Research on cavitation characteristics of water-jet pumps with different blade tip clearances based on entropy production theory

Water-jet pumps are widely used in ship and other power fields. An axial-flow water-jet pump is studied with different size of tip clearance, and the cavitation characteristics are studied based on the theory of entropy production. Modified SST with curvature correction and modified Zwart cavitation model based on vortex identification are adopted, and the numerical calculation method is verified by reference experiments of a model pump. Under normal operating conditions, the results show that when the size of the tip clearance varies from 1.6‰ to 7.9‰ of the impeller diameter, efficiency takes a linear decrease accordingly, and the total entropy production increases linearly. At the same time, the main energy dissipation mode changes from wall dissipation to turbulent dissipation. Under severe cavitation conditions the total entropy production adds 30%. The water-jet pump shows that the entropy production in the impeller section is the highest, which accounts for 40%-50% and is closely related to the vortex and cavitation flow field in the tip clearance of the impeller. The tip leakage vortex region causes cavitation, but significant energy dissipation occurs at the outer edge of leakage vortex and on the nearby wall area, while the attached cavitation on the blade surface is the main source of turbulent dissipation.


Introduction
Water jet propulsion is a power propulsion method in the marine field, and the basic principle is that by doing work on the inhaled water stream, the reaction force of the water jet propeller jet propels the ship forward [1].Water-jet pumps mostly have an axial impellers and the leakage vortex near the gap is the key to the flow characteristic.Increased tip clearance due to wear on the impeller and side walls, manufacturing and equipment deviations and leakage increases as the gap increases [2].The pressure pulsation, energy characteristics and cavitation stability of the pump are negatively affected [3].The internal turbulence field is full of different scales and many types of vortex motion, which can easily induce vortex cavitation.Leakage, vortex and cavitation all lead to energy dissipation in process of the flow, which in turn affects the overall performance of the water-jet pump.The analysis of vortex structure often relies on vortex identification techniques [4][5], and the calculation of energy loss often depends on empirical extrapolation in engineering.
In order to reveal the detailed characteristics of energy dissipation in the internal flow field, entropy production theory has received wide attention [6], and many scholars have applied this theory to analyze energy dissipation in various types of fluid mechanical devices.Kan et al. [7] analysed the effect of leakage flow at the gap of an axial pump, revealing the main components and sources of energy loss based on entropy production.In the meantime, analyzing the energy dissipation distribution at different locations inside the pump and under different operating conditions.Ji et al. [8] conducted an entropy production analysis of energy loss at different tip clearances and compared the different effects of leakage flow on the energy utilization of pump impeller and guide vane.Kan et al. [9] analysed the entropy production in the transition state of axial pumps under bidirectional conditions, and studied four modes of the transition process, and the position of the guide vane has a large influence on the pressure, vorticity and entropy production of the system, and Fei et al. [10] studied the energy loss variations at different cavitation development stages and revealed the intrinsic connection between leakage flow and cavitation vortex.
This paper takes research of a water-jet pump, focuses on the flow field analysis for the different widths of the tip clearances and carries out the numerical simulation research under the different cavitation conditions, studies the energy dissipation characteristics under the different widths of the tip clearance using entropy production analysis.Meanwhile, revealing the patterns of variations and correlation between the leakage flow, vortex flow, cavitation and the energy characteristics.

Computational case
A numerical computational model of the water-jet pump was established by referring to the model experiments of the axial flow propulsion pump carried out by Chesnakas et al. [11].As shown in Figure 1, the computational domain includes the inflow section, impeller section, guide vane section and outflow section.The rotation direction of the runner is -Z axis.Table 1 shows the main parameters of the calculation model of the water-jet pump, the rotor diameter D is consistent with the diameter D1 of the inlet, and the outlet D2=0.7D1.In the paper, the tip clearance is dimensionless and defining as the ratio of the width of the tip clearances to the rotor diameter.In the model experiments of Chesnakas et al. [11] the tip clearance of the water-jet pump is 1.6‰.tip clearances dimensionless.Three other tip clearances of 2.6‰, 5.2‰ and 7.9‰ were set up in the numerical simulations in order to investigate the effect of increasing the tip clearance size on the downstream flow field and energy dissipation.

Numerical calculation methods and boundary conditions
RANS is used to calculate the cavitation flow field of the water-jet pump, the turbulence model uses the SST-CC model with rotation and curvature correction.The cavitation model uses the Zwart model with local rotation correction, and the calculation method has been verified in the literature [5][6].
The inlet is given a total pressure condition, the numerical magnitude reflects different cavitation levels.The outlet is given a flow rate condition, and the numerical magnitude is determined according to different flow conditions.The impeller section is located in the rotating coordinate system and the steady solutions is used with a convergence accuracy of RMS=10 -5 .

Mesh generation and validation
The inflow and outflow sections of the water-jet pump are meshed with a tetrahedral unstructured grid, and the mesh is optimized near the main shaft and the wall of impeller edge.The guide vane and impeller section uses hexahedral structural grid, and the mesh division is completed in TurboGrid software.Mesh encryption is carried out at the tip clearance, and the number of nodes along the radial width of the gap is not less than 50.Grid layers encryption is also carried out near the surface of the guide vane and impeller to ensure the quality of the grid near the wall surface.The local mesh and the y + distribution of the impeller and guide vane section are shown in Figure 2, which meet the requirements of wall function calculation.The grid errors were evaluated by the GCI method [12] based on the Richardson extrapolation values.Based on the energy characteristics parameters of the water-jet pump under optimal operating conditions, three groups of grids with different encryption degrees were set up to obtain the values of parameters related to energy characteristics through numerical calculations of the flow field.Verifying the head, power and efficiency of the three energy characteristics of the parameters, and comparing the results of calculations, its parameters of the grid convergence indicators are less than 3%. is shown in Table 2. Ultimately determining the calculation of the grid scheme selected, and the total number of grids for 8.33 million [5].
Table 2. Influence of grid number on energy characteristic parameters.

Entropy production theory
The entropy production theory is derived from the second law of thermodynamics, and the entropy production is the dissipation effect caused by irreversible factors, mainly including turbulent dissipation and heat transfer dissipation.For the flow field without considering the temperature change, the isothermal model is used to ignore the heat transfer dissipation.The entropy production due to turbulent dissipation D S is calculated as: where T is the fluid temperature.The instantaneous velocity is composed of time-averaged velocity and pulsation velocity, the average term of entropy production rate replaces the velocity in Eq. ( 1) with time-averaged velocity, the pulsation term of entropy production is calculated based on turbulence energy and turbulent vortex frequency in the turbulence model, and the entropy production of the wall is related to the shear stress of the wall and wall mesh velocity, and the specific calculation formulas are shown in the literature [13].The above three items (mean entropy production, pulsation term and wall entropy production) together constitute the total entropy production, and the total entropy of the flow field can be obtained through the integration of each part.

Validation of numerical calculation results
In model experiments with water-jet pump, the cavitation factor N * is used to describe operating conditions with different degrees of cavitation, defined as Where pt1 represents the total propulsion pump inlet pressure (Pa) and pv is the saturation pressure, which is taken to be 3540 Pa in the calculation.
The relevant performance parameters are defined as follows: Where pt represents the total pressure (Pa), subscripts 1 and 2 indicate the inlet and outlet of the water-jet pump, respectively, ρ is the density of water (kg/m 3 ), n is main shaft speed (r/s), Torque is the pump shaft output torque (N•m), Q * is the flow coefficient, defined as Q * =Qv/(nD 3 ), and Qv is the volume flow rate (m 3 /s).
The energy characteristics at different cavitation coefficients N * are studied with reference experiments.Take the variation curve of head coefficient H * with cavitation coefficient N * under the condition of Q * =0.83 optimum flow rate, tip clearance b=2.6 ‰ as an example.Meanwhile, comparing the results of numerical simulations and experimental tests, as shown in Figure 3.Under the experimental conditions, as the cavitation coefficient N * decreases, the head coefficient of the pump basically remains unchanged when N * is large, and the external characteristics of the pump in this stage are not much different from those when no cavitation occurs; while the head coefficient appears to rise slowly when N * becomes small.When N * decreases further, the head curve appears to drop steeply, corresponding to the thrust collapse stage of the water-jet pump.The performance curve obtained by numerical simulation under normal clearance width is basically consistent with the experimental curve law.The thrust collapse belongs to the non-normal operation stage, which does not belong to the research tip clearance flow field in this paper.The cavitation is mainly concentrated in the impeller section.Three working conditions, N * =1.93, 1.46 and 1.19, were selected, and the errors of the water-jet pump energy characteristic parameters from the experimental values were calculated, as shown in Table 3, and the errors are less than 2%.At the same time, the cavitation simulation results of the impeller are compared with the experimental images, as shown in Figure 4.The simulation results use the vapor phase volume fraction iso-surface to characterise the cavitation (as shown in the blue area in the figure), and the cavitation morphology of the tip clearance under the three operating conditions is basically consistent with the experimental images.The reliability of the numerical calculation results is verified by the comparison of Figures 3  and 4.
Table 3. Numerical simulation deviation analysis of the water-jet pump.
N  4. Comparison of cavitation images between numerical simulation and reference experiment. [6]

Analysis of flow losses at different tip clearances
In order to study the flow loss characteristics of the water-jet pump at different tip clearances, the total entropy production value at different tip clearances is compared with the energy characteristics parameters of the pump based on entropy production calculation, with N * =1.931 and tip clearances of 1.6‰、 2.6‰、 5.2‰ and 7.9‰ respectively, as shown in Figure 5. Corresponding to Figure 4 operating condition A, this condition belongs to primary cavitation, which can reduce the influence of downstream instability phenomena due to violent cavitation phenomena on the observation of flow patterns with different gaps.From Figure 5, the total entropy production ΔS (W/K) of the water-jet pump basically increases linearly with the increase of the tip clearance.It means the dissipation and loss of the flow field will also increase linearly with the increase of the tip clearance.However, the dimensionless parameters, head coefficient H * , power coefficient P * and efficiency η show a linear decreasing trend with the increase of the tip clearance, and their change trend corresponding to the increase of the total entropy production value.This is due to the fact that an increase tip clearance makes the tip leakage flow and vortex more intense and unstable, resulting in the deterioration of the downstream flow field and increasing the energy loss in the flow field.4 shows the ratio of the entropy production value of each segment area to ΔS of the flow field.The energy loss is mainly concentrated in the rotor, accounting for 40-50% of the total entropy production in the flow field.At the same time, the loss in the guide vane and the outflow channel should not be neglected.As the tip clearance increases, the entropy production value of the impeller section increases accordingly, while the guide vane section and the outflow channel section corresponding decrease and the inlet channel section remains basically unchanged.The picture and table show that the increase of the tip clearance will lead to an increase in the strength, scale and instability of the tip leakage flow and vortex, and the level of interference with the main stream in the impeller will also increase, leading to an increase in the energy loss of the impeller section.When the strength of the tip leakage vortex increases, making it less likely to rupture in the impeller, and energy is not completely dissipated, and then affecting the flow field in the downstream guide vane section.
In a water-jet pump, the guide vane is the component that converts the kinetic energy of the fluid into potential energy and therefore creates a vortex structure to dissipate the kinetic energy in the fluid and reduce its flow rate.The width of the tip clearance affects the strength of the tip leakage flow, which in turn affects the speed at which the fluid enters the guide vane section, causing the dissipation in that section.Therefore, the energy loss and dissipation in the guide vane is negatively related to the width of tip clearances.6(a) combined with the shares of the components of entropy production value, it can be seen that turbulent dissipation and wall dissipation account for the majority of the total entropy production, while direct dissipation accounts for less than 0.5%, which is basically the same for different tip clearances.As the tip clearance increases, the turbulent dissipation gradually increases and the wall dissipation decreases.Taking b=7.9‰ as an example, Figure 6(b) demonstrates the entropy production data of turbulence dissipation and wall dissipation for different flow channel segments.As seen from the figure, both turbulence dissipation and wall dissipation in the impeller segment account for the largest percentage, and the wall dissipation in the outlet segment also accounts for a high percentage due to the large extension length.As the cavitation region of the water-jet pump is mainly concentrated in the impeller section, and the turbulent dissipation is mainly related to the turbulent kinetic energy and turbulent vortex characteristics.
The cavitation and vortex characteristics of the impeller section will be analysed in the next section.
(a)Entropy production as a percentage for different tip clearance (b)Distribution of entropy production for different calculation domains(b=7.9‰)Figure 6.Distribution statistics of entropy production components.Figure 7(a) shows the total entropy production of the water-jet pump for different tip clearances at different flow conditions, from which the difference in total entropy production at different cavitation coefficients for small clearance(b<5.2‰) is small.At large tip clearances, the total entropy production increases as the cavitation deteriorates, with an increase of 13.9% at b=7.9‰ for cavitation coefficient N * =1.14 compared to 1.931.From the analysis in Figure 7(b)(c)(d), it can be seen that at small clearances, the deterioration of cavitation leads to an increase in the tip leakage flow, while the high dissipation vortex formed and the chaotic region is transferred from the rotor to the downstream guide vane and the outflow.Therefore, the total energy loss and dissipation hardly changes, indicating that cavitation has little effect on the energy loss and dissipation aspect of the overall water-jet pump at small clearances b<5.2‰, and is not sensitive on the change in the cavitation coefficient N * .At large clearances b=7.9‰, the cavitation also causes secondary vortices, backflow vortices, etc., so that the total entropy production increases as the cavitation increases.

Relationship between flow characteristics and energy loss
The impeller is an important part of the energy conversion of the water-jet pump , which is surrounded by a rich vortex structure.Due to phenomena such as the rotation of the fluid and the fusion and generation of different vortices, which makes the energy loss around it larger, while the vortex flow is more important in the physical phenomena related to the evolution of cavitation, especially when judging the flow loss characteristics.Therefore, the reasonable use of specific vortex analysis methods can capture the main features of the vortices within the cavitation flow and make the prediction of entropy production more accurate.Liu [14] proposed a dimensionless vortex identification method, named Ω vortex identification method.The vortex structure is characterised by the ratio of the vortex size in the rotating part of the fluid micromass to the total vortex size, and the size of Ω represents the 'concentration' of vorticity.A reference range of 0.51 to 0.6 is generally recommended for identifying vortex structures, and Ω is calculated as where ε is a small quantity, taken as 1×10 -3 [s^-2] in this paper, and the subscript F is the Frobenius parametrization.
According to the analysis in section 4.2, the impeller section has the highest entropy production as a proportion of the total entropy production.Therefore, it is necessary to further analyse the flow field characteristics near the gap.Taking the circumferential spread at 98% of the spreading direction from the root to the top of the blade, Figure 8 adopts the Ω vortex identification method to demonstrate the vortex structure at different tip clearances.the blank area is the cross-sectional foil of the blade, the blade rotation direction in the figure is from bottom to top and the main flow direction is from left to right, where the area with higher Ω value corresponds to the local vortex range shown in Figure 8.The vortex zone formed near the inlet side of the blade towards the downstream suction side corresponds to the tip leakage vortex range, with the increase of the tip clearance, the leakage vortex zone is extended and the vortex volume increases, which also increases the intensity of the unstable vortex at the impeller inlet.The area of high Ω values adjacent to the suction side of the blade corresponds to the range of the attached cavitation on the blade surface, which extends downstream with increasing tip clearance and sharpens flow separation.A contour of entropy production pulsation term in the 98% cross-section in the blade spread direction is shown in Figure 9.It can be shown that the downstream flow separation and shedding triggered by the attachment vortices on the suction side of the blade are the main causes of the entropy production pulsation, while the location of the leakage vortex itself does not cause significant turbulence dissipation.As the tip clearance increases, the higher entropy production rate and the wider region.As seen from the results in Figure 6, wall dissipation is also higher in the impeller and guide vane sections, and the distribution of wall entropy production both the impeller shell and the single blade surface are shown in Figure 10.From Figure 10(a) can be seen that for different tip clearances, the peak wall dissipation in the impeller shell is mainly concentrated downstream of the blade outlet edge and continues further downstream to the guide vane section, where the wall dissipation in the guide vane shell is greater.However, as the tip clearance increases, the wall dissipation in the guide vane section extends downstream and the high dissipation region gradually decreases.This once again confirms the above conclusion that the energy loss and dissipation in the guide vane is is negatively correlated with the width of tip clearances.

Conclusion
In this paper, cavitation flow field inside a water-jet pump is analysed using numerical simulation method based on entropy production theory.The following conclusions are drawn: (1) Under different tip clearances, the energy characteristic curve of the water-jet pump is correlated with the variation law of the entropy production value, the pump efficiency curve decreases and the energy loss increases, which is consistent with the variation law of the total entropy production of the flow field.The energy characteristics parameters and the total entropy production of the flow field show a linear variation with the change of the tip clearance.
(2) The energy dissipation in the impeller is the largest at different tip clearances, followed by the guide vane and the outflow channel.The flow losses are mainly reflected in the turbulent dissipation and wall dissipation, and the proportion of both is equal.The energy loss and dissipation in the guide vane is negatively related to the size of the tip clearance.At small clearance(b<5.2‰), as the tip leakage vortex is not completely dissipated in the impeller section, and the excess energy is transferred to the downstream guide vane and the outflow channel.As a result, the overall energy dissipation in the water-jet pump does not differ significantly at different tip clearances, while the wall dissipation on the blade is not sensitive to changes in tip clearances.
(3) Axial water-jet pump cavitation region is mainly concentrated in the impeller section, especially the tip clearance cavitation and blade surface attached cavitation, analysis of the nearby of the blade and the tip clearance flow characteristics can be seen, with the tip clearance increases, the two cavitation phenomenon becomes intense, the blade surface attached cavitation vortex is the main source of turbulence dissipation, the tip leakage vortex area triggered by the vortex cavitation.However, the energy dissipation mainly occurs at the outer edge of the leakage vortex and the nearby wall area.

Figure 1 .
Figure 1.Calculation domain structure of water jet pump.In the paper, the tip clearance is dimensionless and defining as the ratio of the width of the tip clearances to the rotor diameter.In the model experiments of Chesnakas et al.[11] the tip clearance of the water-jet pump is 1.6‰.tip clearances dimensionless.Three other tip clearances of 2.6‰, 5.2‰ and 7.9‰ were set up in the numerical simulations in order to investigate the effect of increasing the tip clearance size on the downstream flow field and energy dissipation.Table1.Main parameters of propulsion pump calculation model.

Figure 2 .
Diagram of local grid.

Figure 3 .
Figure 3.Comparison of performance curves between numerical simulation and reference experiment.

Figure 5 .
Figure 5.Total entropy production and pump energy characteristics parameters for different dimensionless tip clearances.To analyse the distribution of the flow field energy losses, the entropy production values of the segmented areas of the water-jet pump (the four segments shown in Figure 1) were compared.At N * =1.931, different tip clearance conditions were analysed.Table4shows the ratio of the entropy production value of each segment area to ΔS of the flow field.The energy loss is mainly concentrated in the rotor, accounting for 40-50% of the total entropy production in the flow field.At the same time, the loss in the guide vane and the outflow channel should not be neglected.As the tip clearance increases, the entropy production value of the impeller section increases accordingly, while the guide vane section and the outflow channel section corresponding decrease and the inlet channel section remains basically unchanged.The picture and table show that the increase of the tip clearance will lead to an increase in the strength, scale and instability of the tip leakage flow and vortex, and the level of interference with the main stream in the impeller will also increase, leading to an increase in the energy loss of the impeller section.When the strength of the tip leakage vortex increases, making it less likely to rupture in the impeller, and energy is not completely dissipated, and then affecting the flow field in the downstream guide vane section.In a water-jet pump, the guide vane is the component that converts the kinetic energy of the fluid into potential energy and therefore creates a vortex structure to dissipate the kinetic energy in the fluid

Figure 7 .
(a) Total entropy production for different tip clearance at different cavitation coefficient (b) b=1.6‰(c) b=2.6‰(d) b=5.2‰Total entropy production for different components at different cavitation coefficients.

ΩFigure 8 .
Vortex identification contours for the blade top sections.

Figure 9 .
Figure 9. Contours of the entropy productionpulsation term for the blade top sections.As seen from the results in Figure6, wall dissipation is also higher in the impeller and guide vane sections, and the distribution of wall entropy production both the impeller shell and the single blade surface are shown in Figure10.From Figure10(a) can be seen that for different tip clearances, the peak wall dissipation in the impeller shell is mainly concentrated downstream of the blade outlet edge and continues further downstream to the guide vane section, where the wall dissipation in the guide vane shell is greater.However, as the tip clearance increases, the wall dissipation in the guide vane section extends downstream and the high dissipation region gradually decreases.This once again confirms the above conclusion that the energy loss and dissipation in the guide vane is is negatively correlated with the width of tip clearances.Figure10(b)shows the wall dissipation distribution on the suction side of a single blade of the impeller.Meanwhile, the tip clearance has little effect on its wall Figure 9. Contours of the entropy productionpulsation term for the blade top sections.As seen from the results in Figure6, wall dissipation is also higher in the impeller and guide vane sections, and the distribution of wall entropy production both the impeller shell and the single blade surface are shown in Figure10.From Figure10(a) can be seen that for different tip clearances, the peak wall dissipation in the impeller shell is mainly concentrated downstream of the blade outlet edge and continues further downstream to the guide vane section, where the wall dissipation in the guide vane shell is greater.However, as the tip clearance increases, the wall dissipation in the guide vane section extends downstream and the high dissipation region gradually decreases.This once again confirms the above conclusion that the energy loss and dissipation in the guide vane is is negatively correlated with the width of tip clearances.Figure10(b)shows the wall dissipation distribution on the suction side of a single blade of the impeller.Meanwhile, the tip clearance has little effect on its wall

Table 1 .
Main parameters of propulsion pump calculation model.

Table 4 .
Proportion of entropy production in each section of propulsion pump under different tip.