 Hz and 26.25 Hz. The MF and SF of the monitoring probes P5 and P6 are 26.25 Hz and 70 Hz, and the pressure pulsation amplitudes of MF and SF are consistent.

Figure 25 shows the pressure pulsation time domain diagram and comparison of pressure pulsation amplitude of the monitoring probes P1–P3 in a period under different conditions. Table 6 lists the main frequency, the secondary frequency, and the corresponding pressure amplitude of the monitoring probes P1–P3 under different conditions.

**Figure 25.** Pressure pulsation time domain diagram and pressure pulsation amplitude (P1–P3). (**a**) Condition A (best efficiency point). (**b**) Condition B (valley point of hydraulic unstable zone). (**c**) Condition C (valley point of hydraulic unstable zone). (**d**) Pressure pulsation amplitude.



Under the same condition, the pressure pulsation amplitudes of the monitoring probes P1–P3 increase sequentially. The pressure pulsation amplitude near the hub of the GVs outlet is higher than it near the shroud of the GVs outlet. Under condition A, B, and C, the pressure pulse amplitude of P1 is 1.02 times of P3. The pressure pulsation amplitude on the same monitoring probe under different conditions is positively correlated with the flow rate.

Under condition A, the MF and the SF of P1–P3 are 26.25 Hz and 40.83 Hz, which are 2.25 times and 3.50 times of the shaft frequency. The pressure pulsation amplitude of the MF is about 3 times of SF. Under condition B, the MF and the SF of P1–P3 are also 26.25 Hz and 40.83 Hz. The pressure pulsation amplitude of the MF is also about 3 times of SF. Under condition C, the MF and SF of P1 are 29.62 Hz and 41.46 Hz. The MF and SF of P2 are 29.17 Hz and 40.83 Hz. The MF and SF of P3 are 29.17 Hz and 40.83 Hz. The MF and SF of P1–P3 are no longer consistent and lacks of regularity. On one hand, the rotation of impeller and shaft affects the flow downstream a little for the outlet of GV is far from the impeller. On the other hand, the flow pattern in the propulsion pump is not stable and the MF and the SF on each monitoring probe are fluctuating in the low frequency range.

#### **5. Conclusions**

(1) The mixed-flow waterjet propulsion device is tested by establishing the double circulation test loop of waterjet propulsion system. The test results are consistent with CFD results both in the trend and values. The CFD method is reliable.

(2) Conditions A, B, and C are marked as characteristic conditions by analyzing the hydraulic performance of the propulsion pump, which are the BEP (best efficiency point), start point of hydraulic unstable zone, and the valley point of hydraulic unstable zone. Thus, unsteady calculation is promoted and the unsteady flow process of the propulsion pump at different times of the same period is discussed. The surface vortex on the blade under condition C is unstable, and the vortex core and shape pattern vary on a small scale as time. Three turbo surfaces are sliced to study the flow features on each spanwise under different conditions. The steady flow characteristic of each turbo surface is obtained under condition A and B; however, the flow characteristic of each turbo surface varies as time under condition C, due to the unstable velocity and pressure field.

**Author Contributions:** Data curation, L.C. and W.J.; Formal analysis, C.W.; Methodology, C.L.; Writing—original draft, H.L.; Writing—review & editing, D.Z.

**Funding:** This research was funded by [Jiangsu Province Science Foundation for Youths] grant number [BK20170507], [Natural Science Foundation of the Jiangsu Higher Education Institutions] grant number [17KJD580003], [Jiangsu Planned Projects for Postdoctoral Research Funds] grant number [1701189B], [Open Research Subject of Key Laboratory of Fluid and Power Machinery (Xihua University), Ministry of Education] grant number [szjj2019-018], [Science and Technology Innovation and Cultivation Fund of Yangzhou University] grant number [2017CXJ047], [National Natural Science Foundation of China] grant number [51779214], [Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD)], [Jiangsu Province 333 high level talents training project] grant number [(2018) III-1827], [Peak plan six talents in Jiangsu province], and [Key project of water conservancy in Jiangsu province] [2018042].

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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