Comparative study on stator corner separation vortex characteristics in axial-flow pump and tubular pump: To understand the effects of guide-vane cone diffusion on horn-like vortices

The guide-vane cone diffusion is a typical structural difference between an axial-flow pump and a tubular pump, which is shown as low diffusivity and high diffusivity respectively, but it is still unclear how this structural difference affects the vortex characteristics in stator corner separation flows. In this paper, a comparative study of stator corner separation flows in an axial-flow pump and a tubular pump was conducted, and the effects of guide-vane cone diffusion on vortical structures were clarified. Firstly, for the apparent vortical features, compared with the guide-vane of axial-flow pump with low diffusivity, the horn-like vortex in the guide-vane of tubular pump with high diffusivity has the features of smaller scale, weaker swirling strength, shorter evolution cycle and lower pressure fluctuations. Secondly, for the vortex dynamics mechanism, the guide-vane cone diffusion of tubular pump can cause additional pressure energy recovery, which leads to higher adverse pressure gradients, so it is easier to induce additional shroud backflow near the suction surface. In this coupled flow field of main flow, namely hub corner separation flow and shroud backflow, the streamwise periodic oscillation of the pressure function gradient ωS ·▽(▽p/ρ) is found, and it induces the unique vortex-street-like distributions of the deformational vorticity ωS and the rigid vorticity ωR . This physical effect causes a mutual competition between the horn-like vortex near the hub corner and the opposite backflow vortex near the shroud side. It is this competition effect originating from the guide-vane cone diffusion that greatly suppresses the development of the horn-like vortex in a tubular pump.


Introduction
Axial-flow pumps and tubular pumps are two commonly used pump types in large-scale/low-head pumping stations [1], frequently employed in China's South-to-North Water Diversion Project.For instance, the Huai'an Pumping Station utilizes a vertical axial-flow pump with a power rating of 5000kW, while the Jinhu Pumping Station adopts a tubular pump with a power rating of 2200kW [2].In hydraulic design, when the nD conditions are similar, both types of pumps often utilize the same impeller model.For example, the Zaohe Pumping Station and the Longzi Lake Pumping Station, which employ a vertical axial-flow pump and a tubular pump, respectively, both utilize the TJ04-ZL-06 impeller model (tested in China Water Resources Beifang Investigation, Design and Research Co. Ltd.).Similarly, the Wannianzha Pumping Station and the Ehu Pumping Station, which use vertical axial-flow pumps and tubular pumps, respectively, employ the TJ04-ZL-20 impeller model (tested in China Water Resources Beifang Investigation, Design and Research Co. Ltd.).This approach offers several advantages, such as providing fast and accurate results and ensuring high reliability in engineering applications [3].However, despite using the same impeller model, axial-flow pumps and tubular pumps still exhibit significant operational and structural differences.Firstly, in terms of operation, mixed-flow pumps have a horizontal arrangement with different gravitational directions compared to vertical axial-flow pumps, as depicted in figure 1(a) and figure 1(b).Recent research has shown that for water pump systems with no free surface and high rotational speed, centrifugal forces have a more pronounced effect, while the flow differences caused by gravity are negligible [4].This finding has been confirmed through model tests.Secondly, regarding structure, the primary structural difference between the two lies in the diffuser vane angles (as shown in figure 1(c)), which significantly impact the internal flow within the diffuser vanes [5][6], thereby influencing pump performance.Therefore, special attention must be given to these differences in both model design and prototype operation.This study focuses on investigating the impact of diffuser vane structural differences on internal flow characteristics from a model design perspective.In the hydraulic design of axial-flow pumps and tubular pumps, the guide vane is an essential and indispensable component for flow passage.It plays a crucial role in pressure expansion, flow deflection, and elimination of swirl, thereby significantly influencing the flow within the impeller and the discharge passage.Its design directly impacts the pump's energy characteristics and operational stability [6][7].Previous research has identified separation in the flow near the hub as one of the most typical adverse flow phenomena in axial-flow impeller guide vanes [7].This phenomenon results in significant energy loss, flow passage blockage, severe pressure fluctuations, vibrations, and even stall issues [8][9].
Currently, research on flow separation with regard to axial-flow compressors is more extensive than that for axial-flow pumps, providing valuable insights for this study.Based on existing research, it is known that flow separation is primarily influenced by structural differences, such as hub gaps, blade thickness, and area ratio [10][11][12].For instance, Lee et al. [10] found that in multi-stage axial-flow compressors, the presence of stator hub gaps reduces the pressure ratio of the compressor, eliminating flow separation in the hub corner and improving compressor performance.Goel et al. [11] discovered that when the thickness of guide vane blades decreases, the pressure changes on the suction side of the blades become smoother, reducing flow separation and enhancing the hydraulic performance of the pump.Liu et al. [12] found that increasing the area ratio of the guide vane causes severe separation of the fluid from the surface of the blade, resulting in complex flow phenomena such as backflow, secondary flow, and vortices.This weakens the constraint ability of the guide vane on the fluid motion, increases hydraulic losses, and subsequently affects the pump's optimal efficiency and high-efficiency region.However, for guide vanes in axial-flow pumps and tubular pumps, the primary structural difference lies in the vane diffusion.Currently, the research on the impact of vane diffusion on flow separation is not clear.From a fluid dynamics perspective, vane diffusion affects pressure gradients, which in turn alters the distribution of turbulent kinetic energy and changes the characteristics of vortex structures [13].Moreover, the structure of the guide vane itself is highly complex, and the evolution of vortex structures exhibits diversity.Therefore, further comparative research is needed in this aspect.
Therefore, this study will focus on conducting a comparative analysis of stator corner separation vortex characteristics resulting from the structural differences in the guide vanes of axial-flow pumps and tubular pumps, under the condition of using the same impeller.The studies include the following two aspects: 1) To quantify the structural characteristics and differences in flow separation vortex in the stator of axial-flow pumps and tubular pumps.This will mainly include vortex core structure, vortex strength, pressure fluctuations, and other parameters.
2) To reveal the reasons and mechanisms behind the differences caused by variations in vane diffusion.This is of significant importance for controlling flow separation in the stator and enhancing the stability of large-scale/low-head pumping stations.
In this study, a comparative analysis is conducted between the vertical axial-flow pump model and the tubular pump model, both using the same impeller model.Although the influence of gravity on the internal flow characteristics of the model can be neglected, the alignment of gravity direction is maintained consistently for emphasizing the impact of vane diffusion on flow separation.

Research object
This study focuses on a vertical axial-flow pump unit model and a tubular pump unit model as the research objects.The flow passage structure mainly includes the inlet channel, pump stage, and outlet channel.In both pump unit models, the pump stage impeller adopts the TJ04-ZL-20 impeller model.The axial height and geometric parameters of the guide vane remain unchanged.However, in the tubular pump unit model, the guide vane hub and vane diffusion angle γ are set at 15.5°, which is greater than that of the vertical axial-flow pump unit model.The axial sectional structure diagrams of the pump stages in both pump unit models are illustrated in figure 2. The relevant geometric and hydraulic parameters of the pump stage are as follows: impeller diameter D2=300mm, hub inlet diameter Dh=140mm, blade tip clearance h=0.15mm, number of blades ZB=4, number of guide vanes ZG=7, blade setting angle β0=0°, rated speed n=1450r/min, and the impeller rotates counterclockwise when viewed in the direction of water flow.The three-dimensional computational domain models for the axial flow pump and tubular pump configurations in this study are shown in figure 3.

Meshing scheme and its verification
For the finite volume spatial discretization, high-quality hexahedral grids are employed to discretize the computational domains (figure 3).The boundary layer meshing of the impeller blades and the guide vanes is considered, while other regions with complex geometry are locally refined.As per the author's earlier research [14][15][16], in order to ensure the reliability of numerical calculations, a grid convergence analysis is conducted using the Grid Convergence Index (GCI) criterion based on the Richardson extrapolation method.As shown in figure 4, we ultimately established an appropriate meshing scheme, with 6,107,864 and 6,599,446 elements.

Solution setup and the proof experiment for validation
Regarding the research methodology, Wang et al. [17][18] previously conducted an examination of the structural characteristics and aerodynamic properties of flow separation vortex phenomena in vertical axial-flow pumps.They established both a computational approach based on the SST k-ω turbulence model and an analytical method centered on vortex decomposition.In the present study, ANSYS CFX computational software is employed to simulate the internal flow fields.Transient simulations are conducted with time steps corresponding to a 2º rotation of the impeller in each iteration.Numerical discretization strategies encompass the use of the second-order backward Euler scheme for the transient term, the second-order upwind scheme for the convective term, and the central difference scheme for the diffusion term.Boundary conditions involve a mass flowrate inlet characterized by medium turbulence intensity, along with the implementation of a static pressure outlet.The wall surfaces strictly adhere to a no-slip condition, and convergence is determined by a residual standard of 1.0×10 -5 .
In previous studies, the operating condition (0.92Q0) where separating the stator corner separation resulted in a 1% efficiency decrease was defined as the 'critical condition' [14].Therefore, in this study, under the condition of ensuring consistent flow rate of the impeller, the absolute flow rate at the 0.92Q0 (298.9L/s)operating condition of the axial flow pump was selected for research.This corresponds to the critical condition of the axial-flow pump and serves as a basis for comparing and conducting research on the tubular pump.

Monitoring scheme
To capture the characteristics such as pressure fluctuations associated with the separation of the flow in the guide vane domain, relevant monitoring points were arranged within the guide vane domain.The streamwise coordinate at the inlet of the guide vane is denoted as l=0, and at the outlet of the guide vane is denoted as l=1.Six cross-sections were determined in the streamwise direction, with streamwise coordinates of l=0.01, l=0.1, l=0.3, l=0.5, l=0.7, and l=0.9.The hub chord coordinate of the guide vane is denoted as s=0, and the shroud chord coordinate is denoted as s=1.Five sections were determined in the chordwise direction, with chordwise coordinates of s=0.1, s=0.3, s=0.5, s=0.7, and s=0.9.For descriptive purposes, a local coordinate system XYZ was established in the guide vane domain, with the flow passages marked as I~VII in a clockwise direction.Based on the aforementioned arrangement, three monitoring points were determined in the circumferential direction from the pressure side to the suction side of each flow passage at each chordwise section of each streamwise cross-section.Thus, there are a total of 630 monitoring points in the entire guide vane domain.Furthermore, each monitoring point is named according to the following format: P + flow passage position (1~7) + streamwise position (1~6) + spanwise position (1~5) + circumferential position (1~3).

Comparison of the vortex tube structures
According to the idea of vorticity binary decomposition, total vorticity ω can be further decomposed into the rigid vorticity ωR and the deformational vorticity ωS.This relationship is expressed as [19]: (1) First, to compare the basic structural characteristics of separated vortices in the guide vane domains of the two pump configurations, the vortical structures in the impeller and guide vane domains are shown in figure 6, obtained using the rigid vorticity identification method (Vorticity=500s -1 ).Analysis reveals that in both pump configurations, there are evident hub-corner separated vortices in the guide vane domain, primarily characterized by horn-like vortices attached to the hub end wall and the suction surface of the guide vanes.However, the key difference is that in the axial flow pump configuration, the development of the corner-separated vortices almost occupies the entire flow passage, whereas in the mixed flow pump configuration, the development of corner-separated vortices is mainly concentrated in the region near the suction surface of the guide vanes, occupying approximately half of the flow passage space.Furthermore, the scale of the vortex structure of the horn-like vortices is significantly smaller in the mixed flow pump configuration compared to the axial flow pump configuration.On the whole, for stator separation flow with larger diffusivity, the scale of the horn-like vortex structure is significantly reduced, and it is more concentrated near the hub.The evolution period is noticeably accelerated, approximately 0.67 times faster compared to nearly zero diffusion.

Comparison of the swirling strength
The time-averaged rigid vorticity magnitude along the water flow direction l in channel I of the two pump configurations is shown in figure 8, with an observation time approximately 30 times the rotor revolution period.The naming convention for the monitoring points is simplified as "P + 'position l' + 'position s' + 'position c'."In both pump configurations, it is observed that the monitoring points near the suction surface of the blades exhibit relatively high values of the time-averaged rigid vorticity magnitude, while the monitoring points near the pressure surface of the blades exhibit relatively low values.This observation is consistent with the structure of horn-like vortices attached to the hub end wall and the suction surface of the blades.The wave peaks traverse along the vortex motion trajectory and the flow direction, and due to dissipation effects, their amplitudes gradually decrease.The horn-like vortex is the dominant vortex structure in the separated flow and exhibits a characteristic of "increasing and then decreasing axial swirling intensity along the flow direction."Overall, the absolute values of the time-averaged rigid vorticity magnitude at various measurement points in the entire guide vane domain are generally higher in the axial flow pump configuration compared to the tubular pump configuration.The maximum absolute value of the rigid vorticity magnitude is 418.29s - in the axial flow pump configuration and 320.77s -1 in the tubular pump configuration, which is approximately 0.77 times that of the axial flow pump.On the whole, the vortex intensity of the horn-like vortices in the guide vane domain of the separated flow decreases overall as the diffusivity increases, with the maximum rigid vorticity magnitude being approximately 0.77 times that of the case without diffusivity.

Comparison of the pressure fluctuations
The distribution curve of pressure pulsation peak-to-peak values along the flow direction in channel I is shown in figure 9. Analysis reveals that the pressure pulsation peak-to-peak values at various monitoring points in the axial flow pump configuration are higher than those in the tubular pump configuration.The maximum value of pressure pulsation peak-to-peak occurs in the region near the suction surface, which is also the region where horn-like vortex cores grow.Figure 10 presents the distribution of maximum pressure pulsation peak-to-peak values at different flow positions.In the axial flow pump configuration, the maximum peak-to-peak value in the blade-free region between the impeller and guide vane (l=0.01) is approximately 0.91, often used to evaluate the maximum magnitude of fluctuation in axial flow pumps.The maximum peak-to-peak value in the entire guide vane domain is about 1.12, which is 1.23 times the value in the blade-free region.The maximum peak-to-peak value near the outlet (l=0.9) is approximately 0.85, which is 0.94 times the value in the blade-free region.In the presence of guide vane hub separation, the pressure fluctuation amplitude in the blade-free region is relatively small.In the tubular pump configuration, the maximum peak-to-peak value in the blade-free region between the impeller and guide vane (l=0.01) is approximately 0.66, and the maximum peak-to-peak value within the guide vane domain is approximately 0.62, which is 0.94 times the value in the blade-free region.The maximum peak-to-peak value near the outlet (l=0.9) is approximately 0.35, which is 0.53 times the value in the blade-free region.In the presence of guide vane hub separation, the pressure fluctuation amplitude in the blade-free region is relatively high.
On the whole, with increasing diffusivity from almost zero to a larger value, the overall magnitude of pressure pulsations decreases within the guide vane, and the maximum peak-to-peak values of pressure pulsations within the flow passage decrease by 44.6%.

Comparison of pressure energy recovery characteristics: Macroscopic view
The flow inside the axial flow pump and the tubular pump devices is a typical pressure-driven flow.For both the axial flow pump and the mixed flow pump, the diffuser design is the main structural difference between them.To analyze the reasons for the differences in the characteristics of the goat's horn vortex mentioned above, the velocity distribution on the expanded surface of the blade was plotted based on the different diffuser designs of the two devices, as shown in figure 11.For a conventional axial flow pump, it generally uses an axial outflow configuration, where the guide vanes primarily recover the axial component of the velocity Vθ, as shown in figure 11(a).In contrast, for a tubular pump, due to the increased inclination angle, there is a new component of velocity Vc in the direction perpendicular to the axial plane, as shown in figure 11(b).As the flow progresses, this component is also eliminated.Compared to axial flow pumps, the guide vanes of mixed flow pumps not only have to recover the kinetic energy generated by the axial velocity (recovering pressure energy), but also have to eliminate the portion of energy on the cone surface caused by the guide vane diffusion (recovering pressure energy).
Therefore, the recovery of pressure energy △p is defined as: Circumferential recycling Cone recycling Where ρ is the density, Va is the water flow velocity, α is the flow angle, and γ is the diffuser divergence angle.
From the equation above, it can be observed that, compared to a vertical axial flow pump, in a tubular flow pump with high diffusion, the hub cone of the guide vane will further generate additional cone pressure recovery on the basis of circumferential pressure recovery.This leads to a faster increase in outlet pressure and a higher adverse pressure gradient (figure 12), making it more prone to induce backflow at the suction surface outlet.The topological structure of this backflow is depicted in figure 13, and spatially it hinders the development of internal flow within the guide vane.

Comparison of deformation vorticity characteristics: Microscopic view
From a micro perspective of vortex dynamics theory, the pressure variation along the vortex line serves as the kinetic foundation for vortex tube deformation [20][21].
[ ] ( ) According to the Eq. ( 3), figure 14 illustrates the distribution of the pressure function gradient term ωS•▽(▽p/ρ), deformational vorticity ωS, and rigid vorticity ωR within the guide-vane region.The pressure function gradient along the deformational vorticity determines the rate of change of the deformational vorticity, which characterizes the ability to describe twisting vortices.The twisting effect of the deformational vorticity is the kinematic reason for inducing the development of rigid rotation.The rigid vorticity describes the rigid rotational motion of the vortices and represents the apparent result of separated vortices in the guide vane flow field.
It was found that in the high diffusivity guide vane, the pressure function gradient ωS•▽(▽p/ρ) exhibits streamwise periodic oscillation, leading to a unique vortex street distribution of the deformational vorticity ωS and the rigid vorticity ωR.There is a competition and constraint relationship between the two (the deformational vorticity of the twisting vortices exhibits a "flame-like" or "vortexstreet-like" oscillation, inducing anisotropic vortex pairs similar to the von Kármán vortex street).This balancing effect directly suppresses the development space of the horn-like vortex, resulting in smaller scale, lower vortex strength, and reduced pressure fluctuation amplitude.The main source of structural differences in the stator corner separation vortex between low diffusivity and high diffusivity is the distribution of the pressure function gradient term ωS•▽(▽p/ρ), which varies from low diffusivity to high diffusivity.On the whole, under high diffusivity, the stator hub cone surface generates additional cone pressure energy recovery, building upon the circumferential pressure energy recovery.This leads to a faster increase in outlet pressure and higher adverse pressure gradient, making it easier to induce backflow at the suction surface outlet.This coupling effect with the horn-like vortex results in a phenomenon of anisotropic vortex attenuation, limiting the development of horn-like vortices.Consequently, the vortex scale becomes smaller, and both the vortex strength and pressure fluctuation amplitude are reduced.

Summary and conclusions
A comparative study on stator corner separation vortex characteristics in the axial-flow pump and tubular pump is conducted to clarify the effects of guide-vane cone diffusion on the horn-like vortices.The conclusions are obtained as follows: (1) The typical vortical characteristics are quantified.Under the same operating conditions and with the same impeller model, as the diffusion increases from almost negligible to a larger extent, the separated flow with greater diffusion within the guide vane exhibits smaller scales in the vortex structure of the horn-like vortex, lower vorticity strength, accelerated development cycle, weaker pressure fluctuations, and lower amplitude of pulsation.
(2) The intuitive flow mechanism is revealed.The diffusion of the tubular pump guide vane cone results in additional pressure energy recovery, leading to a higher adverse pressure gradient and a greater likelihood of inducing backflow at the suction surface outlet.In the main flow region, it was observed that the pressure function gradient ωS•△(△p/ρ) exhibits streamwise periodic oscillation, inducing a unique vortex street distribution of the deformational vorticity ωS and the rigid vorticity ωR.The coupling interaction between the generated anisotropic backflow vortex and the horn-like vortex leads to an anisotropic attenuation.It is this competition effect originating from the tubular pump guide vane cone diffusion that significantly suppresses the development of the horn-like vortex within the tubular pump.

Figure 1 .
Figure 1.Schematic of the arrangement of axial-flow and tubular pumping systems.

Figure 4 .
Figure 4. Finite volume spatial discretization of the computational domains.

Figure 5 .
Figure 5. Schematic diagram of the monitoring points.

Figure 6 .
Figure 6.Comparison of vortex structures under two types of guide-vane cone diffusion.

Figure 7
Figure7illustrates the complete evolution cycle of the horn-like vortex in Channel I.In both the axial flow pump and mixed flow pump configurations, the horn-like vortex undergoes a cycle of birthgrowth-development-decay as the impeller rotates.At the inlet of the guide vane, a separation region is formed at the hub corner.As it progresses downstream, the horn-like vortex tube gradually elongates and deforms, occupying an increasing volume of space.Near the outlet of the guide vane, the vortex tube gradually breaks and leaves the guide vane domain with the mainstream flow, marking the beginning of a new evolution cycle.However, there are differences between the two configurations.The development of the horn-like vortex in the mixed flow pump configuration is faster.Specifically, in the axial flow pump configuration, the evolution cycle of the horn-like vortex is approximately 1.2 times the impeller cycle, whereas in the mixed flow pump configuration, it is approximately 0.8 times the impeller cycle.

Figure 7 .
Figure 7.The evolution process of the horn-like vortex.

Figure 9 .
Figure 9.Comparison of pressure fluctuations peak-to-peak values.

Figure 10 .
Figure 10.Distribution of maximum pressure pulsation peak-to-peak values.

Figure 11 .
Figure 11.Velocity component characteristics corresponding to different diffusivity.

Figure 14 .
Figure 14.Distributions of vorticity and pressure function gradient term.