Numerical investigation of vortex structure and unsteady evolution in multi-stage double-suction centrifugal pump

Multi-stage double-suction centrifugal pump is constructed to handle situations with large flow rates and high head. However, due to the complex internal flow structure, the pump can experience hydraulic excitation caused by the presence of numerous vortical structures. Such excitation can lead to unstable pump operation and increased energy losses. In this study, we aim to analyze the multi-stage double-suction centrifugal pump by combining numerical simulation using detached eddy simulation (DES) and experiments to accurately capture the vortical structure and elucidate the mechanism of the rotor-stator interaction (RSI) formation. The results indicate that the omega vortex identification method can accurately capture the vortex structure in the pump, irrespective of the threshold value and without the influence of wall shear layers. Additionally, based on this identification method, we have analyzed the unsteady evolution characteristics of the vortex structure in the pump. Specifically, we have focused on the shedding of wake vortices and their collision with the tongue. The findings suggest that the rotor-stator interaction primarily arises from the periodic shedding of wake vortices near the impeller outlet. In summary, this study provides valuable insights into the flow dynamics of multi-stage turbomachines.


Introduction
Due to their high flow rate and head, multi-stage double-suction centrifugal pumps are commonly used in large-scale irrigation and long-distance water transport projects.However, the pump's internal flow is complex, and it contains numerous vortex structures that can generate complex hydraulic excitations.This can cause unstable operation and increase energy losses, ultimately leading to lower energy efficiency.As a result, it is critical to study the unsteady vortex characteristics of the pump.
As Computational Fluid Dynamics (CFD) develops, numerical calculation has become increasingly widespread in the field of prediction and optimization of turbomachinery performance.There are many factors that affect the accuracy of numerical simulation, and one crucial factor is the selection of turbulence models.Currently, the technique that is widely employed in practical engineering is the Reynolds-Averaged Navier-Stokes Simulation (RANS) method, which performs Reynolds averaging on the physical quantities of the flow field in the time domain and then solves the time-averaged governing equations.Its most significant advantage is its high computational efficiency.However, the accuracy of the RANS method is relatively low, and it cannot obtain much transient information.Therefore, it cannot anticipate the dynamic changes of unsteady flow patterns, particularly vortex structures.
In 1997, Spalart et al. proposed the Detached Eddy Simulation (DES), one of the most widely used hybrid RANS/LES models.The DES method offers the advantages of both RANS and LES methods and has a simple construction and easy application characteristics, making it increasingly popular for analyzing internal flow in turbomachinery.Liu et al. [1] compared the computational results of a hydraulic torque converter under RANS, DES, and LES turbulence models.The researchers discovered that the LES and DES models could capture the transient vortex characteristics, such as vortex generation, shedding, and breakdown, more effectively than the RANS model.The DES model's computational time was also shorter than the LES model while maintaining computational accuracy.These results suggest that the DES model can efficiently and accurately simulate the internal flow field of the hydraulic torque converter.Yamada et al. [2] employed the DES model to simulate an axial compressor and explained the flow mechanism of the onset of rotational stall based on the computational results.The CFD technology via the DES-based turbulence model also demonstrated good stability and accuracy in the numerical simulation of reversible turbines and single-stage centrifugal pumps, especially in the investigation of pressure pulsation and vortex characteristics [3].The structure of multistage pumps is complex, and the internal flow exhibits apparent turbulent phenomena.Therefore, RANS models cannot accurately solve and predict the flow structure.Although DES is a highly accurate turbulence model, ensuring the convergence and accuracy of numerical simulations of internal flow fields in multi-stage double-suction pumps still poses particular challenges [4].
Vortices are a typical flow structure in the non-stationary flow inside turbomachinery.The process of vortex incipient, growth, and shedding to complete dissipation induces various instability problems in the operation of turbomachinery, such as turbulent motion noise [5], rotational stall [6], and cavitation [7], which will eventually affect the regular operation of the system.Therefore, the accurate identification of vortex structures in fluid machinery is of great significance to mastering various unsteady flow structures and optimizing the flow process in turbomachinery [8].In intuitive understanding, vortices are commonly associated with the rotational motion of fluids, but it is important to note that there is still much controversy in academia over the specific definition of vortices.In 1858, the German physicist Helmholtz defined the concepts of vortex lines, vortex tubes, and vortex filaments and proposed the famous Helmholtz equation-Subsequently, scholars [9] defined vorticity based on these concepts.Nevertheless, as research progressed, some scholars found that vorticity could not be equivalent to vortices, as the correlation between high vorticity regions in turbulent boundary layers and actual vortex structures was very low [10].Therefore, scholars [11][12][13] proposed Q criteria, λ2 criteria, and the regularized helicity Hn criteria depending on the velocity gradient tensor and Galilean invariance.These methods have been widely used in the field of turbomachinery vortex identification [14,15].Despite their widespread use, traditional vortex identification methods lack a clear physical interpretation and are highly sensitive to threshold selection, exhibiting strong dependence on specific examples and time steps.To address these issues, a new approach was presented by Liu et al. [16], which involves decomposing vorticity into two components, namely rotational and non-rotational.The rotational component was represented using the Liutex vector.The resulting dimensionless and normalized omega vortex identification method possesses clear physical meaning and is insensitive to threshold selection.Additionally, this method is capable of capturing both strong and weak vortices, in contrast to traditional Q criteria and λ 2 criteria.Moreover, through rigorous mathematical derivation, Liu et al. [17] demonstrated that the omega method exhibits Galilean invariance.This approach has effectively analyzed and studied various classic flow problems, such as boundary layer transition.Meanwhile, the omega vortex identification method is gaining increasing popularity and application in the field of turbomachinery (e.g., cavitation vortex [18], draft tube vortex rope ), with mounting evidence demonstrating its accuracy and versatility.
To sum up, the analysis of the evolutionary characteristics of vortex structures within a multi-stage double-suction centrifugal pump is of considerable academic importance and practical engineering value, utilizing a high-resolution turbulence model and a novel vortex identification method.Therefore, in this paper, a high-quality structured mesh was generated via the software ICEM-CFD, and high-precision numerical simulations of the prototype pump were performed using the DES turbulence model.The Kolmogorov law and the built-in blending equation in ANSYS-CFX were employed to verify the resolution of turbulence kinetic energy and ensure the reliability of the numerical calculations.The omega identification method was used to investigate the vortex structure evolution inside the pump, focusing on the portrayal of the periodic shedding of the wake vortex and its impingement and interference process with the tongue to analyze the rotor-stator interaction formation mechanism inside the pump.

Identification method of vortical structure
Selecting threshold values for vortex structures is a sensitive issue when using traditional vortex identification methods, as the weights may vary depending on the numerical case and time step.To mitigate this problem, the omega vortex identification method proposed by Liu et al. [16] is employed in this paper.The formula for this method is presented below.
2 2 max 0.001 ( ) Where D and R represent a symmetric tensor (deformation rate tensor) and an antisymmetric tensor (rotation rate tensor), respectively, and δ is a positive number whose purpose is to avoid the denominator being zero.
For practical applications, Ω=0.52 can be generally chosen as a fixed threshold to identify vortex structures, but this is mainly based on the simple flow numerical simulations such as flat channel flow, so the relationship between the vortex identification effect and the threshold value within a multi-stage double-suction pump will be investigated in Section 4.2.

Pump modeling
The pump model under study is a well-established multi-stage centrifugal pump with an axisymmetric and horizontally split design, illustrated in Figure 1.The second-stage impeller adopts a double-suction design, utilizing the exact mechanism as the single-suction impeller.To minimize radial force, a partition structure is included inside the double volute case.Detailed design parameters and geometric specifications can be found in Table 1.   2 illustrates the flow domain used in the numerical simulation.The suction and discharge pipes were extended to a certain length to avoid an overflow error in the numerical calculation, thereby enhancing the simulation precision.

Mesh generation
In this research, a reasonable topology strategy was used to achieve a high-precision hexahedral meshing of the computational domain via the software ICEM-CFD.Moreover, to ensure accurate simulation of vortex structure evolution in the pump and satisfy the DES model's near-wall meshing requirements, the first mesh height of the critical solid surface was less than 0.08mm, while the growth ratio was controlled in the range of 1.15-1.35.The detailed mesh is shown in Figure 3.As can be seen from Figure 4(a), the maximum y + value for each computational domain was less than 20, while the maximum aspect ratio was less than 200. Figure 4(b) separately demonstrates the blade y + distribution on the feature spanwise planes along the streamwise (s=0 represents the blade leading edge; s=1 denotes the blade trailing edge).It can be found that the overall y + values near the blade were between 5-6, and the maximum y + value occurred at the trailing edge.

Turbulence model and boundary conditions
In this study, the DES model depended on the k-ω shear stress transport turbulence model.The turbulent length scale l RANS is replaced by the DES length scale l DES , which serves as a switch in the model.
( ) Where l LES represents the filtering length scale of the LES subgrid-scale model, which is equivalent to the grid size; Δ denotes the maximum grid scale in the three directions of the element; C DES represents the model coefficient calculated by the blending function, which can be directly set to 0.65.
At the inlet and outlet, total pressure (1atm, the reference pressure was set to 0atm) and mass flow rate were used as the boundary conditions, respectively.The near-wall functions were handled by the standard wall functions built into ANSYS-CFX.As the research subject was highly complex rotating machinery, a second-order upwind scheme was employed to discretize the convective term rather than the central difference scheme to ensure the convergence of calculations.Meanwhile, a second-order implicit Euler difference scheme was used to discretize the transient term.The convergence accuracy was set to 10 -5 , the maximum number of iterations was set to 20 steps, and the timestep size was set to 0.00033557 seconds, which corresponds to a one-time step per every 1 degree of rotation of the impeller.The total simulation time was set to 0.604027 seconds, which corresponds to 15 revolutions of the impeller.During the calculation process, specific essential physical quantities were monitored, and the calculation was deemed to have converged when these quantities exhibited stable periodic oscillations.

Convergence analysis and timestep irrelevance analysis 3.4.1. Grid convergence index
The grid convergence index based on the Richardson extrapolation method was used to determine the specific number of grids for each flow domain.A detailed description of the selection of critical variables and grid distribution has been described in the previous study [19].In contrast to the previous study on the optimization of multi-stage double-suction pumps, the fine-grid scheme was chosen as the final solution to obtain more detailed information on the internal flow field.Figure 5(a) depicts the finegrid convergence index using error bars, while Figure 5(b) compares the extrapolated values and the fine-grid scheme.The maximum discretization uncertainty was 4.33%, which is lower than 5%.The fine-grid solution was close to the extrapolated value (the maximum relative error was 3.35%).Based on these findings, it can be concluded that the fine-grid scheme (comprising a total of 29.43 million meshes) meets the criteria of the GCI.

Timestep analysis
The outcomes of the numerical simulation are significantly influenced by the timestep selection.A smaller timestep can lead to more accurate simulations but also higher computational costs.Thus, the variation law of the pressure pulsation coefficient (C p ) at different monitoring points (P1 and P2) at three different timesteps in the last calculation cycle was compared.As shown in Figure 6, the curves at different timesteps exhibited significant peaks and valleys and presented an apparent periodicity.In the rotor, the curves were almost identical.Near the tongue area with intense energy exchange, the results for 3°/timestep were distinct from the other two schemes, while the results for 1°/timestep were consistent with those for 0.5°/timestep.Considering the calculation accuracy and time cost, the chosen 1°/timestep in this paper was reasonable.In addition, the root mean square of the Courant number was less than 20 for all numerical computations at this timestep.

Numerical method reliability verification
Currently, the validation of numerical simulations of multi-stage double-suction centrifugal pumps through visualization tests is challenging.Therefore, this paper ensures the reliability of simulation results by assessing the resolution of the DES model and external characteristics tests.First, a variable (Blending Function for DES model) was invoked in CFX-POST to verify the location of the RANS method and the LES method.This built-in function assigns a value of 1 (red region) in the area where the RANS method is used and a value of 0 (blue region) where the LES method is used.However, if the convection term is discretized using the central difference format, the global value is 1.The midsection of the first-stage impeller at the nominal condition was selected to validate that the mesh resolution can meet the DES model's specifications.As can be seen from Figure 7, the blue area covered almost the whole mainstream area of the impeller passage, except for a small part of the impeller outlet area and near the blade.Meanwhile, the red area near the blade almost overlapped with the area where the blade solid surface mesh was refined, as depicted in Section 3.2.As a result, it can be proved that the RANS method and LES method were used in a suitable area.Additionally, the power spectrum density distribution of the turbulence eddy dissipation (TED) at two different monitoring points (refer to Figure 6) is illustrated in Figure 8.As can be seen, they both satisfied the Kolmogorov law, which indicates that the simulation results via the DES method captured the inertial subrange of turbulence.To further validate the reliability of the simulation methodology, the pump was investigated experimentally using an open-loop test rig at Shandong Shuanglun Co. Ltd., China.A schematic of the test rig is depicted in Figure 9.The experimental configuration comprised an electromagnetic flowmeter with a precision of Â ±0.5% to measure flow rate and a torque meter with an accuracy of ±0.5% to measure shaft torque.Additionally, two piezo-resistive pressure transducers were employed, covering the range of 0-100kPa and 0-2.5MPa, with an uncertainty of ±0.1%.Data acquisition and analysis were carried out using a LabVIEW program, which recorded and analyzed the statistical characteristics of inlet/outlet pressure, flow rate, and shaft torque under various operating conditions at the nominal rotational speed.Experimental measurements were conducted at different flow rates to obtain the pump head, efficiency, and shaft power.The corresponding numerical simulation results were based on the average of the last three cycles of unsteady calculations.Meanwhile, these parameters were non-dimensionalized to obtain the head coefficient Ψ and the shaft power coefficient λ, respectively.The comparison results are demonstrated in Figure 10.The trends of the simulated head coefficient, efficiency, and shaft power coefficient were consistent with the experimental results.At the design point, the relative error between the simulated efficiency and the experimental value was 2.4%, and the relative error of the head coefficient was 1.8%.As the operating conditions deviated from the design point, the relative error of the calculation increased, which might be due to the complex flow conditions inside the pump, such as flow separation and other unstable flow conditions, which were exacerbated at off-design points.In addition, the gray area problem inherent in the DES method itself might also be a factor contributing to this phenomenon.Nevertheless, in general, the agreement between the simulated and experimental characteristic curves was high, and the numerical simulation results of this study can be considered reliable and able to reflect the flow conditions inside the pump realistically.
Where u 2 denotes the impeller outlet circumferential velocity, g represents the gravitational acceleration factor, ρ represents the water density, and R 2 represents the impeller outlet radius.
Figure 10.Hydraulic performance comparison between the numerical and experimental results.

Threshold verification
To investigate the relationship between the threshold value and the vortex identification effect of the omega method in the multi-stage pump, Figure 11 illustrates the overall vortex characteristics inside the pump at different thresholds for three operating conditions.The vortex structures were colored by the vorticity intensity, and all relevant physical variables were averaged over the last cycle of calculations.
As the threshold value increased, the number of vortex structures decreased slightly but not significantly, especially for some characteristic vortices, such as the backflow vortex at the impeller inlet and the passage vortex between the blades.In addition, the regions with high vorticity intensity of the vortex structures inside the pump were mainly located near the wall surfaces of the two-stage impeller blades and in the regions with intense rotor-stator interaction.There was no clear relationship between the vortex scale and the vorticity intensity.For example, the vortex scale inside the inter-stage flow channel was relatively large, but its vorticity intensity was very low. Figure 12 further demonstrates the above point.The vortex surface area hardly differed with the increasing omega value under different operating conditions.The effectiveness of the omega method in analyzing the vortex evolution process of multi-stage double-suction centrifugal pumps has been demonstrated, as it was capable of identifying strong and weak vortices simultaneously and was insensitive to changes in threshold values.Hence, a fixed threshold of omega=0.52 is a reasonable choice for analyzing the vortex evolution process in the pump.

Evolutionary process analysis of impeller vortex structure
Although velocity streamlines can intuitively indicate the presence and location of vortices, they have significant limitations in displaying local vortex structures and do not satisfy Galilean invariance.As a result, velocity streamlines have limited usefulness in displaying vortex structures in moving components such as centrifugal pump impellers.Therefore, this section mainly focuses on analyzing the vortex structure evolution process inside the rotor under the partial-load condition (0.6Q d ).

Vortex structure in first-stage impeller
In order to investigate the relationship between the vortex structure distribution inside the impeller and energy dissipation, turbulence eddy dissipation (TED) was used to color the vortex structures.T denotes the time it takes for the impeller to rotate one revolution.Figure 13 shows the vortex structure evolution in the first-stage impeller.At t 0 , there were many elongated reflux vortices with different scales along the circumferential direction at the impeller inlet.The passage vortices mostly appeared as small-scale longitudinal vortices (vortices along the circumferential direction) and some large-scale streamwise vortices (vortices along the flow direction).Large-scale separation vortices were attached to the blade suction side due to flow separation.At certain positions near the trailing edge of some blade pressure side, due to the symmetric structure of the flow channel, there existed a pair of symmetrically distributed vortex clusters α and α', which had not only similar shapes but also had similar energy dissipation distributions.In general, the TED value of vortices inside the impeller channel was lower than those at the inlet and the interface between the rotor and the stator.As the rotor rotated, distinct vortex bands began to appear near the shroud, and the reflux vortices gradually decreased, even disappearing at t 0 +1/3T.Meanwhile, the number of vortices inside the rotor channel gradually decreased after the process of merging, fragmentation, and dissipation, and the energy dissipation was relatively low, which indicates that the internal flow of the first-stage impeller was relatively stable during this period, and energy loss was also relatively small.When the impeller rotated to t 0 +1/2T, many large-scale vortices appeared inside the channel, blocking the flow passage, and the reflux vortices started to reappear, leading to a poor flow state of the first-stage impeller.When the rotor continued to rotate to t 0 +5/6T, the originally large-scale vortices inside the channel broke up into many small-scale vortices, and there were no prominent vortex bands near the impeller shroud.However, the separation vortices near the blade suction side have developed and extended towards the impeller outlet compared to the first two time periods.

Vortex structure in second-stage impeller
Figure 14 shows the vortex structure evolution in the second-stage impeller.At t 0 , the vortex structure scale in the second-stage impeller was large, and the vortices almost occupied half of the passage.However, the energy dissipation was small, and the high-value dissipation regions were mainly concentrated on the blade suction side.There were evident reflux vortices at the impeller inlet and wake vortices near the tongue, and it was also the place with the highest energy dissipation in the secondstage rotor.As time increased, the large-scale vortices in the impeller flow path gradually broke up and dissipated and then evolved into a variable number of small-scale streamwise vortices and wall vortices.These small-scale vortices continued to dissipate as the impeller rotated, and the vortex structure within the impeller flow domain constantly decreased until the moment t 0 +1/3T.Meanwhile, the scale of the reflux vortex at the impeller inlet in this time period also showed a significant trend of reduction.After that, the flow pattern inside the impeller became worse, and the vortices initially attached to the suction side gradually started to fuse and develop continuously toward the blade trailing edge, and the vortices almost reoccupied the whole flow channel again.Finally, at t 0 +2/3T and t 0 +5/6T, except for the separation vortex attached to the blade suction side and the inlet reflux vortex, almost no other largescale vortex can be seen inside the impeller, and no apparent wake vortex structure can be observed near the blade trailing edge.

Research on the rotor-stator interaction
Figure 15 demonstrates the evolution process near the tongue by tracing the vortex structure V1.At t 1 , the wake vortex V1 was located below the tongue and exhibited an elongated shape.As blade B2 gradually approached the tongue, the vortex structure V1 was gradually squeezed and diffused towards the tongue and the outlet of the volute casing.At t 1 +1/30T, V1 collided directly with the tongue and split into three vortex structures of different sizes.Subsequently, one part of the fragmented vortex system moved toward the left side of the tongue and merged with other vortices inside the volute casing, eventually entering the diffuser section.The other part entered the volute casing on the right side of the tongue and moved in the direction of the rotor rotation, affecting the downstream region of the tongue.Finally, at t 1 +1/8T, blade B2 moved to the position of blade B1 at t 1 , blade B3 moved to the position of blade B2 at t 1 , and a new elongated wake vortex structure V2 appeared below the tongue.It can be inferred from the symmetry of the impeller that the evolution of the vortex structure V2 would be similar to that of V1.Depending on the above analysis, it can be concluded that the shedding process of the wake vortex near the tongue was periodic, with a period of 1/8T, which corresponds to one time the blade passing frequency (BPF).The evolution process of the vortical structure near the tongue.The spectra of pressure, vorticity, and omega value at 0.6Q d for monitoring points I1, I2, and I3 near the tongue are given in Figure 16.It can be found that the pressure spectrum, vortex spectrum, and omega value spectrum of these three measurement points all had the same primary excitation frequency, i.e., one time the blade passing frequency, which was consistent with the frequency of the impeller wake vortex shedding near the tongue, indicating that the rotor-stator interaction between the first-stage impeller and inter-stage flow channel was mainly formed by the impact and interference process between the wake vortex periodically shed at the impeller outlet and the tongue.

Conclusion
The vortex characteristics were investigated using the omega identification method depending on the results of the DES turbulence model.The emphasis was placed on the detailed analysis of the vortex structure evolution process and the RSI formation mechanism.The main findings are summarized as follows.
1. To ensure the reliability of numerical simulation results based on the DES model for multi-stage turbomachinery, it is necessary to verify the turbulent kinetic energy resolution using multiple methods in addition to the external characteristic test verification.2. The omega vortex identification method was highly robust against threshold selection and resistant to the impact of strong wall shear layers inside multi-stage pumps.Therefore, the omega method can be considered the preferred for vortex identification in multi-stage doublesuction centrifugal pumps.3. The vortex structure evolution near the tongue was mainly manifested as the collision and interference between the wake vortex in the first-stage impeller and the stator.The frequency of wake vortex shedding corresponded to the blade passing frequency, consistent with the primary frequency observed in the pressure spectrum, vorticity spectrum, and omega value spectrum at the monitoring point near the tongue.These findings indicate that the periodic shedding of the wake vortices and their collision with the stator were the primary causes of the formation of the rotor-stator interaction.

Figure 2 .
Figure 2. Flow domains of the investigated pump.

Figure 3 .
Figure 3. Mesh details.As can be seen from Figure4(a), the maximum y + value for each computational domain was less than 20, while the maximum aspect ratio was less than 200.Figure4(b) separately demonstrates the blade y + distribution on the feature spanwise planes along the streamwise (s=0 represents the blade leading edge; s=1 denotes the blade trailing edge).It can be found that the overall y + values near the blade were between 5-6, and the maximum y + value occurred at the trailing edge.
(a) y + distribution (b) y + distribution characteristics of the blade

Figure 6 .
Figure 6.Time step irrelevance analysis under the nominal condition.

Figure 7 .
Figure 7. Blending Function for DES model for the first-stage impeller.Additionally, the power spectrum density distribution of the turbulence eddy dissipation (TED) at two different monitoring points (refer to Figure6) is illustrated in Figure8.As can be seen, they both satisfied the Kolmogorov law, which indicates that the simulation results via the DES method captured the inertial subrange of turbulence.

Figure 8 .
Figure 8. Power spectrum of the turbulence eddy dissipation.To further validate the reliability of the simulation methodology, the pump was investigated experimentally using an open-loop test rig at Shandong Shuanglun Co. Ltd., China.A schematic of the test rig is depicted in Figure9.The experimental configuration comprised an electromagnetic flowmeter with a precision of Â ±0.5% to measure flow rate and a torque meter with an accuracy of ±0.5% to measure shaft torque.Additionally, two piezo-resistive pressure transducers were employed, covering the range of 0-100kPa and 0-2.5MPa, with an uncertainty of ±0.1%.Data acquisition and analysis were carried out using a LabVIEW program, which recorded and analyzed the statistical characteristics of inlet/outlet pressure, flow rate, and shaft torque under various operating conditions at the nominal rotational speed.

Figure 9 .
Figure 9. Test rig.Experimental measurements were conducted at different flow rates to obtain the pump head, efficiency, and shaft power.The corresponding numerical simulation results were based on the average of the last three cycles of unsteady calculations.Meanwhile, these parameters were non-dimensionalized

Figure 11 .
Figure 11.Vortex structures identified by the omega method with different thresholds.Figure12further demonstrates the above point.The vortex surface area hardly differed with the increasing omega value under different operating conditions.The effectiveness of the omega method in analyzing the vortex evolution process of multi-stage double-suction centrifugal pumps has been demonstrated, as it was capable of identifying strong and weak vortices simultaneously and was insensitive to changes in threshold values.Hence, a fixed threshold of omega=0.52 is a reasonable choice for analyzing the vortex evolution process in the pump.

Figure 12 .
Figure 12.The quantitative statistic of vortex areas for different omega values.

Figure 13 .
Figure 13.The evolution process of the vortical structure in the first-stage impeller.

Figure 14 .
Figure 14.The evolution process of the vortical structure in the second-stage impeller.

(a) t=t 1 (Figure 15 .
Figure 15.The evolution process of the vortical structure near the tongue.The spectra of pressure, vorticity, and omega value at 0.6Q d for monitoring points I1, I2, and I3 near the tongue are given in Figure16.It can be found that the pressure spectrum, vortex spectrum, and omega value spectrum of these three measurement points all had the same primary excitation frequency, i.e., one time the blade passing frequency, which was consistent with the frequency of the impeller wake vortex shedding near the tongue, indicating that the rotor-stator interaction between the first-stage impeller and inter-stage flow channel was mainly formed by the impact and interference process between the wake vortex periodically shed at the impeller outlet and the tongue.

Table 1 .
Main design parameters.