Investigation of radial force in a high-speed miniature pump

A high-speed miniature pump design is put forth for utilization in the aerospace sector, where the pump’s reliability holds critical importance under microgravity operational conditions. This study focuses on a dynamic bearing-equipped miniature centrifugal pump, chosen for its suitability in investigating radial forces. The prototype of the miniature pump is manufactured and tested by experiments at the rotational speed of 10,000r/min. A good agreement is observed between the numerical predictions and the experimental results. The radial force acting on different blade part has been analyzed for several flow rate conditions using the steady state CFD calculation. It has been found that the radial force exerted on the blade trailing edge exit is the main cause of the total radial force for the impeller. Moreover, using the unsteady CFD simulation, the evolution process of the radial force changing for both the direction and the magnitude with respect to the annular position due to the shaft rotation is discussed in details. The pressure pulsation is also monitored in impeller and volute casing. The results show that the amplitude of the unsteady radial force has a strong relationship with the time it takes for the blade trailing edge exit passing the volute tongue region, while the direction of the radial force hardly changes. The findings of this study contribute to facilitating the broader implementation of the high-speed miniature pump across diverse sectors of society.


Introduction
The utilization of high-speed miniature pumps has become widespread in various cutting-edge industries to facilitate the transport of fluids.The compact nature of these pumps makes them especially adept for deployment in the aerospace sector, where demands for small size and lightweight design are critical.Therefore, in scenarios involving microgravity operations, ensuring the high reliability and efficiency of these high-speed miniature pumps becomes an essential requirement.For the high-speed miniature pump with dynamic bearings, on one hand, the hydraulic force induced by the impeller may cause the severe fluid-induced vibration, which will extremely threaten the safety of the operation [1,2].On the other hand, as a passive suspended designed pump, there is a need to have a clear insight into the dynamic characteristics of the radial force, which will finally exerted on the dynamic bearing and determines the stability of the rotor suspension [3].
In the context of aerospace applications, the high-speed miniature pump exhibits an inherently low specific speed, resulting in its characteristic features of small flow rate and high pump head [4,5].Under the condition of this, the fluid structure inside the pump is far more complicated compared with the ordinary centrifugal pump.Furthermore, changing of the operating conditions as well as unsteady pressure pulsations may strongly affect the fluid force of the impeller, thus leading to the vibrations or even fatigue cracking of the pump [6][7][8].Therefore, it is of vital importance to investigate the performance of the radial force with respect to different working conditions and the dynamic characteristics of the radial force under the design condition.
For the dynamic characteristic of the pump, many factors such as the structure of the impeller blade, the design parameters of the volute casing and the intense rotor-stator interaction are closely related to the radial force, which will further result in the vibration and noise during the operation and even lead to the severe fatigue damage.In terms of the impeller parameters, numerous studies have been carried out by changing the blade profile, such as the blade outlet [9][10][11][12][13][14], wrap angle of the blade [15,16], distribution of blade thickness [17][18][19][20], and splitter blades [21,22], to investigate the relationship between the structure parameters with the flow-induced vibrations both numerically and experimentally.In terms of the influence of blade outlet shape, Nishi [9] et al. studied the unsteady radial force exerted on the centrifugal pump impeller relating to the blade outlet angle.The results show that with the increase of the blade outlet angle, the main frequency component of the pressure fluctuation will increase significantly, which is responsible for the increase of the radial force.Cui B [10,11] et al. investigated the influence of the trailing edge of straight blade on the pressure pulsation of centrifugal pump, and it is concluded that the pressure pulsation and vibration can be effectively reduced by properly cutting the trailing edge of blade.Al-Qutub [12] et al. verified through the experiments that the way of V-shaped cutting on the trailing edge of the blade can effectively alleviate the pressure pulsation of the centrifugal pump under off-design conditions.In terms of the influence of blade wrap angle, Tan L [15] and Tan M [16] studied the influence of different blade wrap angles on the hydraulic characteristics and dynamic characteristics of the single-blade centrifugal pump utilizing both experiments and numerical simulations.The results show that the unsteady radial force gradually decreased with the increase of blade wrap angle.Similarly, as for the blade thickness distribution, Zhang Y [17] found that the impeller blades with the design of airfoil thickness distribution could reduce the jet wake phenomenon at the impeller outlet to a certain extent, thus reducing the pressure pulsation inside the centrifugal pump.Yousefi H [18] and Li X [19] et al. studied the influence of the blade load distribution on internal pressure pulsation and radial force for a low specific speed centrifugal pump.The research results show that for two-dimensional cylindrical blades, front loaded blade design is helpful to reduce the amplitude of pressure fluctuation and the radial force.In addition, particularly for low specific speed centrifugal pumps, impellers with splitter blades are designed to improve the performance of the pump.Zeng G [21] investigated the influence of the circumferential position of the splitter blade on the vibration characteristics in the centrifugal pump both numerically and experimentally.Furthermore, the influence of the starting point of the leading edge and the deflection angle for the splitter blade have also been evaluated through numerical simulation.Ye L [22] et al. revealed that the proper design of the splitter blade structure could help to improve the jet wake structure at the outlet of the centrifugal pump, thus further reducing the pressure fluctuation amplitude in the centrifugal pump.
Apart from the blade profiles, other structural parameters has been proved to be closely related with the unsteady radial force, such as the structure of static parts such as volute and guide vane, the clearance between the impeller and volute and so on.Aiming at the influence of the volute profile, Zhang N [23,24] et al explored the influence of the inclined volute section on the pressure pulsation and compared it with the conventional volute by means of experiments.Furthermore, they also observed the vibration of the centrifugal pump with inclined volute when cavitation occurs.Alemi H [25,26] explored the influence of the volute clearance, the angular position of the volute tongue and the shape of the tongue on the radial force of the centrifugal pump through numerical simulation.
Studies have shown that shortening the impeller clearance, adopting a shorter volute tongue shape and slightly reducing the angular angle of the volute tongue within a certain range can help reduce the radial force of the centrifugal pump under the design condition.Kaupert K A [27] conducted an experimental study on the dynamic characteristics of a double volute high specific speed centrifugal pump, measured and summarized the characteristics of pressure fluctuations in the pump during operation.Yan H [28] and Yan P [29] et al. conducted numerical simulation studies on centrifugal pumps with different specific speeds for the double volute design, and revealed that the double volute design affects the velocity field, pressure field and radial force influence.Al-Qutub A [30] et al. explored the influence of the radial clearance between the impeller and the volute by experiments, and discussed the influence of the rotor-stator interaction on the pressure pulsation and vibration inside the centrifugal pump.González J [31] combined numerical simulations and experiments to deeply analyze the influence of the radial clearance between the impeller and the volute on the steady-state and transient radial force.Moreover, for the centrifugal pumps that are equipped with stationary guide vanes, Zhang M [32] carried out numerical simulations for centrifugal pumps with different numbers of guide vanes.The results show that when the number of guide vanes is the same as the number of blades, the radial force of the centrifugal pump reaches the smallest.However, at the same time, the magnitude the pressure pulsation in the pump will be significantly higher than in other cases.The numerical results of Takamine T [33] et al. showed that for a centrifugal pump with guide vanes, the rotational stall phenomenon in the guide vane area will aggravate the fluctuation of the radial force of the centrifugal pump.
From the study above, it is noted that many of the studies only points out the significant influence of the structure parameters that mainly contributes to the unsteady radial force and pressure pulsation, with little regards to the force component exerted on different part of the impeller under different conditions.For the further improvement of the high-speed miniature pump with dynamic bearing, it is of vital importance to analyze the dynamic characteristics of the unsteady radial force from the view of force components exerted on the different parts on the impeller.
The objective of this study is to introduce a dynamic bearing-equipped high-speed miniature pump tailored for aerospace applications.The miniature pump's prototype was constructed and subjected to experimental testing, followed by a meticulous comparison between the experimental outcomes and numerical simulation results.Drawing upon these findings, an analysis of the internal flow characteristics and radial forces of the high-speed miniature pump is subsequently conducted using steady-state numerical simulations.The effects of different parts contributing to the total radial force under different working conditions and their relationship between the internal flows are investigated.Furthermore, for the dynamic characteristics, the pressure pulsation and the unsteady radial force exerted on different parts with respect to the relative position of the impeller are also investigated.

Test pump
The study mainly focuses on an aerospace-oriented high-speed miniature pump equipped with dynamic bearings.The visual representation of this high-speed miniature pump is depicted in Figure 1.
A key feature of this compact pump lies in its incorporation of two dynamic bearings within the design, which serve to counterbalance the impeller's radial forces.This innovative configuration ensures that the rotor remains free from contact with the stationary pump components.Concurrently, this design significantly diminishes the likelihood of mechanical wear on the pump shaft, consequently leading to a substantial extension in the operational lifespan of the high-speed miniature pump.

Computation domain
In order to enhance the hydraulic efficiency of the high-speed miniature pump, a detailed numerical analysis is conducted to scrutinize the internal flow dynamics.Various geometric and hydraulic parameters under design conditions are provided in Table 1.The computational domain encompasses the entire flow pathway, encompassing the pump inlet, impeller, volute casing, and pump outlet, as illustrated in Figure 2. Notably, the pump is of a centrifugal type and possesses a specific speed of 32.8 r/minm 3 /sm, calculated using the specific speed formula .For the simulation, the inlet of the suction pipe serves as the entrance plane within the computational domain, while the outlet of the discharge pipe functions as the exit plane within the computational domain.

Mesh generation and independence test
A structural hexahedral mesh for the entire domain is generated using the commercial software ANSYS-ICEM, with careful consideration to maintain y+ values within the range of 1.The structural hexahedral mesh for the impeller is displayed in Figure 3, revealing a refinement of grids near the physical walls, particularly in the vicinity of the impeller blades.The mesh independence test is conducted, involving six distinct mesh densities, to compute the steady flow at the design point's flow rate.As depicted in Figure 4, minor fluctuations in head and efficiency are observed with increasing grid count.After reaching approximately 4.88 million grids, the head and efficiency stabilize.Striking a balance between computational time and accuracy, a total of 4.88 million grids is selected as it proves sufficient for simulating the internal flow dynamics within the high-speed miniature pump, while maintaining acceptable computation time.As such, the 5th mesh generation scheme is chosen as the computational mesh for this study.

Numerical methods
Within the pump, the flow is treated as incompressible.The continuity, momentum, and energy equations are expressed in Cartesian coordinates as follows [10]: where ui (i, j=1,2,3) represents the velocity vector component, xi (i, j=1,2,3) stands for the coordinate component, p stands for static pressure, U is the fluid density, and μ is the dynamic viscosity.
The Reynolds-averaged Navier-Stokes equations are solved using the ANSYS-CFX commercial CFD code.At the same time, the k-Z SST turbulence model is selected to capture the complicated internal flow features.Regarding boundary conditions, the total pressure is specified at the inlet plane, while the outlet plane of the pump is defined by the mass flow rate.Furthermore, the solid surfaces within the flow passage are designated as no-slip and characterized by a smooth wall condition.
The impeller's flow passage is established within the rotational coordinate system, while other components such as the volute casing, inlet pipe, and outlet pipe are situated within the stationary coordinate system.The interface linking the rotational and stationary coordinate systems employs the sliding technique.
For steady simulations, the rotor-stator interface employs the mixing-plane model.A time step of 30/(Sn) is chosen for iterations based on the balance between computational accuracy and cost.
To obtain the dynamics of radial force, an unsteady flow computation is conducted for the highspeed miniature pump.This unsteady simulation employs the transient rotor-stator approach, with a comparatively reduced time step corresponding to one degree of impeller rotation per time step.This choice of time step size was made to strike a balance between computer efficiency and obtaining the necessary temporal precision to record unsteady flow phenomena.While advection utilizes the highresolution scheme, the second-order backward Euler scheme is employed for other transient terms.
In both steady and transient simulations, the root mean square (RMS) residuals are maintained below 10 -6 .

Pump performance
The hydraulic performance of the miniature pump is assessed using the test rig depicted in Figure 5.This setup incorporates two static pressure sensors with the uncertainty of ± 0.1% at both the inlet and outlet of the pump, for measuring the pump's pressure values.An electromagnetic flow-meter, offering uncertainty of ± 0.2%, is employed to gauge the flow rate.The entire testing setup encompasses the pump under examination, the pipeline, a data acquisition system, and the aforementioned gauges.Throughout the experiment, water at 25°C serves as the testing fluid.The tested pump operates at a rotational speed of 10,000 r/min.The experimental data regarding hydraulic performance are compared with the results obtained through numerical calculations, as illustrated in Figure 6, where Hexp is the experimentally measured pump head and Hcal represents the pump head numerically predicted.
As depicted in Figure 6, there is a noteworthy alignment between the numerical results and the experimental data, both under the design condition and at higher flow rates.Specifically, at the design condition, the relative error between the pump head calculated through numerical simulation and the value determined experimentally stands at 1.55%.The radial forces and their directions on different blade parts and entire impeller under the design flow rate are shown in Figure 8, while the total radial force of different blade parts and the entire radial force of the impeller are plotted in Figure 9. From Figure 9, it is noted that under the design flow rate condition, the total radial forces acting on the pressure sides and the suction sides of four blades of the impeller are small, and that acting on trailing edge exit of four blades is much larger.Thus, the radial force of the impeller is mainly due to that acting on the blade trailing edge exit, indicating the necessity to optimize the geometry of blade trailing edge.
To investigate the relation between the radial force and flow field, the internal flow pattern is analyzed at the design condition, as shown in Figure 10. Figure 10(a) presents the velocity distribution at the mid-span section of the impeller.The dimensionless velocity is defined by Equation (3).
( 3 ) where v is the velocity and u2 is the peripheral velocity at the impeller exit.
Figure 10(b) presents the static pressure distribution at the mid-span section of the impeller.The dimensionless pressure is defined by Equation (4).
where p is the static pressure, and U is the density of the fluid.
Steady state distribution of vorticity at z direction on mid-span section of the impeller is displayed in Figure 10(c).The flow field shown in Figure 10 is asymmetric though the impeller blades are periodically located around the pump shaft.It is clear that the asymmetric pressure distribution in the pump passage results in the radial forces acting on each blade parts and the entire impeller.The asymmetric distributions of velocity and vorticity are closely related with the unique distribution of static pressure, which is relatively smaller near the TEEs of BLD1 and BLD2.Because the integration of static pressure on each blade part contributes to the radial force, these flow features indicate that the main radial force for the impeller is due to low pressure at TEEs of BLD1 and BLD2.

3.2.2
The radial force on different blade part at several flow conditions.Figure 11 shows the radial force for different parts under the four flow rates.It is clear that the radial force on the impeller gradually decreases from the low flow-rate of 0.5Qd to the large flow-rate of 1.2Qd, and its direction move closer to the volute tongue.Besides, the radial force acting on the TEE is the most important among the blade parts, and especially that is larger than the radial force on entire impeller at 1.2Qd.
In Figure 12, the radial force on different blades is also plotted separately under different flow conditions.It is noted that the direction of radial force acting on each blade changes with the flow-rate, and its asymmetric distribution becomes more obvious from large flow-rate to small flow-rate.The asymmetric feature of radial force is resulted from the asymmetric distributions of velocity and pressure in the flow passage due to the volute casing geometry as shown in Figures 13 and 14. (a It can be seen in Figure 15 that for all flow rate conditions, there is always a high value vorticity region locating near the trailing edge of BLD2, which partially blocks the flow in the volute casing.The blockage at this area makes the fluid in the volute speed up and the pressure reduce accordingly, which finally leads to the unbalanced force on the blade trailing edge exit.Based on the results, the radial force of the impeller seems to be derived from both the asymmetric flow pattern in blade-to-blade channels and the high-speed low-pressure region near the blade trailing edge around the volute tongue under small flow rate condition.As the flow rate increases, the radial force mainly originate from the high-speed low-pressure region in the volute casing.where (x,y,z,t) is the averaged static pressure at the monitoring point, and U2 is the peripheral velocity at the impeller exit.

g g p
The relative distribution of averaged pressure coefficient at different monitoring points is illustrated in Figure 17.It can be seen that the averaged pressure at the impeller exit little changes along the peripheral direction, while for the volute casing, the averaged pressure at the points K2, K3, K4, much change compared with that at other monitoring points.This agrees with the result derived in the former parts that there is a low-pressure region downstream the tongue (with θ in the region from 35 to 135 approximately).Moreover, this area of low-pressure region is responsible for the unbalanced radial force exerted on the impeller blade, which finally has great impact on the radial force of pump impeller.Apart from the averaged pressure distribution of the pump, it is also necessary to quantitatively evaluate the amplitude of the pressure fluctuation.Thus, the standard deviation of pressure fluctuations is introduced.The pressure fluctuation coefficient Cpsd is calculated by the Equation(6).
where p(x,y,z,ti) represents the instantaneous static pressure at time instant ti, and (x,y,z,t) is the averaged static pressure.
Figure 18 shows the relative distribution of standard deviation of pressure fluctuation coefficient at the monitoring points.As for the volute casing, the pressure fluctuation has a relatively regular distribution compared with the monitoring points in the impeller part.In terms of the impeller part, it can be seen that the pressure fluctuation at the outlet of the impeller changes greatly corresponding to the position of the monitor points.The region near to the volute tongue can be detected a larger pressure fluctuation, which is a result of unsteady rotor-stator interaction between the impeller and the volute.Therefore, the region that has lower averaged pressure in the volute casing share a higher pressure fluctuation at impeller outlet (with θ in the region from 35 to 135 approximately), which signs that there will be a larger radial force fluctuation exerted on the blade trailing edge exit.
Moreover, the phenomenon of parity alternation for the pressure fluctuation can be observed both in the volute casing and in the impeller simultaneously.For the monitoring points that share the same  The pressure coefficient Cp is defined to normalize the pressure fluctuations in transient state using the Equation (7).

U p x y z t p x y z t C U (7)
To delve deeper into the analysis of unsteady pressure pulsations within the pump, frequency domain signals of pressure fluctuations are examined using the Fast Fourier Transform (FFT) method.Figure 19a shows the pressure spectra at I2, I4, I6 and I8, and Figure 19b shows the pressure spectra at K2, K4, K6 and K8 in volute casing.Note that the impeller rotating frequency is marked as fn.Examining the frequency spectra of pressure coefficient fluctuations depicted in Figure 12, it becomes evident that the pressure fluctuations at the impeller's monitoring point are notably pronounced at both the impeller's rotational frequency and its higher harmonics.Conversely, for monitoring points located within the volute casing, the primary peaks of pressure fluctuations correspond to the blade passing frequency.

Radial forces in transient state.
To improve the reliability of the high-speed miniature pump with dynamic bearing during its operation, it is necessary to investigate the dynamic characteristics of  Figure 21 shows the radial force fluctuations on different blade part in frequency domain, where "BLD_all" represents for the total radial force of the whole impeller, "all suction side" for the radial force on four suction sides, "PS_all" for the radial force on four pressure sides, and "TEE_all" for the radial force on four blade trailing edge exit.As shown in Figure 21, significant peaks of radial force fluctuations on different blade part occur at the blade passing frequency i.e. 4fn and its higher harmonics.Particularly, for the resultant radial forces on all pressure sides, high amplitude can be detected at 4fn and 8fn.Taking place of the blade passing frequency, two times of impeller rotating frequency i.e. 2fn becomes the dominant frequency for the resultant radial forces on all suction sides, and the second highest amplitude of force for the resultant radial forces on all suction sides is 6fn, which makes it distinguish largely from the other parts.
Figure 22 shows the radial force distribution on single blade and the impeller with the rotation of the impeller.Note that the radial force of single blade here is the transient radial force of BLD1, and the initial position is specified as 0 degree.Figure 22a shows that as the impeller rotates at 35 degrees, there is a drop of radial force on single blade, corresponding that the BLD1 reaches the region near the volute tongue.As mentioned above, the rotor-stator interaction around that region is closely related to the drop of pressure on the blade, thus directly leading to the sharp decrease of the radial force on single blade.Meanwhile, the other three blades that are far from the volute tongue have almost the same value.Therefore, the sharp decrease of the radial force on the blade that near to the volute tongue greatly strike the balance among the four blades.The total radial forces distribution revealed in Figure 22b proves that every time when a single blade passes the volute tongue region, there will be a dramatic increase of the total radial force.
The evolution of the radial force can be concluded by the following steps.Firstly, one single blade gradually approaching the volute tongue region, accompanying the sudden decrease of the radial force on that blade.Meanwhile, the total radial force of the whole impeller gradually increases due to the unbalanced force.At the time, the trailing edge exit of the blade has the shortest distance with the volute tongue, which signs that the single blade reaches the volute tongue region, the radial force on that blade achieves its minimum value and the total radial force of the whole impeller reaches its maximum.It is also the time that the whole impeller has the most disequilibrium state.In the following stage, as the impeller continue to rotate and that single blade turns away from the volute tongue region, the radial force on that blade recovers to the normal level soon.With the radial forces on four blades back into balance, the total radial force drops dramatically along with the recovery of that single blade radial force.Finally, as the time goes on, the next blade becomes the one that is the most close to the volute tongue region.Moreover, the state of radial force back to the first stage again that ultimately form a repeated cycle.
To have a more detailed investigation of the radial force revolution, comparison of radial force distribution on different parts of single blade has been made in the view of rotation frame, as illustrated in Figure 23. Figure 23 reveals that as the single blade gradually approaches the volute tongue region, the force on pressure side is firstly influenced.With the distance between the blade and the volute tongue becomes closer, the radial force on pressure side starts to decrease, thus leading to reducing of the radial force on the blade.Meanwhile, as the single blade reaches the volute tongue region, a violent impact on radial force on the blade trailing edge exit and the suction side can be detected.These effects work together contributes to the radial force on the single blade.
Similarly, the total radial force on entire impeller is also a combination of forces on four blades.Figure 24 shows the comparison of total radial forces distribution on different blade parts with the rotation of the impeller in the view rotating frame.It can be seen from Figure 24 that, every time when there is a single blade is about to reaching the volute tongue region, the total radial force of the whole impeller will increase dramatically.The same tendency can be observed on the pressure side and the blade trailing edge exit.Differently, for one period of blade passing, there exists two peaks of radial force on blade trailing edge exit.The time point when the blade trailing edge exit has the closest distance with the volute tongue shows the maximum value of radial force on all blade trailing edge exit side.It is because that every time the blade trailing edge exit passes by the volute tongue, there will be a sharp decrease of radial force on that, which acutely break the balance between the other three blade trailing edge exit sides.Furthermore, there is a distinctive phenomenon that not every period of blade passing will leads to the increase of the radial force, which is different from the characteristics shown on the blade trailing edge exit and the pressure side.This is corresponding with the results discussed above that dominant frequency for the resultant radial forces on all suction sides is 2fn instead of 4fn like the resultant radial forces on blade trailing edge exit and the pressure side.The reason of this needs to be further investigated in the future study.
Furthermore, it has to be noted that the radial force on the single blade is the composition of radial force vectors, which combines both value and the directions.The radial forces fluctuations above only discuss the absolute value of the radial force, which is only one aspect of influence factors.The direction of the radial forces on different parts will be further discussed in the following paragraphs.
As the radial forces will eventually act on the dynamic bearing, to have a better understanding of the direction of the radial forces on different parts, the forces are transformed into stationary frame in the following study.Figure 25 shows the radial forces distribution on single blade with the rotation of the impeller in the view of stationary frame.Figure 25 reveals that, as the single blade passes by the volute tongue region, both the value and the direction of the force magnitude are affected by the rotor-stator interaction.At the initial state when the starting position of the monitored blade is at 0 degree, the total radial force on that blade is in the third quadrant.As the impeller rotates clockwise, the direction of radial force is changing in the clockwise direction as well.There exists an inflection point in the direction of radial force as the single blade passes by the volute tongue region, from which a low-pressure region has been found in the former discussion.To demonstrate the evolution of the radial force direction in detail, radial forces in x and y direction on single blade with the rotation of the impeller are plotted in Figure 26.As depicted in Figure 26, it can be seen that for the radial force in x direction on the single blade, the fluctuation of it is like a regular sinusoidal function, which means the change of the blade position hardly has any influence on the radial force in x direction.In contrast, the radial force in y direction is extremely sensitive to the rotation of the impeller, which shows an obvious inflection in connection with the relative position between the single blade and the volute tongue.The three turning point of the radial force in y direction is figured out in the picture.It can be seen that the angle of the T1 is when the monitored single blade rotate to the region of the volute tongue.At this time, the trailing edge exit of the blade begin to sweeping pass the volute tongue.As the impeller rotates to the angle of T2, the trailing edge exit of the blade has fully passed the volute tongue and begin to move away from the rotor-stator interaction region.Furthermore, as the single blade rotates to the position that far from the volute tongue, the radial force in y direction becomes regular again like that of radial force in x direction.Lastly, there is also a small deflection of the radial force in y direction at the angle of T3.It is the time that the former blade that is 90 degree ahead of the monitored single blade leaves the volute tongue, which mean the single blade will rotate back again to the rotor-stator interaction region for a next period.Therefore, it can be concluded that the rotor-stator interaction mainly affects the radial force in y direction on a single blade.
To further make out the relation between the direction of the redial force and the relative position of the blade and the volute tongue, radial forces distribution on single blade with the rotation of the impeller is plotted in Figure 27.As depicted in Figure 27, when the monitored individual blade reaches the angle denoted as T1, there is an abrupt alteration both in the magnitude and direction of the radial force exerted on that particular blade.As previously mentioned, T1 corresponds to the point at which the trailing edge exit of the monitored blade commences sweeping past the volute tongue.Given that a finite time is required for the trailing edge exit of the blade to completely clear the volute tongue, this pattern of change persists until the impeller rotates to the angle of T2, marking a second abrupt shift in the radial force.
Furthermore, in conjunction with the radial force trend at the angle T3, it becomes apparent that when the trailing edge exit of a blade departs from the volute tongue's vicinity, a significant influence ensues not only on the radial force of the blade closest to the volute tongue but also on that of the blade positioned 90 degrees behind the former.These results collectively highlight the intrinsic relationship between the rotation of the blade's trailing edge past the volute tongue and the fluctuation in radial force on an individual blade.This dynamic, in turn, influences the overall radial force experienced by the entire impeller.
The radial force on single blade in the stationary frame is shown in Figure 28.It can be seen that no matter how the impeller rotates, the direction of the total radial force nearly remains unchanged that always points to the direction in the second quadrant.In other words, the rotation of impeller only affects the amplitude of the total radial force, hardly changing the direction of the total radial force.The analysis in the former part may explain why the direction of the total radial force points to the upper side of the impeller, the same direction of the outlet of the volute casing.As mentioned before, when a single blade passes the volute tongue region, there is a sharp diversion of the radial force on that blade.The force pointing down side of the impeller that is supposed to make the balance with radial forces on other three blades is greatly diminished due to the rotor-stator interaction.Therefore, this kind of unbalance shows up as the total radial force pointing upside.Total radial forces in x and y direction distribution with the rotation of the impeller are illustrated in Figure 29, which demonstrates the evolution of the total radial force direction in detail.As can be seen from Figure 29, during one period of impeller rotation, there are four cycles for the fluctuation of the total radial force both in x direction and in y direction.Taking a careful look at the angle when the radial force in y direction, we can see that when one single blade reaches the volute tongue region, the radial force in y direction ascends to the highest peak, while there is a slight fluctuation observed for the radial force in x direction.As the trailing edge exit of one single blade gradually rotates passing by the volute region, both the value of radial force in x direction and in y direction reduces accordingly.At the time when the single blade is about to leave the volute tongue, the radial force in x direction and in y direction reaches their minimum respectively.During one period of impeller rotation, there will be four blades pass by the volute tongue region.Therefore, four cycle of fluctuation can be observed for one impeller rotation period, and the phase angle between each cycle is 90 degree.
To have a clearer insight into the relation between impeller rotation angle and the radial force on single blade, the radial force distribution is plotted with the rotation of the impeller in Figure 30.It can be seen that when the single blade leaves the volute tongue region, the total radial force decreases to its minimum simultaneously.As the next single blade gradually approaching the volute tongue, the radial force slowly rises with a slight change in the direction.As long as one single blade reaches the volute tongue region, the total radial force obtains the maximum and then drops quickly.It is also indicated that when the blade trailing edge exit starts to pass the volute tongue, the radial force reaches its maximum.Meanwhile, when the blade trailing edge exit leaves the volute tongue, the radial force reaches its minimum.During the evolution, the radial force direction hardly changes.

Conclusions
In this study, the investigation of the radial force has been carried out for a high-speed miniature pump.Based on the performance predicted by numerical simulation, the prototype of the miniature pump has been manufactured and tested by experiments.The experimentally measured hydraulic performance near the design condition agrees fairly well with the numerical results.Moreover, the contribution of the radial force acting on different blade parts has been discussed based on the steady and the unsteady simulations.For this study, the following conclusions can be drawn: For steady characteristics, the radial force exerted on the blade trailing edge exit is the dominant component for entire impeller.At the operation of relatively low flow-rate, the asymmetric distribution inside the impeller contributes to the unbalanced radial force.As the flow rate increases, especially near the design condition, the rotor-stator interaction effect of the blade trailing edge exit passing the volute tongue becomes vital for the radial force.A high-speed low-pressure area can be observed as the blade trailing edge exit sweeping the volute tongue, which is the cause of the unbalanced radial force; For the dynamic characteristic, the evolution process of the radial force is strongly associated with the relative position between the blade and volute tongue due to the impeller rotation.The results show that the amplitude of the unsteady radial force has a strong relationship with the time it takes for the blade trailing edge exit passing the volute tongue region, while the direction of the radial force is insensitive to impeller rotation.

Figure 2 .
Figure 2. Computation domain of the pump.

Figure 5 .
Figure 5. Schematic diagram of the test rig.

Figure 6 .
Figure 6.Hydraulic performance of the test pump.

3. 2 .
Radial forces predicted by steady simulation 3.2.1.The radial force on different blade part at design condition.For analysis, four blades of the impeller are marked as BLD1~BLD4, and a single blade is marked by three parts, namely pressure side (PS), suction side (SS) and trailing edge exit (TEE) as shown in Figure7.

Figure 7 .
Figure 7. Blades and blade parts of the impeller.

FFigure 8 .
Figure 8. Radial force acting on different blade parts and entire impeller.

Figure 9 .
Figure 9. Contribution of different blade parts to total radial force.

Figure 10 .
Figure 10.Flow on mid-span section under the design condition.

Figure 14 .
Figure 14.Pressure distribution on mid-span section under different flow rates.

Figure 15 .
Figure 15.Vorticity in z direction distribution on mid-span section under different flow rates.

3. 3 . 1 .
Pressure pulsations.To analyze the pressure fluctuation, several monitoring points are set on the mid-span section as shown in Figure16.Monitoring points K1~K8 locate along the volute casing wall with the angle difference of 45 degrees, and monitoring points I1~I8 locate at the impeller exit.

Figure 16 .
Figure 16.Monitoring points in the pump.

Figure 17 .
Figure 17.Averaged pressure coefficient at different monitoring points.
International Conference on Fluid Machinery (AICFM 17 2023) Journal of Physics: Conference Series 2707 (2024) 012040 IOP Publishing doi:10.1088/1742-6596/2707/1/01204015degrees, where the points at impeller outlet has a higher-pressure fluctuation, the corresponding in the volute casing will display a lower pressure fluctuation, and the opposite versa.

Figure 18 .
Figure 18.Standard deviation of pressure coefficient fluctuations at different monitoring points.

Figure 19 .
Figure 19.Frequency spectra of pressure coefficient fluctuations at different monitoring points.
the radial force on the impeller.Figure20shows the radial force fluctuations on different blade part in frequency domain.

Figure 20 .
Figure 20.Radial force fluctuations of different blade parts.As illustrated in Figure20, the peaks of radial forces on different blade parts correlate with the impeller rotating frequency and its higher harmonics, and the amplitude of radial force fluctuation acting on pressure side is the highest.It is found that the radial force fluctuation amplitude on the suction side at blade passing frequency decreases nearly zero.On the contrary, the radial force fluctuation amplitude on the blade trailing edge exit increases to the second highest among all the frequencies.Figure21shows the radial force fluctuations on different blade part in frequency domain, where "BLD_all" represents for the total radial force of the whole impeller, "all suction side" for the radial force on four suction sides, "PS_all" for the radial force on four pressure sides, and "TEE_all" for the radial force on four blade trailing edge exit.

Figure 21 .
Figure 21.Radial force fluctuations on different blade part in frequency domain.

Figure 22 .
Figure 22.Radial force distribution on single blade and entire impeller.

Figure 23 .
Figure 23.Radial force distribution on different blade parts in the view of rotating frame.

Figure 24 .
Figure 24.Comparison of total radial forces distribution on different parts with the rotation of the impeller in the view of rotating frame.

Figure 25 .
Figure 25.Radial force acting on single blade in the view of stationary frame.

Figure 26 .
Figure 26.Radial forces in x direction and y direction distribution on single blade with the rotation of the impeller.

Figure 27 .
Figure 27.Radial force distribution on single blade with the rotation of the impeller.

Figure 28 .
Figure 28.Radial force of all blades in the stationary frame.

Figure 29 .
Figure 29.Total radial forces in x and y direction distribution with the rotation of the impeller.

Figure 30 .
Figure 30.Radial force on all blades with impeller rotation.

Table 1 .
Parameters for the high-speed miniature pump.