Numerical Study of Unsteady Pressure Fluctuation at Impeller Outlet of a Centrifugal Pump

Intense uid-dynamic interaction at the impeller outlet strongly aects the unsteady ow and pressure stability within the centrifugal pump. In order to have a better understanding of the pressure uctuation of centrifugal pumps, a numerical calculation is carried out by using the RNG k-epsilon turbulence model under various ow rates. e numerical calculation results are compared with the experimental results in order to verify the reliability of the calculation model. e amplitude and frequency distribution of pressure uctuation at the impeller outlet is obtained and analyzed in the time and frequency domain. e research results show that the blade passing frequency is the dominant frequency of the pressure uctuation. And the pressure uctuation is a periodic uctuation. As the ow rate decreases, the periodicity of the pressure uctuation decreases. Besides, the amplitude and intensity of pressure uctuation are closely related to ow rate and spatial location. At the low ow rate, the amplitude of pressure uctuation in the time domain and frequency domain is enlarged greatly, especially near the tongue region. e pressure dierence distribution on both sides of the blade surface is extremely uneven, and the pressure changes signicantly.


Introduction
As an important energy conversion equipment, centrifugal pumps have been widely used in industrial processes and modern engineering. Pressure uctuation is a complex phenomenon of uid dynamics in centrifugal pumps, which a ects the stable operation of centrifugal pumps and the life of equipment [1][2][3][4][5][6][7]. Strong rotor-stator interaction is the main source of large pressure uctuation, which has a great impact on the stable and safe operation of the centrifugal pump. In the designing process of the pumps, reducing the pressure uctuation inside the centrifugal pump should be considered. e comprehensive understanding of unsteady pressure uctuation lays a foundation for further reducing the pressure uctuation intensity of the pump.
In recent years, due to the engineering needs of centrifugal pumps, more and more attention has been paid to pressure uctuation characteristics [8][9][10]. Gonzalez et al. [11] studied the unsteady ow e ects due to impeller-volute interaction in a centrifugal pump and found that the pressure uctuation level increases largely at o -design ow rates. Bai et al. [12] studied the pressure uctuation and unsteady ow in a centrifugal pump. Results show that the pressure uctuation amplitude in the di user increases gradually with the increase of the di user vanes numbers. Wang et al. [13] analyzed the unsteady pressure uctuation in a multistage centrifugal pump with the whole ow eld and revealed the main factors which have a large e ect on the intensity of the pressure uctuation within the pump. Liu et al. [14][15][16] investigated the pressure characteristics and internal ow characteristics of the pump with clearance. e pressure uctuation within the pump has been investigated by using experimental and numerical methods [9,17,18]. Spence and Amaral-Teixeira [19] studied the pressure uctuation in a centrifugal pump by using numerical methods under di erent ow rates. Jin et al. [20] investigated the transient characteristics in a double-suction centrifugal pump during the starting period by numerical simulation and experimental validation.
Research on reducing the intensity of pressure uctuation has also been carried out by many scholars. Zhang et al. [21] studied the unsteady pressure uctuation distribution in a centrifugal pump with a slope volute and found that a slope volute pump significantly reduces the intensity of pressure fluctuation. Gao et al. [22] studied the effect of the blade trailing edge profile on the performance and unsteady pressure fluctuation in a centrifugal pump. e results show that the ellipse on the pressure side and ellipse on both sides' profiles contribute significantly to the reduction of pressure fluctuation. Gao et al. [23] studied the pressure fluctuation distribution in a centrifugal pump with different cutting blades. It is found from multiple comparisons that an appropriate cut angle can effectively reduce the intensity of the pressure pulsation in the pump. Song et al. [8] analyzed the effect of impeller arrangement on pressure fluctuation in a double-suction centrifugal pump. Results show that a suitable impeller arrangement can improve pump performance and reduce the intensity of pressure fluctuation.
While numerous studies have investigated the pressure fluctuation and unsteady characteristics in different kinds of pumps, the pressure fluctuation characteristics at the impeller outlet still need further investigation. So, in the present work, the unsteady pressure fluctuation characteristics at the impeller outlet are carried out in a centrifugal pump under various flow rates. e numerical calculation results are compared with the experimental data to verify the accuracy of the calculation results. e amplitudes of pressure fluctuations at monitoring points under various flow rates in the time domain and frequency domain are studied and analyzed in detail.

Parameters of the Centrifugal Pump.
e centrifugal pump studied in the present work is a single-stage singlesuction centrifugal pump. Figure 1 shows the computational domain of the centrifugal pump which consists of four computational domains: impeller, volute, inlet, and outlet. e numerical calculations are carried out with this computation domain under various flow rates. e parameters of the pump are listed in Table 1. e flow rate and the pressure head at the design flow rate are Q d � 551 m 3 /h and H d � 25 m, respectively.

Independence Verification of Mesh
Density. Mesh quality has a great influence on the accuracy of numerical calculation. e structured mesh has better convergent characteristics in the numerical calculation of the pump than the unstructured mesh. Besides, the structured mesh has significant advantages in dealing with key areas such as the volute tongue.
us, the hexahedral mesh is used in the computation domain of the centrifugal pump. Structure mesh cells are generated in order to achieve accurate unsteady flow structures and pressure properties. e flow field near the wall has a large variation in velocity and pressure, so the mesh near the solid surface was mainly refined in order to ensure the calculation accuracy.
Before the numerical calculation, the mesh independence test was carried out. e nondimensional pressure head coefficient is defined as follows: where g stands for gravity, H stands for the pressure head, n stands for the rotating speed and r stands for the outer radius of the impeller. Five sets of meshes ranging from 2.38 million to 11.06 million were applied in the mesh independence test. e results of pressure head coefficient under different mesh numbers are shown in Table 2.
When the number of elements increases to 5.71 million, there is limited influence on the centrifugal pump head coefficient. erefore, considering the factors of computing resources and calculation accuracy, the case with 5.71 million elements is selected for the final calculation. e mesh is fine enough to satisfy y + < 200 near the wall. is    Figure 2.

Numerical Methods.
e Governing equation in this study is the unsteady three-dimensional incompressible Reynolds-averaged Navier-Stokes equations. e numerical calculations are conducted by using the commercial CFD code of ANSYS-Fluent 16.0. e medium used for numerical calculations and experiments is water. RNG k-ε turbulence model is used in the numerical calculation, which is also applied in many studies [24][25][26][27]. e finite volume method (FVM) is used to solve the system and the coupling between velocity and pressure is obtained by the SIMPLE algorithm. e velocity inlet is selected at the inlet of the duct. Neumann boundary condition is applied at the outlet of the domain for velocity and the pressure is given at the outlet. No slip wall is imposed on the solid walls. e transient flow characteristics in the centrifugal pump are obtained by two steps of calculation, namely, steady calculation, and unsteady calculation. First of all, the steady numerical calculation result is first obtained by using the multiple frames of reference (MRF) method. Secondly, the calculation data of steady numerical simulation in the pump is taken as the initial condition of the unsteady computation by using the sliding mesh model. For each impeller revolution, the calculation is performed in a time sequence of 360 time steps. e time step is set as Δt � 1.7 × 10 −4 s, which is corresponding to one rotation degree of the impeller, in order to obtain enough resolution of unsteady flow rates.

Location of Pressure Monitoring
Points. Strong pressure fluctuation induced by rotor-stator interaction has a great impact on the centrifugal pump. In order to obtain the pressure fluctuation characteristic of the pump at the impeller outlet, eight points were equally distributed, as shown in Figure 3. For point 1, the angle with the horizontal axis is 45 degrees, and the angle between two adjacent points is 45 degrees. Figure 4 shows the comparison between the numerical calculation results of the centrifugal pump and the experimental results [28]. It is discovered that the head curve of numerical calculation is in good agreement with the experimental data overall. e maximum of the relative error is within 5%. At the design flow rate, the relative error between the experimental data and the predicted flow rate is about 2.5%. In total, the small difference in head performance illustrates that the calculation model, mesh system, and boundary conditions in the present calculation are valid to predict the centrifugal pump performances. Figure 5 shows the distribution of head and velocity at the pump outlet versus the time. It is observed that the distribution of the pump head shows periodic fluctuations, and it shows a tendency to stabilize. However, the velocity distribution at the pump outlet does not show a uniform distribution trend. e velocity fluctuates greatly and  Science and Technology of Nuclear Installations 3

Validation of the Numerical Simulation.
presents an irregular trend after the impeller rotates. It is noticed that the velocity reaches its steady state after 8 circles. us, the analysis of pressure fluctuation is carried out after this time.

Unsteady Pressure Fluctuation at Impeller
Outlet. e pressure coefficient is used to describe the distribution of pressure fluctuation which is defined as follows: where p stands for static pressure,p stands for time average static pressures of the probes, and u stands for impeller circumferential velocity.
At the initial position, the tongue is corresponding to the middle region of the impeller outlet, as is shown in Figure 3. Figure 6 presents the pressure fluctuation and frequency spectra at selected monitoring points at the design flow rate.
e pressure fluctuation at each monitoring point is closely related to the relative position between blades and volute tongue [29]. As can be seen from Figure 6(a), the pressure fluctuation at the monitoring point has an obvious periodic fluctuation rule.
e pressure fluctuation at p1 near the tongue is relatively larger. Besides, it is found that the distribution of pressure fluctuation can be roughly divided into two categories, which are presented in Figure 6  position between the impeller and volute tongue. e frequency and amplitude of the pressure fluctuation are obtained by fast Fourier transformation (FFT), which is presented in Figure 6(b). e f n stands for impeller rotating frequency. It is found that the main pressure fluctuation components possess the blade passing frequency (f BPF ) and its harmonic frequencies.
As Figure 7 shows, the pressure fluctuation components at f BPF . It is observed that the distribution of pressure spectra at f BPF shows a trend of wavy decline, in general. e maximum value is obtained at p1, which locates near the volute tongue. As the monitoring points move away from the volute tongue, the pressure amplitudes decrease due to the weakening of the pressure fluctuation intensity. e variation between each of the three monitoring points shows a completely opposite distribution trend, as shown by the blue arrow in Figure 7. e pressure spectra at f BPF show some degree of modulation, which is characterized by the four peaks and valleys. is modulation pattern is closely

Influence of Flow Rate on Pressure Fluctuation
Distribution. Figure 8 shows the pressure fluctuation in the time domain at different flow rates in two impeller revolutions. It is noticed that the pressure fluctuation distribution of monitoring points presents a periodic distribution. As is shown in Figure 8, there are four distinct peaks and valleys for each monitoring point in one impeller revolution, which is corresponding to four impeller blades. It is obvious that the pressure fluctuation characteristics show significant differences at different flow rates. At 0.4Q d , the pressure fluctuation signals are irregular and nonuniform. e pressure signal fluctuates largely at p1, which locates near the volute tongue. At 0.7Q d , the pressure fluctuation shows certain regularity, and the range of pressure fluctuation is decreased to a certain extent. At 1.0Q d , the pressure fluctuation distribution is quite regular and pressure fluctuation amplitude in the time domain achieves its minimum value. e pressure signal changes periodically overtime at the  monitoring points. As the flow rate increases to 1.3Q d , the pressure fluctuation amplitude in the time domain increases obviously, especially at p1. Figure 8 mainly presents the distribution regulation of pressure fluctuation over time. However, the data is mainly about qualitative analysis of pressure fluctuation distribution. erefore, the average value and distribution range of pressure fluctuation at each measuring point at different flow rates are analyzed in detail. Figure 9 shows data such as the amplitude and average value of the pressure fluctuation range at each point under different flow rates. Because of the wide range of pressure fluctuations, the average pressure of each measuring point does not change significantly with the space and flow rate, so it is especially given in Figure 10. e range of the pressure fluctuation stands for the amplitude of the pressure fluctuation. At 0.4Q d , the range of pressure fluctuation at monitoring points is relatively large, which reveals the strong unsteady pressure fluctuation. In general, the pressure fluctuation range decreases from p1 to p5 region and increases to p7 then decrease to p8. e maximum pressure fluctuation range is obtained at p1, which was affected by the volute tongue, while the minimum value is obtained at approximately p5. Distribution of pressure fluctuation range indicates pressure fluctuation strength at monitoring p1 and p2 is stronger than at other points.
Internal complex unsteady flow is the main reason for the large pressure fluctuation. As the flow rate increases to 0.7Q d , the pressure distribution is similar to 0.4Q d , while the pressure fluctuation range at the monitoring points is decreased in a small scale. At the design flow rate, the maximum pressure at monitoring points is almost the same, except for p1. At 1.3Q d , the pressure fluctuation range of the remote tongue measuring point does not change much, but the pressure near the tongue measuring point increases to Science and Technology of Nuclear Installations varying degrees. e increase of p1 was most significant near the tongue, indicating that the rotor-stator interaction was the most significant near the tongue. As Figure 10 shows the average pressure distribution at different flow rates. At 0.4Q d , the average pressure at the monitoring points increases obviously from p1 to p8. e pressure value increases from p1 along the counter-clockwise direction, which is in accord with the supercharged principle of the pump volute. e minimum average pressure is achieved at p1, and it is obviously lower than other points. At 0.7Q d , the average pressure increases gently from p1 along the counter-clockwise direction. At 1.0Q d , the average pressure value of the monitoring points is almost the same, which reveals the uniform pressure distribution at the design flow rate. At 1.3Q d , the average pressure of the monitoring points decreased from p1 to p8, which is different from other flow rates. e volute has the function of collecting fluid and is also an energy conversion device. Along the spiral direction of the volute, the cross-sectional area of the volute flow channel gradually expands, so the high-speed fluid thrown out by the impeller can be gradually decelerated, and the kinetic energy can be effectively converted into static pressure energy. e increase in flow rate causes the movement of the low-pressure area in the flow channel towards the inlet of the volute, as shown in Figure 11(d). In addition, the unsteady flow and the large flow rate affect the conversion of kinetic energy into static pressure by the volute to some extent, thus causing the distribution of the impeller outlet pressure under 1.3Q d . Figure 12, the pressure fluctuation in the frequency domain at different flow rates shows discrete spectral characteristics. It was found that the dominant frequency is f BPF and its multiples. e pressure fluctuation amplitude shows a decaying trend in the frequency domain. Evident peaks at harmonic frequency 3f BPF could be identified. However, its amplitude has been greatly reduced.

Pressure Fluctuation in Frequency Domain. As is shown in
Pressure fluctuation amplitude in the frequency domain is relatively small at 1.0Q d , which reveals a good internal flow situation. However, pressure fluctuation amplitude in the frequency domain is increased obviously at off-design flow rates, especially at the small flow rate. e amplitude of pressure fluctuation at different monitoring points also varies greatly. In general, the maximum amplitude of pressure fluctuation in the frequency domain is obtained at p1, which is located near the tongue. e amplitude of pressure fluctuation in the frequency domain is decreased from p1 along the impeller rotation direction. With the increase of the flow rate, the pressure fluctuation amplitude  in the frequency domain first decreases and then increases and the minimum of which was obtained at 1.0Q d . Figure 13 shows the amplitude of pressure fluctuation under f BPF and 2f BPF at the monitoring points. It was found in Figure 13(a) that, with the increase of the flow rate, the amplitude of pressure fluctuation decreases, and its minimum value achieves at the design flow rate. As the flow rate increased from 1.0Q d to 1.3Q d , the amplitude of the pressure fluctuation shows an increasing trend. e amplitude of pressure is relatively large at the partial flow rate, especially at 0.4Q d . For large flow rates, the amplitude distribution is almost consistent with the design flow rate, except for the measuring points near the tongue. In Figure 13(b), the pressure amplitude at the monitoring points has a similar distribution tendency. e pressure fluctuation amplitude at the measuring point near the tongue is relatively large. e amplitude of pressure fluctuation is relatively large from p1 to p3 at 1.3Q d . While the amplitude of pressure fluctuation is relatively large from p5 to p8 under the 0.4Q d flow rate.
It was concluded that the maximum value is obtained at the point p1, which is located near the tongue region at 0.4Q d . e minimum value is obtained at p6, which is located far from the tongue. In general, the amplitude of pressure at monitoring points decreases from the point near the tongue region along the impeller rotation direction. Its minimum value is achieved at the point located farthest from the tongue. Figure 11 presents pressure distribution on the center plane under various flow rates. e pressure coefficient is defined as follows:

Pressure Coefficient Distribution.
where p stands for static pressure, p ref stands for the reference pressure (1 atm), and u stands for impeller circumferential velocity. In general, along the direction from the impeller inlet to the volute outlet, the pressure presents a trend of gentle and continuous increase. e minimum pressure is obtained at the inlet of the impeller, and the maximum pressure is obtained at the outlet of the pump. In addition, the pressure in the pump is larger at a small flow rate. As the flow increases, the pressure gradually decreases. At partial flow rate, low-pressure regions are observed clearly at the leading edge, and pressure coefficient distribution is nonuniform at the inlet of the impeller passages, especially at 0.4Q d . With the increase of the flow rate, the pressure in the impeller passage increases gently. At the Science and Technology of Nuclear Installations design flow rate, the pressure distribution is evenly distributed and increases gradually from inlet to outlet of the impeller. e pressure difference is relatively small between both sides of the impeller passages which reveals a good flow situation. As the flow rate increase to 1.3Q d , the pressure amplitude decreases significantly, and the distribution is still relatively uniform. While enlarged low-pressure areas are captured at the trailing edge in one impeller rotating period with the increase of the flow rate. e interaction between the low-pressure areas near the trailing edge and the highpressure area in the volute and the strong rotor-stator interaction may be the major reason for the increase of pressure fluctuation in the frequency domain near the tongue region. Figure 14 presents the pressure distribution of blade surfaces in impeller passage A, which is shown in Figure 3. L stands for the relative length of the blade. It was found that the pressure coefficient on the pressure side is larger than that on the suction side. In general, the pressure increases gradually along the direction from the impeller inlet to the impeller outlet. e maximum pressure value is achieved near the region at the impeller outlet for the pressure side (PS) and suction side (SS). However, the pressure is decreased to a certain extent at the impeller outlet, which is caused by the lowpressure region near.
Trailing edge (TE). At 0.4Q d , the pressure coefficient on the PS and SS is the worst and nonuniform. Besides, the intersections are found on both sides of the blade surfaces. e pressure coefficient on PS is enlarged significantly after about 0.7 L, which is in good agreement with Figure 11(a). Large adverse pressure gradient and pressure difference on both sides of the blade surfaces are easy to generate the flow separation and secondary flow, which finally results in the unsteady internal flow situation. As the flow rate increases to the design flow rate, pressure increases slightly and the pressure difference on the PS and SS is reduced. Smooth pressure changes on both sides of the flow passage indicate a good internal flow situation. At 1.3Q d , the pressure on the PS increases slowly, and the pressure near the outlet of the flow channel decreases. However, the pressure on the SS is significantly reduced. In addition, it was also found that the low-pressure area at the trailing edge of the blade tends to expand as the flow rate increases.

Conclusions
In the present work, the numerical calculation is performed for the three-dimensional centrifugal pump in order to investigate the characteristics of pressure fluctuation in a centrifugal pump at design and off-design flow rates. e numerical calculation results are in good agreement with the experimental data, which proves the reliability of the calculation results. e conclusions obtained from the numerical simulations are as follows: (1) e periodicity of pressure fluctuation is captured in the time-domain distribution, and the distribution of pressure fluctuation appears irregular under a small flow rate. e average pressure at the impeller outlet increases rapidly at a small flow rate from the tongue along the impeller rotation direction, while the pressure distribution is the opposite at a large flow rate. (2) e dominant frequency is clearly obtained under various flow rates. e amplitude of pressure fluctuation is minimum at the design flow rate, and increases at the off-design flow rate, especially at a small flow rate. In addition, as the flow rate decreases, the pressure distribution on both sides of the flow channel becomes worse, which is mainly reflected in the suction side of the blade. (3) e distribution of frequency spectra is nonuniform at the impeller outlet circumferential direction. Its amplitude decreases along the impeller rotation direction near the tongue region, and a minimum value are achieved near the region far away from the tongue.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.