Effects of Radial Clearance Between Rotor and Casing on Flow Characteristics in a Centrifugal Pump

Abstract

The electrification of the automotive industry and the lightweighting of aerospace equipment demand high-efficiency centrifugal pumps for compact spaces. A novel centrifugal pump incorporates an integrated impeller-motor rotor design, achieving a more compact footprint and higher power density. However, research is scarce on the radial clearance between the rotor and casing. This study presents a comprehensive investigation of the internal flow dynamics, combining numerical simulations with experimental validation. A significant reduction in fluctuation amplitude for pump efficiency, head coefficient, and frictional loss rate occurs when the clearance ranges from 1.0 to 1.5 mm. Within clearances of 0.75 to 1.5 mm, complex vortex systems emerge in the radial clearance, inducing diverse circumferential high-speed zones. Pressure fluctuations within the radial clearance are predominantly governed by the blade passing frequency. At a clearance of 1.5 mm, the rotational harmonic amplitude at monitoring points exceeds the blade passing frequency amplitude by a factor of 1.9, while the average pressure fluctuation intensity at other points increases significantly by 36.9%. An optimal clearance of 1.25 mm achieves a balance between flow characteristics and energy consumption. This research provides practical insights for optimizing pump energy performance and operational stability.

1. Introduction

The electrification of the automotive industry and the lightweighting of aerospace equipment are accelerating. Consequently, it is critical to design high-efficiency and high-power electronic coolant pumps for compact spaces [1,2]. Replacing traditional radial flux motors with axial flux motors can significantly enhance power density within a fixed volume [3,4]. This approach offers distinct advantages in weight reduction and miniaturization [5,6]. Figure 1 illustrates an innovative centrifugal pump integrated with an axial flux canned motor. In this configuration, the stator and rotor are arranged in parallel along the axial direction. Furthermore, the impeller is integrated with the rotor as a single unit. Here, δ_r denotes the specific radial clearance between the rotor and the casing.

Currently, research on pumps with axial flux motors (AFMs) focuses primarily on motor components. Ai et al. [7] analyzed the variation in axial and radial forces with air gap and rotational speed for a pump with an AFM used in cryogenic submerged applications. Since an axial flux permanent magnet synchronous motor (PMSM) provides high power density and high torque at low speeds, Hoffstaedt et al. [8,9] integrated a contra-rotating reversible pump-turbine with an AFM. This design achieves low-head pumped hydro storage in coastal and shallow-water regions. Through experimental comparisons of two PMSMs, Eker et al. [10] found that the axial flux configuration effectively enhances the power density and efficiency of pump systems. Yim et al. [11] retrofitted a liquid hydrogen centrifugal pump with an AFM, resulting in significantly improved torque density and mechanical stability. For a ventricular assist pump with an AFM, Karabulut et al. [12,13,14,15] investigated how stator materials and structural layouts affect motor performance. They also conducted thermal analyses on its rotor and stator. Zhao et al. [16] introduced hybrid excitation technology into an axial flux switching motor for a pump-turbine. This allows the system to control speed and power across multiple operating conditions. Zhang et al. [17] compared a pump with an AFM to a traditional electronic coolant pump. Their results showed that the AFM-based pump exhibits superior electromagnetic torque performance under the same thermal load. Choi et al. [18,19] replaced the stator of an AFM in an electronic water pump with a PCB stator. They subsequently analyzed its efficiency, torque performance, and thermal stability. For a pump with a PCB-stator AFM, Youn et al. [20] proposed a non-adjacent end-winding structure to ensure electrical symmetry. Ma et al. [21] studied an ultracompact micro centrifugal pump with an integrated printed AFM, optimizing both its hydraulic and heat dissipation performance. Song et al. [22] converted a radial flux motor for a pump into an AFM of equivalent volume and optimized its torque ripple and efficiency. Nahin et al. [23] integrated a hydrostatic radial piston pump with an AFM to analyze its volumetric, mechanical, and overall efficiency. Furthermore, Gao et al. [24] proposed a blood pump with an axial flux bearingless hysteresis motor and tested its torque performance. Despite these studies, there is almost no analysis on how structural parameters influence the energy performance and internal flow fields of pumps with AFMs. In particular, the impact of motor-specific parameters remains insufficiently explored.

The design and optimization of various clearance structures in pumps directly affect their internal flow characteristics and energy performance. Therefore, it is essential to study the influence of clearance parameters on pumping characteristics. Carvalho et al. [25] investigated the effects of medium physical properties and clearance on the leakage loss and efficiency of a sucker rod pump. They found that the clearance had the most significant impact. Ranawat et al. [26] analyzed how wear ring clearance increases across two orders of magnitude, along with different eccentricity levels, affect the performance of a centrifugal pump. Lei et al. [27] evaluated the impact of impeller inlet wear ring clearance on the external characteristics, internal flow field, and energy loss of a double-suction centrifugal pump. Si et al. [28] researched the influence of front and rear wear ring clearance sizes on the hydraulic performance and operational stability of a low-specific-speed multistage centrifugal pump. They also analyzed the internal flow loss mechanism and performance evolution. Lin et al. [29] analyzed the effect of seal ring clearance on the performance and energy consumption of a submersible sewage pump. Han et al. [30] modified the rear wear ring of a canned pump. They analyzed the influence of the rear wear ring clearance width on axial force, leakage, and hydraulic performance. Using a combination of computational fluid dynamics (CFD) and the acoustic finite element method, Zhou et al. [31] analyzed the effects of rear wear ring clearance width on a centrifugal pump. Their study covered energy performance, flow field features, and internal noise. Zhang et al. [32] studied how various clearances in the rotor structure affect the geometric characteristics and performance of a vacuum pump. For a gear pump, Zhan et al. [33] analyzed the effects of different radial and end-face clearances on internal leakage and viscous friction losses. Similarly, Liu et al. [34] researched the influence of radial and axial clearances on the efficiency of a gear pump. Zhai et al. [35] conducted a simulation on a miniature gear pump with a canned motor to analyze the impact of clearance flow structures on flow and thermodynamic characteristics. Liang et al. [36] investigated the effects of seven different working clearances on a hydrogen circulation pump. They examined the internal flow field, pressure pulsations, flow pulsations, and radial forces. In a regenerative hydrogen pump, the clearance between the impeller disk and the casing affects performance and disk forces. Yang et al. [37] analyzed the influence of this clearance width on the hydraulic performance, axial force, and disk friction power of the pump. The secondary flow path is crucial in the design of centrifugal blood pumps. Wu et al. [38] used Large Eddy Simulation to compare the flow fields of small and large clearance sizes and analyzed their impact on pump performance. Zhu et al. [39] analyzed the influence of the secondary flow path clearance on the performance and blood compatibility of a centrifugal blood pump. Zhou et al. [40] analyzed a vortex pump under different axial clearances through internal flow field studies. They discussed the influence of clearance size on the external characteristics and clearance flow features. Liu et al. [41] and Chen et al. [42] analyzed the influence of stator-rotor clearance variations on the internal flow patterns within full tubular pumps.

Regarding the axial flux canned motor pump studied in this manuscript, the impeller and rotor feature an integrated structure. Consequently, the fluid within the radial clearance between the rotor and the casing interacts directly with the main flow. However, previous research only analyzed the axial gap between the stator and rotor [43]. The influence of radial clearance on the hydraulic performance and internal flow characteristics of the pump remains unclear.

Section 2 outlines the key parameters of the centrifugal pump with an axial-flux canned motor and details the numerical simulation methodology. Section 3 describes the experimental setup and presents the validation of the numerical method against the experimental results. Then, Section 4 provides a comprehensive discussion on the effects of six radial clearances ranging from 0.25 to 1.5 mm on the hydraulic performance and internal flow characteristics of the axial flux canned motor pump, specifically analyzing the pressure, velocity, vorticity, entropy production, and pressure fluctuations.

2. Numerical Simulation

2.1. Pump Parameters

As shown in Figure 2, the focus of this study is a centrifugal pump driven by an axial-flux canned motor. The pump operates at a design flow rate (Q_d) of 9 m³∙h⁻¹ and a rated speed of 5200 r∙min⁻¹.

The pump primarily consists of an impeller and a volute.

Table 1. Main geometric parameters.

Component	Parameter	Value
Impeller	blade inlet diameter D1	32 mm
	blade outlet diameter D2	53 mm
	blade outlet width b2	7.8 mm
	number of blades z	7
	rotor diameter Dr	72.5 mm
	rotor axial length Lr	11 mm
Volute	inlet width b3	12 mm
	outlet diameter Dd	32 mm

summarizes their main geometric parameters. The suction pipe diameter equals the impeller inlet diameter. Similarly, the discharge pipe diameter matches the volute outlet diameter. Each pipe extension is five times its respective diameter in length.

2.2. Mesh Generation

Figure 3 illustrates the computational domain of the axial-flux canned motor pump. It consists of six parts: the suction pipe, the front chamber, the impeller, the volute, the motor clearance, and the discharge pipe.

ANSYS ICEM CFD 2020 R2 is used to generate hexahedral meshes for the pipes, chamber, clearance, and impeller. Block-structured topologies, such as O-grid, C-grid, and Y-grid, are employed for the flow passages. These structured grids exhibit high quality. They all feature a Jacobian determinant above 0.5 and minimum angles exceeding 18° [44]. However, the volute presents a significant meshing challenge. Its complex 3D geometry and sharp features near the tongue make a fully hexahedral mesh difficult to achieve. Therefore, a hybrid meshing strategy is adopted for this component. A hybrid grid is generated using ANSYS Meshing. It consists of both tetrahedral and hexahedral elements. Local mesh refinement is applied in critical areas. The resulting hybrid mesh exhibits an orthogonal quality above 0.7. Its aspect ratio remains below 3. The average y⁺ value is 2.6 for the impeller domain and 20 for the volute domain, while those of the other domains are all below 28. Overall, the mesh quality satisfies the requirements for a high-fidelity analysis of the internal flow field [45].

2.3. Computational Setup

CFD simulations in this study are performed using ANSYS CFX. The Unsteady Reynolds-Averaged Navier–Stokes (URANS) method is employed for turbulence modeling. Equations (1) and (2) present the governing equations.

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_i\right)=0$$

(1)

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho u_i\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho u_i{u}_j\right)=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[{\mu }_{eff}\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)\right]+f_i$$

(2)

where ρ is the density, t is the time, u is the velocity, x is the coordinate, p is the pressure, μ_eff is the effective viscosity, f_i is the external force. The Shear Stress Transport k-ω model is utilized alongside the URANS equations to close the system. This model accurately captures the primary flow characteristics.

Simulations are performed in an absolute pressure framework. The reference pressure is 0 Pa. A total pressure of 1 atm is specified at the inlet. A mass flow rate corresponding to the operating condition is set at the outlet. The impeller is defined as a rotating domain. All other domains are stationary. Solid surfaces are modeled as no-slip smooth walls. The transient simulation employs the transient rotor-stator interface. The working medium is water at 25 °C, with a density of 997 kg∙m⁻³ and a dynamic viscosity of 8.899 × 10⁻⁴ kg∙m⁻¹∙s⁻¹. The time step corresponds to 0.2 degrees of impeller rotation. This setting resolves unsteady flow phenomena with high fidelity. The global average Courant-Friedrichs-Lewy number is maintained below 0.4 to ensure numerical stability. The convergence criterion is met when the Root Mean Square (RMS) residuals for all equations drop below 10⁻⁵.

2.4. Energy Performance of Simulations

In the numerical simulation, the head of the centrifugal pump is defined as follows:

$$H={\displaystyle \frac{p_{2,tot}-p_{1,tot}}{\rho g}}$$

(3)

where p_1,tot is the total pressure at the pump inlet and p_2,tot is the total pressure at the pump outlet.

The head is non-dimensionalized to eliminate the effects of geometric dimensions and rotational speed. This facilitates performance comparisons across different operating conditions based on similarity laws. The head coefficient is defined as follows:

$$\psi ={\displaystyle \frac{2gH}{u_2^2}}$$

(4)

where u₂ is the peripheral velocity at the impeller outlet.

The pump efficiency is defined as follows:

$${\eta }_e={\displaystyle \frac{\rho gHQ}{\left(T_f+T_r+T_b\right)\omega }}$$

(5)

where Q is the flow rate, T_f is the friction torque on the front shroud of the impeller, T_r is the friction torque on the outer surface of the rotor, T_b is the torque on the blades, and ω is the angular velocity of the impeller.

In addition, the friction loss ratios of the impeller front shroud and the rotor outer surface are defined by Equations (6) and (7), respectively.

$${\eta }_{ffc}={\displaystyle \frac{T_f}{T_f+T_r+T_b}}$$

(6)

$${\eta }_{rfc}={\displaystyle \frac{T_r}{T_f+T_r+T_b}}$$

(7)

The total friction loss ratio of the pump is defined as the sum of these two components:

$${\eta }_{fc}={\eta }_{ffc}+{\eta }_{rfc}$$

(8)

2.5. Mesh Independence Study

A grid independence study is conducted using six mesh schemes. The mesh counts range from 1.42 million to 7.85 million. As shown in Figure 4, the head coefficient and efficiency are the key assessment metrics. Both values begin to plateau when the mesh count exceeds 5.67 million. Specifically, the relative errors for both the head coefficient and efficiency between the 5.67 million and 7.85 million meshes are less than 0.1%. This trend indicates that the results are independent of the grid resolution. Therefore, the 5.67 million mesh scheme is adopted for subsequent simulations. This choice strikes a balance between numerical accuracy and computational cost.

2.6. Monitoring Locations

Figure 5 illustrates the locations of the monitoring points within the radial clearance. Points RP1 through RP8 are located on Plane 1 and are distributed circumferentially at intervals of 45°. Plane 1 represents the middle cross-section of the radial clearance, with an axial distance of 0.5L_r from both the front and rear planes of the rotor.

3. Experimental Setup and Validation

3.1. Experimental Setup

Figure 6 depicts the experimental test rig. It comprises the pump under test, a servo motor (INOVANCE, Shenzhen, China), and two pressure sensors (Stork, Stuttgart, Germany). The system also includes a dynamic torque sensor (DAYSENSOR, Bengbu, China), a turbine flowmeter (KROHNE, Duisburg, Germany), valves, a tank, and a data acquisition system (National Instruments, Austin, TX, USA).

The test rig complies with the Grade 1 accuracy specifications of the ISO 9906:2012 standard [46]. Two absolute pressure sensors measure the inlet and outlet pressures. Each sensor has an accuracy of ±0.1%. A turbine flowmeter measures the flow rate with ±0.3% accuracy. Additionally, a dynamic torque sensor monitors the pump torque with ±0.2% accuracy. The systematic uncertainties for the head and efficiency are ±0.14% and ±0.56%, respectively. All sensors output a 4–20 mA current signal. These signals are fed into a data acquisition chassis. An NI LabVIEW 2016 platform then processes the data. This configuration allows for real-time monitoring of all key performance parameters. During experiments, the inlet valve remains fully open to minimize hydraulic losses. The outlet valve is used to regulate the flow rate.

3.2. Experimental Validation

Figure 7 compares the experimental data with the numerical results. This comparison validates the simulation method. The head coefficient and efficiency are evaluated across various flow rates. The dimensionless flow rate is defined as Q* = Q/Q_d. The test pump geometry is identical to the numerical model. This consistency ensures a direct and valid comparison.

Numerical and experimental data show good agreement between 0.3Q_d and 1.3Q_d. At Q_d, the predicted head coefficient is 0.889. This value represents a relative deviation of only 0.4% from the measured data. The predicted efficiency at this point is 70.3%. This result is 1.8 percentage points higher than the experimental value. Across the entire operating range, the average relative error for the head coefficient is 1.0%. The peak deviation of 2.3% occurs at 1.2Q_d. For efficiency, the average absolute error is 1.6%. The largest discrepancy is a negative deviation of 3.7% at 0.3Q_d. This close agreement confirms the high fidelity of the simulation approach [47,48]. Therefore, the method is suitable for analyzing the pump’s unsteady flow characteristics. Subsequent analysis in this study is conducted at Q_d.

4. Discussion

4.1. Energy Performance

As illustrated in Figure 8, the radial clearance exerts a significant nonlinear influence on the performance of the centrifugal pump with an axial-flux canned motor. Within the range of 0.25 to 1.5 mm, the efficiency exhibits a non-monotonic trend, initially decreasing before increasing. Specifically, the efficiency peaks at 72.3% with a 0.25 mm clearance, while the minimum value of 71.6% occurs at 0.75 mm. In contrast, the head coefficient remains relatively stable and is less sensitive to changes in radial clearance. Notably, within the 1.0 to 1.5 mm range, the pump demonstrates high performance stability, with negligible variations in efficiency, head coefficient, and total friction loss ratio.

4.2. Pressure Field

Figure 9 illustrates the time-averaged pressure contours on the middle annular cross-section, unfolded circumferentially as shown in Figure 5, under different radial clearances. Here, the pressure coefficient is defined as p* = p/(0.5ρu₂²).

A distinct low-pressure region consistently appears within the radial clearance beneath the diffuser inlet. The spatial distribution of the high-pressure zone exhibits a nonlinear relationship with the radial clearance: at a clearance of 0.25 mm, the high-pressure region is most extensive, spanning a circumferential range of approximately 45° to 315°. As the clearance increases to 1.0 mm, this high-pressure zone shrinks to a range of approximately 90° to 225°. At this point, the pressure in the low-pressure zone reaches its minimum. Consequently, the pressure difference between the high- and low-pressure zones is maximized. When the clearance is further increased to 1.25 to 1.5 mm, the circumferential span of the high-pressure region remains largely unchanged, exhibiting only a slight extension in the axial direction.

4.3. Velocity Field

When the fluid flows through the top of the radial clearance, the abrupt change in geometry induces an adverse pressure gradient. This gradient triggers boundary layer separation, leading to a significant reduction in fluid kinetic energy and the formation of a low-velocity zone, as illustrated in Figure 10. The velocity is presented in its non-dimensional form, v* = v/u₂. As the radial clearance increases, the enlarged space allows greater freedom for the low-velocity fluid to expand, thereby extending the axial range of the low-velocity region.

Under different radial clearances, the time-averaged velocity distributions and streamlines of the axial cross-section of the motor clearance are presented in Figure 11. When the radial clearance is between 0.25 and 0.5 mm, the confined space inhibits flow separation. Consequently, no distinct velocity anomalies or vortices are observed within the clearance. As the clearance increases to 0.75 mm, the radial expansion of the high-velocity region near the rotor’s abrupt geometric change triggers shear layer instability. This induces three circumferential high-speed zones accompanied by incipient vortices, while secondary flow separation leads to the formation of an independent vortex structure at the corner.

At a clearance of 1.0 mm, the number of circumferential high-speed zones decreases to two, yet their axial fluctuations intensify. Concurrently, the corner vortex shrinks, and a complex system comprising four coexisting vortices emerges in the middle of the clearance. Under the 1.25 mm condition, the flow topology is reconstructed: a large, symmetrically distributed dual-vortex structure forms centrally, confining a robust circumferential high-speed zone within the shear layer between them. Independent the small vortex also arises at the channel corners due to secondary separation. The small vortex interacts with the large dual-vortex structure to form a three-vortex system, which maintains the stability of the high-velocity zone via shear momentum exchange. As the clearance further increases to 1.5 mm, the small corner vortex is entrained and merged into the large vortex, resulting in a simplified dual-vortex dominant mode. The high-velocity zone remains stably confined between the vortices, although its kinetic energy is slightly reduced.

4.4. Entropy Production

An entropy production method is employed within the URANS framework. This methodology accounts for both viscous and turbulent dissipation. Equations (9)–(12) present the governing equations.

$${\mathrm{\Phi }}_{\overline{D}}={\displaystyle \frac{\mu }{2}}{\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)}^2$$

(9)

$${\mathrm{\Phi }}_{D\prime }=\rho \epsilon$$

(10)

$$\mathrm{\Phi }={\mathrm{\Phi }}_{\overline{D}}+{\mathrm{\Phi }}_{D\prime }$$

(11)

$$P_{EP}={\displaystyle {\int }_V\mathrm{\Phi }\mathrm{d}V}$$

(12)

where ${\mathrm{\Phi }}_{\overline{D}}$ is the viscous dissipation, ${\mathrm{\Phi }}_{D\prime }$ is the turbulent dissipation, ε is the turbulent kinetic energy dissipation rate, Φ is the total entropy production dissipation, P_EP is the total entropy production power, and V is the volume.

As shown in Figure 12 and Figure 13, at a clearance of 0.25 mm, the confined flow space results in a high level of entropy production. As the clearance increases to 0.5 mm, turbulent kinetic energy dissipation within the radial clearance peaks, leading to the maximum observed entropy production. When the clearance further expands to 0.75 mm, the increased flow freedom mitigates the intensity of shear layer instability. Consequently, with the reduction in turbulent dissipation in the middle of the clearance, the overall entropy production decreases to the third-highest level. However, in local regions of the rotor–stator axial clearance, intensified turbulent mixing causes a localized increase in entropy production. As the clearance extends from 1.0 to 1.5 mm, the flow becomes geometrically dominated. Energy dissipation is primarily governed by boundary layer development and wall viscous shear. The regions of high entropy production shift towards the casing wall and the rotor outer surface. Furthermore, the overall entropy production exhibits a decreasing trend as the clearance increases.

4.5. Pressure Fluctuations

Pressure fluctuations are non-dimensionalized to facilitate comparisons across various operating conditions. The pressure fluctuation coefficient, C_p, is employed for this purpose [49]. This coefficient eliminates the influence of dimensional quantities, such as fluid velocity and density. Consequently, C_p provides a common baseline for the analysis. This approach enhances the generality and reliability of the findings. The definition of C_p is as follows:

$$C_p={\displaystyle \frac{p_i-\overline{p_{\theta }}}{0.5\rho u_2^2}}$$

(13)

where p_i is the instantaneous pressure, and $\overline{p_{\theta }}$ is the time-averaged pressure.

As shown in Figure 14, for all radial clearance sizes, the time-domain pressure fluctuation signals at each monitoring point exhibit distinct periodic oscillations that strictly correspond to the number of impeller blades. This indicates that the rotor-stator interaction between the impeller and tongue dominates the pressure fluctuations within the radial clearance. However, a large radial clearance can weaken the stability of the blade passing frequency modulation. When the clearance increases to 1.5 mm, the periodicity of the signals from RP2 to RP6 deteriorates significantly, particularly at RP5.

Table 2. Pressure fluctuation statistics.

δr (mm)	Parameter	Monitoring Point
		RP1	RP2	RP3	RP4	RP5	RP6	RP7	RP8
0.25	Max	0.009	0.006	0.004	0.004	0.004	0.006	0.007	0.008
	Min	−0.011	−0.008	−0.006	−0.004	−0.004	−0.007	−0.010	−0.010
	RMS	0.005	0.003	0.002	0.002	0.002	0.003	0.004	0.004
0.5	Max	0.009	0.006	0.005	0.005	0.005	0.006	0.008	0.010
	Min	−0.014	−0.009	−0.008	−0.006	−0.006	−0.009	−0.012	−0.013
	RMS	0.005	0.003	0.003	0.002	0.002	0.003	0.005	0.005
0.75	Max	0.011	0.007	0.009	0.007	0.006	0.006	0.010	0.012
	Min	−0.012	−0.007	−0.009	−0.007	−0.007	−0.009	−0.010	−0.013
	RMS	0.005	0.003	0.003	0.003	0.003	0.003	0.004	0.006
1.0	Max	0.010	0.006	0.005	0.006	0.006	0.005	0.007	0.011
	Min	−0.013	−0.008	−0.005	−0.005	−0.004	−0.005	−0.009	−0.012
	RMS	0.005	0.003	0.002	0.002	0.002	0.003	0.004	0.006
1.25	Max	0.010	0.006	0.007	0.005	0.005	0.007	0.007	0.013
	Min	−0.012	−0.008	−0.005	−0.005	−0.006	−0.009	−0.011	−0.013
	RMS	0.005	0.003	0.002	0.002	0.002	0.003	0.004	0.006
1.5	Max	0.010	0.007	0.008	0.008	0.010	0.007	0.007	0.012
	Min	−0.012	−0.008	−0.008	−0.009	−0.010	−0.008	−0.009	−0.013
	RMS	0.005	0.003	0.003	0.003	0.004	0.003	0.004	0.006

summarizes the time-domain characteristic parameters of pressure fluctuations at all monitoring points under various radial clearances. The maximum (Max) and minimum (Min) values define the dynamic range of the fluctuations, while the RMS value provides a robust measure of their average intensity. RP1, located below the diffuser inlet, and RP8, below the tongue, consistently exhibit high fluctuation levels. At the 1.5 mm clearance, the RMS value at RP8 reaches its maximum, which is 36.9% higher than that at 0.25 mm clearance. In contrast, the RMS value at RP1 peaks at 0.5 mm clearance, showing a 12.9% increase compared to its value at 0.25 mm.

At clearances of 0.75 to 1.0 mm, RP5 exhibits the lowest average pressure fluctuation intensity, with an RMS value 56% to 61% lower than that of RP1. However, when the clearance increases to 1.5 mm, the RMS value at RP5 surges by 55.3%, indicating intensified energy dissipation driven by enhanced vortex merging under large clearance conditions. At the 0.25 mm clearance, the variation in RMS values across all monitoring points is less than 15%. In contrast, at the 1.0 mm clearance, the relative difference between the extremes at RP5 and RP8 reaches 165%.

The frequency-domain analysis results of pressure fluctuations at all monitoring points within the radial clearance are presented in Figure 15. Here, the blade passing frequency (BPF) corresponds to 7f_s, where f_s represents the shaft frequency. The BPF and its second harmonic are present at all locations. The amplitude at the BPF is notably high, universally acting as the dominant frequency. In comparison, the amplitude at its second harmonic is significantly lower, exerting a minor influence on the overall pressure fluctuations.

As the increasing clearance from 0.25 mm to 1.5 mm intensifies the rotor-stator interaction between the impeller and tongue, the BPF amplitudes at RP1 and RP8 show a monotonic increase of 12.1% and 40.5%, respectively. In contrast, the BPF amplitudes at other monitoring points show non-monotonic variations. The amplitudes at RP2 and RP3 first decrease, then increase, and subsequently fluctuate, with maximum increases of 18.8% and 44.2%, respectively. The BPF amplitude at RP4 fluctuates within a narrow range, peaking at clearances of 0.75 mm and 1.25 mm and reaching its lowest value at 0.5 mm, with a maximum decrease of 5.6%. RP6 also exhibits fluctuating BPF amplitude, with a maximum decrease of 22.5%. The BPF amplitude at RP5 first increases and then decreases, reaching its maximum and minimum values at 0.75 mm and 1.5 mm, respectively, with a maximum decrease of 10.9%. Meanwhile, RP7 shows a trend of initial decrease followed by an increase, with a maximum increase of 12.1%.

At a clearance of 0.25 mm, the f_s component is observed at all monitoring points, with the strongest influence at RP1 and the weakest at RP4, where the amplitude is reduced by up to 79.2% compared to that at RP1. When the clearance increases to 0.5 mm, a third harmonic of the f_s appears in the pressure fluctuations. The variation trend of its amplitude aligns with that of the f_s, peaking at RP1 and reaching a minimum at RP5, with relative extreme differences of 113% and 159%, respectively. As the clearance further expands to 0.75 mm, additional harmonics of the f_s emerge. At this point, the f_s at RP2 is suppressed, while the points are affected by a broadband influence spanning from 2f_s to 5f_s. Notably, the amplitude of 3f_s at RP5 and RP6 even exceeds that of the f_s. At a clearance of 1.0 mm, the amplitude of 2f_s at RP4 matches that of the f_s. Apart from the BPF and its second harmonic, RP5 is only influenced by 3f_s.

At a clearance of 1.25 mm, all monitoring points remain subject to broadband influences from the f_s and its harmonics. However, when the clearance increases to 1.5 mm, RP1 and RP2 are significantly affected by 2f_s and 3f_s. At RP4 and RP5, the influence of the f_s and its harmonics surpasses that of the BPF, with amplitudes reaching up to 1.9 times that of the BPF.

5. Conclusions

This study investigates an axial-flux canned motor pump. It features a novel integrated impeller-rotor design. A numerical simulation methodology is presented. Validation against experimental data confirms the accuracy of the computational model. This manuscript systematically investigates the effects of radial clearances ranging from 0.25 to 1.5 mm between the rotor and the casing on the energy performance and internal flow characteristics of the centrifugal pump with an axial-flux canned motor. The main findings are summarized as follows:

(1)

Efficiency reaches its peak and valley values at radial clearances of 0.25 mm and 0.75 mm, respectively. Head coefficient is relatively less affected by the radial clearance. When the clearance is between 1.0 and 1.5 mm, which corresponds to a ratio of approximately 0.04 to 0.06 relative to the impeller outlet radius, the variations in efficiency, head coefficient, and total friction loss ratio decrease significantly, indicating that the performance tends to stabilize.

(2)

The circumferential extent of the high-pressure region within the radial clearance contracts as the clearance increases. At a clearance of 0.25 mm, the confined space results in high entropy production, whereas at 0.5 mm, turbulent dissipation in the shear mixing zone reaches its peak. A further increase in clearance causes the high entropy production regions to shift toward the wall, leading to a decreasing trend in overall entropy production. For clearances ranging from 0.75 to 1.5 mm, a complex vortex system forms within the radial clearance, inducing the generation of diverse circumferential high-velocity zones.

(3)

Pressure fluctuations are primarily BPF-dominated. However, at 1.5 mm, rotational harmonics at RP4 and RP5 exceed the BPF by 1.9 times, with a 36.9% intensity surge at RP8. When the clearance is 1.25 mm, corresponding to a ratio of approximately 0.05 relative to the impeller outlet radius, the energy loss is reduced, pressure fluctuations remain controllable with stable BPF dominance, achieving an optimal balance between flow stability and energy consumption.