Simulation of Internal Flow Characteristics of an Axial Flow Pump with Variable Tip Clearance

: This study investigated the inﬂuence of the change in blade tip clearance on the internal ﬂow characteristics of a vertical axial ﬂow pump. Taking the actual running vertical axial ﬂow pump of a pumping station as the research object, based on the SST k- ω turbulent ﬂow model, the numerical simulation technology was used to study the effects of different tip clearances on the pressure, turbulent kinetic energy, Z–X section pressure and ﬂow state of the impeller at the middle section. Furthermore, the impact of clearance layer tip leakage was also analyzed. Unsteady calculations of ﬂow characteristics under the design conditions were performed. The research results showed that the variation trend of the pressure in the impeller was basically the same under different tip clearance values. With the increase in the clearance value, the pressure gradient along the water inlet direction of the blade decreased and the leakage vorticity increased. Observing the leakage vorticity distribution of the gap layer under the ﬂow condition of 0.6 Q 0 , it was found that when the tip clearance was smaller than 1 mm, the leakage ﬂow was small and easily assimilated by the mainstream, and the leakage ﬂow and mainstream had a certain ability to compete, which caused adverse effects on the performance of the pump device. The pressure pulsation characteristics showed that the leakage ﬂow caused by the tip clearance caused a high-frequency distribution, and the clearance obviously inﬂuenced the pressure pulsation characteristics.


Introduction
In the areas along the rivers in China, in order to meet the requirements of water transfer and irrigation, there are many vertical axial flow pumping stations [1,2]. These axial flow pumping stations have the characteristics of large flow and low lift [3], have been widely used in the eastern route of the South-to-North Water Diversion Project, and have achieved great social and economic benefits [4]. In the design of the vertical axial flow pump, due to the way water is pumped [5] and in order to prevent the impeller from scraping and colliding with the pump casing when the impeller rotates at high speed, it is necessary to leave a certain gap between the top of the impeller blade and the pump casing, which is the tip clearance. When the impeller is working, there is a pressure difference between the working surface and the back of the blade, and the existence of the pressure difference will lead to the generation of a leakage flow at the clearance position at the top of the blade. The interaction between the leakage flow and the main flow not only affects the pump performance but also induces abnormal vibration and noise, and even causes safety hazards and affects the normal operation of the pumping station [6].
Domestic and foreign experts and scholars studied the influence of tip clearance flow laws and the performance of impeller machinery. Zhang Desheng et al. [7] studied the pressure difference distribution, leakage, tip leakage vortex intensity and inlet axial velocity distribution in the tip clearance area of an inclined flow pump under different tip clearances. Shi Weidong et al. [8] used the SMPLEC algorithm to simulate different clearances under different working conditions and analyzed the axial velocity and circulation distribution of an impeller outlet in detail. Zhang et al. [9] revealed the gap flow field structure and leakage vortex evolution of a semi-open centrifugal pump by changing the gap according to the numerical calculation of the whole flow channel of the semi-open impeller centrifugal pump. Li Hui et al. [10] used the delayed detached vortex method to simulate a dynamic blade with a tip clearance and then studied the turbulent characteristics of the leakage area through the turbulent kinetic energy distribution and Lumley triangle. Then, the POD decomposition method was used to decompose the flow field in the leakage area, and finally, the loss analysis of the tip leakage flow was carried out. Lu Jinling et al. [11] carried out a full flow channel numerical simulation of a semi-open centrifugal pump using Fourier transform (FFT) to convert the time domain value of each monitoring point to a frequency domain value and analyzed the correlation mechanism between leakage vortex trajectory and blade load, as well as the spectral characteristics of the leakage vortex. Li et al. [12] studied the internal flow characteristics of the tip clearance of tubular turbines under off-design conditions and analyzed the pressure and velocity vector distribution of the internal flow field in the tip clearance, as well as the axial velocity and turbulent kinetic energy distribution characteristics in the tip clearance. Liu et al. [13] observed the development and trajectory of the leakage vortex by changing the tip clearance and performed a spectral analysis. When the tip clearance increased from 0 mm to 10 mm, the maximum amplitude of the pressure pulsation in the impeller increased sharply. Due to the increase in leakage flow, the main frequency increased from 145 Hz to 184 Hz. Kan et al. [14] used a standard k-epsilon turbulence model to simulate the flow characteristics of a selfdeveloped helical axial flow pump. Pressure, streamline and turbulent kinetic energy analyses of the leakage flow (TLF) in a helical axial flow pump were undertaken, revealing the effect of tip clearance on the flow behavior and boosting performance of a helical axial flow pump. Shen et al. [15] used a computational fluid dynamics method to study the effect of different tip clearance widths on the tip flow dynamics and main flow characteristics of axial flow pumps. The distribution 0.7Q (BEP) of turbulent kinetic energy, average axial velocity and average vorticity at a specific flow rate was analyzed, and it was found that the flow structure of the tip vortex and its transmission strongly depended on the tip gap width. The efficiency and head of the pump both increase the energy loss as the tip clearance increases. Feng Jianjun et al. [16] studied the pressure pulsation of an axial flow pump under different tip clearances and proposed a new method to determine the pressure pulsation of the grid node in a pump based on pressure statistics. It was found that the existence of a tip clearance enlarged the pressure pulsation from the hub to the rim of the impeller. Li Yibin et al. [17] studied the pressure pulsation characteristics of different tip clearance regions of a diagonal flow pump under the condition of a small flow rate, and the influence on the transient operation stability of the diagonal flow pump was revealed. Through the analysis of the pressure pulsation spectrum, the internal relationship between the R TC and the pressure pulsation in the vicinity of the tip clearance area was understood. The results show that selecting an appropriately small R TC can improve the overall hydraulic performance of the oblique flow pump. Zhang Hua et al. [18] studied the influence degree and mechanism of the blade tip clearance on the internal and external characteristics of the centrifugal pump, a special adjustment mechanism for the blade tip clearance was designed, and the external characteristics test and a pressure pulsation test were carried out at the same time. The results show that the ratio of the tip clearance to the average diameter of the impeller should be between 0.13% and 0.22%. This provides a reference for the vibration and noise reduction of a semi-open screw centrifugal pump. Li Rennian et al. [19] studied the influence of the tip clearance on the inlet pressure pulsation characteristics of the diagonal flow pump, where the results show that with the increase of the tip clearance, the lift of the diagonal flow pump decreases gradually. The amplitude of pressure pulsation in the mainstream area of the impeller inlet is small, while the amplitude of pressure pulsation in the near-wall area is large. The larger tip clearance can reduce the pressure pulsation amplitude at the impeller inlet, which is beneficial for improving the operation stability of the model oblique flow pump. Li Yaojun et al. [20] studied the unsteady flow characteristics of the axial flow pump rim area under different tip clearances, where the results showed that when the tip clearance increased from 0.001D 2 to 0.003D 2 , the pump head and efficiency decreased by 6.2% and 5.6%, respectively. When the clearance value is greater than 0.001D 2 , the primary leakage vortex in the rim clearance developed to the front face of the adjacent blade and a large number of secondary leakage vortices are generated in the clearance area. Guelich, J. and Ulanicki, B. et al. [21,22] studied the efficiency and power characteristics of a pump group, which were simulated by changing the Reynolds number and considering the constant or variable speed of the pump.
In this study, a full flow channel numerical calculations were carried out for the vertical axial flow pump [23], and the influence of the tip clearance change on the total pressure, turbulent kinetic energy distribution, Z-X section pressure, flow state, and the leakage flow and leakage volume of the gap layer were analyzed in detail. Numerical simulations of the unsteady flow in the axial flow pump with different tip clearance sizes were carried out to study the effect of the clearance size on the leakage vortex shape of the rim clearance and to analyze the pressure pulsation characteristics of the clearance layer to obtain the optimal blade rim. The variation law of the side and inlet and outlet pressure pulsations with different rim gap sizes theoretically revealed the gap flow field structure and leakage flow evolution of the vertical axial flow pump with a changing gap. This provides reference significance for more efficient operation and management of vertical axial flow pump stations [24,25].

Vertical Axial Flow Pump Parameters and Research Plan
This study took a vertical bidirectional axial flow pump station as the prototype for the 3D modeling. The pump station adopted the 2500ZLQ-20-3.0 vertical axis open axial flow pump produced by a factory. The designed flow rate of the pump was Q 0 = 20 m 3 /s, the impeller diameter was 2.50 m, the number of impeller blades was 3, the number of guide blades was 7, the specific speed was 1250 and the impeller speed was 150 r/min.
In order to study the change in the internal flow characteristics of the pump under different clearance sizes and to ensure that the external characteristics of the prototype pump remained unchanged, the method of turning the impeller was adopted [26]. Four different tip clearances (d = 1 mm, 2.5 mm, 3.5 mm, 5 mm) were selected to simulate the flow under five working conditions. Through comparative analysis, the internal flow of vertical axial flow pump with different tip clearances was expounded. Figure 1a below is the flow chart of the research plan. Figure 1b is the tip clearance diagram of the vertical axial flow pump.

Mesh Subdivision
As shown in Figure 2a, the physical model of the computational domain in this study included the inlet section, the impeller section, the guide vane section and the outlet section. Considering the control of the overall number of mesh elements and the number of effective nodes, a hexahedral mesh was used for the divisions and the impeller section was refined to control the mesh quality.

Mesh Subdivision
As shown in Figure 2a, the physical model of the computational domain in this study included the inlet section, the impeller section, the guide vane section and the outlet section. Considering the control of the overall number of mesh elements and the number of effective nodes, a hexahedral mesh was used for the divisions and the impeller section was refined to control the mesh quality.
Since the mesh size and quality have an important influence on the calculation results, by adjusting the mesh size of the impeller blade surface and the rim clearance area, the design flow conditions of the pump under different mesh schemes when the tip clearance was d = 2.5 mm were compared. The efficiency of the pump was compared with the test results, and a mesh-independent analysis was conducted. As shown in Table 1, the pump head and efficiency calculated using grid schemes A, B and C were basically the same. Considering the economy of numerical calculation, grid scheme B was adopted in this study. With different tip clearances, the total number of computational domain grids was about 1.3 × 10 7 ; the global mesh and impeller vane mesh are shown in Figure 2b. Since the mesh size and quality have an important influence on the calculation results, by adjusting the mesh size of the impeller blade surface and the rim clearance area, the design flow conditions of the pump under different mesh schemes when the tip clearance was d = 2.5 mm were compared. The efficiency of the pump was compared with the test results, and a mesh-independent analysis was conducted. As shown in Table 1, the pump head and efficiency calculated using grid schemes A, B and C were basically the same. Considering the economy of numerical calculation, grid scheme B was adopted in this study. With different tip clearances, the total number of computational domain grids was about 1.3 × 10 ; the global mesh and impeller vane mesh are shown in Figure 2b.

Governing Equations and Turbulence Model Selection
The internal flow of a pump is unsteady viscous flow, which can be described using Navier-Stokes equations [27]. The Reynolds-averaged Navier-Stokes (RANS) method was used to decompose the various characteristic variables of the turbulent flow into timeaveraged Reynolds equations and fluctuating values, and then the turbulent viscosity coefficient was introduced to establish the turbulent model. For a general incompressible Newtonian fluid, the following control equations are used.
The mass conservation equation, also called the continuity equation, is as follows:

Governing Equations and Turbulence Model Selection
The internal flow of a pump is unsteady viscous flow, which can be described using Navier-Stokes equations [27]. The Reynolds-averaged Navier-Stokes (RANS) method was used to decompose the various characteristic variables of the turbulent flow into time-averaged Reynolds equations and fluctuating values, and then the turbulent viscosity coefficient was introduced to establish the turbulent model. For a general incompressible Newtonian fluid, the following control equations are used.
The mass conservation equation, also called the continuity equation, is as follows: Momentum equation: where t is time, f f is the volume force vector, ρ f is the fluid density, v is the fluid velocity vector and τ f is the shear force tensor given by where p is the static pressure, µ is the dynamic viscosity and e is the velocity stress tensor: The numerical simulation and experimental results showed that under the optimal conditions, the predicted external characteristic curve of the SST k-ω turbulence model was in good agreement with the experimental curve, where the head error was 4.688% [28]. Therefore, the SST k-ω was selected as the turbulence model to conduct the numerical simulation of the whole flow channel in the computational domain of the axial flow pump, where k is the turbulent kinetic energy, and the transport equation is: ω is the turbulent dissipation rate, and the related equation is In this formula, U j is the vector velocity (m/s), P k is the turbulent generation rate and µ t is the turbulent viscosity (m 2 /s).

Boundary Conditions and Arrangement of the Pressure Fluctuation Monitoring Points
The boundary conditions in this study were the velocity inlet and free outlet. The flow at the pump outlet was fully developed. The impeller speed was n = 150 r/min and the rotor-stator dynamic-static interface was a frozen rotor. The wall of each flow component adopted a smooth non-slip wall, The impeller surface and hub surface were set as moving walls without heat transfer, and the convergence accuracy was 10 −4 . In the unsteady calculations, the time step was 1/40 of the impeller rotation period of 0.4 s, and the sampling time was 8 impeller rotation periods. Detailed parameters are shown in Table 2. In order to analyze the pressure fluctuation characteristics of a blade near the flange side, as well as the inlet and outlet, under the influence of different flange clearance leakage flow, the pressure fluctuation monitoring points were arranged as shown in Figure 3.
In order to analyze the pressure fluctuation characteristics of a blade near the flan side, as well as the inlet and outlet, under the influence of different flange clearance lea age flow, the pressure fluctuation monitoring points were arranged as shown in Figure   Figure 3. Schematic diagram of the monitoring point layout. The test situation represented the head when the gap was 1 mm and evaluated t relationship between the efficiency and flow rate. It can be seen from Figure 4 that the t head value was generally higher than the numerical simulation head value, and decreas with the increase in the flow rate. However, when the flow rate increased, the drop r was larger than the numerical model head, and there was a cross-over with the numeri model head value. The gap between the test and simulation was only about 0.2 m. Figu 5 shows that the efficiency of the test was lower than the simulation efficiency under t condition of a small flow rate, but the efficiency growth rate was larger than the simu tion value. At 0.6Q0, the test efficiency gradually intersected with the simulation efficien and began to exceed it. At 0.9Q0, the efficiency gap between the test and the simulati was the largest, reaching less than 2%. It shows that the efficiency gap was within t controllable range, and the numerical simulation was basically consistent with the exp imental situation.

Head-Flow Curve and Head-Efficiency Curve
This shows that the digital-analog calculation was closer to the actual situation can be seen from Figure 5 that there was no linear correlation between the drop in the and the increase in the clearance. In the process of increasing the clearance from 1 mm 5 mm, with the increase in the clearance, the change in the lift and efficiency was relative small. When the gap was small, the leakage was small; the leakage was very small relati to the mainstream, and the impact on the mainstream was negligible, making the e ciency higher. It can be seen from Figure 5 that with the gradual increase in the gap, t leakage had a certain influence on the mainstream and the influence on the flow fie increased, which caused the head and efficiency drops to be significantly larger. When t clearance continued to increase (referring to the 3.5 mm-5 mm stage), the leakage volum also continued to increase, but the increase in the tip clearance was moderate compar with the previous one. The difference in leakage flow was limited; therefore, the drop the head and efficiency was also smaller than before.  The test situation represented the head when the gap was 1 mm and evaluated the relationship between the efficiency and flow rate. It can be seen from Figure 4 that the test head value was generally higher than the numerical simulation head value, and decreased with the increase in the flow rate. However, when the flow rate increased, the drop rate was larger than the numerical model head, and there was a cross-over with the numerical model head value. The gap between the test and simulation was only about 0.2 m. Figure 5 shows that the efficiency of the test was lower than the simulation efficiency under the condition of a small flow rate, but the efficiency growth rate was larger than the simulation value. At 0.6Q 0 , the test efficiency gradually intersected with the simulation efficiency and began to exceed it. At 0.9Q 0 , the efficiency gap between the test and the simulation was the largest, reaching less than 2%. It shows that the efficiency gap was within the controllable range, and the numerical simulation was basically consistent with the experimental situation.

Head-Flow Curve and Head-Efficiency Curve
This shows that the digital-analog calculation was closer to the actual situation. It can be seen from Figure 5 that there was no linear correlation between the drop in the lift and the increase in the clearance. In the process of increasing the clearance from 1 mm to 5 mm, with the increase in the clearance, the change in the lift and efficiency was relatively small. When the gap was small, the leakage was small; the leakage was very small relative to the mainstream, and the impact on the mainstream was negligible, making the efficiency higher. It can be seen from Figure 5 that with the gradual increase in the gap, the leakage had a certain influence on the mainstream and the influence on the flow field increased, which caused the head and efficiency drops to be significantly larger. When the clearance continued to increase (referring to the 3.5 mm-5 mm stage), the leakage volume also continued to increase, but the increase in the tip clearance was moderate compared with the previous one. The difference in leakage flow was limited; therefore, the drop in the head and efficiency was also smaller than before.    Figure 6 shows the total pressure distribution of the middle section of the impelle under different tip clearances. It can be seen from Figure 6 that the pressure changed in the same trend under different tip clearances. The diffusion and the pressure at the outle were small, and the degree of pressure diffusion was relatively uniform in this section but at the edge of the front end of the impeller, affected by the gap leakage vortex, a loca high-pressure area appeared. There were high-pressure areas on both the working fac and the back of the blade, which were mainly due to the existence of the tip clearance. Th high-pressure fluid on the working face of the blade passed through the tip clearance and directly acted on the back of the blade. With the increase in the clearance value, the pres sure gradient along the water inlet direction of the blade decreased. As the leakage vorti city increased, the work done by the water on the blade decreased. When the tip clearanc increased from 1 mm to 5 mm and the tip clearance was small, the leakage vorticity wa small, and it was easier to generate a local high pressure at the edge of the inlet direction When the tip clearance value was large, the high-pressure fluid crossed the gap. It acted on the back of the blade, which reduced the workability of the water flow for the blade.    Figure 6 shows the total pressure distribution of the middle section of the impelle under different tip clearances. It can be seen from Figure 6 that the pressure changed in the same trend under different tip clearances. The diffusion and the pressure at the outle were small, and the degree of pressure diffusion was relatively uniform in this section but at the edge of the front end of the impeller, affected by the gap leakage vortex, a loca high-pressure area appeared. There were high-pressure areas on both the working face and the back of the blade, which were mainly due to the existence of the tip clearance. The high-pressure fluid on the working face of the blade passed through the tip clearance and directly acted on the back of the blade. With the increase in the clearance value, the pres sure gradient along the water inlet direction of the blade decreased. As the leakage vorti city increased, the work done by the water on the blade decreased. When the tip clearance increased from 1 mm to 5 mm and the tip clearance was small, the leakage vorticity wa small, and it was easier to generate a local high pressure at the edge of the inlet direction When the tip clearance value was large, the high-pressure fluid crossed the gap. It acted on the back of the blade, which reduced the workability of the water flow for the blade.  Figure 6 shows the total pressure distribution of the middle section of the impeller under different tip clearances. It can be seen from Figure 6 that the pressure changed in the same trend under different tip clearances. The diffusion and the pressure at the outlet were small, and the degree of pressure diffusion was relatively uniform in this section, but at the edge of the front end of the impeller, affected by the gap leakage vortex, a local high-pressure area appeared. There were high-pressure areas on both the working face and the back of the blade, which were mainly due to the existence of the tip clearance. The high-pressure fluid on the working face of the blade passed through the tip clearance and directly acted on the back of the blade. With the increase in the clearance value, the pressure gradient along the water inlet direction of the blade decreased. As the leakage vorticity increased, the work done by the water on the blade decreased. When the tip clearance increased from 1 mm to 5 mm and the tip clearance was small, the leakage vorticity was small, and it was easier to generate a local high pressure at the edge of the inlet direction.

Effect of Clearance Variation on Total Pressure Distribution of Impeller Middle Section
When the tip clearance value was large, the high-pressure fluid crossed the gap. It acted on the back of the blade, which reduced the workability of the water flow for the blade.

Effect of the Clearance Variation on the Turbulent Kinetic Energy Distribution in the Middle
Section of Impeller Figure 7 shows the distribution of the turbulent kinetic energy in the middle section of the impeller under the design conditions. It can be seen from the figure that the area with a large turbulent kinetic energy was concentrated at the edge of the impeller, and the turbulent kinetic energy was particularly large near the tip of the blade. This was because the high-pressure fluid leaked from the gap and crossed the working face to the back to form a mixing loss zone. This affected the normal flow of the fluid in the flow channel, where a region of high turbulent kinetic energy was created. In the four different tip clearance cases, the tip leakage flow increased with the increase in the tip clearance size, and the extreme value of the leakage flow increased when the turbulent kinetic energy region increased. Combined with the analysis of the total pressure distribution in the middle section of the impeller in Figure 6, it can be seen that the pressure gradient in the flow channel along the water flow direction was reduced and the influence of the leakage flow changed, resulting in the formation of unstable turbulent pulsations inside the flow channel. The interaction of fluid molecules, friction and collisions aggravated the internal energy loss of the fluid such that a part of the internal energy was converted into heat energy and conducted out, thereby reducing the efficiency of the pump device.

Effect of the Clearance Variation on the Turbulent Kinetic Energy Distribution in the Middle
Section of Impeller Figure 7 shows the distribution of the turbulent kinetic energy in the middle section of the impeller under the design conditions. It can be seen from the figure that the area with a large turbulent kinetic energy was concentrated at the edge of the impeller, and the turbulent kinetic energy was particularly large near the tip of the blade. This was because the high-pressure fluid leaked from the gap and crossed the working face to the back to form a mixing loss zone. This affected the normal flow of the fluid in the flow channel, where a region of high turbulent kinetic energy was created. In the four different tip clearance cases, the tip leakage flow increased with the increase in the tip clearance size, and the extreme value of the leakage flow increased when the turbulent kinetic energy region increased. Combined with the analysis of the total pressure distribution in the middle section of the impeller in Figure 6, it can be seen that the pressure gradient in the flow channel along the water flow direction was reduced and the influence of the leakage flow changed, resulting in the formation of unstable turbulent pulsations inside the flow channel. The interaction of fluid molecules, friction and collisions aggravated the internal energy loss of the fluid such that a part of the internal energy was converted into heat energy and conducted out, thereby reducing the efficiency of the pump device.

Effect of the Clearance Variation on the Turbulent Kinetic Energy Distribution in the Middle
Section of Impeller Figure 7 shows the distribution of the turbulent kinetic energy in the middle section of the impeller under the design conditions. It can be seen from the figure that the area with a large turbulent kinetic energy was concentrated at the edge of the impeller, and the turbulent kinetic energy was particularly large near the tip of the blade. This was because the high-pressure fluid leaked from the gap and crossed the working face to the back to form a mixing loss zone. This affected the normal flow of the fluid in the flow channel, where a region of high turbulent kinetic energy was created. In the four different tip clearance cases, the tip leakage flow increased with the increase in the tip clearance size, and the extreme value of the leakage flow increased when the turbulent kinetic energy region increased. Combined with the analysis of the total pressure distribution in the middle section of the impeller in Figure 6, it can be seen that the pressure gradient in the flow channel along the water flow direction was reduced and the influence of the leakage flow changed, resulting in the formation of unstable turbulent pulsations inside the flow channel. The interaction of fluid molecules, friction and collisions aggravated the internal energy loss of the fluid such that a part of the internal energy was converted into heat energy and conducted out, thereby reducing the efficiency of the pump device.   Figure 8 shows the pressure distribution of Z-X section with different tip clearances under the design conditions. It can be seen that under different tip clearance values, highpressure fluid entered from the bottom inlet, passed through the impeller guide vane in turn and flowed out from the upper outlet. The total pressure of the Z-X section increased gradually along the inlet direction and reached the maximum at the outlet, and there was  Figure 8 shows the pressure distribution of Z-X section with different tip clearances under the design conditions. It can be seen that under different tip clearance values, highpressure fluid entered from the bottom inlet, passed through the impeller guide vane in turn and flowed out from the upper outlet. The total pressure of the Z-X section increased gradually along the inlet direction and reached the maximum at the outlet, and there was a local high-pressure area at the outlet section of the impeller. The total pressure fluctuation at the inlet position of the impeller was not obvious. When the high-pressure fluid reached the blade working face, the isobaric line began to change significantly and was accompanied by large fluctuations. The pressure fluctuations were mostly concentrated in the impeller channel, which intensified the instability of the fluid flow. It can also be seen from Figure 8 that the pressure of the blade's working face was significantly greater than that of the back surface, and even the pressure of some regions was close to zero at the back surface of the blade. This was because the high-pressure fluid near the tip clearance of the impeller blade working face side leaked to the back of the blade through the clearance, and the working efficiency of the blade was reduced, which showed the trend of pressure reduction on the back of the blade.

Effect of the Gap Change on the Pressure and Flow State of the Z-X Section
Water 2022, 14, 1652 10 of 18 a local high-pressure area at the outlet section of the impeller. The total pressure fluctuation at the inlet position of the impeller was not obvious. When the high-pressure fluid reached the blade working face, the isobaric line began to change significantly and was accompanied by large fluctuations. The pressure fluctuations were mostly concentrated in the impeller channel, which intensified the instability of the fluid flow. It can also be seen from Figure 8 that the pressure of the blade's working face was significantly greater than that of the back surface, and even the pressure of some regions was close to zero at the back surface of the blade. This was because the high-pressure fluid near the tip clearance of the impeller blade working face side leaked to the back of the blade through the clearance, and the working efficiency of the blade was reduced, which showed the trend of pressure reduction on the back of the blade.  As shown in Figure 9, due to the existence of the tip clearance, there was a significant backflow at the top of the impeller blade, flow around the outlet, flow pattern disorder and so on. The flow entered the impeller, and with the increase in tip clearance, the backflow in the impeller channel increased significantly. When the clearance was small, the leakage was small, resulting in a large pressure difference on both sides of the blade, which led to the formation of a flow vortex in the impeller passage. With the gradual increase in the tip clearance, the leakage increased, causing part of the fluid gathered in the working face to cross the clearance to form backflow. The larger the clearance was, the more backflow there was. By observing the above figure, it can be seen that the backflow caused by a certain gap (Figure 9b) could make the fluid flow smoother, which was sufficient to demonstrate that a certain gap (Figure 9b) could improve the flow pattern of the impeller channel.   As shown in Figure 9, due to the existence of the tip clearance, there was a significant backflow at the top of the impeller blade, flow around the outlet, flow pattern disorder and so on. The flow entered the impeller, and with the increase in tip clearance, the backflow in the impeller channel increased significantly. When the clearance was small, the leakage was small, resulting in a large pressure difference on both sides of the blade, which led to the formation of a flow vortex in the impeller passage. With the gradual increase in the tip clearance, the leakage increased, causing part of the fluid gathered in the working face to cross the clearance to form backflow. The larger the clearance was, the more backflow there was. By observing the above figure, it can be seen that the backflow caused by a certain gap (Figure 9b) could make the fluid flow smoother, which was sufficient to demonstrate that a certain gap (Figure 9b) could improve the flow pattern of the impeller channel.
Water 2022, 14, 1652 10 of 18 a local high-pressure area at the outlet section of the impeller. The total pressure fluctuation at the inlet position of the impeller was not obvious. When the high-pressure fluid reached the blade working face, the isobaric line began to change significantly and was accompanied by large fluctuations. The pressure fluctuations were mostly concentrated in the impeller channel, which intensified the instability of the fluid flow. It can also be seen from Figure 8 that the pressure of the blade's working face was significantly greater than that of the back surface, and even the pressure of some regions was close to zero at the back surface of the blade. This was because the high-pressure fluid near the tip clearance of the impeller blade working face side leaked to the back of the blade through the clearance, and the working efficiency of the blade was reduced, which showed the trend of pressure reduction on the back of the blade.  As shown in Figure 9, due to the existence of the tip clearance, there was a significant backflow at the top of the impeller blade, flow around the outlet, flow pattern disorder and so on. The flow entered the impeller, and with the increase in tip clearance, the backflow in the impeller channel increased significantly. When the clearance was small, the leakage was small, resulting in a large pressure difference on both sides of the blade, which led to the formation of a flow vortex in the impeller passage. With the gradual increase in the tip clearance, the leakage increased, causing part of the fluid gathered in the working face to cross the clearance to form backflow. The larger the clearance was, the more backflow there was. By observing the above figure, it can be seen that the backflow caused by a certain gap (Figure 9b) could make the fluid flow smoother, which was sufficient to demonstrate that a certain gap (Figure 9b) could improve the flow pattern of the impeller channel.   Figures 10 and 11 show the relative velocity vectors and leaky vortex 3D shapes under different tip clearances at a 0.6 flow rate. When the tip clearance was 1 mm, the  Figures 10 and 11 show the relative velocity vectors and leaky vortex 3D shapes under different tip clearances at a 0.6Q 0 flow rate. When the tip clearance was 1 mm, the leakage of the tip head relative to other positions was large and the pressure difference between the working face and the back of the blade was also the largest. Since the tip clearance was very small relative to the impeller diameter and the collision between the leakage flow and the pump shell caused loss, the actual leakage flow was a small part relative to the mainstream and had little effect on the flow. Under the influence of the mainstream after the formation of the leakage flow, the leakage flow was mixed into the mainstream. The pump device had good performance under this small tip clearance.

Influence of Clearance Change on Tip Leakage of Clearance Layer
leakage flow and the pump shell caused loss, the actual leakage flow was a small part relative to the mainstream and had little effect on the flow. Under the influence of the mainstream after the formation of the leakage flow, the leakage flow was mixed into the mainstream. The pump device had good performance under this small tip clearance.
As the tip clearance gradually increased, the leakage also gradually increased. Compared with Figure 11a, a relatively obvious leakage flow was formed in Figure 11b. At this time, the influence of the leakage flow on the mainstream also increased, but the leakage flow at this time was still relatively small and was still mixed into the mainstream, and thus, the leakage vortex disappeared. As the tip clearance was further expanded to 5 mm, the leakage volume was further increased and the leakage vortex intensified. The leakage flow and the mainstream had a certain ability to resist and interact. In the encounter position, the entrainment phenomenon occurred, resulting in the intensification of the vortex in the flow channel. At this time, the larger tip clearance had a negative impact on the performance of the pump device and caused concomitant damage to the blade.  relative to the mainstream and had little effect on the flow. Under the influence of the mainstream after the formation of the leakage flow, the leakage flow was mixed into the mainstream. The pump device had good performance under this small tip clearance. As the tip clearance gradually increased, the leakage also gradually increased. Compared with Figure 11a, a relatively obvious leakage flow was formed in Figure 11b. At this time, the influence of the leakage flow on the mainstream also increased, but the leakage flow at this time was still relatively small and was still mixed into the mainstream, and thus, the leakage vortex disappeared. As the tip clearance was further expanded to 5 mm, the leakage volume was further increased and the leakage vortex intensified. The leakage flow and the mainstream had a certain ability to resist and interact. In the encounter position, the entrainment phenomenon occurred, resulting in the intensification of the vortex in the flow channel. At this time, the larger tip clearance had a negative impact on the performance of the pump device and caused concomitant damage to the blade.  As the tip clearance gradually increased, the leakage also gradually increased. Compared with Figure 11a, a relatively obvious leakage flow was formed in Figure 11b. At this time, the influence of the leakage flow on the mainstream also increased, but the leakage flow at this time was still relatively small and was still mixed into the mainstream, and thus, the leakage vortex disappeared. As the tip clearance was further expanded to 5 mm, the leakage volume was further increased and the leakage vortex intensified. The leakage flow and the mainstream had a certain ability to resist and interact. In the encounter position, the entrainment phenomenon occurred, resulting in the intensification of the vortex in the flow channel. At this time, the larger tip clearance had a negative impact on the performance of the pump device and caused concomitant damage to the blade.

Characteristics of the Pressure Pulsation in the Tip Clearance Region
In order to further reveal the influence of the tip clearance on the flow characteristics in the rim area of the vertical axial flow pump under the design conditions, the time domain and frequency domain characteristics were analyzed by monitoring the pressure pulsation characteristics in the tip clearance area. The amplitude of the pressure pulsation was described by the relative value of the double-amplitude peak-to-peak value of the mixing frequency in the time domain, and the spectral characteristics of the pressure pulsation were obtained by using the fast Fourier transform (FFT). In order to improve the frequency resolution of the FFT analysis, the sampling time was continuously calculated for four cycles, starting from the fourth impeller rotation cycle, and the static pressure data of the monitoring points in these four cycles were collected to analyze the flow field spectrum. The amplitude of the pressure pulsation was represented by the pressure pulsation coefficient Cp, which is expressed as follows: In these formulae, ∆P is the difference between the instantaneous pressure and the average static pressure at the monitoring point (Pa); ρ is the fluid density (kg/m 3 ); D is the impeller diameter (m); U is the peripheral speed (m/s) and n is the pump speed (r/min). Table 3 is the amplitude table of the monitoring points at the impeller under different tip clearances. From the data shown in this table, it can be seen that the main frequency of the monitoring points at the impeller was a multiple of the rotational frequency of the impeller. Because of the influence of the periodic rotation of the three blades of the impeller and the mutual interference of the static flow field at the inlet, the main frequency of most of the monitoring points at the impeller was the same as the blade frequency. In particular, the main frequency of the measuring points at the impeller rim and the inlet hub was the blade frequency. The outlet of the impeller was connected to the inlet of the guide vane; therefore, in addition to being affected by the main frequency of the impeller, some frequencies were also affected by the guide vane. Under different tip clearances, with the increase in the clearance, the amplitude first increased and then decreased, while the change trend of the position of the inlet and outlet and the axial monitoring position was basically the same; therefore, the three points at the rim were further analyzed. Figure 12 shows the pressure pulsation characteristics of the monitoring point P3 near the impeller inlet for different tip clearances. It can be seen that the pressure pulsation at the monitoring point of the pressure pulsation at the impeller inlet had a relatively obvious law, which was similar to a sinusoidal waveform. There was a complete peak and trough in one cycle. When the tip clearance was small, the pressure variation range was large because the impact of the leakage flow was large under the same working conditions. When the tip clearance was small, there were many complex high-frequency components in the frequency domain. Since P3 was located at the leading edge of the tip, the pressure here was affected by the direct work done by the blade and also by the leakage flow, which caused the flow here to be more stable. It was complicated and exacerbated the instability of the flow.   Figure 12 shows the pressure pulsation characteristics of the monitoring point P3 near the impeller inlet for different tip clearances. It can be seen that the pressure pulsation at the monitoring point of the pressure pulsation at the impeller inlet had a relatively obvious law, which was similar to a sinusoidal waveform. There was a complete peak and trough in one cycle. When the tip clearance was small, the pressure variation range was large because the impact of the leakage flow was large under the same working conditions. When the tip clearance was small, there were many complex high-frequency components in the frequency domain. Since P3 was located at the leading edge of the tip, the pressure here was affected by the direct work done by the blade and also by the leakage flow, which caused the flow here to be more stable. It was complicated and exacerbated the instability of the flow.    Figure 13 shows the pressure pulsation characteristics of the monitoring point P2 in the middle of the blade tip. It can be seen that the four different blade tip clearances showed similar regular changes, but the amplitudes were slightly different; its frequency domain characteristics also had complex components, but they were not as chaotic as the inlet end, and the middle of the blade tip was affected by the periodic rotation of the blade and leakage flow. Figure 14 shows the pressure pulsation characteristics of the monitoring point P1 at the tip outlet. It can be seen that the regularity was not as obvious as before in the case of different tip clearances since there were no obvious periodic peaks and valleys, showing a certain amount of randomness; its frequency domain also had a large number of high-frequency parts, but it was not as good as the inlet and the middle of the tip, and as the tip clearance increased, the high-frequency part decreased, which was caused by interference from the leakage flow. Figure 13 shows the pressure pulsation characteristics of the monitoring point P2 in the middle of the blade tip. It can be seen that the four different blade tip clearances showed similar regular changes, but the amplitudes were slightly different; its frequency domain characteristics also had complex components, but they were not as chaotic as the inlet end, and the middle of the blade tip was affected by the periodic rotation of the blade and leakage flow.   Figure 14 shows the pressure pulsation characteristics of the monitoring point P1 at the tip outlet. It can be seen that the regularity was not as obvious as before in the case of different tip clearances since there were no obvious periodic peaks and valleys, showing a certain amount of randomness; its frequency domain also had a large number of highfrequency parts, but it was not as good as the inlet and the middle of the tip, and as the tip clearance increased, the high-frequency part decreased, which was caused by interference from the leakage flow.

Conclusions
(1) As the tip clearance increased, the lift and efficiency decreased, but there was no linear relationship between the degree of the decrease and the change in the clearance. Furthermore, in the process of increasing the flow rate, the difference in the head under the size of the tip clearance first decreased and then increased, but the difference in the head under the condition of a small flow was larger than that under the condition of a large flow. In the relationship between the efficiency and flow rate, from a small flow condition to a large flow condition, the efficiency first increased and then decreased, forming a saddle region, and the pump performance dropped sharply after the flow rate exceeded the design condition.
(2) In the Z-X section, there was an obvious backflow at the top of the impeller blade, and there was a flow around the outlet and flow pattern disorder. When the gap value was 2.5 mm, a

Conclusions
(1) As the tip clearance increased, the lift and efficiency decreased, but there was no linear relationship between the degree of the decrease and the change in the clearance. Furthermore, in the process of increasing the flow rate, the difference in the head under the size of the tip clearance first decreased and then increased, but the difference in the head under the condition of a small flow was larger than that under the condition of a large flow. In the relationship between the efficiency and flow rate, from a small flow condition to a large flow condition, the efficiency first increased and then decreased, forming a saddle region, and the pump performance dropped sharply after the flow rate exceeded the design condition. (2) In the Z-X section, there was an obvious backflow at the top of the impeller blade, and there was a flow around the outlet and flow pattern disorder. When the gap value was 2.5 mm, a certain amount of reflux caused the fluid flow to be smoother, indicating that a certain gap could improve the flow pattern of the channel. Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Data on the analysis and reporting results during the study can be obtained by contacting the authors.