Flow Characteristics and Energy Loss of a Multistage Centrifugal Pump with Blade-Type Guide Vanes

: Multistage pumps with blade-type guide vanes are widely used in offshore oil production, the petrochemical and coal-chemical industries, and nuclear power fields for its advantages of large flow rate, high pressure, and excellent operation stability. However, the internal flow of this kind of pump is complex; in particular, the hydraulic, flow, and pressure pulsation characteristics of the different stages are quite different, which has a great impact on the design and performance predictions of this kind of pump. Thus, in this paper, the hydraulic performance, unsteady flow characteristics, evolution of vortex structures and pressure pulsation characteristics in a 10 stage centrifugal pump are investigated numerically. The results show that inverse flow, jet-wake flow, and rotor-stator interaction flow are the key factors causing energy loss and efficiency decline at every stage and in the whole pump. The vortex evolution at the rotor–stator interaction regions is actually the process that the vortex structures fall off and impact on the pressure surface at the leading edge of the guide vane blade at a frequency that equals to the impeller blade passing frequency. Furthermore, under the actions of the guide vane with confluence cavity, the pressure pulsation within the final-stage guide vane contains low-frequency components with large bandwidths, which mainly results from the confluence flow disturbance at the outlet of the cylindrical guide passage.


Introduction
Multistage centrifugal pumps are widely used in the aerospace, petrochemical, and other fields due to their advantages of high head and strong reliability.Under the working conditions of high pressure and high rotational speed, flow instability phenomena, such as invers flow, jet-wake flow, and rotor-stator interactions, easily occur within these pumps, causing excessive structural vibration and even damage.Furthermore, researchers have carried out much work on the relationships between the unstable flow structures and the characteristics of multistage centrifugal pumps.Kimura et al. [1] investigated the relationship between the geometric parameters of the inlet pipe and the upstream flow structure of the inducer through experiments and numerical simulations.Fukao et al. [2] conducted a simulation work of a model inducer using the LES (Large Eddy Simultion) method and found that the backflow gradually extends upstream to the impeller inlet with the decreasing flow rate.Xin et al. [3] analyzed the jet-wake flow structure at the impeller outlet and proposed that these structures would induce unstable vortices and produce periodic pressure and velocity fluctuations around the outflow of the impeller.Zhang [4] proposed that the jet wake phenomenon around the impeller outlet was gradually weakened through PIV (Particle Image Velocimetry) experiments on a centrifugal model pump.Wuibaut [5] compared the velocity distribution and pressure field pulsation characteristics within the impeller passage through a PIV test and concluded that the jet wake phenomenon occurred in the dynamic-static gap between the impeller outlet and the guide blade inlet, which verified that the unsteady flow within the guide blade was closely related to the jet wake phenomenon.Through comparisons of the energy loss and velocity distributions in a pump, Meng et al. [6] found that the flow separation caused by the asymmetry of the jet wake in the guide vane and the volute was the main factor contributing to the energy loss within the guide vane.Gao et al. [7] experimentally measured the dynamic pressure pulsation at the impeller outlet, and found that the pressure pulsation amplitude near the wake vortex was large by combining their measurements with numerical simulation results.
The pressure pulsation and excitation force characteristics caused by rotor-stator interactions are also an area of research focus.Zhang et al. [8] effectively reduced the contact between the wake vortex and the pump tongue by changing the tilt degree of the volute outlet, thus reducing the pressure pulsation around the tongue.Zhang et al. [9] investigated the unsteady flow field of a centrifugal model pump with ultra-low specific speed and found that eddies of different sizes and quantities appeared in the impeller flow passage and that the pressure fluctuation on the volute decreased along the flow passage.Jia et al. [10] studied the unsteady flow structure and vibration caused by pressure pulsation in a centrifugal pump by using entropy generation and energy loss methods.Their results showed that there was large flow loss near the volute tongue passage, and the pressure pulsation at the shaft frequency was the main element affecting vibration stability of pump rotor.Liu et al. [11] obtained the conclusion that the rotor-stator interaction in a centrifugal pump was the main cause of the pressure pulsation, and that the pressure pulsation was correlated with the intensity of the secondary flow within the volute.Zhao [12] found that the flow separation appeared around the leading edge of the impeller pressure side and moved along the flow passage in the form of vorticity, and that the interaction between stall vorticity and the volute enhanced the rotor-stator interaction.Zeng [13] conducted a hydraulic performance study of a high-speed centrifugal pump with a diffuser and the results showed that the pressure pulsation near the tongue was the highest, while the pulsation between two diffuser blades was the lowest.Win [14] proposed a nonlinear frequent-domain method for solving the vibration of array turbines and found that the distribution characteristics of pressure coefficient and skin friction coefficient on the blade surface of the downstream turbines were significantly affected by the upstream turbine wakes, especially the far wakes.Nakhchi [15] used the spectral-hp element method to directly investigate the flow instability, pressure fluctuation, and vortex generation on the compressor blade surface.The results showed that the blade vibration had a great influence on the vortex generation along the suction side of the blade and the laminar flow separation of the bubbles.
Fluid flow is accompanied by the movement of the vortex structure.It has been found that unsteady pressure pulsation is related to the vortex shedding effect.The accurate identification of the vortex structure is of great significance for understanding the turbulent flow mechanism, solving the problem of hydro-mechanical instability, and flow optimization.Thus, recently, Wang et al [16] used a model test, high-speed imaging, and an SST (Shear Stress Transport) k-ω turbulence model to analyze the evolution of a suctionside vertical cavitation vortex (SSPCV).The results showed that the evolution of vertical cavitation vortexes on suction side could be divided into three stages: generation, development, and fragmentation.During the generation stage, the turbulent kinetic energy, velocity gradient, and vortex kinetic energy continued to increase, and the vortex expansion term was the main factor causing the vortex degree.Subsequently, the vortex gradually dissipated and entered the development stage, and the vortex was mainly caused by the deformation of fluid micelles.Finally, under the influence of the next blade, it entered the fracture stage, accelerating the dissipation of the vortex.Zhang et al. [17] analyzed the influence of eddy current on pump performance by installing a vortex generator at the impeller inlet of an axial flow pump.It was found that the inlet vortex reduced the uniformity of the axial velocity distribution at the impeller inlet and increased the radial velocity, and that the vortex induced the pressure pulsation at the impeller inlet at 0-1 times the rotation frequency.Guo [18] used three-dimensional PIV to study the time-domain variations of free surface vorticity and vorticity in the closed pump inlet under different pressure conditions.The results showed that the occurrence of vorticity was closely related to pressure, and the number of vorticities increased with the increase of absolute pressure value.Yang et al. [19] investigated the pressure pulsation of the impeller and the inducer in different time series and found that the pulsation in the region of rotor-stator interaction and downstream region was closely related to the shedding of the unstable vortex.Ni et al. [20] proposed that pressure pulsation was related to the vorticity distribution and unsteady vortex shedding effects.Zhang [21] investigated the complex flow structure around the volute tongue region based on PIV technology.The results showed that under large flow-rate conditions, there were positive and negative vorticities located on the pressure sides and suction sides of the blade, but under small flow-rate conditions, the negative vorticities on the suction side of the blade disappeared due to the separation and reverse flow structures.
Due to the gradual expansion of vortex research, different kinds of vortex recognition methods are widely used in the field of hydraulic machinery.Among these methods, the Q criterion and λ2 criterion are highly dependent on the selected Q value and λ2 value, i.e., the unreasonable selection of these parameter values leads to some vortex structures not being recognized [22].Recently, Liu [23] proposed a new Omega vortex recognition method and clearly proved that the method has Galileo invariance.Compared with the traditional Q criterion and λ2 criterion, this method gives clear physical meaning to the selection of parameter value Ω. Gui [24] analyzed the direct numerical simulation data of a rotating jet using different vortex recognition methods and concluded that the Omega method was superior to other methods at identifying small-scale vortices.Zhang et al. [25] used this method to analyze the shedding vortex located between the impeller and the tongue under different operating conditions and pointed out that this method could accurately identify the vortex structure, especially the small vortex located at the rotor-stator interaction positions.
In this study, an unsteady flow simulation of a multistage centrifugal pump with a radial guide vane in each stage was conducted by using the LES method.Based on the simulation results, the external characteristics, velocity distribution and vorticity distribution characteristics of the pump were analyzed under different flow conditions.Furthermore comparative analyses between the pressure pulsation characteristics at different monitoring points in the guide vane passage and the periodic variance of the vortex structure in the rotor-stator interaction region were performed using the Omega method.Based on these analyses, the influence of the vortex structure near the monitoring points on the pressure pulsation is discussed and the correlation between the frequency characteristics of the pressure pulsation and those of the vortex shedding is evaluated.

Geometric Model and Mesh Generation
The research object in this paper is a 10 stage centrifugal pump with a design flow rate Qd of 290 m 3 /h, a head Hd of 2369 m, and a rotating speed nd of 3550 rpm.The computational domain of the 10 stage centrifugal pump is shown in Figure 1, which includes the pump inlet, the inhalation chambers, the front chamber, the wear rings, the impellers, the guide vanes, the back chambers, the extrusion chamber, and the pump outlet.Table 1 shows the geometric parameters of the main flow components of the centrifugal pump.It is noted that, except for the inlet diameter of the impeller, other parameters from the second impeller to the tenth impeller are the same as those of the first impeller.The structure and parameters of the first nine-stage guide vane are identical, while the structure of the last-stage guide vane is obviously different from that of the first nine-stage guide vane.The flow field in the multistage centrifugal pump was meshed by Ansys-Icem CFD 18.0 (Ansys, Canonsburg, PA, USA).Structural grids were used for front chambers, back chambers, wear rings, impellers, inlet and outlet domains, and unstructured grids with good adaptability were used for the domains with complex structures, such as inhalation chambers and guide vanes.As shown in Figure 2, six groups of grids were established to verify the grid independence to ensure the simulation results were accurate and credible enough.It can be seen that when the total number of grids exceeds 91.3 million, the dimensionless head fluctuation does not exceed 0.02% and the efficiency fluctuation does not exceed 0.12%.Thus, in order to save computing resources and better capture the flow details of the multistage centrifugal pump, the model of grid 91,286,861 was selected for numerical calculation.In this case, the minimum y+, average y+, and maximum y+ values are 5.78, 11.32, and 38.49, respectively.The minimum and maximum element sizes are 3.83×10 −4 mm and 3.91×10 −2 mm, respectively.Since the parameters from the second impeller to the final impeller are the same as those of the first impeller, except for the inlet diameter, the first impeller was selected as an example.The grid division details of the first-stage impeller and its guide vane and of the last-stage impeller and its guide vane are shown in Figure 3 as examples.

Numerical Simulation Method
In this study, the large eddy simulation (LES) method was employed based on the commercial software ANSYS-Fluent 18.0 in the unsteady simulation of this 10 stage centrifugal pump.It is important to note that, before the calculations, the results based on the standard K-ε turbulence model were used as the initial conditions for the unsteady simulation cases.The wall-adaptive Eddy-Viscosity Model (WALE) was adopted as the sublattice model of incompressible turbulence.The SIMPLEC algorithm was adopted to deal with the velocity-pressure coupling.Second-order upwind discretization was used for the convection terms.Non-slip wall was adopted for the wall boundary condition.Meanwhile, the pressure inlet and the mass flow outlet with a 128.88 kg/s flow rate were set as the inlet boundary condition and the outlet boundary condition, respectively.A frozen rotor model was adopted in the steady-state calculation for the dynamic and static interface, and the slip grid was set in the unsteady-state transient calculation.The transient calculation was conducted with a time step of 4.7×10 −5 s, which is the time required for the impeller to rotate one degree.

Monitoring Points
In order to analyze the pressure pulsation characteristics of the flow field inside the multistage centrifugal pump, three monitoring points, including Point a, Point b, and Point c, were set up in the guide vane passages of every stage.As shown in Figure 3a, Point a was located at the inlet of the guide vane passage; Point b was located near the middle of the guide vane passage; and Point c was located at the outlet of the guide vane.In addition, considering that the guide vane structure's 10th stage was quite different from those of the other stages, monitoring Point d was added at the exit of the cylindrical guide vane flow passage of the last stage, as shown in Figure 3b.

Experimental Verification
The hydraulic characteristic experiments on this 10 stage multistage centrifugal pump were conducted in a closed-loop test system, which complied with the National A Level accuracy standard, as shown in Figure 4. Two piezoelectric pressure sensors were respectively set at the inlet and outlet of the pump.The measuring range of the inlet pressure sensor was -0.1~0.5 MPa, the measuring range of the outlet pressure sensor was 0~30 Mpa, and the sensitivity of both sensors was 1000 mV/MPa.The head curves and efficiency curves of the multistage centrifugal pump obtained from the numerical simulation and hydraulic experiments are shown in Figure 5.It can be seen that the efficiency curves coincided well, i.e., both curves increased first and then decreased with the increasing flow rate, and the maximum error of efficiency was 6%.Furthermore, the calculated head and experimental head of the multistage pump decreased gradually with the increases in flow rate, with a maximum error of 2.4%, which validate that the applied turbulence model and numerical simulation method are suitable for this pump.

Head Characteristics of Every Stage within the Pump
According to the numerical simulation results, the head variations of every stage within the 10 stage model pump under different flow rate conditions are shown in Figure 6.It can be observed that with the increase in the flow rate, all the head curves of every stage gradually decreased.Under all the flow rates from 58 m 3 /h to 348 m 3 /h, the head of the first stage remained the highest, and that of the last stage remained the lowest.Under the designed flow rate of 290 m 3 /h, the head gap of stage 2 to stage 9 was the smallest.The gap gradually increased when the flow rate deviated from the designed flow rate.

Unsteady Flow Structures
The streamline distributions of the middle section of typical stages within the model pump under different flow conditions are indicated in Figure 7.It was observed that when the centrifugal pump operated at a low-flow-rate condition, the flow angle of the fluid entering the impeller passage was relatively large, resulting in different degrees of reverse flow at the impeller inlet and outlet (shown as Position 1 and Position 2 in Figure 7a-c).It is obvious that under 0.2 Qd conditions, there were more vortices at the inlet passages of the second stage and the last stage compared with the first-stage inlet, which significantly blocked the flow channel and raised the flow loss.This is also the reason why the heads from the second stage to the last stage were all lower than that of the first stage, as shown in Figure 6.As the flow rate increased, the number of vortex structures at the inlet of every stage decreased, i.e., the unstable flow at the inlet improved gradually.
The fluid within the radial guide vane underwent a radial-axial-radial alternate change process in the limited space.Different degrees of vortices were generated around the inlets of the guide vanes and the flow passages (shown as Position 3 in Figure 7a-c).The velocity streamline distributions within the first-stage and the second-stage guide vanes were basically the same, but were quite different from the distribution within the last-stage guide vane, whose structure had a large cavity in which to gather and transport the liquid to the outlet.Compared with the first and second guide vane passages, there were much more reverse flow and vortices in the back cavity of the last-stage guide vane passage and the impact loss, diffusion loss, and along-flow loss of this stage increased significantly.As a result, the head of the last-stage over-current component was the lowest, as shown in Figure 6.In recent years, researchers introduced axial vorticity distribution to analyze the flow characteristics and energy loss of single-stage centrifugal pumps [23,24]; thus, the axial vorticity distributions of different stages are drawn in Figure 8 to investigate the unstable flow structures in this multi-stage model pump.As shown in the figure, high vorticity regions always appeared in the cross sections of the first-stage, second-stage, and laststage flow passages under both large-flow-rate and small-flow-rate conditions.However, the high-vorticity area under 0.2 Qd was the largest and the high vorticity regions showed obvious asymmetric distribution (such as Region A and Region B).This is because, with the rotation of the impeller, the fluid within the impeller channel had a shear layer with the velocity gradient, especially under small-flow-rate conditions.As the velocity gradient of the shear layer at the trailing edge of the impeller blades was the largest, the high vorticity area was the largest and the flow loss was also the largest, shown as Region B in Figure 8.
Moreover, it can also be concluded from Figure 8 that the vorticity distribution characteristics of the three stages differed little under large-flow-rate conditions.However, under small-flow-rate conditions, the high-vorticity area of the last-stage passage was obviously larger than that of the first stage and the second stage, with more obvious asymmetric characteristics, which indicates that the rotor-stator interaction between the impeller blades and the guide vanes of the last stage was more pronounced.The axial vorticity distribution of the first-stage rear guide vane is drawn in Figure 9 and compared with the distribution of the last-stage guide vane, shown in Figure 10.It can be seen from Figure 9 that there was a large number of unstable vortex structures at the impeller inlet.The fluid between the outlet of the rear guide vane and the inlet of the next-stage impeller interfered with each other, which induced obvious inverse flow in the downstream flow channel (impeller inlet).Furthermore, as shown in the figure, under small-flow-rate conditions, the actual liquid-flow angle entering the impeller channel and the inlet speed increased, which raised the fluid impact loss and caused the flow capacity to drop sharply.Figure 10 shows the axial vorticity distribution of the last-stage guide vane under different flow-rate conditions.It can be seen that under small-flow-rate conditions, the vorticity near the cylindrical diversion exit distributed unevenly, i.e., Regions 1, 2, and 3, which were far away from the pump outlet, had low vorticity, while Regions 5, 6, and 7, which were close to the outlet, have high vorticity.As the flow rate increased, regions with high vorticity behaved with a certain degree of symmetry and distributed more evenly, which is consistent with the distribution of internal backflow and rotor-stator interaction in the last-stage guide vane passage under the low-flow-rate conditions shown in Figure 7.
Based on the above analysis, it can be concluded that there were unstable flow phenomena in the multi-stage model pump, such as inverse flow, jet-wake flow, and rotorstator interaction flow.Meanwhile, these unstable flow structures aggravated energy loss and induced fluid pressure pulsations within the pump, which obviously reduced the head, efficiency, stability, and operational reliability of the model pump.Thus, the energy dissipation caused by backflow, jet-wake, and rotor-stator interference can be weakened by optimizing the structure parameters of the impeller and the guide vane themselves or by optimizing the fit parameters of the impeller and guide vane, especially the clearance between them.

Distribution and Evolution of Vortex Structures in Rotor-Guide Vane Region
In this paper, a new Omega vortex identification method is introduced to explore the influence of rotor-stator interaction flow field within the pump, whose specific formula is shown below: where R is the vortex vorticity and Ω is the ratio of local vortex vorticity to the total vorticity.
Figure 11 shows the distribution characteristics of the vortex structure in the firststage rotor-stator interaction region (shown as Position B of Figure 8) under different flow conditions.It was noted that under the flow rate of 0.2 Qd, more vortex core structures on the pressure side of the blade's leading edge were generated.These core structures were densely distributed in a single guide vane channel, which lad to much stronger pressure pulsations under low-flow-rate conditions.However, under large-flow-rate conditions, the vortex cores were sparsely distributed and the pressure pulsations were much weaker.The regeneration-growth-shedding-rupture process of the vortexes in the rotor-stator interaction regions within two impeller-blade rotation periods, i.e., 1/7 of the impeller rotation periods, are shown in Figure 12.As shown in Figure 12a, at the beginning, the vortex structures formed during the previous period (shown as Region A) had fully grown, and the vortexes began to generate near the leading edge of the guide blade (shown as Region B).With the rotation of the impeller, the vortex in Region B gradually stretched and lengthened, while the vortex in Region A started to fall off and moved gradually towards to the leading edge of the downstream guide vane.As time passed, the vortex in Region B was further stretched and extended, but the vortex in Region A completely fell off, impacting the guide-blade leading edge downstream, and began to break.Finally, at the end of this period, shown in Figure 12d, the vortex in Region A was completely broken and moved into the downstream region.At this point, the vortex in Region B was fully developed, and a new vortex (shown as Region C) began to form near the leading edge of the guide blade.
Thereafter, the regeneration-growth-shedding-rupture process of the vortex structures occurred periodically with the rotation of the impeller (as shown in Figure 12e-h), i.e., the same process occurred in Region B and Region C as in Region A and Region B. Thus, it can be concluded that the development of the vortices occurs completely during one impeller-blade-rotation period, which indicates that the rotor-stator interaction effect is induced by the periodic process of the vortex shedding and impacting on the pressure surface at the leading edge of the guide vane blade.

Pressure Pulsations within the Rotor-Stator Interaction Regions
A dimensionless pressure coefficient Cp is introduced in this paper to analyze the pressure pulsation characteristics at the monitoring points of the first-stage and last-stage flow channels: where p represents the static pressure value at the monitoring point at a certain time;p represents the average pressure at the monitoring point within a period; ρ represents the fluid density; and U2 represents the circumferential velocity at the impeller outlet.
Figure 13 shows the time-domain diagram of the pressure pulsation coefficient of the monitoring points in the first-stage guide vane.As the fluid entered the diffuser section of the guide vane (from a to c), the kinetic energy generated by the fluid velocity gradually transformed into pressure energy and the fluid flow gradually became stable in the diffuser section of the guide vane, which caused the coefficient amplitudes from Point a to Point c to decrease gradually.Furthermore, the pressure pulsation coefficients in the firststage guide vane showed periodic changes, with seven wave peaks and seven wave troughs, corresponding to the number of the impeller blades.Under the flow rate of 0.2Qd, the maximum amplitudes of the pressure pulsation coefficient at Points a, b, and c decreased by at least 30%, with the flow from 1.0Qd to 0.2Qd.In terms of the degree of decline in the amplitude, the closer the monitoring point to the rotor-stator interaction region, the greater the pressure pulsation coefficient amplitude.

Conclusions
In this study, an unsteady numerical simulation and experiments of a 10 stage centrifugal pump based on the LES method is conducted and the internal flow characteristics, including the vorticity distributions, the evolution of vortex structures, and the pressure pulsation characteristics, were analyzed.The results indicate that the inverse flow, jetwake flow, and rotor-stator interaction flow are the key factors causing energy loss and efficiency decline in the pump.The vortex-cluster evolution at the rotor-stator interaction region is the process through which the vortex structures periodically fall off and impact on the pressure surface at the leading edge of the guide vane blade, which means that the dominant pulsation frequency in every stage guide vane is the impeller blade passing frequency 7fn.Furthermore, under the actions of the guide vane with confluence cavity, the pressure pulsation within the final-stage guide vane contains low-frequency components with large bandwidths, which mainly results from the confluence flow disturbance at the outlet of the cylindrical guide passage.It is concluded that the energy dissipation caused by backflow, jet-wake, and rotor-stator interference can be weakened by optimizing the structure parameters of the impeller and the guide vane themselves or by optimizing the fit parameters of the impeller and the guide vane, especially the clearance between them.

Figure 5 .
Figure 5. Hydraulic performance curves of the model pump.

Figure 6 .
Figure 6.Head characteristics of every stage within the model pump at different flow rates.

Figure 7 .
Figure 7. Streamline distributions of the middle section within the model pump under different flow rate conditions: (a) streamline distribution of the first stage; (b) streamline distribution of the second stage; (c) streamline distribution of the last stage.

Figure 14 Figure 14 .Figure 15 .
Figure14compares the pressure pulsation frequency characteristics of Point a in the first-stage and last-stage guide vane under 0.2 Qd and 1.0 Qd flow-rate conditions.The figure indicates that the dominant frequency of the first-stage coefficient was 7fn, which is equal to the blade passing frequency.Except for the 7fn component, there was still a dominant component with the frequency of fn and low-frequency components with a wide frequency band (12~120 Hz) in the last-stage guide vane.Furthermore, with the increase in the flow rate, the amplitude of the low-frequency components decreased obviously, indicating that the vortex structure under small flow was the main factor causing lowfrequency pressure pulsation.Combined with the vorticity distribution analyses shown in Figure11, it can be inferred that at the flow rate of 0.2 Qd, the vortex core structure in a single guide vane channel is densely distributed and the internal flow field is much more turbulent, leading to large low-frequency pressure pulsations in the pump.

:
Methodology and validation, C.L., Z.Z. and J.G.; formal analysis, L.Z., C.L. and B.C.; writing-review and editing, L.Z. and C.L. All authors have read and agreed to the published version of the manuscript.Funding: This research was funded by the National Natural Science Foundation of China (Grant No. 51976199 and Grant No. U1709209), Key Research and Development Program of Zhejiang Province (Grant No. 2020C01027 and Grant No. 2022C03036), Zhejiang Province High-Level Talent Special Support plan (2019R51002) and Advanced Space Propulsion Laboratory of BICE and Beijing Engineering Research Center of Efficient and Green Aerospace Propulsion Technology (LabASP-2021-08).

Table 1 .
Geometrical parameters of the 10 stage model pump.
Figure 1.Computational domain of the multistage centrifugal pump.