Time–frequency signal analysis of flow pulsation in siphon outlet conduit based on HHT considering the pump‐conduit interaction

The unsteady three‐dimensional numerical simulation calculation of the vertical axial flow pump device is performed based on CFD to examine the pressure pulsation and energy distribution features of the water flow within the siphon outlet conduit (SOC) under the hydraulic coupling of the pump and the flow conduit. The pressure pulsation signals (PPS) of monitoring points are decomposed using the Hilbert–Huang approach via empirical mode decomposition (EMD) and Hilbert spectrum analysis. The results show that the PPS of monitoring points of the SOC has no obvious periodicity. The low‐frequency range below 20 Hz serves as the primary frequency, and the energy ratio of the high‐frequency signal above 700 Hz is less than 1%. Under the condition of a small flow rate 0.3Qbep, there is obvious high‐frequency pulsation above 500 Hz at the inlet of the upstream section of the SOC, and there are periodic components distributed around 200 Hz. The energy of the pressure pulsation is primarily focused in the low‐frequency range below 40 Hz at the SOC outlet section. The PPS at the top and bottom monitoring points of the hump section are consistent, and the pressure pulsation energy (PPE) is mainly concentrated in the low frequency below 40 Hz. In both the big flow condition (1.2Qbep) and the optimal flow condition (1.0Qbep), the PPE of the monitoring points at the top of the hump section is distributed in the middle and low‐frequency band below 40 Hz, and the energy of monitoring points at the bottom of the hump section is concentrated in the low‐frequency band below 20 Hz, accounting for more than 70%. When the flow rate increases from 0.3Qbep to 1.2Qbep, the peak value of the pressure pulsation coefficient at the main frequency of each monitoring point at the top of the hump section changes little, and the peak value of the pressure pulsation coefficient at the main frequency of each monitoring point at the bottom increases first and then decreases. From the upstream section to the hump section of the SOC, the average frequency of each intrinsic mode function (IMF) of the PPS under different flow conditions shows an overall increasing trend. From the hump section of the SOC to its outlet, the average frequency of each IMF of the PPS decreases under the conditions of 1.0Qbep and 1.2Qbep, and there is no obvious change under the condition of 0.3Qbep.

The inlet and outlet conduits are an important part of the low-lift pump device. The outlet conduit is the transition section between the pump and the outlet pool. It aims to enhance kinetic energy recovery without creating a vortex or destabilizing the water as it flows from the pump outlet into the outlet pool by improving the water's turn and diffusion. The energy performance of the pump device is, therefore, significantly impacted by the hydraulic loss of the outlet conduit. Due to the hydraulic coupling between the pump and the outlet conduit, the water flow obtains energy through the impeller rotation and flows out from the outlet conduit. However, the uniformity and stability of the flow are also destroyed while the water flow obtains energy, resulting in complex flow phenomena such as flow separation, vortex and secondary reflux in the outlet conduit, and the pressure pulsation signals (PPS) of the flow in the outlet conduit is also very complex. There are many types of outlet conduits for low-head pumping stations, such as siphon outlet conduit (SOC), straight pipe outlet conduit, low hump outlet conduit, and so forth. SOC are one of the several types of outlet conduits that are appropriate when the outlet pool's water level does not vary much. The simplicity, reliability, and economy of the SOC flowstopping approach are benefits. Among the 14 vertical pumping stations newly built in the first phase of the eastern route of the South-to-North Water Diversion Project in China, seven pumping stations have adopted SOC, accounting for 50%, such as Liulaojian Second Station, Siyang Pumping Station, Suining Second Station, and Denglou Pumping Station in Jiangsu Province. In the actual pumping station project, if the design is not considered properly or the influence of external factors such as water level and flow rate, the SOC may take a long time to form a stable siphon flow or cannot realize the siphon. There may also be a large hydraulic loss of the flow channel during the siphon process, a high head during the pump operation, causing vibration of the unit, excessive noise, and other adverse hydraulic phenomena, which cause great harm to the safety and stability of the project. 1 The internal flow pulsation of the SOC under the hydraulic coupling of the pump and the flow conduit is thus of great theoretical relevance and engineering practical benefit.
There are many studies on the outlet conduit of the pump device. Yang et al. 2 and Wang et al. 3 optimized the three-dimensional (3D) structure of the wellbore outlet passage of the vertical axial flow pump device (VAFPD) and the volute of the centrifugal pump device. Xu et al. 4 and Wang et al., 5 respectively, analyzed the internal flow field of the SOC of an axial flow pump and the S-shaped outlet conduit of a slotted axial flow pump. In addition, the analysis of the pressure pulsation of the water flow in the passage of the pump is particularly rich; for example, Yang et al. 6 combined a physical model test and numerical simulation to explore the variation rule of the flow field in the hump section of the SOC, the pressure pulsation and the influence of pier separation on the pressure and velocity distribution on the inner wall of the straight outlet duct and its vicinity. Zhang et al. 7 studied the centrifugal pump's properties for pressure pulsation and unsteady flow structure using a hybrid Reynolds-averaged Navier-Stokes/large eddy simulations method. Wang et al. 8 analyzed the frequency domain features of pressure pulsation in a mixed flow pump as a turbine based on Fourier frequency transform (FFT). The time and frequency domains at each monitoring point on the diffuser and outlet elbow were investigated by Wang et al. 9 He et al. 10 explored the pressure pulsation features of a pump turbine in a draft tube. It can be seen that many achievements have been made in the research on the pulsation features of the internal flow of the pump device, which has a high reference value. However, the analysis methods of the pulsation time-frequency signal mainly focus on the FFT, 11-13 short-time Fourier transform, [14][15][16][17] and wavelet transform (WT). [18][19][20] Fourier transform is only suitable for analyzing stationary signals with a fixed frequency and is not suitable for analyzing nonstationary and nonlinear complex pulsating signals. The WT is essentially a Fourier transform with an adjustable window, which does not free it from the constraints of the Fourier transform. The choice of the wavelet basis function has a significant impact on the decomposition effect. Hilbert-Huang transforms (HHT) 21 not only has the features of multiresolution but also has adaptability, which can well solve the time-frequency analysis problem of nonlinear and nonstationary signals. At the same time, unlike other analysis methods, only the amplitude of pressure pulsation can be obtained, while the Hilbert-Huang method can clarify the energy proportion of different frequency bands in PPS, which is conducive to the analysis of the influencing factors of PPS. Considering the complex PPS of internal water flow caused by the hydraulic coupling between the pump and the outlet conduit, this paper uses HHT to analyze the flow fluctuation in the SOC, which complements the cognition of the pump station's flow conduit engineering hydraulics, is beneficial to reducing the hydraulic loss of the SOC, improving the efficiency of the pump device, and provides some reference data for the efficient, safe and stable operation of the pump station.

| RESEARCH OBJECT AND TEST DEVICE
The VAFPD is the subject of this paper. There are five parts to the flow passage: the elbow inlet conduit, impeller, guide vane, 60°elbow, and SOC. The 3D model is shown in Figure 1, and the main design parameters are shown in Table 1. The SOC mainly comprises four parts: upstream section, hump section, downstream section, and an outlet section. Figure 2 is the geometric dimension diagram of the SOC. The main control parameters of the SOC are: the inclination angle of the upstream section is 30°, the inclination angle of the downstream section is 44°, the conduit's horizontal projection is 7.05D in length., the conduit is 2.94D in height, the diameter of the inlet is 1.17D, the outlet of the conduit is 1.33D in height, and the outlet is 1.60D in width, the hump section is 0.75D in height, the hump section is 1.60D in width, and D is the nominal diameter of the impeller.
The model test of VAFPD is carried out on the closed cycle test bench of hydraulic machinery in the Key Laboratory of Universities in Jiangsu Province. The VAFPD model is shown in Figure 3A, and the 3D model of the test bench is shown in Figure 3B. The test bed consists of a closed inlet tank, elbow inlet conduit, axial flow pump, 60°elbow, SOC, main pump motor, closed outlet tank, PVC pipe, auxiliary pump, electromagnetic flowmeter, and butterfly valve.
T A B L E 1 Main design parameters of VAFPD.

Parameters Value
Nominal diameter of impeller (D) 120 mm The fluid motion in the pump device complies with the principles of conservation of mass, momentum, and capacity. It may be roughly described as incompressible 3D viscous turbulence. Without considering the heat exchange between the bodies, the fluid inside the pump can be described by continuity and momentum equations. The RNG k-ε turbulence model can better deal with the problems of high strain rate and streamline bending and improve the accuracy of simulating the rotating and vortex flow in the average flow. The RNG k-ε turbulence model has good adaptability in the numerical calculation of the flow field in the pump device. 22,23 This paper uses the RNG k-ε model to simulate the VAFPD with the SOC. The boundary conditions are specified in the style depicted in Table 2 concerning References 24,25 to more accurately simulate the real flow inside the pump device.
The interface on both sides of the impeller is set to transient rotor stator for unsteady calculation. The determination of the time step for unsteady computation satisfies the following criterion of Courant number: where   v is the absolute value of the estimated average velocity, and l is the minimum size of the mesh.

| Division of computational domain grids
An unstructured grid is widely used because of its advantages of convenient generation and automatic adaptation to various geometric figures. However, a structured grid still has the advantages of fast generation speed, high grid quality, and effective capture of the physical structure surface. Based on the advantages of a structured grid, this paper uses ANSYS TurboGrid software to mesh the calculation domain of the impeller and guide vane, the H/J/L-Grid topology structure was used for the impeller, and the H-Grid topology structure was used for the guide vane. ICEM CFD software was used to perform structural grid topology on the inlet and outlet flow channels and elbows and used O-type topology for encryption. The wall grid of each flow structure is shown in Figure 4. The mesh quality of the impeller and guide vane is evaluated from five aspects: minimum face angle, maximum face angle, maximum element volume ratio, maximum edge length ratio, and connectivity number.
The parameters are shown in Table 3, and there is no negative volume mesh or bad mesh.
The mesh of other flow passage components is evaluated by calculating the ratio of the 2 × 2 × 2 Jacobi determinant. When the result is 1, the mesh element is completely regular, and 0 means that the mesh element has one or more degenerated edges. Generally, it is considered that the mesh quality is better if it is higher than 0.4. The mesh quality check result is shown in Figure 5, and all the mesh quality is above 0.5.
It is vital to confirm the independence of grid number to exclude the impact of grid size and the number of grids on the results of calculations. When the grid is refined to a specific degree, the solution result sensitivity to the number of grids is reduced, indicating that the solution results will not change much with the change in grid number. At this time, it can be considered that the numerical calculation is independent of the number of grids. Under the optimal flow condition (1.0Q bep ), the results of different mesh numbers on the efficiency of the VAFPD are selected for mesh number independence analysis. A total of eight mesh schemes are used for comparison, and the results are shown in Figure 6. As can be seen from the figure, when there are 5.1 million grids instead of 3.9 million, the efficiency change of the pump device is slowed down, and the absolute deviation is less than 0.5%, reaching 0.32%, which is stable enough to fulfill the requirements of the calculation. When the grid is refined to 5.47 million, the efficiency change value of the pump device is further reduced to 0.14%, and then the efficiency change rate of the pump device tends to be stable. Therefore, this paper selects the scheme with about 5.47 million grids number to numerically calculate the VAFPD of the SOC. Utilizing the grid convergence index (GCI) to confirm the grid convergence of the pump device, and finally, the number of grids that can enable the result of numerical simulation to reach the pseudo steady state condition is selected. 26 The device efficiency under the optimal flow condition 1.0Q bep is selected as the parameter to calculate the GCI results. The results are shown in Table 4. CGI decreases gradually with the grid densification, which is less than 1%, indicating that the dispersion error is small. 22 After mesh independence and convergence analysis, the numerical calculation of the VAFPD was finally determined under the total mesh number of 5.47 million.

| Experimental verification
The energy performance test of the pump device was performed following the specifications of SL 140-2006 "Code for model pump and its installation acceptance tests." The efficiency and head test values of the VAFPD at 2200 r/min speed were tested. The results are shown in Figure 7, which displays that the variation trend of the external characteristic curve obtained by the numerical simulation and the experimental method is consistent. The numerical calculation results and the experimental results accord well in the region of high efficiency. The head curve is slightly different in the small flow region, mainly due to the complex flow pattern. The efficiency curve is slightly different in the large flow region. In the vicinity of the optimal conditions (1.0Q bep ), the relative error of the head is 2.36%, the absolute error of efficiency is 1.87%; in the range of 0.3Q bep -1.2Q bep , the relative error of the head and the absolute error of efficiency are not more than 5%. In summary, through comparison, it can be concluded that the results of the numerical calculations and the findings from the tests on the physical model correspond well. The numerical calculation results are true and credible and can accurately reflect the internal flow features of the pump device.

| CALCULATION METHOD AND PRINCIPLE
The components of HHT are the Hilbert transform (HT) and empirical mode decomposition (EMD). EMD is the core part of this method, and Huang no longer thinks T A B L E 3 The mesh quality of impeller and guide vane. that signals are combined based on sine signals. Instead, it is believed that any complex signal can be decomposed into a limited number of intrinsic mode functions (IMF), and then HT is performed on each IMF component to obtain the instantaneous frequency with physical significance, corresponding Hilbert spectrum (HS) and energy distribution, which more intuitively presents the time-frequency characteristics of each component of the original signal.

| Intrinsic mode function
The IMF needs to fulfill two requirements 21 : 1. In the whole data segment, the number of extreme points N e and zero-crossing N 0 must be equal or not more than one, namely, |N 0 − N e | ≤ 1. 2. At any time t, the mean value of the upper envelope, determined by the local maximum point, and the lower envelope, determined by the local minimum point, is 0; that is, concerning the time axis, the upper and lower envelopes are locally symmetric.
The above two conditions are the important basis for judging whether the signal is an IMF component. The concept of IMF lays the foundation for EMD and HS analysis.

| Empirical mode decomposition
Usually, most signals can not meet the conditions of IMF, so the EMD method is used to decompose a signal into several simple oscillation functions which meet the conditions of IMF components. After the EMD shown in Figure 8, the original signal x(t) is decomposed into the sum of a series of IMFs x n and residual r, that is: The EMD method can adaptively decompose a complex signal into a series of IMF according to the original signal. Compared with other decomposition methods, it does not need to select the corresponding basis before decomposition, which can reduce the influence of human operation factors on the decomposition quality and avoid false harmonic components in the result processing. 27 It is better suited for nonstationary signal analysis.

| Empirical mode decomposition
For a given IMF component function x(t), we can get his HT results: where τ is the integral variable of HT. With x(t) as the real part and y(t) as the imaginary part, an analytic signal z(t) can be constructed: where a(t) is the instantaneous amplitude of the analytic function: θ(t) is the corresponding instantaneous phase: The instantaneous frequency ω(t) corresponding to the HT is expressed as a derivative of the instantaneous phase:

| HS and marginal spectrum
The corresponding modal energy distribution E i is: Define the IMF energy ratio E p : Based on HS H ω t ( , ), the marginal spectrum (MS) is further defined, and the MS h ω ( ) is obtained by integrating the HS in time: The difference between the Hilbert MS and the traditional Fourier time-frequency diagram is that in the Fourier spectrum, the presence of energy at a certain frequency means that a sine or cosine wave with that frequency exists throughout the signal. When energy is present at a certain frequency in the MS, it indicates that a wave with that frequency is more likely to appear at a certain time during the entire signal duration. The MS displays the cumulative value of the corresponding components of different frequencies in the signal over the entire time, and the specific occurrence time is given in the HS. If the energy of a certain frequency appears in the signal, there must be an oscillation of the frequency. In other words, the MS can more correctly depict the signal's true frequency component. 28

| Pump device unsteady calculation pulsation monitoring point arrangement
To investigate the PPS features inside the SOC of the VAFPD, a monitoring point is arranged at the center of the inlet of the SOC, numbered P01, which is 0.833D away from the center line of the impeller. Three monitoring locations with the numbers P02-P07 are placed at the top and bottom of the hump section of the SOC; a monitoring point is arranged near the outlet surface of the SOC, numbered P08, 7.25D from the center line of the impeller. The arrangement of pressure monitoring points is shown in Figure 9.

| Verification of numerical simulation results
The pressure pulsation test adopts CY302 highprecision digital pressure sensor. The synchronous data acquisition of the two pressure sensors is realized through the 485-20 digital sensor hub and data acquisition software, as shown in Figure 10. The parameters of the CY302 high-precision digital sensor are shown in Table 5. A pressure sensor is positioned at the top and bottom of the hump section of the SOC, and the arrangement position of the pressure sensor and the number of each measuring point are shown in Figure 11. The measuring points P03 and P06 are situated, respectively, in the center of the top and bottom of the hump section.
When analyzing the PPS monitoring points inside the pump device, the pressure pulsation coefficient C p F I G U R E 9 Pressure pulsation monitoring point arrangement of pump device.  features the PPS. 22 The pressure pulsation coefficient is defined as: where p is the instantaneous pressure value of the PPS monitoring point, p is the average pressure value, and u is the circumferential velocity of the impeller.
To increase the reliability of the calculation results, as the rotation time increases, the calculation of the flow field motion will tend to be steady, so the PPS data of the last five rotation cycles of the numerical simulation are taken to analyze the pump device. Taking the data of five cycles of P03 and P06 at the top and bottom of the hump section under the condition of 1.0Q bep , the PPS physical model test results and the unsteady PPS numerical calculation results are fast Fourier transformed, and the PPS frequency domain diagram is drawn and compared. The results are shown in Figure 12.
As shown in Figure 11, the main frequency of PPS in the model test and numerical simulation of P03 and P06 is at 14.6 Hz, and the error is small in the low-frequency band. The C p of the test is higher than that of the numerical simulation. On the whole, the unsteady results of the SOC of the axial flow pump are consistent with the numerical results. The unsteady numerical results are credible and can be used for further HHT analysis. respectively. The remaining component energy accounts for less than 5%.

| Analysis of pressure pulsation features
HS and MS analysis of PPS for the inlet monitoring point at the upstream section of SOC ( Figure 14). It can be seen from the results that under the condition of 0.3Q bep , the high-frequency pulsation of monitoring point P01 above 500 Hz is more obvious than the design flow and large flow conditions, and there are periodic components distributed near 200 Hz; under the condition of 1.2Q bep , the frequency of pulsation signal decreases and the amplitude increases in the third to fourth cycles. On the fluctuation amplitude, in response to an increase in flow rate, the peak value of C p at the main frequency shows a downward trend, and it is the largest at a small flow rate. The peak value of C p at 0.3Q bep is 3.6 and 3.9 times of 1.0Q bep and 1.2Q bep , respectively.  were around 715, 199, 100, 65, 10, and 6.9 Hz, respectively. The energy was concentrated in IMF4 and IMF5, accounting for about 65% and 27%, and the remaining components accounted for less than 5%. Figure 16 is the EMD result of the PPS of the monitoring point P02-P07 at the hump section under 1.0Q bep . Table 8 is the average frequency and energy ratio of each IMF.

| Analysis of pressure pulsation at hump section
It can be seen from the results that the pressure pulsation energy (PPE) of the monitoring points at the top of the hump section is distributed in the middle and low-frequency bands below 40 Hz, but the distribution features are slightly different: the monitoring point P02 is evenly distributed in IMF4-IMF6, and the average frequencies are 36. 8 Figure 17 shows the EMD results of the PPS at the monitoring points P02-P07 at the interface of the hump section at 1.2Q bep . Table 9 shows the average frequency and energy ratio of each IMF.
It can be seen from the results that the PPE at the top monitoring point of the hump section is distributed in the middle and low-frequency band below 40 Hz, but the distribution features are slightly different: the monitoring point P02 is more evenly distributed in IMF4 and IMF5, with an average frequency of 36.0 and 8.7 Hz, respectively, accounting for 41.9% and 39.9%, respectively; the monitoring point P03 is more concentrated in IMF4, with an energy ratio of 89.3% and an average frequency of 22.2 Hz; the monitoring point P04 is concentrated in IMF4, with an average frequency of 16.5 Hz and an energy ratio of 92.9%.
The amount of the three monitoring points at the bottom is concentrated in the low-frequency component below 20 Hz, accounting for more than 70%. The average T A B L E 7 Average frequency distribution and energy ratio of each IMF (0.3Q bep ).   Figure 18. According to the result, the PPE of the monitoring point P03 is mostly focused in the low frequency below 40 Hz, but there is a certain amount of medium and high-frequency signals at 1.0Q bep and 1.2Q bep , and the occurrence time can be reflected in the HS. In response to an increase in flow rate, the peak value of C p at the main frequency is smaller than that at other monitoring points. The peak value of C p at 0.3Q bep is 0.93 and 1.19 times that of 1.0Q bep and 1.2Q bep , respectively.
HS and MS analysis of PPS for the central monitoring point P06 at the bottom of the hump section is shown in Figure 19. According to the result, the PPE of monitoring point P06 is mostly focused on the low frequency below 40 Hz, but there is a certain amount of medium and high-frequency signals in the range of 50-200 Hz at 1.0Q bep , and the occurrence time can be reflected in the HS. Regarding pulsation amplitude, in response to an increase in flow rate, the peak value of C p at the main frequency increases first and then decreases. The peak value of C p under 0.3Q bep is 0.67 and 0.94 times that of 1.0Q bep and 1.2Q bep , respectively. Figure 20 shows the EMD results of the PPS of the monitoring point P08 at the outlet section of the SOC under the conditions of 0.3Q bep , 1.0Q bep , and 1.2Q bep . Table 10 shows the average frequency and energy ratio of each IMF. Under the condition of 0.3Q bep , the main energy of the PPS at the monitoring point P08 is concentrated in IMF4 and IMF5, the proportion of energy is about 65.0% and 27.2%, and the proportion of other components is less than 5%. Under the condition of 1.0Q bep , the PPE of monitoring point P08 is distributed in IMF4-IMF6, and the average frequency is below 30 Hz, accounting for 18.3%, 18.1%, and 61.9%, respectively. Under the condition of 1.2Q bep , the PPE of monitoring point P01 distributes in IMF4 and IMF6, T A B L E 8 Average frequency distribution and energy ratio of each IMF (1.0Q bep ).   Figure 21. The results show that the PPE of the monitoring point P08 is mostly focused in the low frequency below 40 Hz, but there are more medium and high-frequency signals at 1.0Q bep and 1.2Q bep , and the occurrence time can be reflected in the HS. Regarding pulsation amplitude, in response to an increase in flow rate, the peak value of C p at the main frequency increases first and then decreases. The peak value of C p under 0.3Q bep is 0.67 and 0.94 times that of 1.0Q bep and 1.2Q bep , respectively.

| Analysis of pressure pulsation along path
The inlet monitoring point P01, the outlet monitoring point P08 and the two monitoring points P03 and P06 in the center of the hump section of the SOC are selected to analyze the PPS along the way under different flow conditions. The values of the pressure pulsation in the hump section take the numerical average of the monitoring point P03 and the monitoring point P06.
Under the condition of 0.3Q bep , the IMF average frequency distribution and energy ratio of the pressure pulsation along the SOC are shown in Figure 22. At 0.3Q bep , the energy at the inlet of the upstream section of the SOC is mostly focused in IMF4, while the energy at the hump section and the outlet section of the SOC is concentrated in IMF4 and IMF5. From the inlet of the SOC to the hump section, the average frequency of each IMF shows an overall increasing trend, while the average frequency distribution of the IMF at the hump section of the SOC is similar to that at the outlet of the conduit. The energy ratio at the inlet of the upstream section of the SOC is higher than that at the hump section and outlet of the SOC in IMF1-IMF3, but it is the opposite in IMF4-IMF6.
Under the condition of 1.0Q bep , the IMF average frequency distribution and energy ratio of the pressure pulsation along the SOC are shown in Figure 23. Under the condition of 1.0Q bep , the energy at the inlet of the upstream section of the SOC is mostly focused on IMF3 and IMF4, and the energy at the hump section and outlet of the SOC is mostly focused on IMF4, IMF5, and IMF6. From the inlet of the SOC to the hump section, the average frequency of each IMF shows an increasing trend, while from the hump section to the outlet of the SOC, the average frequency of each IMF shows a decreasing trend. From the inlet to the outlet of the T A B L E 9 Average frequency distribution and energy ratio of each IMF (1.2Q bep  SOC, the energy distribution of the water flow is gradually transferred from IMF3 and IMF4 to IMF5 and IMF6. Under the condition of 1.2Q bep , the IMF average frequency distribution and energy ratio of the pressure pulsation along the SOC are shown in Figure 24. Under the condition of 1.2Q bep , the energy at the inlet and hump section of the SOC is mostly focused on IMF4 and IMF5, and the energy at the outlet of the SOC is mostly focused on IMF4, IMF5, and IMF6. From the inlet of the SOC to the hump section, the average frequency of each IMF shows an increasing trend, while from the hump section to the outlet of the SOC, the average frequency of each IMF shows a decreasing trend. From the inlet to the outlet of the SOC, the energy distribution of the flow pressure pulsation is gradually transferred from IMF4 and IMF5 to IMF5 and IMF6. The water flow is affected by the impeller rotation at the inlet of the SOC, and the impact of the impeller rotation on the water flow is reduced at the hump section, and it almost disappears at the outlet of the flow conduit. From the inlet to the outlet of the SOC, the F I G U R E 20 Empirical mode decomposition of pressure pulsation signal at the inlet of the upstream section of SOC (P08).
T A B L E 10 Average frequency distribution and energy ratio of each IMF (P08).

| CONCLUSION
Considering the hydraulic coupling between the pump and the flow conduit, the 3D steady and unsteady numerical calculation of the SOC of the pump device is carried out, and the energy performance of the pump device is predicted. The results are compared with the experimental results to verify the validity of the numerical calculation. The Hilbert-Huang method is used to analyze the time-frequency features of the PPS of the SOC, and the pressure pulsation spectrum distribution features of the characteristic points at different positions of the SOC are obtained. The following conclusions were reached: 1. There is no obvious periodicity in each monitoring point of SOC, and the main frequency is maintained below 20 Hz. A small amount of high-frequency signals above 700 Hz can be decomposed at each characteristic section of the upstream section, hump section, and outlet section of the SOC, but the energy ratio is less than 1%. 2. Under the condition of 0.3Q bep , there is obvious highfrequency pulsation above 500 Hz at the inlet of the SOC, and there are periodic components distributed around 200 Hz, and the peak value of C p at the main frequency decreases in response to an increase in flow rate, which is 3.6 and 3.9 times under the condition of 0.3Q bep and 1.2Q bep , respectively. The PPE of the outlet section of the SOC is mostly focused on the low frequency below 40 Hz, but there are more medium and high-frequency signals under 1.0Q bep and 1.