Article

Transient Flow Characterization of Rotor–Stator Cavities in Two

Through-Flow Modes: Centrifugal and Centripetal
Yulong Yao 1,2 , Chuan Wang 1,2,3,4, *, Yitong Wang 2 , Jie Ge 5 , Hao Chang 1 , Li Zhang 6 and Hao Li 7, *

1 International Shipping Research Institute, GongQing Institute of Science and Technology,

Jiujiang 332020, China; yyl199876@126.com (Y.Y.); changhao@ujs.edu.cn (H.C.)
2 College of Hydraulic Science and Engineering, Yangzhou University, Yangzhou 225009, China;
wangyitong@yzu.edu.cn
3 State Key Laboratory of Ecohydraulics in Northwest Arid Region, Xi’an University of Technology,
Xi’an 710048, China
4 Institute of Fluid Engineering Equipment, JITRI, Zhenjiang 212013, China
5 SHIMGE Pump Co., Ltd., Taizhou 317525, China; gj@shimge.com
6 Institute of Precision Manufacturing, Suzhou Vocational Institute of Industrial Technology,
Suzhou 215100, China; zhanglizhengjiang@126.com
7 Institute of Farmland Irrigation, Chinese Academy of Agricultural Sciences, Xinxiang 453002, China
* Correspondence: wangchuan198710@126.com (C.W.); lihao01@caas.cn (H.L.)

Abstract: This study investigates the influence of roughness on the transient flow behavior in the
chamber based on the performance requirements of the pump rotor–stator chamber, aiming to
elucidate the mechanism of roughness in real operating conditions. Three-dimensional models under
two types of flow (centrifugal and centripetal) are developed, and transient numerical analyses
are performed through numerical simulation and experimental validation. The results show that
roughness significantly accelerates turbulence development in centrifugal through-flow, particularly
in the middle- and high-radius regions, increasing the turbulent kinetic energy by approximately 18%
compared to smooth surfaces. Transient flow analyses indicate that roughness leads to an overall
pressure drop of around 10% within the cavity while facilitating the formation of high-pressure zones
near the rotor. In centrifugal flow, high-pressure regions develop rapidly in the high-radius area,
resulting in a stepped pressure distribution with a peak pressure increase of 12% at the outermost
Citation: Yao, Y.; Wang, C.; Wang, Y.; radius. In centripetal flow, the pressure distribution remains more uniform, yet significant pressure
Ge, J.; Chang, H.; Zhang, L.; Li, H. rise trends emerge over time, with pressure increasing by 8% due to the presence of roughness. This
Transient Flow Characterization of study presents a systematic analysis of the effects of roughness on transient flow characteristics
Rotor–Stator Cavities in Two in rotor–stator cavities across two flow modes for the first time, providing valuable insights for
Through-Flow Modes: Centrifugal optimizing pump design and performance under real-world conditions.
and Centripetal. Water 2024, 16, 3678.
https://doi.org/10.3390/w16243678 Keywords: rotor–stator cavity; roughness; through-flow pattern; flow characteristics; numerical simulation
Academic Editor: Giuseppe Pezzinga

Received: 29 October 2024

Revised: 10 December 2024
1. Introduction
Accepted: 18 December 2024
Published: 20 December 2024
The rotor–stator chamber is a popular structure in many machines, such as pumps [1],
gas turbines [2], and aero-engines [3]. The disk chamber in the pump is an important part of
secondary return, helping to re-convey leaking flow from the top of the impeller to the inlet
and preventing backflow [4]. The rotor–stator chamber in a gas turbine is a key element of the
Copyright: © 2024 by the authors. secondary air system, playing a vital role in cooling high-temperature components while also
Licensee MDPI, Basel, Switzerland. preventing the backflow of high-temperature gasses from the cooling air discharged by the
This article is an open access article compressor [5]. The intricate flow dynamics present in the rotor–stator cavity directly influence
distributed under the terms and flow performance, playing a crucial role in the reliability and efficiency of these machines.
conditions of the Creative Commons Research on the flow properties of rotor–stator chambers has gained significant atten-
Attribution (CC BY) license (https://
tion, with scholars exploring the complexities of the internal flow field using theoretical
creativecommons.org/licenses/by/
analysis, experimental methods, and numerical simulations. Nield et al. [6] presented
4.0/).

Water 2024, 16, 3678. https://doi.org/10.3390/w16243678 https://www.mdpi.com/journal/water

Water 2024, 16, 3678 2 of 14

an early model of the effect of turbulent structure and axial gap on the flow field in disk
cavity flow, which considered the interaction of turbulence with the boundary layer and
helped to predict the flow behavior with different gaps. The boundary layer theory of
Schlichting [7] has likewise had a significant impact on the theoretical study of rotor–stator
cavity flows, particularly in the prediction of turbulence development near the walls. Fac-
chini et al. [8], based on a lot of test data, investigated the spin ratio, vortex structure, and
boundary layer separation phenomena in the cavity at different Reynolds numbers and
pointed out that the spin pattern is a contributory factor affecting the flow characteristics.
Nguyen et al. [9] utilized Particle Image Velocimetry to accurately measure the transient
properties of the internal flow in a disk cavity, uncovering the formation and evolution
of vortex structures at high Reynolds numbers. Using computational fluid dynamics and
entropy generation doctrinal, Zhang et al. [10] examined the effect of pump chamber con-
figuration (guide vanes and volute) on the hydraulic loss space distribution of mixed flow
pumps. Geis et al. [11] employed laser Doppler velocimetry (LDV) to study the axial flow
characteristics within the disk cavity, providing insights into the flow behaviors under
various inlet conditions. Through numerical simulations, Raisee et al. [12] revealed the
turbulent pulsation characteristics of cooling air in the rotor–stator cavity, showing that
varying flow patterns have a direct impact on vortex formation and development. In their
study, Shen et al. [13] applied large eddy simulation (LES) to study the influence of various
axial gaps on flow fields and vortex structures. Their findings, which are consistent with
the experimental data, highlight the enhancement of vortex characteristics under small
gap conditions. Using the RANS and LES methods, Fernando et al. [14] modeled flow
properties affected by roughness and found that rough walls have a significant effect on
core flow and vortex structure changes.
Furthermore, the study of heat transfer characteristics is also crucial. Ekkad et al. [15]
utilized a transient liquid crystal technique to determine the local convective heat transfer
coefficients in a disk cavity in the radial direction and found that there was a strong correla-
tion between the temperature distribution, the heat transfer efficiency, and the flow pattern.
Karabay et al. [16] explored the heat transfer behavior at different inlet temperatures and
cooling flow rates, revealing that higher flow rates enhance the heat transfer efficiency of the
cooling gas. Hu et al. [17,18] analyzed the implications of the axial gap on heat transfer using
numerical simulation and an experiment, showing that as the gap decreases, heat transfer is
enhanced, but the mobility model becomes more complex. Additionally, Petkovic et al. [19]
simulated transient heat transfer characteristics in a disk cavity using CFD techniques, demon-
strating the dual effects of wall temperature gradient and cyclonic structure on heat transfer. In
their experimental study, Chang et al. [20] introduced a new concept of measurement, show-
ing that the rotor system’s relative temperature can be determined from the stator system’s
absolute parameters. The results highlight that speed and the inlet mass flow rate are the
primary factors affecting the flow characteristics in the disk cavity. Poncet et al. [21] performed
numerical predictions based on single-point mathematical modeling of second-order full
strain transport closure at low Reynolds numbers and presented comparisons with empirical
data from the literature, which showed a robust correlation between the numerical results
and the experimental observations. Luo et al. [22] utilized the transient thermochromic liquid
crystal (TLC) technique to survey pressure distribution and heat transfer in the disk cavity,
determining the convective heat transfer characteristics of rotating disk surfaces. Concurrently,
Liu et al. [23] presented research on flow and heat transfer characteristics in disk cavities using
conjugate CFD and fully coupled FE/CFD, comparing their findings with steady-state flow
research predictions and test results.
While the aforementioned studies have elucidated the flow and heat transfer charac-
teristics of disk cavities under various operating conditions, most have assumed smooth
wall surfaces and neglected the impact of roughness [24–27]. In practical applications, wall
roughness is prevalent, particularly under high temperatures and complex stress environ-
ments, where it can obviously affect the fabric of the boundary layer, flow distribution,
and heat transfer performance [28–31]. The turbulence-enhancing effects of roughness
Water 2024, 16, 3678 3 of 14

and flow separation phenomena can notably alter the heat transfer efficiency of cooling
air and the evolution of vortex structures. Recently, Yao et al. [32] demonstrated that rotor
roughness accelerates boundary layer separation on the stator, an effect that becomes more
pronounced under weak through-flow conditions. Regarding heat transfer, rotor roughness
not only contributes to an overall temperature increase in the cavity but also decreases the
thickness of the thermal boundary layer.
Given the oversight of roughness effects in existing studies, it is crucial to investigate
how roughness impacts flow and heat transfer characteristics in disk cavities. This research
examines the transient flow characteristics in both centrifugal and centripetal through-
flow disk cavities using numerical simulations and test validation under varying surface
roughness conditions. Notably, the selected flow modes encompass the majority of disk
cavity flows, offering valuable insights for future pump design.

2. Numerical Method
2.1. Theoretical Equation
This research uses numerical simulations which are based on the Reynolds-Averaged
Navier–Stokes (RANS) equations, which depict the mass, momentum, and energy conserva-
tion of an incompressible turbulent fluid. The control equations are formulated as follows:
Continuity equation:
∂ρ
+ ∇ · (ρν) = 0 (1)
∂t
where ρ represents the fluid density, v represents the velocity vector, ▽ denotes the scatter-
ing operator, and t represents the time.
Momentum equation:

∂(ρν)
+ ∇ · (ρνν) = −∇ p + ∇ · τ + f (2)
∂t
where p represents the static pressure, τ represents the stress tensor, and f represents the
volumetric force.
The expression for the stress tensor τ is given by
h i 2
τ = µ ∇ν + (∇ν)T − µ(∇ · ν) I (3)
3
where µ represents the dynamic viscosity of the fluid, and I represents the unit matrix.
Energy equation:

∂(ρE)
+ ∇ · (ν(ρE + p)) = ∇ · (k∇ T ) + Φ (4)
∂t
where E represents the total energy, T represents the temperature, k represents the thermal
conductivity, and Φ is the viscous dissipation term.
The total energy E is expressed as follows:

p v2
E = h− + (5)
ρ 2

where h represents the enthalpy and v2 /2 represents the kinetic energy term.
The Shear Stress Transport (SST) k-ω turbulence model has been used in this study
because of its advantages in predicting secondary flow characteristics, re-attachment, and
separation. The turbulent kinetic energy transport term (k) and the turbulent dissipation
rate (ω) in the SST k-ω turbulence equation can be calculated using Equations (6) and (7).
  
→ µ
∇ · (ρk ν ) = ∇ · µ + t ∇k + µt S2 − β∗ ρkω (6)
σk
Water 2024, 16, 3678 4 of 14

  
→ µt 1
∇ · (ρ ν ω ) = ∇ · µ+ ∇ · ω + 2(1 − F1 )ρ ∇ · ω + γρS2 − βρω 2 (7)
σω σω ω
where µt S2 and γρS2 represent the generation terms derived from the turbulence kinetic
energy and turbulence dissipation rate, respectively. µt is the turbulent viscosity. The
constants γ, β, σk , and σω can be obtained from the mixing factor F1 .

a1 k
ut = (8)
max( a1 ω, SF2 )

where a1 is a model constant, S is the magnitude of the mating rate, and F2 is the mixing
function. " √ ! !#4 
k 500ν 4σω2 k
F1 = tanh min max , ,  (9)
β∗ ωy y2 ω CDkω y2
" √ !#2 
2 k 500ν
F2 = tanh max ,  (10)
β∗ ωy y2 ω

where y is the distance to the wall and CDkω is the cross-derivative term of k and ω. Other
important model constants are σk = 0.85, σω = 0.5, β* = 0.09, α1 = 5/9, β1 = 0.075, and
σω2 = 0.856.
The SST k-ω turbulence model was chosen for this study due to its superior perfor-
mance in capturing key flow phenomena in rotor–stator cavities, particularly in regions
with high shear stresses and complex turbulence induced by surface roughness.

2.2. Computational Model

In this work, the transient flow properties of a disk cavity are investigated at different
Reynolds numbers and roughness levels while excluding the impact of special structures
such as seals in the side chambers. For this purpose, a disk cavity model with axial in-flow
and out-flow is constructed for a typical rotor–stator configuration, and the corresponding
fluid domains are modeled in three dimensions using UG, as shown in Figure 1. The model
encompasses two types of flow, centrifugal and centripetal, with separate examinations of
how roughness impacts flow characteristics. Notably, a full-annulus model is employed
Water 2024, 17, x FOR PEER REVIEW 5 of 16
rather than an axisymmetric model to capture the complete circumferential flow patterns.
Axial diagrams of the rotor–stator chamber for both flow types are presented in Figure 2
alongside their key structural parameters in Table 1.
model encompasses two types of flow, centrifugal and centripetal, with separate
Table 1. Key structural
examinations of howparameters
roughnessof disk cavity.flow characteristics. Notably, a full-annulus
impacts
model is employed rather than an axisymmetric model to capture the complete
b (mm) s (mm) so (mm) ro (mm)
circumferential flow patterns. Axial diagrams of the rotor–stator chamber for both flow
200 20 5 20
types are presented in Figure 2 alongside their key structural parameters in Table 1.

Figure 1. Calculation model.

Water 2024, 16, 3678 5 of 14
Figure 1. Calculation model.

Figure
Figure 2. A schematic
2. A schematic representation
representation of
of the
the axial
axial surface
surface of
of the
the centrifugal
centrifugal and
and centripetal
centripetal through-
through-
flow in the rotor–stator cavity.
flow in the rotor–stator cavity.

Water 2024, 17, x FOR PEER REVIEW2.3. Grid 6 of 16

Table 1. Key structural parameters of disk cavity.
To ensure mesh independence in the simulation results, we employ multiple meshing
techniques
b (mm) and conduct mesh independence tests. sStructured
s (mm) o (mm) meshing is performed
ro (mm) using
ICEM, with grid refinement
200 regions where flow applied at the inlet, outlet, and walls, as
20parameters experience5significant variations, well as in 20
other regions
as in other as illustrated
where flow parameters experience significant variations, as illustrated in Figure 3. Because
in Figure
this study3.focuses
Because onthis
thestudy
effect focuses
of surface on roughness
the effect ofon surface roughness
heat transfer on interior
in the heat transfer
wall
2.3. Grid
of the disk cavity, near-wall meshing is critical. In order to accurately captureaccurately
in the interior wall of the disk cavity, near-wall meshing is critical. In order to the effect
of To the
capture ensure
roughness, themesh
effect independence
of roughness,
thickness in layer
theinitial
of the the of
thickness simulation
ofthe
theinitial results, of we
layermesh
near-wall employ
theisnear-wall multiple
set to exceedmeshall
is
meshing
equivalent techniques
set to exceedroughness and conduct
all equivalent
values. roughness mesh
values.
Additionally, independence
since +
Additionally, tests.
a high y value sincecanStructured
a lead
highto meshing
y+ inaccuracies
value is
can lead
in
performed
mixing andusing ICEM,
conversion
to inaccuracies in withand
during
mixing grid refinement
numerical
conversion applied
during at
simulations, thestudy
this inlet,implements
numerical outlet, and walls,
simulations, thisasstudy
well
a refinement
process in the near-wall region, maintaining the y + value at around 1 to ensure the adequate
implements a refinement process in the near-wall region, maintaining the y+ value at
resolution
around 1 to ofensure
near-wallthe turbulence in the SSTofmodel
adequate resolution [17].turbulence in the SST model [17].
near-wall

Figure 3. Grid
Figure 3. Grid of
of rotor–stator
rotor–stator cavity.
cavity.

A grid independence study was performed to ensure the accuracy and reliability of
the numerical simulations. This assessment involved progressively refining the grid
density and monitoring the variations in key flow parameters, including the flow rates at
both the inlet and outlet, as shown in Figure 4. The relative difference in flow rates
 decreased with an increasing grid density, indicating the convergence of the numerical
solution.
For all grids used in this study, the maximum deviation across simulations was less
than 0.15% once the grid count exceeded 2.289 million elements. This deviation is well
Water 2024, 16, 3678 6 of 14
within acceptable limits for engineering applications, demonstrating that further
refinement would have a negligible impact on the results. Consequently, a grid with 2.289
million elements
A grid was selected
independence studyfor theperformed
was simulations, balancing
to ensure computational
the accuracy efficiency
and reliability of and
the
solution accuracy.
numerical simulations. This assessment involved progressively refining the grid density
Additionally,
and monitoring the all grids used
variations in keyin flow
this study exhibited
parameters, a quality
including rating
the flow exceeding
rates 0.8,
at both the
inlet and outlet,
ensuring as shown
that the in Figure 4.
grid resolution andThequality
relativewere
difference in flow
sufficient forrates decreased
capturing the with
flow
an increasing grid
characteristics density, indicating the convergence of the numerical solution.
accurately.

Figure 4. Grid
Figure 4. Grid independence
independence test.
test.

For all grids used in this study, the maximum deviation across simulations was less than
0.15% once the grid count exceeded 2.289 million elements. This deviation is well within
acceptable limits for engineering applications, demonstrating that further refinement would
have a negligible impact on the results. Consequently, a grid with 2.289 million elements was
selected for the simulations, balancing computational efficiency and solution accuracy.
Additionally, all grids used in this study exhibited a quality rating exceeding 0.8, ensuring
that the grid resolution and quality were sufficient for capturing the flow characteristics
accurately.

2.4. Boundary Conditions

ANSYS CFX is employed for the numerical simulation of the disk cavity. The computa-
tional model comprises a cavity formed by the rotor and stator, along with inlet and outlet
sections. Water is set as the working fluid, and the SST k-ω turbulence model is selected for
its high accuracy in predicting flow behavior within the disk chamber, effectively handling
both near-wall flow and shear flow farther from the wall. The rotor is provided with walls
that rotate at a specified rotational speed, while the remaining walls are no-slip and adiabatic.
The inlet to the computational model is configured as a mass flow inlet, specifying both the
mass flow rate and inlet temperature. This mass flow simulates the cooling gas generated
by the compressor. The outlet is designated as a static pressure wall. Both centripetal and
centrifugal through-flow rotor–stator cavities are analyzed separately by varying the inlet and
outlet boundary conditions. All calculations are transient, using a time step of 0.0001 s over a
total of 20,000 steps. The simulation begins with the assumption of a uniform flow field and
addresses any unphysical phenomena through an initial calculation period. A convergence
accuracy of 10−5 is set to ensure stable convergence under all conditions.
In ANSYS CFX, the critical parameter of roughness is primarily addressed through
turbulence models and wall functions, which account for the impact of surface roughness
on flow behavior and heat transfer. The rough wall function used by CFX is based on the
Nikuradse equivalent sand roughness model [33], which is a classical roughness model
that describes the effect of wall roughness through sand roughness (ks ). The wall function
introduces roughness into the simulation in the form of a modified logarithmic velocity
profile, which affects frictional resistance and heat transfer. The revised logarithmic velocity
profile is expressed as
Water 2024, 16, 3678 7 of 14

 +
1 + y
u = ln + + Br (11)
κ ks
where u+ represents the dimensionless velocity, y+ represents the dimensionless distance,
ks + represents the dimensionless roughness height, κ represents the Kármán constant, and
Br represents the modified wall constant.
CFX influences heat transfer characteristics indirectly by altering the velocity profile
and turbulence behavior of the boundary layer. When using the logistics law for wall
temperature for a smooth wall, the wall heat flux density may be possibly underestimated.
Thus, the logarithmic law for wall temperature is adjusted as below when performing
numerical analog simulations.
 2
+
Tlog = 2.12 ln(Pr · y∗ ) + 3.85Pr1/3 − 1.3 − ∆Bth (12)

∆Bth = (1/0.41) ln 1 + C0.3Prk+

s (13)
where the energy correction factor C is 0.2.
Ultimately, the range of parameters studied in this paper is shown in Table 2. With
the above numerical simulation setup and methodology, the transient flow behavior in
the disk chamber can be accurately captured under different Reynolds numbers and
roughness conditions. These results provide valuable references for turbulence model
selection and roughness treatment in practical engineering applications. The selected
surface roughness range was determined based on typical values observed in practical
applications, such as rotor–stator systems in gas turbines, where surface roughness often
results from manufacturing processes, wear, or coating degradation. Additionally, the
chosen range ensures a representative investigation of its impact across different flow
conditions, encompassing values that are both realistic and challenging for heat transfer
and flow behavior. The Reynolds number (Re) used in this study is calculated as

ωb2
Re = (14)
ν
where ω is the angular velocity of the disk, rad/s; b is the outer radius of the disk, mm; and
ν is the kinematic viscosity of air, m2 /s.

Table 2. Range of research parameters.

Parameter Range
Reynolds number Re 8.85 × 105 –1.77 × 106
Through-flow coefficient Cw 843–1966
Axial gap ratio G 0.1
Inlet total temperature Tin 300 K
Rotor surface temperature Tr 350–432.75 K
Roughness ks + 0–200 µm

2.5. Test Validation

For the verification of the robustness of the numerical results, we compare them with
experimental measurements based on the radial temperature distribution. Due to the
consistency of the model, detailed experimental procedures and data are provided in our
previous work [18]. The results demonstrate a strong correlation between the two, with
discrepancies primarily arising from the assumption in the numerical simulation that the
temperatures on the rotor wall and stator wall are constant, and there is a temperature
grade during the test. As such, the simulation setup in this research can be justified by a
combined comparison of the numerical analyses and the experimental data.
Water 2024, 16, 3678 8 of 14

Water 2024, 17, x FOR PEER REVIEW 9

3. Results and Discussion
Water 2024, 17, x FOR PEER REVIEW 9 o
The total pressure distribution (TP) provides insights into flow characteristics, includ-
ing flow uniformity In andcentripetal
the formation and evolution
through-flow, whileofthe vortices.
pressure By analyzing
distributionthewithin total the cav
pressure variation, it
relativelyis possible to identify
uniform, through-flow, areas
the high-pressure of high and low pressure present in the
In centripetal whileregion at largerdistribution
the pressure radii primarily withinresults from
the cavity
flow and thus understand the flow pattern of the fluid within the cavity. Understanding the
interplay
relatively of flow structure,
uniform, boundary
the high-pressure layerat effects,
region kinetic
larger radii energyresults
primarily conversion
from
trend of the total pressure under different working conditions can help designers to make
interplay of
localized flow structure,
vortices. The flow boundary
structure layerof effects, kineticthrough-flow
centripetal energy conversion, is rela
more reasonable decisions in improving the cooling system and enhancing the performance
localized vortices.
homogeneous, Thefluidflowflowingstructure of centripetal through-flow stableisflow
relativ
of the equipment. Figures 5 andwith 6 givethethe total pressure towards the center,
distributions under forming
the twoa types fie
homogeneous,
of through-flow,this flow pattern,
respectively. with
The the the
fluid
results fluid flowing
movement
show that theistowards
more
pressure thein
regular, center, forming
resulting
the lower in aamore
cavity stable flow field
balanced
of the o
this
pressureflow pattern,
distribution. the fluid movement
However, at is
high more
centrifugal through-flow is distributed in a stepwise manner, with high-pressure zones regular,
radii, localizedresulting
areas in a
of more
high balanced
pressure ove
ma
pressure
existing at a highform
radius due distribution.
and to forming However,
changeswithin at high
in thea direction
short period radii,
of of localized
motion
time. areas
of centrifugal
In the of high
fluid and pressure mayf
centrifugal
through-
form
flow, the fluid isSignificantdue
subjected to to changes in
centrifugal forces
pressurization the direction
occursthat move
in the of motion
the over
cavity fluidtime.of the
particles fluid and through-flowfor
centrifugal
to the outside,
Centripetal us
resulting in higherSignificant
centrifugal pressurization
forces on occurs
the fluid in the
layerscavityat over
higher time. Centripetal
radii.
involves the contraction and acceleration of the fluid towards a central region. As the This through-flow
centrifugal usu
force encourages involves
the fluid
duration the tocontraction
accumulate
increases, andat acceleration
high radii,
the contraction ofofthe
effectcreatingthefluid towards
a region
fluid ofahigh
between central region.
the pressure.
rotor and As the rfl
stator
The kinetic energy of
duration the fluid
increases, is converted
the into
contraction hydrostatic
effect of the energy
fluid as
between
in a gradual rise in the flow rate, with kinetic energy being converted into static pre the flow
the changes.
rotor and stator res
At high radii, where the fluid
in a gradual risevelocity
in the flow is comparatively
rate, inwith high, thebeing
conversion of kinetic
leading to a significant increase thekinetic
chamber energy pressure. converted into static
In this context, wallpress
roug
energy into hydrostatic
leading energy
to a causes aincrease
significant rapid increase
in the in pressure
chamber in thatIn
pressure. region.
this However,
context, wall roughn
enhances the pressure rise tendency within the cavity. The roughness heighten
when the wall surface
enhances is rough, the increased
the pressure frictional resistance
rise tendency due toThe roughness leads
frictional resistance between the fluidwithin and the the cavity.
wall, resulting roughness
in greaterheightens
energy
to greater energyfrictional
losses during flow, resulting in an overall pressure drop.
resistance between the fluid and the wall, resulting in greater energy los The roughness
during
of the surface leads flow.
to increased While there is an overall pressure drop, the effects of roughness ne
during flow. Whileinteraction
there is anbetween the fluiddrop,
overall pressure and the
the effects
wall, creating
of roughnessa near
thicker boundary rotor
layer, andwhich, in high-pressure
in turn, increases regions can lead to In
the pressure localized pressure increases a
rotor and in high-pressure regions can leadloss. addition,
to localized roughness
pressure increases an
propensity
affects the formation of flow forstructures,
pressure buildup.especially Collectively,
in the vicinity these analyses
of the rotor.not only
The provide
flow a scie
propensity for pressure buildup. Collectively, these analyses not only provide a scien
foundation for the design and optimization
characteristics in the rotor region are enhanced by turbulence due to high-speed rotation, of pumps and related applications bu
foundation for the design and optimization of pumps and related applications but a
creating a high-pressure
highlightregion. potential However, at locations
future research awaycarrying
directions, from thesignificant
rotor, the flow may and pra
theoretical
highlight potential future research directions, carrying significant theoretical and pract
be attenuated byvalue.
the influence of rough surfaces, resulting in a pressure drop.
value.

(a)Smooth
(a) Smooth wall
wall Roughwall
(b)Rough
(b) wall

Figure
Figure 5. Transient 5.
5. Transient
evolution
Figure of TP inevolution
Transient of TP
centrifugal
evolution of TP incentrifugal
centrifugal
through-flow
in through-flow
in through-flow
rotor–stator ininrotor–stator
rotor–stator
cavity. cavity.
cavity.

(a)Smooth
(a) Smooth wall
wall (b)
(b)Rough
Roughwall
wall
Figure 6. TransientFigure 6. Transient
evolution evolution
of TP inevolution of TP
centripetal in centripetal
in through-flow in rotor–stator cavity.
Figure 6. Transient of through-flow
TP in centripetal rotor–stator
through-flow cavity.
in rotor–stator cavity.
In centripetal through-flow, while
Turbulent kinetic the pressure
energy distribution
(TKE) plays within
a vital role the cavityfluid
in determining is rela-
energy loss
tively uniform, the Turbulent kinetic
high-pressure energy
region at (TKE)
larger plays
radii a vital role
primarily in determining
results from the fluid energy los
interplay
overall system efficiency. In a disk cavity, the distribution and transfer pathways of T
overall system efficiency. In a disk cavity, the distribution and transfer pathways o
govern the energy conversion processes occurring within the cavity. A thoro
govern the energy conversion processes occurring within the cavity. A thor
investigation of the TKE distribution can lead to optimized hydrodynamic designs
investigation of the TKE distribution can lead to optimized hydrodynamic design
Water 2024, 16, 3678 9 of 14

of flow structure, boundary layer effects, kinetic energy conversion, and localized vortices.
The flow structure of centripetal through-flow is relatively homogeneous, with the fluid
flowing towards the center, forming a stable flow field. In this flow pattern, the fluid move-
ment is more regular, resulting in a more balanced overall pressure distribution. However,
at high radii, localized areas of high pressure may still form due to changes in the direction
of motion of the fluid and centrifugal forces. Significant pressurization occurs in the cavity
over time. Centripetal through-flow usually involves the contraction and acceleration of
the fluid towards a central region. As the flow duration increases, the contraction effect of
the fluid between the rotor and stator results in a gradual rise in the flow rate, with kinetic
energy being converted into static pressure, leading to a significant increase in the chamber
pressure. In this context, wall roughness enhances the pressure rise tendency within the
cavity. The roughness heightens the frictional resistance between the fluid and the wall,
resulting in greater energy losses during flow. While there is an overall pressure drop,
the effects of roughness near the rotor and in high-pressure regions can lead to localized
pressure increases and a propensity for pressure buildup. Collectively, these analyses not
only provide a scientific foundation for the design and optimization of pumps and related
applications but also highlight potential future research directions, carrying significant
theoretical and practical value.
Turbulent kinetic energy (TKE) plays a vital role in determining fluid energy loss and
overall system efficiency. In a disk cavity, the distribution and transfer pathways of TKE govern
the energy conversion processes occurring within the cavity. A thorough investigation of the
TKE distribution can lead to optimized hydrodynamic designs that minimize energy losses
and enhance system efficiency. The temporal variation in TKE in the centrifugal flow sub-
stator cavity is illustrated in Figure 7. The distribution of turbulent kinetic energy exhibits
axial stratification, indicating significant differences in turbulence intensity within different
regions. Overall, the values of turbulent kinetic energy exhibit a hierarchical pattern with respect
to axial position, closely tied to the rotational velocity gradient and shear stress within the
cavity. Notably, turbulence energy is significantly elevated near the rotor, likely due to the high
rotational speed creating a strong shearing effect that induces enhanced turbulence generation in
this area. In contrast, lower turbulent kinetic energy is observed near the stator, consistent with
its smaller velocity gradient and relatively stable flow conditions. Additionally, the temporal
variation in turbulent kinetic energy reflects the transient characteristics of the flow field within
the cavity. The trends of turbulent kinetic energy at different axial positions over time may
indicate a gradual development or adjustment of the flow, particularly in relation to boundary
layer formation and the evolution of vortex structures. Under rough walls, the turbulent kinetic
energy in the cavity rises overall and keeps moving towards the stator. Roughness disrupts
the laminar flow state within the boundary layer by increasing the flow resistance near the
wall, leading to the fluid having greater pulsation and vortex generation, thus increasing the
turbulent kinetic energy. The rough surface promotes the early development of turbulence
and causes it to be enhanced first in the region close to the rotor. This is due to the high-speed
rotation of the rotor which generates strong shear forces, which, together with the effect of
roughness, makes the generation and development of turbulence more intense in this region.
The variation in TKE with time for the centripetal flow disk cavity is shown in Figure 8.
A notable difference is that the region of high turbulent kinetic energy occurs at the lower
radius of the disk cavity, specifically at the exit. In centripetal flow, fluid moves from the outer
edge of the cavity towards the inner edge. The shortening of the flow path and reduction
in radius increase the fluid’s velocity gradient at the lower radius, resulting in greater shear
and stronger turbulence generation. Consequently, the area near the outlet (low radius)
exhibits higher turbulent kinetic energy. This elevated turbulent energy is primarily attributed
to the contraction effect of the centripetal flow and the flow disturbances caused by wall
roughness. The roughness amplifies the turbulent kinetic energy within the disk cavity and
intensifies the turbulence’s extension towards the mid-radius. This phenomenon highlights the
specific turbulence generation mechanisms present in centripetal flow and the enhancement
in turbulence intensity by rough walls.
 outlet (low radius) exhibits higher turbulent kinetic energy. This elevated turbu
energy is primarily attributed to the contraction effect of the centripetal flow and the
disturbances caused by wall roughness. The roughness amplifies the turbulent ki
energy within the disk cavity and intensifies the turbulence’s extension towards the
Water 2024, 16, 3678 radius. This phenomenon highlights the specific turbulence generation 10 of 14 mechan
present in centripetal flow and the enhancement in turbulence intensity by rough wa

(a) Smooth
Water 2024, 17, x FOR PEER REVIEW wall (b) Rough wall 11

Figure
Figure 7. Transient 7. Transient
evolution of TKEevolution of TKE
in centrifugal in centrifugal
through-flow in through-flow in rotor–stator cavity.
rotor–stator cavity.

(a) Smooth wall (b) Rough wall

Figure
Figure 8. Transient 8. Transient
evolution of TKEevolution of TKE
in centripetal in centripetal
through-flow in through-flow in rotor–stator cavity.
rotor–stator cavity.

The core swirl ratio K (Equation

The core swirl ratio (11)), a crucial (11)),
K (Equation parameter influencing
a crucial parameteroverall system
influencing overall sys
performance, is performance,
defined as theis ratio of the angular velocity of the rotating fluid to
defined as the ratio of the angular velocity of the rotating fluid to th that of
the rotor. This ratio
the directly reflects
rotor. This ratiothe characteristics
directly of the
reflects the rotating flowofwithin
characteristics the cavity.
the rotating flow within
The radial distributions of the K for the two passages are illustrated
cavity. The radial distributions of the K for the two passages are illustrated in Figures 9 and 10, in Figur
respectively, where x is defined by Equation (12). For both passages, the core swirl ratio
and 10, respectively, where x is defined by Equation (12). For both passages, the core s
increases along the flow direction. In centrifugal flow, the fluid radially moves outward from
ratio increases along the flow direction. In centrifugal flow, the fluid radially m
the center of the disk cavity. As the fluid interacts with the rotating rotor wall during this
outward from the center of the disk cavity. As the fluid interacts with the rotating r
outward movement, the high rotational speed of the rotor surface leads to a gradual increase
wall during
in the fluid’s angular velocity. this
Asoutward
the fluidmovement,
progresses the high the
towards rotational speed of the
outer diameter, the rotor
shearsurface l
effect intensifies, further increasing the rotational velocity and resulting in a gradual rise in towards
to a gradual increase in the fluid’s angular velocity. As the fluid progresses
outer
the core swirl ratio diameter,
along the shear
the radial effectIn
direction. intensifies,
centripetal further increasing
flow, the the rotational
fluid flows from the velocity
outer diameter towards the inner diameter. Although the fluid flows inward, its rotationalIn centrip
resulting in a gradual rise in the core swirl ratio along the radial direction.
velocity near theflow,
rotorthewallfluid flowsincreases
steadily from thedue outer diameter
to the towards
continuous the inner
influence diameter.
of the rotatingAlthough
surface. As the fluid
fluidflows
approaches
inward,the its inner diameter,
rotational velocitythenear
effect
theofrotor
the wall
rotorsteadily
wall becomes
increases due to
more pronounced, causing a influence
continuous gradual increase in the fluid’s
of the rotating angular
surface. As the velocity and consequently
fluid approaches the inner diam
raising the K in the radial direction. Notably, wall roughness enhances
the effect of the rotor wall becomes more pronounced, causing a gradual the K within the cavity.
increase in
However, it is more significant only at high radii under centrifugal through-flow.
fluid’s angular velocity and consequently raising the K in the radial direction. This may be Nota
due to the fact that
wall inroughness
the initial stage of centrifugal
enhances the K within flow, the
the fluid However,
cavity. is not yet fully developed
it is more significant on
and the role of roughness in perturbing the fluid is relatively weak. However, as the fluid
high radii under centrifugal through-flow. This may be due to the fact that in the in
flows towards the outer radius, the turbulence gradually strengthens and the role of roughness
stage of centrifugal flow, the fluid is not yet fully developed and the role of roughne
increases, eventually showing a significant core swirl ratio enhancement in the high radius
perturbing the fluid is relatively weak. However, as the fluid flows towards the o
region. The rise in the Reynolds number results in a marked increase in the core swirl ratio for
both centrifugal radius, the turbulence
and centripetal flows. This gradually
phenomenon strengthens
indicatesand the role influence
the amplified of roughness of increa
the Reynolds number on the turbulence intensity and fluid rotation characteristics within theradius reg
eventually showing a significant core swirl ratio enhancement in the high
The riseinto
cavity. Gaining insights in the
the Reynolds number results
effects of roughness in a marked
can facilitate increase
the further in the coreofswirl ratio
optimization
both centrifugal
flow and heat transfer designs inand centripetal
disk flows. This phenomenon
cavities, particularly for applications indicates the amplified
involving high influ
of the Reynolds number
Reynolds numbers and rough surface conditions. on the turbulence intensity and fluid rotation characteri
within the cavity. Gaining insights into the effects of roughness can facilitate the fur
optimization of flow and heat transfer designs in disk cavities, particularly
applications involving high Reynolds numbers and rough surface conditions.
Ω
K =
Ωf
Water 2024, 16, 3678 11 of 14

Ω
K= (15)
Ωf
where Ω represents the angular velocity of the disk, and Ωf represents the angular velocity
of the fluid.
Water 2024, 17, x FOR PEER REVIEW r 12
Water 2024, 17, x FOR PEER REVIEW x= (16) 12
b

(a)
(a)Re
Re==8.85
8.85××10
105
5 (b)
(b) Re
Re == 1.33
1.33 ×× 10
6
106 (c) Re
(c) Re == 1.77
1.77 ×× 10
6
106.
Figure 9. Effect ofFigure 9.
9. Effect
roughness
Figure of
of roughness
on core
Effect on
onincore
swirl ratio
roughness swirl
swirl ratio
a centrifugal
core in
in aa centrifugal
ratiothrough-flow through-flow
rotor–stator
centrifugal rotor–stator cav
cavity.
through-flow rotor–stator ca

(a)
(a)Re
Re==8.85
8.85××10
105
5 (b)
(b) Re
Re == 1.33
1.33 ×× 10
6
106 (c)
(c) Re
Re == 1.77
1.77 ×× 10
6
106
Figure
Figure 10.
10. Effect
Effect of
of roughness
roughness on core swirl
swirl ratio in
in aa centripetal through-flow rotor–stator ca
ca
Figure 10. Effect of roughness on core swirl ratioonincore ratiothrough-flow
a centripetal centripetal through-flow
rotor–stator rotor–stator
cavity.

To further explore To
To further
further explore
exploreflow
the transient the
the transient
transient flow
characteristicsflow characteristics
characteristics
in the disk cavity in
in the
the disk
disk the
under cavity unde
cavity unde
influence
influence of roughness,
influence we of roughness,
present thewe
of roughness, we present
transient the
presentvariations transient
the transient variations
in velocity
variations in
at in velocity
three at three
radialat three
velocity
positions, illustrated
positions,in Figures
positions, illustrated
illustrated11 and in 12. For11
in Figures
Figures 11centrifugal
and
and 12. 12. For through-flow,
For centrifugal the effect of the
centrifugal through-flow,
through-flow, the eff
eff
roughness on velocity
roughness is more
on limited
velocity at
is
roughness on velocity is more limited the
more low-radius
limited at location
the near
low-radius the inner
location diameter
near the inner dia
(x = 0.25). This is(xbecause
(x ==0.25). at this
0.25). This
This is location,
is because
because at the
at this
thisrotational
location,velocity
location, of the velocity
the rotational fluid is low, and
of the theis low, an
fluid
perturbations created by
perturbations the wall roughness have not yet significantly affected the main
perturbations created created by by thethe wall
wall roughness
roughness have not yet significantly affected the
flow of the fluid, leading to a smaller increase in velocity. The effect of roughness reaches
flow
flow of the fluid, leading to a smaller increase in velocity. The effect of roughness re
of the fluid, leading to a smaller
its maximum at the mid-radius position (x = 0.5). Figure 11 illustrates that at this location,
its maximum at
at the
the mid-radius position (x = 0.5). Figure 11 illustrates that at this loc
wall roughness its maximum
markedly enhances mid-radius
the momentum position exchange of the fluid by increasing
wall roughness markedly enhances the momentum exchange of the fluid by incre
the friction and wall roughness
turbulent kinetic markedly
energy at enhances
the fluid–wall interface, leading to a velocity
the
increase of 41.89%.the friction
friction
This and
and turbulent
phenomenon turbulent kinetic
suggestskinetic energy
energy
that at the of
the impact fluid–wall
roughness interface,
on the flowleading is to a ve
increase
most significantincrease of 41.89%.
of 41.89%. likely
at the mid-radius, This phenomenon
due to the moderate relative velocity of the fluid on the
This phenomenon suggests that the impact of roughness
in this region andis
is most
most significant at
at the
significantinteraction
the intensified the mid-radius,betweenlikely
mid-radius, due to
the shear the and
layer moderate relative
turbulence. Atvelocity
the high-radiusfluid in
in this
location
fluid region
close
this to the
region and the
the intensified
andouter diameter interaction
intensified (x = 0.75), the between
effectthe shear layer is
of roughness and turbu
somewhat diminished,
At
At the and despite
the high-radius
high-radius location the higher
location close rotational
close to speed of the fluid, the increase
to the outer diameter (x = 0.75), the effect of roughn in
velocity at the high radius diminished,
somewhat
somewhat fails to be asand
diminished, significant
and despite as
despite thethat at the
higher mid-radius,
rotational speedasofthe thefluid
fluid,isthe incre
already strongly perturbed
velocity
velocity at
at the
theathigh
the mid-radius.
high radius
radius fails fails totoHowever,
be the roughness
be as significant as that at still
theexerts some as the fl
mid-radius,
enhancement onalreadythe fluid velocity, which is manifested as a sustained increase in velocity.
already strongly
strongly perturbed
perturbed at at the
the mid-radius. However, the roughness still exerts
Under centripetal through-flow conditions, the influence of disk roughness on the
enhancement
enhancement on the fluid velocity, which is manifested as a sustained increase in vel
on the fluid velocity,
velocity changes across the three radial locations displays a more uniform character, con-
Under centripetal through-flow conditions, the influence of disk roughness o
trasting slightly withUnderthe trend centripetal
observedthrough-flow
in centrifugal through-flow. However, the largest
velocity
velocity changes
changes across across the the three
three radial locations displays a more uniform char
contrasting
contrasting slightly
slightly with with the the trend
trend observed in centrifugal through-flow. Howeve
largest
largest increase
increase is is still
still exhibited
exhibited at at the mid-radius location (x = 0.5). This indicate
despite
despite the the different
different flow flow directions,
directions, the effect of roughness is still most prominent
mid-radius location,
mid-radius location, reflecting
reflecting the the sensitivity of the flow characteristics to roughn
Water 2024, 16, 3678 12 of 14

increase is still exhibited at the mid-radius location (x = 0.5). This indicates that despite the
different flow directions, the effect of roughness is still most prominent at the mid-radius
Water
location, reflecting the sensitivity of the flow characteristics to roughness at the mid-radius.
Water2024,
2024,17,
17,xxFOR
FOR PEER
PEER REVIEW
REVIEW 13
13
The analysis of velocity, turbulent kinetic energy, and core swirl ratio distributions for
different flow modes in the disk cavity can provide important guidance for design and
optimization in engineering applications. These findings can help to improve the cooling
effect and enhance equipment, as
as well
the stability
equipment, andas
well the
heat
as reliability
thetransfer and
and performance
efficiency
reliability of
of the
the equipment,
of the equipment,
performance by
as well as by
equipment, controllin
thecontrollin
roughness
reliability and performance effect.
of the equipment, by controlling the roughness effect.
roughness effect.

(a) xx == 0.25
(a) 0.25 (b) xx == 0.5
(b) 0.5 (c) x
(c) x == 0.75
0.75
Figure
Figure
Figure 11. Velocity 11.
11. Velocity
transientVelocity transient
transient
evolution evolution
evolution
for different for different
different radial
for positions
radial radial positions in centrifugalin
positionsthrough-flow
in centrifugal in centrifugal through-flo
through-flo
rotor–stator
rotor–stator cavity. cavity.
rotor–stator cavity.

(a)
(a) xx == 0.25
0.25 (b) x = 0.5 (c) x = 0.75

Figure 12. Velocity transient

Figure
Figure 12. evolution
12. Velocity
Velocity for different
transient
transient radial
evolution for positions in centripetal
different radial positionsthrough-flow
in centripetalin
through-flo
rotor–stator cavity.
rotor–stator
rotor–stator cavity.
cavity.
4. Conclusions
4.
4. Conclusions
Conclusions
This paper investigates the transient flow characteristics of the disk cavity under both
This
This paper
centrifugal and centripetal paper investigates
investigates
through-flow the transient
conditions flow characteristics
utilizing of the diskand
numerical simulations cavity unde
centrifugal
experimental validation. and
centrifugalItand centripetal
focuses through-flow conditions utilizing
on the effects of roughness on the turbulent kinetic
centripetal numerical simulation
energy, core swirl ratio, total pressure
experimental
experimental validation.
validation. distribution,
It focuses and
on thetheeffects
transient evolution of
of roughness onvelocity
the turbulent k
at the rotor surface within
energy, corethe cavity.
swirl
energy, core swirl The
ratio, findings
total of
pressure the study
distribution,are as
and follows:
the transient evolution of ve
1. The analysisat
at the
the rotor
rotor surface
indicates surface within theroughness
that increased cavity. The findings
leads to anofoverall
the study are as follows:
pressure drop
within the1. cavity,
1. The while
The analysis high-pressure zones form in the high-radius region
analysis indicates that increased roughness leads to an overall pressure under
centrifugal flow.within
In centripetal
within the
the cavity, flow,
cavity, the pressure
while within zones
high-pressure the cavity
form exhibits
in the ahigh-radius
significant region u
rising trend over time, with roughness
centrifugal further enhancing this phenomenon. These
centrifugal flow. flow. In centripetal flow, the pressure within the cavity exhib
findings offer valuable insights into the mechanisms driving changes in pressure
significant
significant rising trend over time, with roughness further enhancing
distribution across various flow modes.
phenomenon.
phenomenon. These findings offer valuable insights into
2. The transient flow analysis indicates that roughness significantly influencesthe themechanisms
distri- dr
changes
changes in
in pressure
pressure distribution across various flow modes.
bution of turbulent kinetic energy within the cavity. In centrifugal flow, the roughness-
2.
2. The
induced turbulenceThe transient
transient
enhancement flow isanalysis indicates
particularly evidentthatatroughness
the mid-radius significantly
positioninfluence
(x = 0.5), wheredistribution
distribution
the local velocityof turbulent
increasekinetic
reachesenergy within the
its maximum. Thiscavity.
increase Indirectly
centrifugal flow
roughness-induced
influences the evolution turbulence enhancement is particularly
of flow structures, promoting the generation and develop-
roughness-induced evident at the mid-r
ment of vortices. In centripetal
position
position (x
(x = 0.5), flow,
wherealthough the distribution
the local of TKE
velocity increase is relatively
reaches its maximum.
increase
increase directly
directly influences the evolution of flow structures, promoting
generation
generation and development of vortices. In centripetal flow, although
distribution
distribution of TKE is relatively uniform, the increase in flow velocity d
roughness
roughness is is notably pronounced at the mid-radius position.
Water 2024, 16, 3678 13 of 14

uniform, the increase in flow velocity due to roughness is notably pronounced at the
mid-radius position.
3. The increase in the Reynolds number leads to a significant impact on the transient
flow characteristics in both flow modes. In centrifugal flow, higher Reynolds numbers
result in a marked increase in turbulence intensity, with the turbulent kinetic energy
rising by approximately 22% as the Reynolds number increases from 8.85 × 105 to
1.77 × 106 . This enhancement in turbulence intensity promotes the conversion of
kinetic energy into static pressure, resulting in the development of high-pressure
zones, where pressure peaks increase by up to 18% at the outermost radius. Similarly,
in centripetal flow, an increase in the Reynolds number brings about notable changes
in the pressure distribution within the cavity, with pressure increasing by about
12% in the high-radius region as the Reynolds number increases from 8.85 × 105
to 1.77 × 106 . These findings highlight the critical role of the Reynolds number in
influencing both flow behavior and pressure distribution in rotor–stator cavities under
varying operating conditions.
4. Roughness significantly accelerates the transient response of flow in both centrifugal
and centripetal flows. In centrifugal flow, roughness enhances turbulent mixing,
resulting in a marked increase in local velocity and TKE, especially in the high-radius
district, where the local turbulent kinetic energy experiences a dramatic rise. This
phenomenon accelerates fluid flow rates and has a significant impact on pressure
distribution and flow structure within the cavity.
In conclusion, this study provides a thorough analysis of the transient flow character-
istics of the disk chamber under different flow modes and roughness conditions. The influ-
ence of roughness on the turbulent kinetic energy and flow structure is more pronounced
in centrifugal flow, while its effects on centripetal flow are more localized. These results
offer valuable theoretical insights for the design and optimization of rotor–stator cavity
systems under transient conditions, which are particularly relevant for high-performance
equipment such as pumps.

Author Contributions: Conceptualization, C.W. and Y.Y.; methodology, J.G. and Y.W.; software, Y.Y.,
C.W., H.L., Y.W. and H.C.; writing—original draft preparation, Y.Y.; writing—review and editing,
C.W. and L.Z.; visualization, Y.Y. and H.L. All authors have read and agreed to the published version
of the manuscript.
Funding: This research was funded by the Science and Technology Program of Jiangxi Education
Department (GJJ2209004), the Research project on Teaching Reform of Yangzhou University in 2022
(YZUJX2022-D17), the China Postdoctoral Science Foundation (2023M732828 and 2024T170726), and the
Open Foundation of Key Laboratory of Water-Saving Agriculture of Henan Province (KLWSAHP-2023-01).
Data Availability Statement: All data are contained within the article.
Conflicts of Interest: Author Jie Ge was employed by the company SHIMGE Pump Co., Ltd. The
remaining authors declare that the research was conducted in the absence of any commercial or
financial relationships that could be construed as a potential conflict of interest.

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