Research on Variable Circulation Design Method and Internal Flow Characteristic of the Axial Flow Pump

Abstract

To investigate the influence of circulation distribution on axial-flow pump performance, this study integrates numerical simulation and theoretical analysis methods, establishing a simulation framework based on MATLAB and CFX. By adjusting the circulation distribution function from the hub to the tip of the impeller, various design models were constructed. Three-dimensional parametric modeling of the blades was achieved through MATLAB programming, generating key parameters such as blade profile coordinates. Subsequently, the geometric data were imported into CFX to establish a full-flow passage numerical model. The simulation employed the RANS equations with the k-ε turbulence model to analyze flow field characteristics and hydraulic performance under different circulation schemes. Numerical results indicate that under identical circulation distributions, the head–flow curve exhibits a monotonically decreasing trend, while the efficiency curve demonstrates a distinct single-peak characteristic. Notably, under specific design conditions, variations in design parameters can shift the best efficiency point while simultaneously improving efficiency. Cavitation performance analysis reveals that as the hub-to-tip ratio increases, the velocity circulation initially rises rapidly before gradually stabilizing. This distribution pattern effectively optimizes the pressure gradient at the impeller outlet, thereby significantly enhancing cavitation resistance across a wide operational range. The proposed circulation control methodology provides critical theoretical support and engineering guidance for the hydrodynamic optimization of low-head axial flow pumps.

1. Introduction

Pump stations are the core infrastructure of water resource allocation systems, and their operational efficiency directly impacts regional water security and sustainable development. In pump station engineering, axial flow pumps, centrifugal pumps, and mixed flow pumps are the primary hydraulic machinery. Axial flow pumps, characterized by their low head, large flow rate, and high energy conversion efficiency, have become the preferred pump type in engineering fields [1]. According to statistics, the annual power consumption of axial flow pumps accounts for approximately 33.3% [2] of the total industrial electricity usage. Therefore, improving their overall performance is a critical research topic in fluid machinery.

Chen Huicheng et al. [3] investigated the influence of guide vane design parameters on the bidirectional operation performance of bidirectional axial flow pumps using a turbulence model in finite element software. The results showed that the performance of bidirectional axial flow pumps is significantly affected by the outlet angle of the guide vanes. Yang Fan et al. [4] conducted a full-flow passage three-dimensional numerical simulation to study the flow loss characteristics and impeller fatigue reliability of vertical axial flow pump devices. The results indicated that the internal flow loss is primarily caused by turbulent flow-induced turbulent dissipation entropy production, which is much higher than direct dissipation entropy production, with the guide vane body exhibiting the highest turbulent dissipation entropy production. Ding Jingyi et al. [5] explored the effect of blade placement angles and found that as the blade installation angle increases, the axial force on the impeller increases, and as the angle decreases, the axial force also decreases.

Yang Fan et al. [6] studied the influence of adjustable inlet guide vanes on the hydraulic performance of axial flow pumps and impellers. The results showed that as the guide vane angle changes from positive to negative, the performance curve shifts toward a larger flow rate. With an increase in the positive guide vane angle, the optimal operating condition of the axial flow pump moves closer to a smaller flow rate, but with lower hydraulic efficiency. Fan Yunyi et al. [7] investigated the cavitation and energy loss characteristics of an axial flow pump with a specific speed of 1500. Numerical simulation results revealed that cavitation bubbles first appear on the backside of the blade inlet edge, and as cavitation intensifies, the distribution range of cavitation bubbles expands, potentially filling the entire impeller passage under severe conditions. Li Wei [8] employed a lift-based method combined with variable circulation and variable meridional velocity design and, through strength verification, identified the main factors affecting the operational stability of large axial flow pumps. This systematic approach was applied to a fully adjustable axial flow pump with a diameter of 1800 mm, achieving excellent results.

Wang Li et al. [9] used CFD (computational fluid dynamics) numerical simulations and model experiments to study the impact of guide vane angles on the high-efficiency operating range of axial flow pumps. The results demonstrated that optimizing the guide vanes leads to more uniform and reasonable internal flow fields under both design and large flow rate conditions, significantly expanding the high-efficiency operating range of the axial flow pump. Wan Tao et al. [10] investigated the parametric design and numerical simulation of axial flow pumps. Their research showed that based on CFD calculations, the head and efficiency of the predictive model can be determined, and the velocity and pressure distribution patterns within the flow can guide the direction of structural optimization for axial flow pumps. Shi Gaoping et al. [11] conducted numerical simulations to study the effect of different blade installation angles on the performance of axial flow pumps, analyzing static pressure, relative velocity distribution, and head changes. The results indicated that blade installation angles of 0° and −4° result in better velocity distribution without back flow, while an angle of +4° causes liquid back flow on the suction side of the guide vanes, leading to increased hydraulic losses.

In the field of hydraulic optimization research for axial flow pumps, numerous scholars have achieved performance improvements through multidimensional approaches, including increasing the outlet angle of guide vanes to enhance flow characteristics, optimizing the hub ratio and hub contour to reduce hydraulic losses, and exploring cavitation characteristics to improve operational stability. Although the aforementioned study achieved relatively good results, the influence of the circulation distribution pattern from the hub to the rim on axial-flow pump performance was either not considered or only limited to linear circulation correction during the optimization process [12]. As a result, the optimization outcome only represents a partial modification of the circulation distribution, failing to identify the optimal circulation distribution law. Using computational fluid dynamics (CFD) numerical simulation methods [13,14], different circulation distribution models from the hub to the shroud are constructed to systematically study the impact of circulation variation patterns on the hydraulic performance of axial flow pumps. This study aims to reveal the quantitative relationship between circulation distribution and axial flow pump performance parameters, providing a theoretical basis for the optimal design of axial flow pumps.

2. Methods

This study focuses on an axial-flow pump operating at a rotational speed (n) of 1450 rpm, featuring an impeller hub diameter (d_h) of 120 mm and an outer diameter (D) of 300 mm, with design specifications including a flow rate (Q) of 380 L/s, a head (H) of 6 m, and 4 impeller blades (Z). Detailed pump parameters are provided in Table 1.

2.1. Governing Equations

The internal flow within a water pump can be regarded as a three-dimensional, incompressible, viscous turbulent flow without heat exchange. It adheres to the laws of mass conservation, momentum conservation, and energy conservation. Typically, this flow can be described using the continuity equation and the momentum equations.

(1) Continuity equation:

(1)${\displaystyle \frac{\partial \rho }{\partial \mathrm{t}}}+{\displaystyle \frac{\partial (\rho u_i)}{\partial x_i}}=0$

(2) Momentum equation:

(2)$\begin{array}{l}{\displaystyle \frac{\partial (\rho u_i)}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial (\rho u_i{u}_j)}{\partial x_j}}=-{\displaystyle \frac{\partial P}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}[\mu ({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}})+S_i]\\ \end{array}$

where $\rho$ is the density; $u$ is the velocity; $p$ is the pressure; $t$ is the time; $\mu$ is the dynamic viscosity; S is the source term; and $i$ and $j$ represent the components along each coordinate axis, following the rules of tensor operation.

2.2. Turbulence Models

The current Reynolds-averaged Navier–Stokes (RANS) turbulence models [15] mainly include the standard k-$\mathrm{\epsilon }$ model, the renormalization group (RNG) k-$\mathrm{\epsilon }$ model, and the shear stress transport (SST) k-ω model. Each model has its corresponding advantages and disadvantages, as detailed below.

The standard k-$\epsilon$ model introduces the transport equations for turbulent kinetic energy k and turbulent dissipation rate $\epsilon$ to characterize the turbulent viscosity, achieving a two-equation model for Reynolds-averaged simulation. It features a simple model, high computational stability, and high computational efficiency.

The RNG k-$\epsilon$ model is a modification of the standard k-$\epsilon$ model. It incorporates coefficients calculated precisely from theory, taking into account the influence of vortex on turbulence, which improves the simulation accuracy of vortex flows and has a broader range of applications.

The SST k-ω model is a modification of the standard k-$\epsilon$ model. This model can calculate a more accurate eddy viscosity, so it is widely used in the near-wall free stream with relatively high accuracy.

The hydraulic performance in this study demonstrates relatively low sensitivity to model selection. Comparative analysis of various models reveals that the RNG k-$\epsilon$ turbulence model offers superior predictive accuracy, with smaller relative errors in computational results, and the calculated flow field transition patterns align well with experimental observations. Furthermore, during axial-flow pump operation, the fluid surrounding the blades exhibits complex characteristics, including high-curvature flow and intense rotational motion. Conventional turbulence models often struggle to accurately simulate these phenomena. In contrast, the RNG k-$\epsilon$ model effectively refines turbulent viscosity corrections [16], precisely captures fluid flow trajectories, and clearly reveals complex flow features such as secondary flows generated in strong rotational motions. These capabilities enable the model to provide more realistic simulation results, demonstrating distinct advantages in this numerical study. After comprehensive consideration, the RNG k-$\epsilon$ turbulence model was consequently adopted for the numerical simulations of the axial-flow pump in this research.

The equation for turbulent kinetic energy (k) is as follows:

(3)${\displaystyle \frac{\partial (\rho k)}{\partial t}}+{\displaystyle \frac{\partial (\rho k{u}_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}\left[{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}{\displaystyle \frac{\partial k}{\partial x_j}}\right]+2{\mu }_t{E}_{\mathrm{ij}}{E}_{\mathrm{ij}}-\rho \epsilon$

The equation for turbulent dissipation rate ($\epsilon$) is as follows:

(4)${\displaystyle \frac{\partial (\rho \epsilon )}{\partial t}}+{\displaystyle \frac{\partial (\rho \epsilon u_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}\left[{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}{\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]+C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}2{\mu }_t{E}_{\mathrm{ij}}{E}_{\mathrm{ij}}-C_{2\epsilon }\rho {\displaystyle \frac{{\epsilon }^2}{k}}$

where u_i represents the velocity components in each direction; E_ij represents the components of the rate of strain; and µ_t is the turbulent (eddy) viscosity.

In general, the relevant parameters can be set as follows in Table 2.

2.3. Establishment of the Axial-Flow Pump 3D Model

This paper focuses on a hydraulic model of an axial flow pump with a specific speed of 800. As shown in the three-dimensional schematic diagram in Figure 1, the main design parameters of the axial flow pump are as follows: the design speed is 1450 r/min, and the design head is 6 m. The impeller diameter is 300 mm, and the tip clearance is 0.2 mm. The impeller adopts a four-blade structure, and the guide vane section is configured with 7 blades.

Based on the geometric design parameters, a three-dimensional computational model was constructed using numerical modeling methods: First, the geometric parameters of the guide vane blades were imported into MATLAB software, and a two-dimensional airfoil diagram was generated through numerical calculations. Second, Turbo-Grid software was used to create a parametric model of the impeller section, generating a high-precision three-dimensional impeller model. Figure 2 shows the complete three-dimensional model of the axial flow pump impeller and its computational domain schematic.

2.4. Meshing

Mesh generation, as a critical preprocessing step in computational fluid dynamics (CFD) analysis, directly influences the accuracy and computational efficiency of numerical simulations [17]. High-quality mesh generation not only effectively reduces numerical errors but also enhances the convergence of CFD solutions and the reliability of computational results. Therefore, establishing a reasonable mesh generation scheme is of significant importance for ensuring the accuracy of numerical simulations.

Based on the established geometric model, this study employs the finite element discretization method to divide the computational domain into multiple computational cells, enabling precise resolution of flow field characteristics (including hydraulic parameters such as velocity and pressure fields) within each mesh element. The mesh density exhibits a significant impact on simulation results. When the mesh count falls below 150,000, all monitored parameters demonstrate considerable deviations compared to calculations using higher mesh densities. However, when the mesh count exceeds 200,000, the variation amplitude of simulation results diminishes markedly [18]. The final grid size was determined to be no fewer than 150,000 elements, ensuring both computational accuracy and high efficiency. A comprehensive mesh quality assessment for all components within the domain confirmed compliance with standards. The orthogonality angle was 23.8239°, with 99.0972% of the grids rated as “OK” (acceptable) and 0.0000% classified as “Bad” (unacceptable). The skewness value was 2.95952, with 99.9472% of the grids rated as “OK” and only 0.0093% deemed “Bad”. Overall, the mesh exhibited exceptionally high quality.

Prior to mesh generation, preprocessing of the geometric data of the rotating machinery is required. This paper simplifies the rotating machinery into three main components: the hub, blades, and casing. Based on axial-flow pump design principles [19], a four-blade configuration was adopted. This design achieves optimal flow stability while effectively minimizing vortex-induced losses and flow resistance, thereby enhancing the pump’s hydraulic efficiency. This paper defines the rotational axis as the z-axis and sets parameters such as the geometric file sources, coordinate systems, and line types for the hub, casing, and blades. Considering the adjustable blade angle characteristic of axial flow pumps during operation, a reasonable tip clearance is set between the impeller blades and the casing [20]. By editing the topology settings, the geometric model is approximated using a topological structure, and local mesh refinement is applied to key regions such as boundary layers and flow separation zones, thereby optimizing the mesh structure and enhancing the accuracy of fluid simulations (see Figure 3, Figure 4 and Figure 5).

2.5. Boundary Conditions

In this experiment, a steady-state simulation was conducted using a k-$\epsilon$ turbulence model with the following computational settings: the total number of iteration steps was set to 800, and the convergence criterion was defined as residuals of all equations being less than 1 × 10⁻⁵. The computational domain employed the multiple reference frame (MRF) method, where the impeller region was designated as the rotating domain with a rotational speed of 1450 r/min, while the remaining regions were set as stationary (see Table 3).

The blade boundaries, along with the impeller’s inlet and outlet regions, were carefully refined to meet the non-dimensional wall distance (y+) requirements. The y+ parameter represents the dimensionless distance from the wall to the center of the first grid layer adjacent to the surface, which is functionally related to flow velocity, viscosity, and shear stress. An appropriate y+ value indicates optimal mesh arrangement for accurate near-wall flow resolution. The y+ values were maintained below 100 throughout the computational domain, satisfying the fundamental requirements for the selected turbulence model implementation [21] (see Table 4).

The inlet boundary was set at the entrance of the straight conical inlet flow channel, using a static pressure inlet with the pressure set to 1 atm. The outlet boundary was set at the exit of the straight pipe-type outlet flow channel, using a mass flow outlet. All wall boundaries were treated with a no-slip boundary condition [22]. The components included the inlet and outlet flow channel walls, impeller blades, impeller hub, impeller shroud, guide vane blades, guide vane hub, and guide vane shroud. The boundary type for these components was set as “wall”, and the outer wall of the impeller was designated as a counter-rotating wall.

3. Scheme Design

3.1. Design Principle of Variable Circulation

Circulation is the path integral of fluid velocity along a closed curve, and its basic formula with velocity is as follows:

(5)$\mathrm{\Gamma }=V_u\times 2\pi R$

where Γ represents the blade circulation; $V_u$ represents the circumferential velocity; and R represents the radius of the impeller.

According to the radial distribution of flow parameters, the radial equilibrium equation must be satisfied.

(6)${\displaystyle \frac{\mathrm{dp}}{\mathrm{dr}}}={\displaystyle \frac{\rho {\mathrm{v}}_{\mathrm{u}}^2}{\mathrm{r}}}$

where p represents the pressure; r is the blade radius; and $V_u$ is the circumferential velocity.

Because the velocity distribution on the impeller is different, different impeller design methods can be obtained. At present, the design of an axial flow pump is basically based on equal circulation design, and its airfoil on each flow surface produces the same pump high. Based on the design law of equal circulation, this paper will adopt the design method of variable circulation for axial flow pump, and the circulation Γ will change regularly by changing according to it. For the variable circulation design method [23], the most important thing is to change the circulation distribution along the radial direction according to the actual flow situation in the axial flow pump; that is, the load at the hub and rim of the blade is small and the load at the middle of the blade is larger than that at the outer edge. Therefore, this paper will artificially change the distribution on the impeller, control the change of circulation γ, and make its corresponding theoretical head increase along the radius.

The motion of the fluid in the impeller can be divided into relative motion along the blade and circular motion rotating with the impeller, and the velocity components produced by it are axial velocity and circumferential velocity, respectively v_u. However, the axial velocity is generally constant during the whole movement of the fluid. Therefore, the corresponding axial flow pump model can be designed according to the relationship between speed and chord placement angle. The corresponding formula is as follows:

(7)${\beta }_1^{\prime }=\arctan {\displaystyle \frac{V_{m1}}{u}}$

(8)${\beta }_2^{\prime }=\arctan {\displaystyle \frac{V_{m2}}{u-V_{u2}}}$

(9)${\beta }_L=({\beta }_1+{\beta }_2)/2$

where ${\beta }_1^{\prime }$ represents the inlet flow angle of the cross-section; ${\beta }_2^{\prime }$ represents the outlet flow angle of the cross-section; ${\beta }_L$ represents the chord line installation angle; $V_{m1}$ represents the inlet axial velocity; u represents the circumferential velocity; $V_{m2}$ represents the outlet axial velocity; $V_{u2}$ represents the outlet circumferential component velocity; ${\beta }_1$ represents the blade inlet angle; and ${\beta }_2$ represents the blade outlet angle.

3.2. Design Schemes

This paper primarily discusses the influence of different circulation distribution patterns from the hub to the shroud on the performance of an axial flow pump under the condition that the circulation at the hub and the hub-to-shroud ratio remain constant. Therefore, when considering the design scheme, the focus is on the impact of the circulation distribution pattern from the hub to the shroud on the characteristics of the axial flow pump. During the three-dimensional blade modeling using MATLAB, except for slight adjustments to the blade installation angle by altering the circumferential velocity v_u, other parameters such as the number of blades and the impeller diameter remain unchanged.

The design parameters of the prototype pump used in this paper are presented in Table 1.

This study investigates the influence of variable circulation design on the performance of an axial flow pump. Numerical simulations were conducted for five comparative schemes based on the same design operating point. From the hub to the shroud, a radial cross-section was selected at intervals of every 0.06 radius, resulting in a total of 10 sections. The design parameters of each section were substituted into fluid dynamics equations for calculation, ultimately obtaining the velocity fields and corresponding circulation values for each section. Based on these results, the radial distribution curves of circulation and circumferential velocity variation were plotted, as shown in Figure 6 and Figure 7.

The design schemes were constructed as follows: Scheme 1 employs conventional constant circulation design as the baseline reference; Schemes 2–4 establish distinct circulation gradient distributions—Scheme 2 features a nonlinearly increasing gradient along the radial direction, while Scheme 3 demonstrates a linear increasing gradient; Scheme 4 exhibits a nonlinearly decreasing gradient, all maintaining identical circulation values at the blade tip; and Scheme 5 adopts a linear decreasing gradient achieved through symmetric transformation of Scheme 3’s circulation distribution. By conducting coaxial comparative analysis of flow field characteristic curves from all five schemes, this research systematically reveals the influence patterns of different circulation gradient distributions on the hydraulic performance of axial flow pumps. Through importing numerical simulation results of all schemes into the MATLAB numerical analysis platform, three-dimensional blade geometries under different circulation distribution patterns were reconstructed based on parametric modeling methodology. Among them, Figure 8 displays the two-dimensional airfoil profile of the blade for the typical case (Scheme 3). By comparing the spatial surface characteristics of the blades under different circulation distribution patterns, this provides a visual analytical basis for optimized design.

Figure 6 and Figure 7, respectively, illustrate the circulation distribution curves and the circumferential velocity (outlet of the blade) distribution curves from the hub to the shroud for the five design schemes.

4. Variable Circulation Design Results

4.1. Result Analysis

Based on the results of numerical simulations, this paper systematically analyzes the hydraulic performance characteristics of an axial flow pump under different flow conditions and circulation distributions. The flow–efficiency curves and flow–head curves for various blade angles are plotted, as shown in Figure 9.

As can be seen from Figure 9, the numerical simulations demonstrate that under the design flow rate of 380 L/s, all four optimized schemes exhibit significant improvements in both efficiency and head compared to the initial design scheme, validating the effectiveness of the optimization strategies.

The efficiency–flow rate curves display a unimodal characteristic, characterized by a gradual increase followed by a rapid decline. The peak efficiency values of the conventional schemes are concentrated near the design condition (380 L/s), with Schemes 1–3 achieving peak efficiencies exceeding 85%. The efficiency curves of Schemes 1–3 exhibit similar trends below the design flow rate, while Scheme 3 achieves the highest efficiency under high-flow-rate conditions. Scheme 4 shows lower efficiency at both the design and high flow rates but demonstrates slight advantages in low-flow-rate regions. The reverse-designed Scheme 5 (featuring concave blades) shifts its peak efficiency toward low-flow-rate zones due to altered flow field characteristics.

The head of all schemes monotonically decreases the with an increasing flow rate. Scheme 3 achieves the highest head across all flow conditions, followed by Scheme 2. Despite Scheme 5’s superior efficiency, it exhibits the lowest head at the design flow rate and a pronounced downward trend in head.

The linear circulation distribution pattern (Scheme 3) and the gradual-to-steep circulation distribution pattern (Scheme 2) outperform the uniform circulation distribution scheme (Scheme 1) in both efficiency and head characteristics. Notably, the linear distribution scheme demonstrates exceptional performance under supra-design flow rate conditions (Q > 380 L/s). These findings indicate that different circulation distribution patterns are suitable for distinct operational scenarios, necessitating further research to establish universal criteria for optimal scheme selection.

4.2. The Efficient Range of Different Schemes

Based on the aforementioned scheme, this paper plotted the flow–head characteristic curves and flow–efficiency relationship curves under different circulation designs, with the specific results shown in Figure 10.

The high-efficiency region of an axial flow pump refers to the operational range within which the pump unit can maintain optimal hydraulic performance under specific working condition parameters. In the pump’s performance characteristic curve, the high-efficiency region typically corresponds to the peak interval of the efficiency–flow rate curve, i.e., the working interval where the optimal energy conversion efficiency is achieved under specific flow rate-head combinations. This paper employs the relative efficiency method to determine the range of the high-efficiency region; using the lowest peak efficiency among all schemes (Scheme 4; η = 83%) as the benchmark, the interval between the two flow rate points corresponding to a 5% drop in efficiency (η = 78%) is defined as the high-efficiency region. Based on the performance curve analysis results shown in Figure 11, the high-efficiency regions of the schemes are ranked as follows: Scheme 3 > Scheme 1 > Scheme 2 > Scheme 5 > Scheme 4. This result indicates that the optimization schemes employing different circulation distribution laws have a significant impact on the high-efficiency operational range of the axial flow pump.

4.3. The Internal Flow Characteristics of Different Schemes

This paper selects the hub-to-shroud streamline distribution, impeller cross-sectional pressure field, and velocity field as evaluation indicators for the internal flow characteristics of the axial flow pump. By maintaining the circulation distribution at the hub and systematically altering the circulation distribution patterns at other cross-sections, the influence mechanism of the optimization scheme on the internal flow characteristics of the axial flow pump is thoroughly analyzed.

This paper selects three characteristic flow conditions (Q = 240 L/s, 380 L/s, and 440 L/s, corresponding to 0.6 Q, Q, and 1.2 Q, respectively) for comparative analysis. Figure 11 illustrates the pressure distribution on the back of the impeller, the velocity field distribution, and the hub-to-shroud streamline distribution characteristics under each condition. The pressure field analysis results indicate that under low-flow conditions, the inlet region exhibits a significant high-pressure characteristic, while the pressure in the outlet region decreases markedly, leading to an increased pressure difference between the inlet and outlet. Although this enhances the head characteristics, it causes the operating point to deviate from the design point, reducing operational efficiency. Under both the design flow and high-flow conditions, the pressure distribution shows a decreasing trend from the sides of the blade toward the center. The high static pressure maintained in the inlet region on the back of the impeller helps suppress cavitation, while the high velocity and uniform pressure distribution in the central region of the impeller ensure stable flow. The slight pressure reduction in the outlet region reflects the conversion of kinetic energy to pressure energy. Additionally, the significant pressure difference between the high-pressure zone at the leading edge and the low-pressure zone at the trailing edge of the blade provides the primary driving force for fluid flow.

As the flow rate increases, the range of the low-pressure zone gradually narrows, and cavitation phenomena improve. This is mainly attributed to the sufficient back pressure maintained on the back of the impeller, preventing the liquid from reaching the saturation vapor pressure and thereby inhibiting bubble formation. However, when the flow rate increases from 380 L/s to 440 L/s, a slight pressure attenuation occurs in the high-pressure zones on both sides of the blade, leading to a decline in anti-cavitation performance and an increased risk of cavitation.

The analysis of the blade velocity distribution based on Figure 12 reveals significant differences in the internal flow field of the axial flow pump under different flow conditions. Under low-flow conditions (Q = 240 L/s), the deterioration of the inlet flow field conditions caused by the deviation of the inlet flow angle from the design value leads to a highly non-uniform flow field distribution. Specifically, local flow rotation and severe flow separation occur on the back of the guide vanes. These flow anomalies not only increase the hydraulic losses in the impeller but also disrupt the outlet flow field distribution, ultimately reducing the efficiency of the pump section. As the flow rate increases to the design condition (Q = 380 L/s), the inlet flow angle of the impeller gradually approaches an optimal range, significantly improving the inlet flow conditions and enhancing the uniformity of the flow field distribution. The vortex regions on the back of the guide vanes essentially disappear, and flow separation is effectively suppressed, resulting in smooth streamline distribution. However, under high-flow conditions (Q = 440 L/s), although the flow separation on the back of the guide vanes improves, the inlet flow angle of the impeller exceeds the reasonable range, causing new flow separation regions to appear on the front of the guide vanes. This leads to a reduction in flow field uniformity once again.

4.4. Variable Circulation Programs Hydraulic Loss

In the study of the hydraulic performance of axial flow pumps, whether employing constant circulation design or variable circulation schemes, hydraulic loss analysis is a critical aspect for evaluating system energy characteristics and operational efficiency. This research systematically investigates the hydraulic loss characteristics of the impeller, guide vanes, and bend sections, as well as their impact mechanisms on system performance, using numerical simulation methods. Additionally, potential sources of error are thoroughly analyzed.

The hydraulic losses in an axial flow pump system primarily include local losses (bend losses, valve losses, and inlet/outlet losses) and frictional losses (which are not considered in this paper). Local losses mainly arise from the localized resistance encountered by the fluid as it passes through bends and valves, as well as energy dissipation caused by poor design or flow separation at the inlet and outlet.

Based on CFX numerical simulation results, a comparative analysis of the losses in the guide vane and bend sections under different flow conditions for five design schemes was conducted. The results indicate that all five schemes exhibit minimal guide vane losses at the design flow condition (Q = 380 L/s), and the patterns of loss variation are consistent across the schemes. To provide a detailed explanation of the loss characteristics, Scheme 1, which has the lowest guide vane loss, is selected as a representative case. The loss data for Scheme 1 under different flow conditions are presented in Table 5.

The numerical simulation results indicate that the hydraulic loss in the guide vane section of the axial flow pump exhibits significant operational dependency. At the design flow condition (Q = 380 L/s), the guide vane loss reaches its minimum value of 0.096 m. As the operating condition deviates from the design point, the loss gradually increases, with a particularly notable rise under low-flow conditions. This phenomenon is primarily attributed to differences in flow states under varying flow conditions; at the design condition, the guide vane section achieves minimal hydraulic loss due to optimal fluid flow conditions and reasonable geometric design. When the flow rate is below the design value, the reduction in fluid velocity leads to decreased flow stability, potentially inducing flow separation and thereby increasing energy loss. Conversely, under high-flow conditions, the increased velocity triggers intense vortex motion and impact effects, further exacerbating hydraulic losses. A comparison of the hydraulic losses in the guide vane section for each scheme at the design condition is presented in Table 6, providing critical insights for further optimization of the guide vane design.

Under the condition of equal hub ratio design, Scheme 4, despite having the largest velocity circulation distribution, theoretically achieves the optimal match between flow rate and head at the design condition (Q = 380 L/s), maintaining a smooth flow state as the fluid passes through the guide vanes. This effectively suppresses turbulence and flow separation, resulting in superior head characteristics. However, a comparative analysis of guide vane hydraulic loss across different schemes reveals that Scheme 4 exhibits the highest guide vane hydraulic loss at the design condition, leading to an actual head lower than the theoretical expectation. This phenomenon can be attributed to the following factors: First, there is a mismatch between the geometric design parameters of the guide vanes and the inflow angle of the fluid, causing flow separation and vortex generation, which increases energy losses. Second, deviations exist between actual operating conditions and theoretical design parameters, including changes in fluid properties, fluctuations in inflow conditions, and other uncontrollable factors. These factors may collectively contribute to increased hydraulic losses in the guide vane section, ultimately affecting the overall performance of the axial flow pump.

5. Results and Discussion

This study investigates an axial-flow pump system through three-dimensional numerical simulations to systematically examine the effects of variable circulation design schemes on both the external performance characteristics and internal flow patterns. The optimized designs significantly improve the hydraulic flow field within the pump, leading to enhanced operational efficiency with all design variants achieving peak efficiencies above 86.2%. The main research findings are as follows:

(1). Performance characteristic analysis of the axial-flow pump reveals that under identical circulation distribution schemes, the head curve demonstrates a monotonic decreasing trend with increasing flow rate, while the efficiency curve exhibits a distinct single-peak characteristic.

(2). For specific design configurations featuring linearly decreasing circulation distribution, the best efficiency point (BEP) shifts toward lower flow rates while achieving a measurable 0.2% enhancement in peak efficiency compared to conventional designs.

(3). The study on the influence of circulation distribution patterns on the high-efficiency operating range demonstrates that a linearly increasing circulation distribution achieves a broader efficiency range, whereas a delayed-then-accelerated increasing distribution induces secondary flow losses, thereby degrading the pump’s efficiency performance.

(4). By analyzing cavitation performance under different designs, it is observed that as the hub ratio increases, the velocity circulation exhibits an initial rapid growth followed by a gradual leveling-off trend. This optimization significantly improves the pressure distribution at the impeller exit, thereby boosting resistance to cavitation over a broad operational range.

Author Contributions

Conceptualization, X.Y. and Q.G. methodology, X.Y. and Q.G.; software, Q.G., Z.L. and J.G.; validation, X.Y., Q.G. and J.G.; formal analysis, X.Y., J.T. and Z.L.; data curation, Q.G. and J.G.; writing—original draft preparation, X.Y. and Q.G.; writing—review and editing, L.S.; visualization, X.Y., Z.L. and J.T.; supervision, L.S. and J.T.; project administration, Z.L. and J.G.; funding acquisition, X.Y. and Q.G. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Nomenclature

ρ	fluid density (kg·m−3)
u	velocity (m·s−1)
p	pressure (N·m2)
t	time, s
µ	dynamic viscosity (N·s·m−2)
S	source term
ui	velocity components in each direction (m·s−1)
Eij	components of the rate of strain (m·s−1)
µt	turbulent (eddy) viscosity (N·s·m−2)
Г	circulation (m2·s−1)
Vu	circumferential velocities (m·s−1)

Footnotes

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Figures and Tables

Figure 1 Axial flow pump model diagram.

Figure 2 Impeller model diagram.

Figure 3 Mesh diagram of the inlet pipe.

Figure 4 Mesh diagram of the outlet pipe.

Figure 5 Impeller grid diagram.

Figure 6 Variation curve of circulation from hub to rim.

Figure 7 Curve of circumferential speed changes from hub to rim.

Figure 8 Typical-configuration (Scheme 3) 2D airfoil profile of the blade.

Figure 9 Flow–efficiency and flow–head curves at each blade placement angle.

Figure 10 Axial flow pump efficiency zone for each variable circulation scheme.

Figure 11 Pressure contour plot on the blade suction side.

Figure 12 Two-dimensional streamline pattern at the hub section.

Table 1

Design parameters.

Design Flow Q/(L·s−1)	Design Lift H/m	Rotational Speed N/(r·min−1)	Impeller Hub Diameter dh/mm	Number of Blades z	Impeller Outer Diameter D/mm
380	6.0	1450	120	4	300

Table 2

Related parameters table.

${\mathit{C}}_{1\mathit{\epsilon }}$	${\mathit{C}}_{2\mathit{\epsilon }}$	${\mathit{C}}_{\mathit{\mu }}$	${\mathit{\sigma }}_{\mathbf{k}}$	${\mathit{\sigma }}_{\mathit{\epsilon }}$
1.44	1.92	0.09	1.0	1.3

Table 3

Parameter settings.

The Number of Iterations	Convergence Criterion	Rotating Domain Speed (r/min)	Rotor-Stator Interface	Boundary Conditions
800	Equation residuals below 1 × 10−5	1450	Stage types	No-Slip Condition

Table 4

Blade y+ values of different schemes under design conditions Schemes.

	Scheme 1	Scheme 2	Scheme 3	Scheme 4	Scheme 5
y+	93.7299	58.1224	58.0355	68.4969	88.8594

Table 5

Losses at each flow rate for Scheme 1.

Volumes (L/s)	Hydraulic Loss at Guide Vane (m)	Hydraulic Loss at Bend (m)
240	0.873	1.292
320	0.510	0.634
340	0.318	0.442
360	0.190	0.288
380	0.096	0.152
400	0.128	0.155
420	0.211	0.201
440	0.324	0.241

Table 6

Hydraulic loss at the guide vane under each scenario design condition.

Scenario	Scheme 1	Scheme 2	Scheme 3	Scheme 4	Scheme 5
Hydraulic loss at the guide vane [m]	0.096	0.096	0.098	0.187	0.097

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