Numerical investigation on vortex dynamics characteristics of a large vertical centrifugal pump operating in the hump region under cavitation conditions

The large vertical centrifugal pump (LVCP) is widely used in major hydraulic engineering such as cross-region water transfer. This paper focuses on the hydraulic stability of LVCP under extremely operating conditions. The transient numerical simulations of cavitation in LVCP in the hump region are conducted based on delayed detached eddy simulation coupled with a homogeneous cavitation model. The main research is the effect of cavitation on the flow pattern and the vortex dynamics of the LVCP under stall conditions. The results of the numerical investigations show that the variation of the pump performance parameters is consistent with the experimental data under cavitation conditions. Under deep stall conditions, the H drop due to cavitation in the pump can happen earlier. Under the critical cavitation of the stall condition, backflow vortex cavitation, sheet cavitation and separation vortex cavitation can be observed at the impeller inlet. The vortex structures generated at the cavitation tail can result in worse flow patterns in the impeller. The vortex dynamics analysis shows that severe cavitation in the impeller deteriorates the inlet conditions of the vaned diffuser and induces sheet cavitation at the LE of vanes. The cavitations in the vaned diffuser accelerate the production and development of stall vortices and result in increased flow instability. The research in this paper can be helpful to improve the operation stability of LVCP.


Introduction
With the widespread application of major water conservancy projects such as inter-basin water transfer, large vertical centrifugal pumps (LVCP) are required to have a wider range of operation and meet more demanding operating conditions.Therefore, hydraulic stability has become a key indicator that needs improvement for this type of pump, especially under extreme operating conditions such as hump region and cavitation condition.The hump characteristics of LVCP under part-load conditions are closely related to the flow instability in the pump, and the greater the threat to the safe operation of the pump as the power increases.Cavitation can further worsen the flow instability in the pump, leading to even more severe hump characteristic [1][2].Therefore, studying the hydrodynamic characteristics of LVCP in the hump region under cavitation conditions is of crucial significance for improving the hydraulic stability of this pump.
In recent years, some scholars have investigated the potential association between cavitation and hump characteristics of large pumps through experiments and numerical simulations.U et al [3].studied the cavitation phenomenon of a pump-turbine in pump mode under stall conditions by experiments and simulations.Both experiments and simulations were able to capture cavitation vortices in the vaned diffuser flow channel with severe stall.Liu et al [4].investigated the hump characteristics of a pumpturbine based on an improved cavitation model.The investigation indicated that the hump characteristics are closely related to the cavitation flow in the pump-turbine, and the cavitation leads to a significant reduction in the head of the pump-turbine.Lu et al [2].investigated the effect of cavitation on the hump characteristics of a pump-turbine by experiments.The cavitation coefficient had an influence on the rotating stall in the vaned diffuser resulting in the change of the hump characteristics of the pump-turbine.Li et al [5].examined the pressure pulsation in the hump region of a pump-turbine based on large eddy simulations under cavitation conditions.The cavitation generation increased the amplitude of the stall vortex frequency in the vaned diffuser.
Based on the above analysis, most of the current research is mainly focused on pump-turbines with double-layer vaned diffuser.The investigations are more concerned with the effects of cavitation on the hump characteristics and pressure pulsations.The analysis of the effect of cavitation on the flow pattern and vortex dynamics of LVCP under stall conditions in the hump region is still scarce.Consequently, the unsteady numerical simulations of cavitation flow under stall conditions in LVCP have been carried out in this paper based on delayed detached eddy simulation (DDES) and Zwart homogeneous flow cavitation model.The vortex dynamics characteristics in the impeller and vaned diffuser under cavitation conditions are investigated with emphasis.The research content of this paper can provide valuable insights and assistance in improving the operational stability of LVCP.

Governing equations
In this paper, DDES was employed to predict the unstable turbulence of LVCP in the hump region.Detached eddy simulation (DES) was a cost-effective turbulence prediction method which combined Reynolds Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES).The RANS method was used to solve turbulence in the near-wall regions, and the LES method was used in the main flow separation regions, so further computational resources could be saved.Spalart et al [6] proposed the DDES method to improve the traditional DES method and provided limiter protection.The DDES method was able to overcome the mesh-induced separation based on the governing Eqs. ( 1) and (2).
( ) In the DDES governing equations, the ω term was consistent with that in the SST k-ω model, while k in the SST k-ω model was replaced by the length scale criterion for the transformation from the RANS to the LES l DDES term.The l DDES term was defined by Eq. 3.

Cavitation model
In this paper, the flow pattern of LVCP under cavitation conditions in the hump region was numerically simulated.The homogeneous flow cavitation model was used to predict the transfer rate between liquid and vapor phase.The Zwart cavitation model [7] was a typical homogeneous flow model, which assumes that the gas-liquid phase was a uniform incompressible fluid and no slip movement between the two phases.The transport equation of the Zwart cavitation model was defined as Eq. 4.
( ) ( ) The source terms m + , m − in the transport equation represent the mass transport between the vapor growth and collapse phases, also known as the liquid vaporization rate and vapor condensation rate, respectively.The vaporization and condensation processes can be described by using mass transport between gas and liquid, as defined in Eq. 5. (5)

Geometry model
The geometry model in this research was the LVCP with a one-layer stay vane.In order to adapt to the operating characteristics of large flow rate and high head, the impeller with high specific speed n s =215 was designed.The LVCP also included hydraulic components such as an elbow inlet pipe and a volute.The primary parameters of the LVCP were listed in table 1.The mass flow rate was 214kg/s, the head H was 18m and the speed was 1150r/min under the design condition.The computational domain of the geometry model was shown in figure 1.

Mesh details and verification
Refined hexahedral meshing was performed for each domain to meet the computational requirements of the DDES model.The results of the mesh discretization of the key hydraulic components inlet pipe, impeller, stay vane and volute were shown in figure 2. O-block topological structures were applied around the impeller blades and stay vanes and mesh refinement of the boundary layer was completed.
The effect of mesh nodes on the simulation accuracy was evaluated in this study.Three mesh schemes, coarse (11.11 million), medium (7.18 million) and fine (2.99 million) were created to validate the mesh independence.Based on the Grid Convergence Index (GCI) criterion proposed by Celik et al [8], the mesh uncertainty analysis was completed with the simulated H and efficiency η as the evaluation index.
The extrapolated relative errors of the fine mesh scheme η and H were 0.221% and 0.387% and the grid convergence index were 0.286% and 0.473%, respectively.As a result, the fine mesh scheme was finally chosen in this paper for the LVCP simulation under cavitation conditions.The y + distributions of the impeller blades and stay vanes were shown in figure 3, and the average y + values were able to be less than 5.

Experimental validation
It is crucial to verify the reliability of the numerical simulation method in this paper and to capture the hump region of LVCP before investigating the flow characteristics under cavitation conditions in the hump region of LVCP.Firstly, the performance test of LVCP under non-cavitation condition was completed on a closed test bench.The experimental measurements were taken with reference to the International Electrotechnical Commission (IEC) standards, and the LVCP model pump test bench was shown in figure 4. The experimental H and η (EXP) in the range of 0.2Q des to 1.2Q des were compared with the simulated values (CFD) obtained based on the cavitation model, and the results are shown in figure 5.
The CFD H and η were in good agreement with the EXP values and the maximum relative error was less than 5%.The η increased and then decreased with increasing flow rate, and the maximum η appeared near the Q des .The H gradually increased with decreasing flow rate.However, a sudden drop in H was observed in the range of 0.6Q des to 0.8Q des for both EXP and CFD.This corresponded to the positive slope of the Q-H curve, thus this range was defined as the hump region of the LVCP.It could also be seen from the figure that the minimum value of both EXP and CFD H in the hump region happened under 0.65Q des , which was defined as the deep stall condition.The conclusion was that the simulation methodology in this paper was worthy to be trusted.Additionally, experimental measurements of the cavitation performance of the LVCP under the design condition were performed, where internal cavitation was caused by decreasing the pump inlet pressure.The EXP and CFD cavitation performance curves were compared in figure 6.It can be seen from the figure that the CFD H is always higher than the EXP H, which was due to the fact that losses such as friction and leakage were neglected in the simulation.The H of LVCP was not significantly changed at first as the NPSH a decreased, and the cavitation in the pump was just beginning to happen and did not influence the performance of the pump.When NPSH a decreased to 4m, the EXP H was first to show a significant decrease and reached the critical cavitation point at NPSH a =2.6m, when the H drop reached 3%.The CFD H similarly started to decrease with a critical cavitation point of NPSH a =2.2m.When NPSH a fell below the critical cavitation point, a sudden drop in H occurred in both the EXP and CFD because the cavitation in the pump blocked most of the flow paths and thus seriously impacted the working capacity of the pump.It is concluded that the EXP and CFD trends of cavitation performance curves of LVCP are in good agreement.The critical cavitation point NPSH c of the EXP is slightly higher than that of CFD, which is due to more cavitation cores in the EXP medium and are more likely to cause the production of LVCP cavitation.In general, the cavitation simulation method in this paper is reliable.

LVCP cavitation performance analysis
To investigate the flow characteristics of LVCP under cavitation conditions in the hump region, the cavitation performance curves were firstly compared under design condition 1.00Q des , critical stall condition 0.74Q des and deep stall condition 0.65Q des .It was shown in figure 7 that as the flow rate decreased, the H of the pump increased for different NPSH a .However, H under 0.65Q des increased very little compared to that under 0.74Q des , which was due to the unstable flow pattern in the pump such as stall.In addition, when NPSH a was reduced below 3m, the H of LVCP decreased more rapidly under 0.74Q des and 0.65Q des in the hump region.This was attributed to the combined effect of unstable flow pattern and severe cavitation in the pump further causing the working capacity of LVCP to diminish rapidly.The NPSH c decreased gradually with decreasing flow rate to 2.15m, 1.73m, and 1.51m, respectively.It could be observed that the NPSH c under 0.65Q des decreased little compared to that under 0.74Q des , suggesting that the unstable flow pattern in the pump may induce the premature development of cavitation.

Vortex dynamics characteristics in the impeller under cavitation conditions
The cavitation pattern and pressure distribution in the impeller domain at NPSH c under design and stall conditions were shown in figure 8, and the cavitation structures were indicated by iso-surfaces with the vapor volume fraction α v =0.1.Under the design condition (1.00Q des ), it could be observed that the sheet cavitation was obvious on the suction surface (SS) of the blade inlet and showed an inverted triangular shape attached to the blade from the shroud to hub.The pressure distribution revealed that the sheet cavitation at the blade inlet was due to a large low-pressure region on the blade SS.As the flow rate was reduced to the critical stall condition (0.74Q des ), the sheet cavitation at the blade inlet further expanded to the middle of the flow channel.The upwardly raised cavitation cloud could be observed at the leading edge (LE) of the blade near the shroud.In the hump region, the impeller inflow angle was mismatched with the blade settling angle and caused significant backflow phenomenon [9].Therefore, the backflow vortex cavitation due to the low-pressure region caused by the backflow vortex was observed.Under the deep stall condition (0.65Q des ), the sheet cavitation on the SS of the blade inlet covered a wider and thicker region.The severe backflow phenomenon at the impeller inlet led to further development and growth of the backflow vortex cavitation cloud near the shroud.Significant cavitation was also observed near the hub, which was significantly different from the phenomenon under other conditions.The pressure distribution indicated that this was due to the hub separation vortex cavitation caused by the low-pressure region created by the flow separation vortex near the hub.The main conclusion in figure 8 was that the deterioration of the flow pattern at the impeller inlet in the hump region contributed to a more complex cavitation pattern, with backflow vortices and separation vortices inducing more cavitation structures to be created.To further investigate the vortex evolution and dissipation mechanism in the LVCP impeller under cavitation conditions, the interaction between cavitation and vortex as well as vortex dynamics characteristics were evaluated based on the vortex transport equation [10] in a rotating coordinate system.The relative vortex transport equation in the rotating coordinate system was defined by equation 7. The terms one to five on the right side of equation 7 were the relative vortex stretching (RVS), the relative vortex dilation (RVD), the coriolis force (CORF), the baroclinic torque (BT) term and the viscos diffusion (VISD) term, respectively.The internal flow of the LVCP was a high Reynolds number flow, so the VISD was neglected.
The distribution of vorticity and vortex transport term in z direction on the expanded surface of the impeller near the shroud under the design and stall conditions were given in figure 9.The results showed that the location distribution of the high vorticity coincided well with the distribution of vortex transport term.The core region of the sheet cavitation had no significant vorticity, but the vorticity values were significantly elevated at the tail of the sheet cavitation.Under the design condition (1.00Q des ), RVS, RVD, and BT showed high levels at the cavitation tail on the SS of the blade.It indicated that the flow field at the cavitation tail on the SS of the blade inlet was extremely unstable and accompanied by significant vortex deformation and stretched.The vortex generation at the cavitation tail was closely related to the expansion and contraction of the vapor volume during the vapor-liquid exchange as well as the change of the density gradient.
Under the critical stall condition (0.74Q des ), the intensity of the cavitation tail vortex further increased and the backflow vortex at the blade inlet started to appear, which corresponded to that in figure 8(b).The RVD term of the cavitation tail vortex was obviously higher than that under the design condition, which indicated that the expansion and contraction of the cavity volume at the cavitation tail was more intense and dominated the vortex generation.In addition, noticeable separation vortices were generated at the PS of the blade inlet due to the mismatch between the inlet flow and the blade settling angle.The separation vortex produced significant deformation and distortion, so the RVS term was extremely high.The RVS, RVD and CORF terms of the backflow vortex contributed more.This illustrated that besided the vortex stretching and volume change, the inertial force of the impeller rotation also promoted the production of backflow vortices.
The intensity of the backflow vortex, separation vortex and cavitation tail vortex were further enhanced under the deep stall condition (0.65Q des ).In particular, the RVD and CORF of the backflow vortex, the RVS of the separation vortex and the RVD of the cavitation tail vortex were significantly strengthened.Hence, it could be seen from figure 9 that the cavitation generation also further deteriorated the flow pattern at the impeller inlet under stall conditions and led to a further increase in the intensity of various vortices in the impeller.

Vortex dynamics characteristics in the vaned diffuser under cavitation conditions
As the LVCP operated in the hump region, unstable flow structures in the vaned diffuser were frequently generated, such as flow separation and stall phenomena.In order to study the effect of cavitation on the stall flow field in the vaned diffuser, the distribution of vorticity and vortex transport terms in the midsection of the vaned diffuser under NPSH c was presented in figure 10.Under the design condition (1.00Q des ), no significant vortices were observed in the diffuser flow path, and only slight trailing edge flow separation vortex (TEV) was observed at the TE of the vane.As a result, each of the vortex transport term in the diffuser flow path was kept low, and RVD and CORF were the main sources of TEV.
Under the critical stall condition (0.74Q des ), the vorticity in the diffuser flow path increased significantly, which was mainly manifested in the diffuser leading edge impingement vortex (LEV), pressure surface separation vortex (PSV), and TEV.Especially the LEV was more intense, and combined with figure 9 revealed that this was due to the cavitation in the impeller which provided even more mismatched inlet conditions to the diffuser inlet.The stretch and twist of the LEV was evident and certain volume variations were present, resulting in high levels of RVS and RVD.PSV was mainly caused by stretching and twisting of the main flow after flow separation.Compared to the design condition, TEV not only had volume variation but also had flexural deformation.More specifically, slight sheet cavitation could be observed near the LE of the vane, which was caused by separation vortex cavitation due to the high intensity LEV.The generation of sheet cavitation resulted in a larger density gradient and pressure gradient near the LE of the vane, resulting in an elevated BT value.
When the flow rate was further reduced to deep stall conditions (0.65Q des ), the stretch and twist of LEV, PSV and TEV were all more severe, and the coriolis force also had a significant effect on LEV and TEV.The further development and enlargement of the sheet cavitation at the LE of the vane was attributed to the sudden pressure drop in the vortex center due to the enhanced intensity of the LEV.The enhancement of sheet cavitation also caused the flow field at the cavitation tail to be more unstable, which induced a broader influence of the LEV and fused with the PSV, which was the precursor of the stall vortex generation.The main conclusion was that cavitation in the impeller deteriorated the stall flow field in the vaned diffuser, especially enhanced the intensity of the LEV and induced the creation of sheet cavitation.The sheet cavitation at the LE of the vane can further complicate the flow pattern in the diffuser flow path and induce the generation of stall vortex in advance under the effect of RVS, RVD and CORF.

Conclusion
In this paper, the transient numerical simulation of LVCP in the hump region under cavitation conditions was completed based on DDES with Zwart cavitation model.The distribution of cavitation pattern of LVCP in the hump region was elucidated, and the vortex dynamics characteristics in the impeller and vaned diffuser under cavitation conditions were investigated.The numerical investigation results indicated that the simulated performance parameters of LVCP under cavitation conditions were in good agreement with the experimental values.The cavitation performance curves of the LVCP revealed that the pump head decreased more rapidly with decreasing NPSH a in the hump region, and the cavitation in the pump would occur earlier under deep stall condition.Under stall conditions, the unstable flow structures at the impeller inlet could induce the generation of backflow vortex cavitation and separation vortex cavitation.Meanwhile, various cavitation structures and the cavitation tail vortices induced by them could further deteriorate the flow pattern at the impeller inlet.The severe cavitation in the impeller provided worse inlet conditions to the vaned diffuser and led to an increase in the LEV intensity in the diffuser inducing sheet cavitation at the LE of the vanes.The sheet cavitation at the LE of the vanes also resulted in more disturbed flow pattern in the diffuser flow path.The LEV and PSV would prematurely induce the generation of stall vortex under the effect of RVS, RVD and CORF.

Figure 3 .
Figure 2. Mesh of the LVCP.Figure 3. y + distribution.3.3.Numerical simulation setup The total pressure p s was set as the inlet boundary condition of the inlet pipe, and different effective cavitation margin for LVCP NPSH a could be obtained by varying the p s .The uniform mass flow rate at the outlet of the volute was established as a constant value of 214 kg/s, corresponding to the design condition Q des .All solid surfaces such as impeller blades and stay vanes as well as the volute walls were defined as no slip wall.The impeller domain was defined as a rotating domain with a fixed speed of 1150r/min.The SST k-ω turbulence model and the Zwart cavitation model were used for the simulation of the LVCP steady cavitation flow.The initial flow field for the unsteady cavitation simulation of LVCP was the result of the steady simulation and the DDES model and the Zwart cavitation model were used.The time step for the unsteady simulation was 2.89×10 -4 s, which corresponded to the time of 2° of impeller rotation.The maximum number of inner iterations was 15 per time-step.The total time duration of the LVCP unsteady simulation was 12 impeller rotation cycles and the data from the last 6 cycles were used for time-dependent analysis.The evaluation standard for the computed solution convergence was all set as 10 -5 root mean square (RMS).

Figure 6 .
Figure 6.Cavitation performance by CFD and EXP under 1.00Q des .

Figure 7 .
Figure 7. Cavitation performance of LVCP under different conditions.

Figure 8 .
Figure 8. Cavitation patterns in the impeller at NPSH c identified by iso-surface of α v =0.1.

Figure 9 .
Figure 9. Distribution of vortex transport terms near the shroud of the impeller.

(a) 1 Figure 10 .
Figure 10.Distribution of vortex transport terms of the vaned diffuser.