Application of three cavitation models in centrifugal pump cavitation flow simulation

In order to solve the problems of difficult and inaccurate cavitation simulation in centrifugal pumps, this paper adopts the commercial software CFX and realizes the same-platform comparison of Zwart-Gerber-Belamri model, Kunz model and Schner-Saer model in the simulation of centrifugal pumps’ cavitation by using its own language. The research results show that the Schner-Saer model can predict the external characteristics of the pump more accurately without considering the coefficient correction; By adjusting the empirical coefficients, different cavitation models can achieve the same effect; The Kunz model can predict the pressure distribution in the impeller more accurately, and the error is basically kept within 10% except for the local complex flow area; The change of the pump head is more obviously affected by the volume of the vapour in impeller, but weakly affected by the morphology of the vapour.


Introduction
Cavitation is a commonly natural phenomenon caused by local pressure being lower than the saturated vapor pressure, resulting in the formation of steam pockets.These pockets aggregate and collapse with the flow of fluid, not only disrupting the stable flow state but also generating considerable noise and causing damage to nearby surfaces [1,2].When cavitation occurs in a centrifugal pump, it leads to a decline in the pump's overall performance, accompanied by vibrations, noise, and other unfavorable effects [3,4].Hence, delving into the cavitation flow within the centrifugal pump holds paramount importance as it guides the design and enhances the pump's performance.
With the rapid development of computing technology, more and more scholars began to use the method of numerical simulation to study the cavitation phenomenon in centrifugal pumps.How to accurately simulate the cavitation phenomenon in centrifugal pumps under different states has been the focus of research.The key to capturing cavitation in numerical simulation is to establish a cavitation model.Common cavitation flow models include Kubota model [5,6], Merkle model [7], Kunz model [8], Zwart-Gerber-Belamri (ZGB) model [9], Schnerr-Sauer (S-S) [10] and Singhal (Full Cavitation Model) model [11].The Kubota model is an earlier empirical model.On the basis of Kubota's hypothesis, Merkle introduced the characteristic velocity and characteristic time of the flow field to control the cavitation effect.Kunz improved the Merkle model and proposed the Kunz model.The Singhal model is a semi-theoretical equation derived by introducing the simplified Rayleigh-Plesset equation on the basis of the incompressible continuity equation.The ZGB model and S-S model are similar to the Singhal model in that the simplified Rayleigh-Plesset equation is introduced, with the difference in the treatment of bubble size change in the three models.
On the whole, there are enough cavitation models to realize the numerical simulation of cavitation.However, it can also be found from the previous section that various cavitation models are established with certain assumptions and simplifications, so their applicability and accuracy under different conditions are worth discussing.Therefore, this paper selects three models commonly used in rotating machinery cavitation simulation, ZGB model [12], S-S model [13] and Kunz model [14], to investigate its applicability in the simulation of centrifugal pump cavitation.In this paper, the cavitation performance of centrifugal pumps at the design flow rate is firstly calculated and compared with the experimental values.Then the flow fields calculated by the three different models are carefully compared and discussed.

Mathematical mode
Continuity equations： ( ) 0 ) 3 Vapour phase volume fraction transport equation： ( ) ( ) Where: t is represents time, s; uj is the velocity component, the subscript i,j represents the coordinate direction.δij is the Kronecker number; v α is the vapour phase volume fraction; m ρ is rthe mixture density, kg/m 3 ; m µ and t µ are respectively the dynamic viscosity and turbulent viscosity of the mixed medium, kg/(m•s); R is the phase mass transfer rate, kg/(m 3 •s), which is calculated using a cavitation model; Re and Rc are the evaporation rate of liquid phase and condensation rate of vapor phase, respectively.[9] e B

3
(1 ) 2 , 3 Where: Ce, Cc are the evaporation coefficient and condensation coefficient, which are 50 and 0.01 respectively; nuc α is the non-condensable gas volume fraction; RB is the vapor bubble radius, take 1.0×10 -6 m; v ρ and l ρ are vapor phase density and liquid phase density, respectively, kg/m 3 ; Pv is saturated vapor pressure.The saturated vapor pressure of water at 25°C is selected as 3574 Pa in this paper.

Kunz model
(1-)max( , 0) (0. 5 ) Where: Cdest and Cprod are empirical coefficients, both of which are taken as 100; U ∞ is the characteristic speed, and this paper takes the average speed of the inlet; t ∞ is the characteristic time scale;

Geometric modeling and meshing
In this paper, a closed centrifugal pump is used as the object of study [15,16].The model consists of inlet pipe, impeller, front and rear back chambers, and diffuser.Its specific structure is shown in figure 1.The calculation domains are divided into structured hexahedron mesh, and the area near the wall is locally intensified.Under single-phase conditions, the maximum y+ value on the surface of the blade does not exceed 50, which meets the requirements of the turbulence model.The specific mesh division structure of the impeller is shown in figure 2. Five sets of meshes are selected for the trial calculation of the design points under single-phase conditions.It is found that when the number of meshes is greater than 5.06 million, the range of variation of the simulated head is less than 1%.So Mesh3 is selected as the final calculation mesh.

Numerical simulation setup
Under single-phase conditions, the inlet boundary condition is given as the total inlet pressure (1atm), the outlet boundary condition is the mass flow rate.The turbulence model selects the SST k-ω model.For cavitation calculation, water and water vapour at 25°C are selected as the medium, the homogeneous flow model is selected as the two-phase flow model, the SST k-ω model is used as the turbulent flow model.The total pressure is given as the inlet boundary condition, and the specific value is calculated using equation (12).The outlet boundary condition is still given as mass flow.
Where: Pin is the total pressure at inlet; Pv is the saturated vapor pressure; NPSH is Net Positive Suction Head.

Cavitation performance curve and numerical method verification
The comparison between the experimental results and the Simulation results of different cavitation models is shown in figure 3. It can be seen from the figure that the H calculated by the ZGB model and S-S model are closer to the test values.But, the results obtained by the Kunz model using the original empirical coefficients have a large error.Therefore, the optimized empirical coefficients in reference [17] are used to further simulated by modifying the empirical coefficients Cdest and Cprod, from the original 100 to 4400 and 440 respectively.The model after modifying the coefficients is defined as the KunzM model.From the figure 3, it can be found that the accuracy of the simulation results of Kunz model after the modification of the empirical coefficients is significantly improved, and the H obtained are basically consistent with the ZGB model.The applicability of the above three models at different flow rates was similarly discussed in reference [18], where larger empirical coefficients (Cprod = 9×10 5 , Cprod = 3×104) were applied to the Kunz model.It is found that the Kunz model is more accurate at small flow rates and rated conditions due to the consideration of feature lengths and incoming flow conditions.The reference [19] normalized the current common cavitation model forms, and pointed out that all cavitation models can achieve the same effect by adjusting different parameters.Many researchers have also improved the cavitation model, but after comprehensively improving the model, it can be found that although the simulation accuracy has been improved, it still cannot completely coincide with the result obtained by the experiment, which may be due to the accuracy of the calculation method used.Insufficient, it is not able to completely capture the flow phenomenon in the centrifugal pump.In general, the S-S model does not require coefficient correction and has high calculation accuracy, so it can be used as the first choice for the cavitation trial calculation under the condition of no experimental results.

The pressure distribution in impeller
From the equations of the three cavitation models, it can be seen that the accurate prediction of the pressure distribution is also crucial to the accuracy of the simulation.In order to facilitate the discussion of the change of the pressure distribution in the impeller, the pressure coefficient Cp is defined as follows: Where: u2 is the circumferential speed of the impeller outlet.
Figure 4 shows the distribution of Cp at 0.5 span of the impeller at three cavitation margins corresponding to point A, point B, and point C (H=Hd, H=0.98Hd, H=0.97Hd, respectively) in figure 3. It can be found that, except that the prediction at the blade head and outlet, the pressure distribution obtained from the three models are basically in agreement with those from the experiment.The reason for the errors at the blade head and impeller outlet may be due to the fact that these two locations are affected by head impact and static and dynamic interference, resulting in a more complex local flow field, which is difficult to be simulated accurately by the existing simulation means.Comparing the Cp distribution predictions of the three models, it can be found that the Cp predictions of the KunzM model are more accurate than those of the other two models due to the consideration of the influence of the inlet conditions.

Vapour volume distribution in impeller
The volume of Vapours in the impeller and the distribution of Vapour will have a serious impact on the flow field.Therefore, this section mainly studies the prediction of the Vapour field in the impeller by three models.The variation curves of the Vapour volume inside the impeller with the NPSH are shown in figure 5.It can be found that the Kunz model of original coefficients underpredicts the Vapour volume inside the impeller, which is the reason why its H differ greatly from the other models.The S-S model predicts a larger Vapour volume, followed by the ZGB and KunzM, and the latter two are basically close to each other.Figure 6 shows the vapour distribution morphology in the impeller under incipient NPSH, (Point C).It can be seen that the distribution of vapours in the impeller obtained by the three models is basically the same.The difference is that the vapour length predicted by the S-S model is different from that of ZGB and KunzM, and it can be found that the bubble morphology predicted by the S-S model has a small fracture at the blade head position, and its development along the suction side of the blade is not sufficient compared with that of the other two models in the flow channel.There are also some differences between the KunzM model and the ZGB model in terms of some local details, such as the bubble edge shape in the circumferential direction.In general, the prediction of the vapour volume in impeller is the key to the prediction of the H, while the shape of the vapour has little influence on the H. Taking a comprehensive view, there are no significant differences among the three models.According to previous research [17][18][19], it can also be observed that by adjusting the empirical coefficient, similar effects can be achieved among different aeration models.However, upon closer comparison, differences can still be found in the local details of the vapor morphology obtained by different models.The reason for this phenomenon is likely to be attributed to computational methods, as pressure prediction is coupled with vapor morphology prediction.Variations in mass transfer equations among different models can lead to differences in data during the solving iterative process.These differences may be the reason for the variations in pressure distribution and bubble morphology obtained by different models.

Conclusion
This paper compares the application of ZGB, S-S and Kunz cavitation models in the cavitation simulation of centrifugal pumps, and draws the following conclusions.
The S-S model is more accurate in predicting the H, followed by ZGB, and the Kunz model with original coefficients has the worst effect.By modifying the relevant coefficients, different cavitation models can achieve the same effect on the prediction of the H of the centrifugal pump.The S-S model is more suitable for calculation without comparison of test results because it does not need to consider the correction coefficient.
Due to the consideration of the influence of incoming flow and geometric characteristics, the KunzM model can more accurately predict the distribution of the pressure distribution in the impeller.In addition to the flow of complex at blade head and impeller outlet, the simulation obtained pressure and the experimental value of the error basically stays within 10%.
The prediction of the external characteristics of the centrifugal pump is mainly affected by the volume of the vapours in the impeller, and has little correlation with the distribution of the vapours in the impeller.The vapours distribution in the impeller obtained by the three models is basically the same, but there are some differences in the local details.

Figure 3 .
Figure 3.Comparison of H of simulation and experiment.

Figure 4 .
Figure 4. Comparison of simulated value and test value of Cp distribution in the impeller under different working conditions.

Figure 5 .
Figure 5. Variation curve of vapour volume in impeller with NPSH.Figure6shows the vapour distribution morphology in the impeller under incipient NPSH, (Point C).It can be seen that the distribution of vapours in the impeller obtained by the three models is basically the same.The difference is that the vapour length predicted by the S-S model is different from that of ZGB and KunzM, and it can be found that the bubble morphology predicted by the S-S model has a small fracture at the blade head position, and its development along the suction side of the blade is not sufficient compared with that of the other two models in the flow channel.There are also some differences between the KunzM model and the ZGB model in terms of some local details, such as the bubble edge shape in the circumferential direction.In general, the prediction of the vapour volume in impeller is the key to the prediction of the H, while the shape of the vapour has little influence on the H.

Figure 6 .
Figure 6.Isosurface distribution of vapour volume fraction= 0.1 in different cavitation models.Taking a comprehensive view, there are no significant differences among the three models.According to previous research[17][18][19], it can also be observed that by adjusting the empirical coefficient, similar effects can be achieved among different aeration models.However, upon closer comparison, differences can still be found in the local details of the vapor morphology obtained by different models.The reason for this phenomenon is likely to be attributed to computational methods, as pressure prediction is coupled with vapor morphology prediction.Variations in mass transfer equations among different models can lead to differences in data during the solving iterative process.These differences may be the reason for the variations in pressure distribution and bubble morphology obtained by different models.