Study on the Influence of Tip Clearance on Cavitation Performance and Entropy Production of an Axial Flow Pump

Abstract

The clearance existing between the impeller rim and the adjacent shroud within the pump configuration establishes conducive circumstances for the initiation of cavitation. The bubbles generated by cavitation will flow forward with the water, blocking the channel, and result in the degradation of the pump performance. When the cavitation is severe, vibration and noise will be generated. The impact formed by the collapse of the bubbles will seriously erode the blades and form pits on the blade surfaces. Drawing upon the outcomes derived from numerical simulations, this paper investigates the relationship between tip clearance and cavitation in an axial flow pump, with a specific focus on energy dissipation characteristics. The principal findings indicate that the dimensions of the tip clearance predominantly influence the spatial distribution of the tip leakage vortex (TLV) cavitation. The entropy production rate distribution at the tip correlates with both the cavitation level of the pump and the extent of the tip clearance. The shedding phenomenon of the TLV becomes more evident when analyzing the distribution of entropy production rates. During cavitation, an increased tip clearance is associated with a reduction in the dissipation of viscous entropy production within the impeller domain, and the entropy production resulting from turbulent dissipation significantly surpasses that arising from viscous dissipation.

1. Introduction

Axial flow pumps are commonly employed in coastal pump stations as well as in water jet propulsion systems for boats. The impeller stands as the central component of the axial flow pump. To meet operational specifications and mitigate potential friction issues, it is unavoidable that a clearance exists between the tip and the casing [1]. The intricacy of cavitation at the rim is heightened by the presence of tip clearance. A wide variety of types of cavitation interfere with each other, block the flow channel, and result in vibrations, noise, and performance degradation [2].

Many scholars investigated the cavitation and tip clearance of hydraulic machinery through numerical simulation and experiments. Ma et al. [3] conducted an analysis of the flow characteristics within the tip clearance of a Kaplan turbine through the utilization of ANSYS CFX simulations. Song et al. [4] elucidated the significance of the tip leakage flow through an examination of the flow field in the tip region. Han et al. [5,6] applied large eddy simulation in conjunction with the Zwart–Gerber–Belamri cavitation model to numerically analyze turbulent cavitation flow near the tip clearance of a propulsion pump. In the experiment, Liu et al. [7]. used high-speed photography and pressure pulsation sensors to study tip clearance cavitation in a bidirectional axial flow pump station. Zhang et al. [8] studied the tip clearance effect on the open-water characteristics and the flow of the ducted propeller through simulations and revealed the complete life cycle progression of the TLV across various tip clearance dimensions. Zhang et al. [9] carried out experimental investigations to study the influence of uneven tip clearances on the aeroacoustic characteristics of centrifugal compressors. Jahani et al. [10] utilized unsteady three-dimensional analysis to study the tip injection mechanism for stability enhancement of a transonic centrifugal impeller with different tip clearances for a clean or distorted inflow. Chung et al. [11] clarified the impact of varying tip clearance on the aerodynamic loss in a gas turbine. The study involved adjusting the tip clearance and analyzing the total pressure field for each specific case. Xiang et al. [12] performed a sequence of experiments to investigate the effect of tip clearance size on cavitation characteristics in a turbopump inducer and found that large tip clearance distinctly reduces both the non-cavitation and cavitation performance. Guo et al. [13,14] studied the effect of tip clearance on the cavitating flow within a shunt inducer, revealed the flow mechanism of tip clearance, and found that the tip clearance has a certain influence on the critical cavitation coefficient. Zhang et al. [15] applied numerical simulations to study the turbulence caused by cavitation in an axial flow pump and analyzed the origin and the evolution of the perpendicular cavitation vortex in detail. Sun et al. [16] used the large eddy method combined with the Schnerr–Sauer cavitation model to study the dynamic phenomenon on the free surface of the NACA66 hydrofoil. The effect of tip clearance width on the flow in tubular turbines was investigated by Wu et al. [17] who found that the increase in tip clearance width decreases the lowest pressure at the tip and leads to cavitation in advance. Wang et al. [18] employed the large eddy simulation model and the Schnerr–Sauer cavitation model to simulate tip leakage cavitating flow and found that the tip clearance size has a large impact on the cavitation. Karakas et al. [19] studied the inducer tip clearance and its impact on the cavitating and non-cavitating performance of centrifugal pumps by simulations. Shen et al. [20] created and numerically studied a scaled axial flow pump model through the combination of an improved SST k-w turbulence model and a homogeneous cavitation model, and analyzed the effect of blade tip gap size on the TLV dynamics and cavitation. Nichik et al. [21] conducted an experimental study on the inception and development of tip clearance cavitation and found that an increase in the gap size caused the tip clearance cavitation to be initiated at higher cavitation numbers. Wang et al. [22] studied the evolutionary characteristics of a suction-side-perpendicular cavitating vortex in an axial flow pump operating under low flow conditions. Mousmolis et al. [23] examined the cavitation characteristics of a radial flow centrifugal pump equipped with a semi-open impeller and analyzed the cavitation phenomenon based on the numerical results. Certain researchers employed the entropy production theory in the examination of hydraulic machinery. Shen et al. [24] introduced the entropy production theory to investigate the cavitation flow in axial flow pump, and unit performance was studied through numerical simulation. Shu et al. [25] applied enstrophy dissipation theory innovatively to provide guidelines for controlling energy dissipation associated with the TLV and to optimize the design of the multiphase pump. Lai et al. [26] employed a local entropy production method to analyze cavitation flow with varying NPSH. Yu et al. [27] studied the inducing mechanism of entropy production, revealed the influence of cavitation on entropy production, and found that velocity gradient and the cavitation process are the key reasons for entropy production. Li et al. [28] simulated three-dimensional, unsteady cavitating flows of organic fluid R245fa in a diaphragm pump and clarified the entropy production rate during cavitation. Fei et al. [29] studied the cavitating flow in a slanted axial flow pump based on the entropy production theory and vortex dynamics. Yang et al. [30] studied the flow loss characteristics of a 30-degree oblique axial flow pump model according to the method of the entropy production theory and revealed the hydraulic loss characteristics during the internal flow of the oblique axial flow pump. Kan et al. [31] investigated the flow dynamics and the corresponding energy loss characteristics of axial flow pumps as turbines (PAT) using the numerical simulation method coupled with the entropy production theory and proposed the spatial variation pattern for energy losses in pump and turbine operating modes.

Previous research on the relationship between tip clearance and cavitation mainly focuses on the evolution analysis of TLV, but there are few studies on the energy dissipation. This paper combines the entropy production theory with the cavitation characteristics of an axial flow pump and analyzes the influence of different tip clearances on cavitation performance. For the axial flow pumps with different tip clearances, the distribution of vapor volume fraction and entropy production rate on the blade sections and cascade surface at the tip is shown, and the relationship between the entropy production and cavitation coefficient is analyzed. Finally, the influence of the tip clearance on the axial force of the impeller during the cavitation process is analyzed, and the conclusions of this paper are summarized. This study can provide some theoretical guidance for the influence of tip clearance on the cavitation performance and energy dissipation of axial flow pumps.

2. Calculation Model and Scheme Parameters

2.1. Geometric Model

This paper delves into the analysis of an axial flow pump, wherein numerical simulations are executed utilizing the UG NX 12.0 modeling software, the meshing software, and the numerical calculation commercial software ANSYS 19.0. The computational domain is depicted in Figure 1. The impeller possesses a diameter of 300 mm with three blades. The guide vane blades amount to seven, and the impeller speed is maintained at 1400 revolutions per minute.

The structure of this paper encompasses the following main sections: 1. Utilizing the RNG k-ε turbulence model in conjunction with the Z–G–B (Zwart–Gerber–Belamri) cavitation model, the investigation is conducted under the specified design flow rate of 1.0 Qd. CFX 19.0 software is employed to model the cavitation dynamics in the axial flow pump, considering various tip clearances (1.0 mm, 2.0 mm, and 4.0 mm) under different inlet pressure (0.3~0.9 atm). 2. The impact of varied tip clearances on the cavitation behavior in the axial flow pump is analyzed. 3. In accordance with the entropy production theory, the correlation between tip clearance and cavitation distribution is scrutinized from the standpoint of energy dissipation in this investigation.

2.2. Grid Division

In this paper, ANSYS ICEM 19.0 is utilized for the partitioning of unstructured grids pertaining to the inlet and outlet channels, along with the guide vanes. This process aims to discretize the grids. Figure 2 depicts the mesh subdivision within the computational domain of the pump section.

Since the calculation content in this paper involves different tip clearances, the impellers with different tip clearances were meshed by Turbo Grid 19.0, and the mesh was refined at the tip. The refinement effect is visually represented in Figure 3.

The Y Plus values for three tip clearances employed in the numerical analysis are outlined below (Figure 4).

2.3. Grid Independence Verification

For fluid calculation, the precision of the computational outcomes is significantly influenced by the quality of the grid. Given the dimensions of the axial flow pump and the constraints imposed by computer capabilities, this study employs diverse schemes for partitioning the model. Specifically, a 1 mm tip clearance for the pump section was used as an illustration, and nine division schemes with the number of grid cells of 0.54 million, 1.26 million, 2.31 million, 3.26 million, 4.29 million, 5.87 million, 7.28 million, 8.81 million, 9.13 million, 10.17 million, and 13.49 million were obtained, respectively. The pump efficiency of the calculation model is used as the characteristic parameter of the grid independence analysis.

Figure 5 indicates the pump efficiency of the calculation model under different grid partitioning schemes. Upon surpassing 7.28 million grid cells, the pump section attains an efficiency of 82.91%, and the aberration is within ±2%, which meets the requirements. When the number of grid cells exceeds 8.81 million, the efficiency is almost unchanged, and this grid partitioning scheme can better meet the research requirements. Finally, the number of grid cells in the case of structural division is 9.13 million.

3. Numerical Simulation and Boundary Conditions

3.1. Entropy Production Theory

There is irreversible dissipation in various energy conversion processes in nature, and the rise in entropy resulting from dissipation constitutes entropy production. For fluid research, entropy production serves as a viable approach for examining the spatial distribution of hydraulic dissipation within fluids. Mechanical work dissipation leads to entropy increment, and entropy production generally comes from heat transfer or dissipation. Taking into account that temperature change is usually not considered in hydraulic machinery, and that this paper mainly studies the impact of the tip clearance, the calculation process does not include heat transfer. The energy in the fluid is irreversibly converted by the viscous force within the boundary layer, resulting in entropy production. Turbulent fluctuation induced by the high Reynolds number flow in the pump will also cause hydraulic loss. The formula expressing the rate of entropy production due to viscous force dissipation is as follows:

$${\dot{S}}_{\overline{D}}‴=\frac{\mu }{T}\left[{\left(\frac{\partial \overline{u}}{\partial y}+\frac{\partial \overline{v}}{\partial x}\right)}^2+{\left(\frac{\partial \overline{u}}{\partial z}+\frac{\partial \overline{w}}{\partial x}\right)}^2+{\left(\frac{\partial \overline{v}}{\partial z}+\frac{\partial \overline{w}}{\partial y}\right)}^2\right]+\frac{2\mu }{T}\left[{\left(\frac{\partial \overline{u}}{\partial x}\right)}^2+{\left(\frac{\partial \overline{v}}{\partial y}\right)}^2+{\left(\frac{\partial \overline{w}}{\partial z}\right)}^2\right]$$

(1)

The formula expressing the rate of entropy production due to turbulent fluctuating dissipation is as follows:

$${\dot{S}}_{D^{\prime }}‴=\frac{\mu }{T}\left[{\left(\frac{\partial u^{\prime }}{\partial y}+\frac{\partial v^{\prime }}{\partial x}\right)}^2+{\left(\frac{\partial u^{\prime }}{\partial z}+\frac{\partial w^{\prime }}{\partial x}\right)}^2+{\left(\frac{\partial v^{\prime }}{\partial z}+\frac{\partial w^{\prime }}{\partial y}\right)}^2\right]+\frac{2\mu }{T}\left[{\left(\frac{\partial u^{\prime }}{\partial x}\right)}^2+{\left(\frac{\partial v^{\prime }}{\partial y}\right)}^2+{\left(\frac{\partial w^{\prime }}{\partial z}\right)}^2\right]$$

(2)

In the numerical calculation using RANS, the direct reflection of turbulent fluctuation velocity is not feasible but can be reflected by the ε equation. Equation (2) can be written as follows:

$${\dot{S}}_{D^{\prime }}‴=\frac{\rho \epsilon }{T}$$

(3)

where $\overline{u}$, $\overline{v}$, and $\overline{w}$ are the components of the average velocity in the direction of x, y, and z. $u^{\prime }$, $v^{\prime }$, and $w^{\prime }$ are the components of the pulsation velocity in the x, y, and z direction. $\mu$ is the fluid dynamic viscosity. $T$ is the temperature, which takes a constant room temperature of 298 K in the calculation.

The total entropy production rate in each calculation domain is expressed as the summation of the particle entropy production dissipation rate ($\mathrm{W}\cdot {\mathrm{m}}^{-3}\cdot {\mathrm{K}}^{-1}$):

$$S_D{}^{\prime }={\dot{S}}_{\overline{D}}‴+{\dot{S}}_{D^{\prime }}‴$$

(4)

The total entropy production value can be calculated by the formula as follows:

$$S_D={\displaystyle \underset{V}{\int }{\dot{S}}_{\overline{D}}‴\mathrm{d}V}+{\displaystyle \underset{V}{\int }{\dot{S}}_{D^{\prime }}‴}\mathrm{d}V$$

(5)

where $S_D$ is the total entropy output value, and V is the volume of the computational domain.

3.2. Turbulence Model

The numerical simulation is based on the Reynolds time-averaged N-S control equation and the RNG k-ε turbulence model, which is aimed to simulate the flow pattern. The discrete equations are solved by the separation solver. Euler single-fluid method stands as a commonly employed numerical approach for calculating cavitation flow. According to the method, the mixture is regarded as a single fluid in this paper by using the Euler single-fluid method, and its density is related to the vapor volume fraction α_v:

$${\rho }_m=(1-a_v){\rho }_l+a_v{\rho }_v$$

(6)

The momentum of the mixed fluid can be calculated as follows:

$${\rho }_m{u}_{mi}=(1-a_v){\rho }_l{u}_{li}+a_v{\rho }_v{u}_{vi}$$

(7)

The two phases have the same pressure, and the velocity difference between the two phases can be considered in the model by additional source terms. The derivative terms with respect to time ‘t’ in the following equations are essential during computations before the flow stabilizes. Therefore, these derivative terms with respect to time are retained in both the turbulence model and the Z–G–B cavitation model. The solvability is limited to the continuity equation and momentum equation for the mixed fluid:

$$\frac{\partial {\rho }_m}{\partial t}+\frac{\partial }{\partial x_i}({\rho }_m{u}_{mi})=0$$

(8)

$$\frac{\partial }{\partial t}({\rho }_m{u}_{mi})+\frac{\partial }{\partial x_j}(\rho u_{mi}{u}_{mj})=\frac{\partial {\tau }_{ij}}{\partial x_j}+\rho f_i+S$$

(9)

where m denotes the relevant field quantity of the mixed fluid, and S represents the source term considering the velocity difference between two phases.

3.3. Cavitation Model

The Z–G–B cavitation model is employed for simulating the cavitation phenomenon in the axial flow pump. The model is based on the simplified Rayleigh–Plesset equation and ignores the second derivative of the cavity radius with time, focusing on the influence of the change in the cavity volume in the initial and development stages of cavitation.

The Rayleigh–Plesset equation is as follows:

$$\rho (R\ddot{R}+\frac{3}{2}{\dot{R}}^2)=P_V-P_{\infty }(t)+P_{g0}{(\frac{R_0}{R})}^{3\gamma }-\frac{2S}{R}-4\mu \frac{\dot{R}}{R}$$

(10)

The equation expressing the rate of mass change for an individual cavity in this model is as follows:

$$\frac{\partial m_B}{\partial t}={\rho }_V\frac{\partial V_B}{\partial t}=4\pi R_B^2{\rho }_V\sqrt{\frac{2\left(P_V-P\right)}{3{\rho }_l}}$$

(11)

The vapor volume fraction equation is as follows:

$$a_V=N_B{V}_B=\frac{4}{3}\pi R_B^2{N}_B$$

(12)

In the formulas above, $P_V$ is the saturation pressure of vapor. $\rho$ is the density of water, 1000 kg/m³. $N_B$ represents the quantity of bubbles per unit volume. $R_B$, taking the value of 106 m generally, is the radius of bubbles. The formula of total interphase mass transport rate per unit volume is as follows:

$$m_l=N_B\frac{\partial m_B}{\partial t}=\frac{3a_V{\rho }_V}{R_B}\sqrt{\frac{2\left(P_V-P\right)}{3{\rho }_l}}$$

(13)

The collapse of bubbles is described by the equation as follows:

$$R_e=F\frac{3a_V{\rho }_V}{R_B}\sqrt{\frac{2\left|P_V-P\right|}{3{\rho }_l}}sign\left(P_V-P\right)$$

(14)

The equation of the Zwart–Gerber–Belamri cavitation model is as follows:

$$m=\{\begin{array}{ll}{F}_{vap}\frac{3a_{nuc}\left(1-a_V\right){\rho }_V}{R_B}\sqrt{\frac{2\left(P_V-P\right)}{3{\rho }_l}},& P\le P_V\\ F_{cond}\frac{3a_V{\rho }_V}{R_B}\sqrt{\frac{2\left(P-P_V\right)}{3{\rho }_l}},& P>P_V\end{array}$$

(15)

where $a_{nuc}$ is the volume fraction of the cavity core, 5 × 10⁻⁴. Considering that the evaporation process is much faster than the condensation process, the vaporization empirical correction coefficient $F_{vap}$ is taken as 50, and the condensation empirical correction coefficient $F_{cond}$ is 0.001.

The cavitation coefficient σ is used to characterize the cavitation conditions of the pump. The calculation formula of the cavitation coefficient σ is as follows:

$$\sigma =\frac{P-P_V}{0.5\rho u^2}$$

(16)

where $P$ is the total pressure at the impeller inlet, Pa. $u$ is the circumferential velocity of the impeller outlet section, m/s.

3.4. Boundary Conditions

To enhance the precision and convergence of the numerical simulation, the initial values for cavitation calculation are derived from the steady results obtained in the absence of cavitation. The solid walls are set as no-slip walls. The exit of the outlet channel is configured as a mass flow outlet, while the entrance of the inlet channel is designated as a total pressure inlet. The mass flow rate is set to 320 kg/s. The reference pressure is established at 0 atm. In cavitation scenarios, the liquid phase at the inlet is defined as pure water with a temperature of 25 °C. The dynamic viscosity of water at 25 °C is set to $8.89\times {10}^{-4}$ kg/m s. The water volume fraction is set at 1, while the gas phase is specified as vapor at 25 °C, with a vapor volume fraction of 0 and a saturation pressure of 3574 Pa. The dynamic viscosity of vapor at 25 °C is set to $9.86\times {10}^{-6}$ kg/m s. In the steady calculation, the dynamic and static interfaces are set as frozen rotors, and the method of gradually reducing the inlet pressure is adopted to simulate the pump at different cavitation stages.

4. Results and Analysis

4.1. Experimental Device

The experimentation is conducted on a high-precision hydraulic machinery test bench. The tip clearance δ for the axial flow pump used in the experiment is 1.0 mm. The experimental device is shown in Figure 6.

In the test, the flow rate is measured by the DN400 electromagnetic flowmeter. The head of the pump device is equal to the overall energy variation between the pressure measuring points located at the inlet and outlet pipes. The torque meter is used to measure the input torque of the pump. The absolute pressure transmitter is used to measure NPSH. The uncertainty of the experimental device is listed in

Table 1. Experimental uncertainty.

Measuring Instrument	Measurement Interval	Uncertainty
Differential pressure transmitter	0~200 kPa	$U_H=\pm 0.15\%$
Electromagnetic flowmeter	DN400 mm	$U_Q=\pm 0.1\%$
Torque meter	500 N·m	$U_T=\pm 0.15\%$
Absolute pressure transmitter	0~130 kPa	$U_N=\pm 0.01\%$

.

The uncertainty of the system is as follows: $U_S=\sqrt{U_H^2+U_Q^2+U_T^2+U_N^2}=0.235\%$.

4.2. Comparison of External Characteristics

In order to further analyze the numerical simulation results, the numerical results are compared with the experimental data. The tip clearance of the impeller used in the numerical simulation of the external characteristics is 1.0 mm. The head–flow rate curve and efficiency flow curve are shown in Figure 7a. The calculated simulation value of head H closely aligns with the experimental value, and there is no hump phenomenon. Since the numerical study does not consider the mechanical loss, the calculated simulation values of head H and efficiency η are generally higher than the experimental values. According to Figure 7b, the relative uncertainty of head H is less than 3%, and that of efficiency η is less than 4%, which verifies the validity of numerical simulation. The relative uncertainty is mainly caused by the installation uncertainty of the experimental device, the uncertainty of the experimental measurements, and the wall roughness, which is not considered in numerical simulation.

4.3. Comparison of Cavitation Characteristics

During the experiment, a vacuum pump is employed to decrease the inlet pressure, so as to achieve the purpose of changing the net positive suction head (NPSH) of the pump. When the vacuum pump is running, we open the vacuum valve and close the inlet and outlet valves. When the vacuum indicator almost reaches the required value, the vacuum valve should be closed first, then the vacuum pump is stopped, and finally the inlet and outlet valves are opened for the experiments. The tip clearance δ of the impeller used in the experiment is 1.0 mm, and the tip clearance δ of the numerical calculation model is set as 1.0 mm, 2.0 mm, and 4.0 mm, respectively. The cavitation calculation is carried out to create a comparative study of the cavitation performance of the axial flow pumps with different clearances under the design condition, and the curve is shown in Figure 8. Compared with the experimental values, the trends of the curves exhibit a high degree of similarity, confirming the dependability of the numerical simulation.

As depicted in Figure 8, as the cavitation coefficient σ decreases, the head is basically stable at first, then rises slightly, and finally decreases rapidly. The main reason for the stable phenomenon is that there are few cavitation bubbles in the impeller domain at the initial stage of cavitation, which has little effect on the performance of the pump device. As the cavitation deepens, a mass of cavitation bubbles are attached to the blade surface, and a hydraulic smooth zone is formed near the blade surface which reduces the flow resistance loss near the wall and increases the head slightly. When the cavitation develops from the critical cavitation (head decreases by 3%) to the full cavitation, a large number of cavitation bubbles fall off from the blade, gathering and hindering the water flow. As a result, the head decreases rapidly. It can be seen from the figure that the critical cavitation coefficient of the impeller with the tip clearance of 1 mm is near 0.16, and the cavitation head curve of the impeller used in the experiment is slightly lower than that of the impeller with a tip clearance of 1 mm in the numerical simulation, which is mainly caused by the hydraulic loss caused by the wall roughness. For the axial flow pumps of the same model with varying tip clearances, when the cavitation coefficient is constant, the primary reason for the discrepancy in head arises from the increase in the tip clearance, leading to a corresponding augmentation in leakage flow through the clearances. The tip leakage flow causes a reduction in the pressure differential between the suction side and pressure side of the blade. This results in a decrease in the work performed by the impeller on the water flow, consequently lowering the head.

4.4. Distribution of Vapor Volume Fraction

To delve deeper into the flow characteristics within pumps featuring distinct tip clearances while maintaining a consistent cavitation coefficient, this paper takes the cavitation coefficients of 0.16 and 0.20 as examples to analyze the numerical simulation results of the pumps characterized by varying tip clearances. In particular, cavitation distribution characteristics and cavitation vortex are analyzed.

Figure 9 shows the isosurface of the vapor volume fraction of 10% in the impeller domain. The figure illustrates that at a consistent cavitation coefficient, the cavitation distribution locations vary for impellers with different tip clearances. As the tip clearance increases, the main distribution position of cavitation shifts from the leading edge to the trailing edge. For the impeller with a certain tip clearance, with the deepening of cavitation, a cavitation distribution area extending from the tip to the hub appears on the suction surface of the blade. It can also be seen from the figure that the cavitation in the tip clearance is more serious, which is because there is a pressure difference between the pressure surface and the suction surface of the blade, and there is inevitable relative motion between the tip of the blade and the wall surface when the impeller rotates. The flow velocity at the tip of the blade is larger, so the cavitation at the tip of the blade is more severe.

To investigate the vapor volume fraction distribution on the blade sections, five circumferentially uniformly distributed sections are selected on the blades of impellers with different tip clearances. The distribution of vapor volume fraction on the blade sections is shown in Figure 10.

It can be seen from Figure 10 that for the impellers with three kinds of tip clearances simulated, the cavitation is mainly distributed on the suction surface and the tip clearance of the blade at the stage of critical cavitation and full cavitation, and the cavitation becomes more and more severe from the leading edge to the trailing edge. Comparing the diagrams of the same cavitation coefficient in Figure 10, it is evident that impellers featuring varying tip clearances exhibit discernible differences. As the tip clearance increases, there is a corresponding intensification in the concentration of cavitation within the tip clearance section. Additionally, the regions with higher vapor volume fraction values within the tip clearance are also more extensive, whereas the thickness of sheet cavitation on the suction surface decreases. This phenomenon is also reflected in the TLV, as shown in Figure 11.

Vorticity is utilized for delineating the rotational movement of fluid particles. In accordance with the Cauchy–Stokes decomposition, it is defined as twice the angular velocity when a fluid particle experiences rigid body motion revolving around its central axis.

$$\nabla V=A+B=\frac{1}{2}(\nabla V+\nabla V^{\mathrm{T}})+\frac{1}{2}(\nabla V-\nabla V^{\mathrm{T}})$$

(17)

where the velocity gradient tensor, symbolized as $\nabla V$, can be decomposed into two parts: its symmetric component, designated as A, and its antisymmetric component, denoted as B.

Given that vorticity alone is insufficient to fully characterize fluid rotation, a more detailed decomposition involves separating it into two components: vorticity R related to rotational motion and vorticity S of the non-rotational component.

$$\omega =R+S$$

(18)

The dimensionless parameter $\Omega$ is employed to signify the proportion of vorticity attributed to rotational motion in relation to the total vorticity.

$$\Omega =\frac{{\Vert B\Vert }_{\mathrm{F}}^2}{{\Vert A\Vert }_{\mathrm{F}}^2+{\Vert B\Vert }_{\mathrm{F}}^2+\epsilon }$$

(19)

where ${\Vert B\Vert }_{\mathrm{F}}^2$ signifies vorticity linked to rotational motion, whereas ${\Vert A\Vert }_{\mathrm{F}}^2$ denotes the vorticity of non-rotational component. $\epsilon =C_K\times {\left({\Vert B\Vert }_{\mathrm{F}}^2-{\Vert A\Vert }_{\mathrm{F}}^2\right)}_{\max }$, where $C_K$ is a constant, and $\epsilon$ approaches infinitesimally small values. The value of $\Omega$ provides a dependable gauge of the rotational intensity within the local fluid [32].

Figure 11 shows the isosurface distribution of vapor volume fraction and the Omega vortex identification diagram of the TLV of the impellers with three different tip clearances at the stage of full cavitation (σ = 0.11). As evident in the figure, an increase in tip clearance results in a larger TLV, and the TLV position shifts closer to the suction surface of the blade. The gray part of the diagram is the isosurface of vapor volume fraction of 60%, and the part marked by the red ellipse is the TLV cavitation area. Under the same cavitation coefficient σ, the TLV cavitation becomes more serious with the increase in tip clearance. For the impeller with a large tip clearance (δ = 4.0 mm), the TLV cavitation is highly integrated with the shear layer cavitation, which results in a further expansion of the isosurface at the clearance.

To investigate the impact of tip clearance on the cavitation distribution across the cascade surface at the tip clearance, the cascade surfaces are selected at the tip position of the impellers with different tip clearances. The impellers, each featuring a distinct tip clearance, share a common radius of 150 mm. Therefore, for the impeller with a tip clearance of 1.0 mm, the cascade surface of 0.993 span is selected. For the impeller with a tip clearance of 2.0 mm, the cascade surface of 0.987 span is selected. For the impeller with a tip clearance of 4.0 mm, the cascade surface of 0.973 span is selected.

Figure 12 indicates the distribution of vapor volume fraction on the cascade surfaces of three kinds of impellers with different tip clearances under different cavitation coefficients. PS represents the pressure surface of the blade, and SS represents the suction surface. For the impeller with a certain tip clearance, the TLV cavitation develops along the tip region as the cavitation coefficient decreases, and the TLV cavitation region gradually approaches the suction surface, resulting in the enhancement of cavitation at the tip of the trailing edge and the formation of cloud cavitation A. At the stage of full cavitation (σ = 0.11), there is an obvious cavitation region B below the suction surface of the blade, which is caused by the TLV cavitation shedding of the previous blade. Comparing Figure 12a–c, it can be seen that under the same cavitation coefficient, as the tip clearance enlarges, there is a reduction in the vapor volume fraction at the tip, while the cloud cavitation A at the tip of the trailing edge increases. The cavitation region B beneath the suction surface is adjacent to the casing. The chosen cascade surface is more distant from the shroud as a result of the increased tip clearance, so the cavitation region B is less obvious.

4.5. Distribution of Entropy Production

When analyzing the hydraulic loss during the operation of the pump, the total hydraulic loss is often indirectly evaluated by the efficiency formula, and the specific distribution of energy loss in a certain part cannot be directly judged. Entropy production is an effective tool that can intuitively reflect the distribution of irreversible loss energy consumption inside the fluid. To investigate the impact of the tip clearance on the energy dissipation within the impeller domain when cavitation occurs in the pump, this paper uses the entropy production theory to analyze the energy dissipation distribution of the impeller and the change in the entropy production value.

Figure 13 shows the entropy production rate distribution on the blade sections of the impellers with different tip clearances when the cavitation coefficient is 0.16.

In Figure 13, it is shown that the high-value region of the entropy production rate at the tip of the blade section is continuously expanding from the leading to the trailing edge. With the augmentation in tip clearance δ, the concentrated high-value region shifts toward the suction surface, which is also consistent with the distribution of vapor volume fraction shown in Figure 10 and the identification of Omega vortex shown in Figure 11. On the section of the same chord position of the blade, the high-value region of the entropy production rate of the impeller with a tip clearance of 2.0 mm is larger than that of the impeller with a tip clearance of 1.0 mm, as depicted in Figure 13a,b. From Figure 13c, it can be seen that under the large tip clearance of 4.0 mm, the leakage at the tip clearance is large, the turbulence trend is small, and the turbulent dissipation caused by water flow through the clearance is small. Therefore, there is a low-value region of the entropy production rate between the tip and the pump shell.

To study the entropy production rate distribution at the tip more comprehensively, the entropy production rate distribution on the cascade surfaces in the tip region of the impellers with different tip clearances is analyzed and compared with the distribution of the vapor volume fraction. As shown in Figure 14, the selection of the position of the cascade surface is consistent with the selection method of Figure 12 in the previous section.

From Figure 14, it can be seen that under the same tip clearance and different cavitation coefficients, with the decrease in the cavitation coefficient, the high-value regions of entropy production rate distribution on the cascade surfaces decrease, and in Figure 14a, the distribution change in the high entropy production rate caused by the process of TLV cavitation transferring to the trailing edge and falling off below the suction surface of the blade can be clearly observed. Compared with Figure 12a, it shows that the entropy production rate distribution can more clearly reflect the movement of the TLV on the cascade surface at the tip than the distribution of vapor volume fraction on the cascade surface. In the purple outlined region, as depicted in Figure 14, it can be seen that for the impeller with a large blade tip clearance, the high entropy production zones at the tip are predominantly distributed in the vicinity near the trailing edge. This is because an increased tip clearance facilitates the transfer of TLV to the trailing edge, making it more prone to detachment, which is also consistent with the distribution change in cavitation on the cascade surface at the tip in Figure 12.

Figure 15 demonstrates that during the cavitation process of the pump, the entropy production caused by turbulent dissipation is much higher than that caused by viscous dissipation, and the alteration in entropy production resulting from turbulent dissipation with variations in cavitation coefficient is more intense than that caused by viscous dissipation. From Figure 15a, the observation reveals that as the tip clearance decreases, there is an increase in entropy production attributed to viscous dissipation. The difference in entropy production in the chamber of the impeller domain with a tip clearance of 2.0 mm and 4.0 mm is smaller than that in the impeller domain with a tip clearance of 1.0 mm and 2.0 mm. For impellers with different tip clearances, as the cavitation coefficient decreases, the entropy production generated by viscous dissipation shows a trend of being initially stable, then slightly decreasing, and finally slowly rising. Figure 15b indicates that the entropy production caused by turbulent dissipation shows a trend of being stable first, then rising, and finally declining in the impeller domain. The turbulent dissipation entropy production experiences the most pronounced change in the impeller domain featuring a tip clearance of 1.0 mm. The entropy production value computed within the impeller domain featuring a tip clearance of 1.0 mm is initially lower than that of the impeller domain with a tip clearance of 2.0 mm and 4.0 mm. When the pump is close to the critical cavitation, the entropy production value of the impeller domain with a tip clearance of 1.0 mm increases rapidly. When the cavitation coefficient is less than 0.15, the entropy production value of the impeller domain with a tip clearance of 1.0 mm is higher than that of the impeller domain with a tip clearance of 2.0 mm and 4.0 mm and decreases rapidly. This is because as the tip clearance becomes smaller, the flow pattern at the tip is in more disarray. With the deepening of cavitation, more bubbles in the impeller domain are attached to the blade surface, thus forming a hydraulic smooth zone, improving the flow pattern and reducing the hydraulic loss.

4.6. Axial Force Analysis

The service life of pump bearing is related to the axial force. A large axial force is not conducive to protecting the bearing and prolonging its service life. Therefore, considering the characteristics of the axial force acting on the impeller during the cavitation process is essential.

The fluctuation in the axial force of the impeller in relation to the cavitation coefficient is shown in Figure 16. It shows that the axial force acting on the impeller is basically stable at first and then rises slightly, finally falling rapidly. It can be found that the change in trend of the axial force acting on the impeller with the cavitation coefficient is consistent with the trend of the head variation. Upon comparing the axial force on impellers with varying tip clearances, it is evident that a reduced tip clearance correlates with an increased axial force as cavitation intensifies. The main reason is that the larger the tip clearance, the greater the fluid leakage, and the leakage flow diminishes the pressure disparity across the pressure and suction surfaces, resulting in a decrease in the axial force. Therefore, augmenting the tip clearance exerts a mitigating influence on the axial force experienced by the bearing. In practice, the size of the tip clearance should be determined by comprehensively considering the axial force, the head loss caused by the tip leakage, the effect of the tip clearance on the cavitation performance, and the friction between the pump shell and the tip caused by the swing of the pump shaft.

5. Conclusions

Numerical simulation based on ANSYS CFX is used to study the effect of tip clearance on the cavitation performance in an axial flow pump. The simulation is carried out by varying the tip clearance size. Combined with the entropy production theory, the results are compared and analyzed. In summary, the research findings can be outlined as follows:

(1)

Under the same cavitation coefficient, with the broadening of tip clearance, the TLV cavitation accumulates toward the trailing edge and approaches the suction surface. The vapor volume fraction within the TLV cavitation region experiences an escalation, concurrent with a reduction in the thickness of sheet cavitation on the suction surface.

(2)

When cavitation occurs in an axial flow pump, the high-value region of the entropy production rate at the tip of the blade section expands from the leading to the trailing edge. The magnitude of tip clearance directly influences the leakage amount, with larger tip clearances resulting in increased leakage. On the contrary, the smaller the turbulence trend at the tip clearance, the smaller the turbulence dissipation caused by the water flow through the tip clearance. Therefore, there is a low-value region of the entropy production rate between the tip and the pump shell.

(3)

The larger the blade tip clearance, the greater the distribution area of vapor volume fraction below the tip on the cross-section of the blade. Moreover, the region of vapor volume fraction distribution near the trailing edge of the tip increases with the size of the tip clearance. The shedding phenomenon of TLV cavitation can be more clearly observed through the entropy production rate distribution on the cascade surface.

(4)

As the cavitation deepens within the pump, the entropy production resulting from turbulent dissipation is much higher than that caused by viscous dissipation. As the tip clearance increases, there is a decrease in viscous entropy production. Meanwhile, the relationship between the turbulent entropy production and the tip clearance is not obvious.

(5)

In the cavitation process of the axial flow pump, the larger the tip clearance is, the smaller the axial force on the impeller is. Increasing the tip clearance can alleviate the axial force on the bearing. In practical application, the size of the tip clearance should be determined by considering the axial force, the head loss attributed to tip leakage, the impact of tip clearance on cavitation performance, and the friction between the casing and the tip resulting from the oscillation of the pump shaft. For the axial flow pump used in this study, a blade tip clearance of 1 mm is recommended.