Numerical Investigation of Clocking Effects on the Hydraulic Performance of Pump–Turbine in Pump Mode

Abstract

This study numerically investigates clocking effects on pump–turbine hydraulic performance in pump mode. Analyzing the influence of clock position on pressure loss characteristics under three flow conditions and its correlation with internal flow. By integrating local hydraulic loss theory and vortex evolution analysis, the operational mechanism is elucidated. Key results show that the stay vane clock position significantly impacts off-design conditions, causing maximum efficiency differences of 0.855% at 0.8Q_d and 0.805% at 1.2Q_d. At the design condition, guide vane clocking position has a more pronounced effect, yielding a maximum inter-scheme efficiency difference of 0.330%. The optimal scheme positions the tongue at the guide vane trailing edge and 1/4 of the stay vane flow path, minimizing time-averaged losses and enhancing flow stability. The clocking effect alters the scale and intensity of volute dual-vortex structures, significantly increasing energy loss at vortex interfaces, with volute loss identified as the primary factor in performance variation. This work provides a theoretical foundation for applying clocking effects in pump–turbine engineering.

1. Introduction

As key hydraulic machines, pump–turbines hold a significant position in water conservancy and energy systems. As a mature commercial energy storage technology [1], they are widely used for grid frequency regulation and load balancing [2]. Their performance directly impacts the efficiency and stability of power system operation [3,4,5]. With growing global emphasis on sustainable energy, improving pump–turbine performance has become a research focus in the field of hydraulic machinery. Optimal pump-mode performance is of great importance for enhancing the unit’s rapid response to grid frequency regulation demands, as well as improving the flexibility and reliability of pumped storage power stations.

In the field of pump-mode performance optimization, scholars have conducted extensive research. Yang et al. [6] achieved more accurate numerical simulations based on a weakly compressible water model, ultimately finding that the water injection approach could reduce hydraulic losses and suppress unstable flow structures, thereby delaying the onset of the hump region to lower mass flow conditions and expanding the safe and stable operating range of pump–turbines in pump mode. Zeng et al. [7] employed a partially averaged Navier–Stokes turbulence model for numerical simulations, ultimately elucidating the internal flow mechanism behind strong pressure pulsations during rotating stall in pump mode, which is crucial for assessing the stability of pump–turbine units. Wei et al. [8] improved the flow stability in the draft tube and runner under partial-load pump conditions by adding splitter blades, consequently reducing pressure pulsation intensity. Li et al. [9] introduced entropy production theory to investigate the causes of hump characteristics in pump mode, discovering that losses in the runner primarily originate from the suction side of the blades, while losses in the guide vanes and stay vanes stem from the pressure side of the blades and wake effects. Yuan et al. [10] studied cavitation characteristics in the tongue region under varying flow rates and rotational speeds, revealing that rotor–stator interaction and jet-wake effects significantly influence cavitation inception. With advancements in computer technology and multi-objective optimization algorithms, new approaches have emerged for pump–turbine optimization. Li et al. [11] employed a particle swarm optimization algorithm to refine guide vane airfoil parameters, ultimately improving turbine efficiency by 0.76% and pump efficiency by 0.14%. Ji et al. [12] employed an RBF neural network combined with the NSGA-II genetic algorithm to conduct multi-objective optimization on a PAT runner. The optimized model demonstrated efficiency and head improvements of 5.74% and 4.85%, respectively, compared with the original prototype. Current research on pump–turbines primarily focuses on establishing theoretical foundations for safe and stable equipment operation, with special emphasis on abnormal operating conditions such as the hump region and cavitation phenomena. Research on efficiency improvement has predominantly focused on the optimal design of individual flow components. While these studies are of significant importance for performance enhancement, they have overlooked the impact of synergistic interactions among multiple components. To minimize energy losses during pump mode operation, careful consideration must be given to the relative positioning of various flow passage components. This phenomenon, where modifications to the relative positions of components influence both the external performance characteristics and internal flow field behavior of rotating machinery, is termed the clocking effect [13,14].

The clocking effect theory has been widely applied in performance enhancement research for rotating machinery such as compressors and pumps. Schennach et al. [15] experimentally demonstrated that clocking effects significantly alter the secondary flow and loss coefficients in the second vane of a 1.5-stage high-pressure transonic turbine. Jiang et al. [16] found that in a vaned centrifugal pump, when the tongue is positioned at the mid-passage of guide vanes, the impeller experiences reduced radial force, weaker pressure pulsation intensity, and optimal efficiency. Ye et al. [17] investigated clocking effects in centrifugal pumps under stall conditions, revealing that when downstream blade leading edges consistently interact with upstream wake flow, the mixing of low-energy wake flow with boundary layers results in minimized pressure pulsations and maximum head. Furthermore, an appropriate clocking position can effectively mitigate vibration and noise during centrifugal pump operation. Tan et al. [18] found that the clocking effect between impeller stages in multistage centrifugal pumps has a significant influence on the unit’s vibration characteristics. Proper relative positioning of impellers can substantially reduce radial forces acting on the rotor. Under optimal clocking arrangements, the unit’s maximum vibration intensity was reduced by 23.1%, with average vibration intensity decreasing by 17.3%. Zhang et al. [19] demonstrated significant impacts of guide vane clocking positions on pump–turbine efficiency and head. Compared with conventional pumps, pump–turbines possess a greater number of flow passage components, resulting in more complex manifestations of clocking effects.

Based on the research background, clocking effects significantly impact turbomachinery performance. This study introduces clocking effects to pump–turbines, investigating how stay/guide vane positions affect hydraulic performance to identify optimal configurations. By combining localized hydraulic loss rates with flow field diagnostics, we reveal the underlying mechanisms. This approach enables cost-effective performance enhancement, providing a theoretical framework for pump–turbine optimization.

2. Methodology

2.1. Local Hydraulic Loss Rate Theory

Qin et al. [20] introduced the concept of local hydraulic loss rate (LHLR) based on the kinetic energy equation, under the assumption of incompressibility. This concept accounts for both dissipation and transport effects. According to the divergence theorem, the rate of change in the average velocity of Newtonian fluid per unit volume in turbulence can be expressed as:

$${\displaystyle \frac{DU}{Dt}}=\nabla \left({\displaystyle \frac{-p{\delta }_{ij}+2\mu S_{ij}-\rho \overline{u_i^{\prime }{u}_j^{\prime }}}{\rho }}\right)$$

(1)

$$S_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial U_i}{\partial x_j}}+{\displaystyle \frac{\partial U_j}{\partial x_i}}\right)$$

(2)

where U represents the mean velocity of the turbulent flow, the molecule on the right side of Equation (1) denotes the stress tensor, μ is the viscosity, S_ij is the shear strain tensor, and δ_ij represents the Kronecker delta. Based on the above equation, the formula for the hydraulic energy transfer term is derived as follows:

$$\rho {\displaystyle \frac{\partial }{\partial x_j}}\left[\left({\displaystyle \frac{p}{\rho g}}+{\displaystyle \frac{U_i{U}_i}{2g}}\right)g{U}_i\right]={\displaystyle \frac{\partial (2\mu U_i{S}_{ij})}{\partial x_j}}-\rho {\displaystyle \frac{\partial (\overline{u_i^{\prime }{u}_j^{\prime }}{U}_i)}{\partial x_j}}-2\mu S_{ij}{S}_{ij}+\rho \overline{u_i^{\prime }{u}_j^{\prime }}{S}_{ij}$$

(3)

The first and third terms on the right-hand side represent viscous transformation (TRNS-VIS) and viscous dissipation (DIS-VIS), respectively. The second term, related to Reynolds stress transport, is considered an additional effect of turbulent viscosity transport, while the fourth term, related to turbulent kinetic energy production, is regarded as an additional effect of viscous dissipation. The additional terms are calculated as follows:

$$-\rho \overline{u_i^{\prime }{u}_j^{\prime }}={\mu }_t\left({\displaystyle \frac{\partial U_i}{\partial x_j}}+{\displaystyle \frac{\partial U_j}{\partial x_i}}\right)$$

(4)

where μ_t is the turbulent viscosity. Equation (5) is the final formula for calculating the hydraulic energy transfer term.

$$\rho {\displaystyle \frac{\partial }{\partial x_j}}\left[\left({\displaystyle \frac{p}{\rho g}}+{\displaystyle \frac{U_i{U}_i}{2g}}\right)g{U}_i\right]={\displaystyle \frac{\partial (2{\mu }_{\mathrm{eff}}{U}_i{S}_{ij})}{\partial x_j}}-2{\mu }_{\mathrm{eff}}{S}_{ij}{S}_{ij}=TRNS-DIS$$

(5)

where μ_eff represents the effective viscosity, which is the sum of viscosity and turbulent viscosity. Equation (5) represents the local hydraulic loss rate of fluid particles; integrating it yields the total energy dissipation in the mainstream region (LHLR).

Kock et al. [21] proposed entropy wall functions, which are used in this study to calculate losses in the near-wall region. In the entropy wall functions, the dimensionless wall distance (y⁺) is the only independent variable, and the computational domain includes the log-law region and the linear region adjacent to the wall. Melzer [22] modified the entropy wall functions using wall shear stress as a reference value, ultimately providing dimensionless entropy wall functions for both mean flow and turbulent conditions as follows:

$$(T{\dot{s}}_{\mathrm{pro},\mathrm{wall},\overline{D}}{)}^{+}=e^{-{\displaystyle \frac{8}{9}}({\displaystyle \frac{y^{+}}{y_{ln}^{+}}}{)}^2}$$

(6)

$$(T{\dot{s}}_{\mathrm{pro},\mathrm{wall},{\mathrm{D}}^{\prime }}{)}^{+}=\{\begin{array}{c}0.15\\ (\kappa y^{+}{)}^{-1}\end{array}\begin{array}{cc}& \begin{array}{c}\mathrm{for}\\ \mathrm{for}\end{array}\end{array}\begin{array}{cc}& \begin{array}{c}{y}^{+}\le y_{ln}^{+}\\ y^{+}>y_{ln}^{+}\end{array}\end{array}$$

(7)

where κ denotes the Von Karman constant. Integrating Equations (6) and (7) over the cell height and averaging via characteristic length yields the average value of the integral expression for energy dissipation density in wall-adjacent cells.

$$(T{\dot{s}}_{\mathrm{pro},\mathrm{wall}}{)}^{+}={\displaystyle \frac{2}{y_1^{+}}}{\int }_0^{y_1^{+}}[(T{\dot{s}}_{\mathrm{pro},\mathrm{wall},\overline{D}}{)}^{+}+(T{\dot{s}}_{\mathrm{pro},\mathrm{wall},{\mathrm{D}}^{\prime }}{)}^{+}]\mathrm{d}{\mathrm{y}}^{+}$$

(8)

By substituting Equations (6) and (8) into Equation (8), the explicit expression for energy dissipation density is obtained as follows:

$$T{\dot{s}}_{\mathrm{pro},\mathrm{wall}}=T{\dot{s}}_{\mathrm{pro},\mathrm{wall},\overline{D}}+T{\dot{s}}_{\mathrm{pro},\mathrm{wall},{\mathrm{D}}^{\prime }}$$

(9)

$$T{\dot{s}}_{\mathrm{pro},\mathrm{wall},\overline{D}}={\displaystyle \frac{3\sqrt{\pi }}{2\sqrt{2}}}{\displaystyle \frac{y_{\ln }^{+}}{y_1^{+}}}{\displaystyle \frac{{\tau }_w^2}{\mu }}erf({\displaystyle \frac{2\sqrt{2}}{3}}{\displaystyle \frac{y_1^{+}}{y_{\ln }^{+}}})$$

(10)

$$T{\dot{s}}_{{\mathrm{pro},\mathrm{wall},}^{{\mathrm{D}}^{\prime }}}={\displaystyle \frac{2}{y_1^{+}}}{\displaystyle \frac{{\tau }_w^2}{\mu }}(0.15\min [y_1^{+},y_{\ln }^{+}]+{\kappa }^{-1}\max [(ln(y_1^{+})-ln(y_{\ln }^{+})),0])$$

(11)

Integrating Equation (11) over the wall surface yields the energy dissipation T in the near-wall region. The calculation formula for energy dissipation of the pump–turbine is as follows:

$$LHL{R}_1=LHLR+T$$

(12)

2.2. Model and Grid

To investigate the clocking effect on pump–turbine hydraulic characteristics, computational fluid dynamics analysis was conducted under pumping operation conditions. As shown in Figure 1, the computational domain was divided into six distinct flow sections. The pump–turbine principal geometrical parameters were detailed in

Table 1. Parameters of the pump–turbine.

Parameters	Symbol	Value
Runner blade number	Zr	8
Guide vane number	Zg	15
Stay vane number	Zs	11
Runner inlet diameter	D1	350 mm
Runner outlet diameter	D2	465 mm
Guide vane height	b0	84.2 mm

. Simulation conditions comprised a rated flow rate of 0.472 m³/s, 25.5 m head, and rotational speed of 1000 r/min, yielding a specific speed of 221.

The computational domains are discretized using structured grids generated with ANSYS ICEM 18.0, as depicted in Figure 2. To achieve proper near-wall resolution, the boundary layer mesh is refined to maintain y⁺ values below 35 on average. Particular attention is given to mesh refinement on blade surfaces and the tongue region to enhance simulation fidelity. For the flow field analysis, ANSYS CFX 18.0 is employed as the numerical solver. The numerical simulation employed the following boundary conditions: total pressure (101,325 Pa) with 5% turbulence intensity at the inlet and mass flow rate at the outlet (0.378 m³/s, 0.472 m³/s, 0.566 m³/s), while all wall surfaces were treated as no-slip boundaries. For transient analysis, the rotor–stator interface method was implemented between rotating and stationary domains, with steady-state results serving as initial conditions. This study employs the frozen rotor method at rotating/stationary interfaces, which assumes instantaneously fixed relative positions between components. This approach offers computational cost advantages. The unsteady computation spanned 10 complete impeller revolutions, utilizing a time step of 1.67 × 10⁻⁴ s (for the impeller rotating 1°) [23,24,25,26]. To guarantee solution accuracy, the RMS convergence criterion was maintained at 10⁻⁷ throughout the simulation. There are currently multiple turbulence models [27], and this study employs the SST k-ω model. By applying distinct formulations in near-wall and freestream regions, it delivers high accuracy throughout the domain while maintaining computational efficiency.

To ensure mesh-independent solutions, a systematic grid convergence study was performed using five progressively refined meshes ranging from 8.0 to 17.5 million elements. All grid schemes herein meet pump–turbine simulation requirements, with quality exceeding 0.3 and average orthogonality of 0.8. As illustrated in Figure 3, pump–turbine performance characteristics were evaluated based on hydraulic head and efficiency variations across different grid schemes [28,29,30,31]. The analysis demonstrates diminishing returns in head improvement beyond scheme 3 (12 million), with merely a 0.01 m difference observed between schemes 4 (15.4 million) and 5 (17.5 million). When the number of grids is greater than 12 million, the hydraulic efficiency changes very little. After balancing numerical accuracy with computational efficiency, the 15.4-million-element configuration (scheme 4) was adopted for all subsequent simulations.

2.3. Design Schemes and Experiment Verification

As illustrated in Figure 4, where θ₁ and θ₂ represent the circumferential angle between two stay vanes and guide vanes, respectively, θ₁ = 32.73° and θ₂ = 24°. Figure 4 shows the baseline scheme, identified as C0, features an angle of 8° between the tongue and the trailing edge of stay vane 1 (θ_S), and an angle of 23° between the trailing edge of guide vane1 (θ_G). According to the requirement of defining a new scheme for every 8.18° decrease in θ_S, five schemes (C0–C4) were generated due to θ₁ = 32.73°. Among them, the C0 and C4 schemes are the same. Seven guide vane schemes (C0, C5–C10) employed 8° rotational increments while maintaining fixed impeller-guide vane alignment to isolate the influence of the impeller on the inlet flow field of the guide vane. The C10 scheme differs from C0 only in minor impeller positioning.

In order to ensure the reliability of the numerical simulation, the calculated values of the head at different flow rates are compared with the experimental values. Figure 5 shows the test rig and model impeller. During experiments, torque was measured using a JN388 torque sensor with a range of 1000–4000 N·m and measurement accuracy of ±0.2%. Flow rate was measured by an electromagnetic flow sensor with a maximum range of 2650.5 m³/s and accuracy of ±0.5%. Pressure measurements were obtained using pressure sensors covering the −0.1~3.5 MPa range with ±0.5% accuracy. Due to the exclusion of volumetric and mechanical losses in numerical simulations, experimental measurements were slightly lower than calculated values, though both exhibited consistent trends. At design flow conditions, the discrepancies in head and efficiency were merely 1.1% and 0.15%, respectively, demonstrating that the mesh scheme, turbulence model, and computational settings employed in this numerical simulation ensure result reliability.

3. Result and Discussion

3.1. The Clocking Effect of the Stay Vane

3.1.1. Analysis of Hydraulic Performance

Figure 6 demonstrates the variation trends of hydraulic performance under three flow conditions through differential analysis of various clocking schemes. To highlight the clocking effect, performance differentials are calculated relative to the C0 scheme. The total pressure loss in differential performance diagrams is determined using the total pressure method, with calculation formulas for pressure losses in stationary and rotating components expressed as follows:

$$h_{\Delta p}={\displaystyle \frac{{\int }_{In}{P}_{TP}d\dot{m}-{\int }_{Out}{P}_{TP}d\dot{m}}{\rho \dot{m}g}}$$

(13)

$$h_{\Delta p}={\displaystyle \frac{W_S-({\int }_{Out}{P}_{TP}d\dot{m}-{\int }_{In}{P}_{TP}d\dot{m})}{\rho \dot{m}g}}$$

(14)

where h_Δp is the head loss, P_TP represents the total pressure of fluid particles, $\dot{m}$ indicates the mass flow rate of the pump–turbine, subscripts In and Out denote the inlet and outlet sections, and Ws stands for the impeller input power.

Figure 6 reveals that the clocking effect exerts more pronounced impacts under off-design conditions. As shown in Figure 6a, with the adjustment of stay vane positions, the hydraulic efficiency at 1.0Q_d and 1.2Q_d conditions follows a “decrease-then-increase” trend, where the C0 achieves peak efficiency. The minimum efficiency points across all flow conditions consistently occur in the C2 scheme, with maximum efficiency differentials of 0.855% (0.8Q_d) and 0.802% (1.2Q_d), while the 1.0Q_d condition exhibits a negligible difference of 0.090%. The head under varying flow conditions exhibits a “decline-then-rise” pattern with clocking position changes (Figure 6b). The maximum head deviation occurs at the 1.2Q_d—C2 scheme, reaching 0.883%. Comparatively, 0.8Q_d and 1.0Q_d conditions show smaller deviations of 0.321% and 0.05%, respectively. At 0.8Q_d (Figure 6c), the maximum power differential between schemes approaches 900 W, where the C1 yields minimal consumption and C2 incurs peak demand. For 1.2Q_d and 1.0Q_d conditions, power consumption first increases then decreases with clocking angle, recording maximum differentials of 60.7 W and 124.6 W, respectively. Total loss profiles across flow conditions shown in Figure 6d indicate that stay vane clocking positions predominantly affect large-flow operations (1.2Q_d), with strong correlations observed between efficiency variations and internal hydraulic losses.

For the 1.0Q_d condition, the C2 reduces hydraulic losses and increases head, but at the cost of higher power consumption and efficiency degradation. The C3 exhibits the lowest efficiency with excessive power draw, while the C1 consumes more power than the baseline C0 scheme. Consequently, C0 remains the optimal clocking scheme with balanced energy metrics. Under 1.2Q_d condition, all alternative schemes (C1–C3) demonstrate reduced efficiency, lower head, and elevated impeller power draw compared with C0. For the 0.8Q_d condition, the C1 achieves peak efficiency but with a corresponding head reduction. Both C2 and C3 schemes show compromised performance, whereas the C0 scheme maintains better hydraulic characteristics.

3.1.2. Analysis of Transient Characteristics

Figure 7 presents the transient evolution of efficiency, head, and total pressure loss for the C0 across three flow conditions within one impeller revolution cycle, where impeller rotational angle replaces temporal coordinates to investigate the clocking effect’s impact on pump mode transient characteristics. For 1.0Q_d and 1.2Q_d conditions, all performance parameters exhibit strong periodic fluctuations with eight complete cycles per revolution (corresponding to impeller blade count), demonstrating rotor–stator interaction dominance. The 0.8Q_d condition shows consistent peak/valley timing with other conditions, but the cyclical characteristics are disrupted, confirming its classification as an unstable operating regime. The maximum difference in efficiency within one impeller cycle under 0.8Q_d condition is 1.5%. This phenomenon originates from enhanced turbulent flow structures under low-flow conditions, whose spatiotemporal disorder disrupts hydraulic regularity. Notably, loss profiles across all conditions display perfect symmetry with head curves, where loss maxima precisely align with efficiency minima. This inverse correlation verifies that internal loss critically governs both efficiency and head variations under clocking effects. Consequently, time-domain analysis of component-wise total pressure losses is essential for deciphering the mechanism of the clock effect.

Figure 8 illustrates the time-domain and frequency-domain evolution of impeller total pressure loss across operating conditions within one impeller revolution cycle. Under stable conditions (1.0Q_d/1.2Q_d), the impeller loss increases with flow rate, reaching 1.2 times higher at 1.2Q_d than at 1.0Q_d. The loss exhibits eight-period oscillatory behavior per cycle, each containing a subsidiary loss peak. Clocking schemes alter loss peak phasing; C0 achieves maximum loss at cycle initiation (0°), but other schemes (C1–C3) show peak shifts to a 27° rotational angle. Time-averaged losses remain minimally affected by clocking effects, with maximum inter-scheme differences of 0.008 m (1.0Q_d) and 0.015 m (1.2Q_d). To investigate the frequency-domain characteristics of total pressure loss, the frequency spectrum is plotted with shaft frequency multiples as the abscissa, where fb denotes the shaft frequency. Under stable operating conditions (1.0Q_d/1.2Q_d), the dominant loss frequencies for all schemes are 8 fb (blade-passing frequency) and its harmonics. Notably, the C3 scheme shifts its dominant frequency to 16 fb. Additionally, under the 1.2Q_d operating condition, Scheme C1 exhibits unstable peak and trough values, resulting in alterations to its dominant frequency characteristics. This phenomenon originates from the tongue alignment with the stay vane trailing edges, which amplifies flow unsteadiness. For the 0.8Q_d condition, the total pressure loss within the impeller exhibits stochastic fluctuations due to internal flow instabilities. The C0 achieves minimal amplitude deviation, while the C1 configuration yields the lowest time-averaged loss.

Figure 9 presents the time-frequency characteristics of total pressure loss in guide vanes. The guide vane losses exhibit more pronounced flow-dependent variations, with losses at 1.2Q_d increasing to 1.8 times those at the 1.0Q_d condition. For the 1.0Q_d condition, all schemes maintain the characteristic eight-period fluctuation pattern. However, the C1 scheme demonstrates unstable loss fluctuations during the impeller cycle, with its dominant frequency shifting to shaft frequency. This instability stems from the near-zero angle (≈0°) between the stay vane trailing edge and tongue in C1. Under 1.2Q_d conditions, elevated turbulence intensity disrupts the periodic stability of total pressure loss. Consequently, all schemes exhibit significant peaks at shaft frequency, with C1 showing the most pronounced peak and C3 the smallest. Comparing the C3 and C1 schemes, it is found that the C3 scheme had a tongue located in the middle of the fixed guide vane channel. The C0 and C2 schemes have similar peak values with the tongue located on both sides of the stay vane trailing edge. For the 1.0Q_d and 1.2Q_d conditions, the C0 consistently achieves the lowest time-averaged losses, with differentials of 0.004 m and 0.027 m, respectively, compared with the worst-performing schemes. In the 0.8Q_d condition, the C0 scheme exhibits minimal fluctuation amplitude, while C3 achieves the lowest time-averaged loss.

Figure 10 illustrates the time-frequency variations of total pressure loss in the stay vane. The stay vane losses demonstrate more significant flow-dependent characteristics, with losses at the 1.2Q_d condition increasing to 1.9 times those at the 1.0Q_d condition. Due to inherent instability in stay vane loss fluctuations during impeller cycles across all operating conditions, the frequency spectrum exhibits prominent peaks at the shaft frequency. Under stable conditions (1.0Q_d/1.2Q_d), the maximum and minimum peak values consistently occur in C1 and C3, respectively. For the 1.0Q_d and 1.2Q_d conditions, the C0 maintains the lowest time-averaged losses, with differentials of 0.019 m and 0.015 m compared with the maximum time-averaged loss schemes. In the 0.8Q_d condition, loss fluctuations exhibit enhanced periodicity with dominant frequencies at 4fb and 5fb, where the C2 scheme demonstrates both minimal fluctuation amplitude and the lowest time-averaged loss values.

Figure 11 shows the time-frequency variations of total pressure loss in the volute. The total pressure loss increases with the flow rate, and the time-averaged total pressure loss at 1.2Q_d is approximately 1.8 to 2.5 times higher than at 1.0Q_d. For stable operating conditions, small fluctuations occur again in each fluctuation cycle, which results in the main frequency being twice the blade frequency except for the 1.0Q_d condition C1 scheme. At 1.2Q_d, influenced by stronger turbulence, significant peaks still exist at the blade passing frequency for all schemes, with the largest peak in the C1 and the smallest in the C3. The minimum time-averaged loss at both 1.0Q_d and 1.2Q_d occurs in the C0, with differences of 0.048 m and 0.219 m from the maximum time-averaged loss, respectively. For the 0.8Q_d condition, the timings of loss peaks and valleys are consistent across schemes, showing enhanced regularity. The dominant frequency of loss is twice the blade passing frequency for all schemes, with the maximum and minimum time-averaged losses occurring in C3 and C0, respectively, differing by 0.145 m.

In summary, although C1 demonstrates favorable hydraulic performance under the 0.8Q_d condition, its guide vanes, stay vanes, and volute exhibit significant fluctuation amplitudes in energy losses during rotational cycles under stable conditions. The C3 improves the stability of losses within the impeller cycle but results in much higher losses than the C0 at 0.8Q_d, leading to a notable reduction in efficiency. The C2 consistently shows higher time-averaged losses than the C0 under stable conditions. Therefore, the C0 scheme, which demonstrates superior performance across all operating conditions, is identified as the optimal solution.

3.2. The Clocking Effect of the Guide Vane

3.2.1. Analysis of Hydraulic Performance

Figure 12 shows the hydraulic performance variations of different guide vane position schemes. Compared with the stay vane clocking effect, the influence of guide vane clocking position on off-design performance is weaker. At the 1.2Q_d condition (Figure 12a), the maximum efficiency variation range is only half of that caused by the stay vane, while the 0.8Q_d condition also shows a reduction. However, significant performance fluctuations occur at the 1.0Q_d condition, with amplitude similar to 1.2Q_d. The C10 scheme exhibits minimal numerical differences from C0 because their only slight distinction lies in the impeller’s initial position. The optimal hydraulic efficiency for both 1.0Q_d and 1.2Q_d conditions appears in C5, whereas the 0.8Q_d condition achieves higher efficiency with C0. The maximum efficiency difference (0.641%) occurs between the C0 and C8 schemes at 0.8Q_d. For both 1.0Q_d and 1.2Q_d conditions, the C8 scheme yields nearly minimum efficiency values, with maximum differences of 0.330% and 0.377%, respectively. As shown in Figure 12b, the head variations with guide vane clocking positions follow similar patterns to efficiency changes. Under 1.0Q_d and 1.2Q_d conditions, maximum heads both occur in the C5 scheme, increasing by 0.077% and 0.132%, respectively, compared with C0, while the 0.8Q_d condition achieves the highest head with C0. Minimum heads for both 0.8Q_d and 1.2Q_d conditions appear in the C8 scheme, decreasing by 0.671% and 0.310%, respectively, relative to C0. Figure 12c shows that the impact of guide vane clocking position on impeller power consumption also diminishes with increasing flow rate. At the 0.8Q_d condition, the maximum power difference between schemes reaches 880W. For the 1.0Q_d condition, impeller power consumption first increases then decreases with clocking angle, showing a 236 W difference between maximum power (C8) and C0. At 1.2Q_d condition, the maximum power difference reduces to merely 132 W. The total loss variation curves under different conditions shown in Figure 12d indicate that guide vane clocking position exerts a greater influence at low-flow conditions, with the C0 scheme consistently demonstrating smaller losses across all conditions.

For the 1.0Q_d condition, although the C5 scheme shows improvements in efficiency and head, this comes at the cost of increased power consumption and losses, resulting in poorer economic performance when meeting head requirements. All other schemes exhibit lower efficiency and head compared with the C0 scheme, making C0 the optimal scheme for this condition. For the 1.2Q_d condition, the C7 scheme demonstrates performance parameters close to those of C0, with slightly better performance, but its efficiency deteriorates significantly at the design flow condition. For the 0.8Q_d condition, compared with C0, all other schemes exhibit lower efficiency and head, higher power consumption, and greater total losses. Thus, the C0 scheme delivers the best hydraulic performance for this condition, while the C8 scheme performs the worst.

3.2.2. Analysis of Transient Characteristics

Figure 13 presents the time-domain and frequency-domain variations of impeller total pressure loss. The guide vane position also affects the occurrence timing of peak loss in the impeller. For the 1.0Q_d condition, the C0 scheme exhibits its maximum loss peak at the beginning of the cycle, whereas the C6 and C8 schemes reach their peak losses at 9° and 27° angles, respectively. Under 1.0Q_d and 1.2Q_d conditions, the dominant frequencies of loss for different schemes are 8fb and its harmonics, with peak values very close. In addition, the time-averaged loss of the C0 scheme is consistently the highest, while the C8 scheme yields the smallest loss, with differences of 0.002 m and 0.014 m, respectively. In the 0.8Q_d condition, the total pressure loss of the impeller still exhibits irregular fluctuations, and the fluctuation amplitude of loss over time is significantly larger than in other conditions. Additionally, the difference between the maximum time-averaged loss (C6) and the minimum (C8) is 0.111 m, indicating that the clocking effect has a stronger impact on low-flow conditions. Regarding impeller total pressure loss, the time-averaged losses of all schemes under 1.0Q_d and 1.2Q_d conditions are similar, with the C8 scheme consistently demonstrating the lowest time-averaged loss across all conditions.

Figure 14 shows the time-domain and frequency-domain variations of guide vane total pressure loss. The position of guide vanes also affects the amplitude at shaft frequency under 1.0Q_d and 1.2Q_d conditions. For the 1.2Q_d condition, the dominant frequency shifts to shaft frequency in both the C0 and C6 schemes, indicating that altering the guide vane clocking position influences the fluctuation stability of guide vane losses—a phenomenon particularly pronounced under high-flow conditions. For both 1.0Q_d and 1.2Q_d conditions, the maximum time-averaged loss occurs in the C6, while the C0 exhibits the minimum loss, with differences of 0.002 m and 0.009 m, respectively. Under the 0.8Q_d condition, the C6 still shows the highest time-averaged loss, with the C8 scheme yielding the minimum value (difference 0.131 m). In terms of guide vane losses, the C0 scheme has the smallest time-average loss at 1.0Q_d and 1.2Q_d conditions, while the C8 scheme has the smallest time-average loss at the 0.8Q_d condition.

Figure 15 presents the time-domain and frequency-domain variations of the stay vane total pressure loss. Under 1.0Q_d and 1.2Q_d conditions, the instability of stay vane loss fluctuations is significantly more pronounced compared with previous components, resulting in the dominant frequency shifting to shaft frequency for all schemes except C8. This indicates that the C8 scheme helps enhance the stability of both the stay vane and the guide vane losses. For 1.0Q_d and 1.2Q_d conditions, the C8 scheme exhibits the highest time-averaged loss, while the C0 scheme shows the minimum time-averaged loss with differences of 0.005 m and 0.012 m, respectively. In the 0.8Q_d condition, the difference between the maximum time-averaged loss (C6) and the minimum (C0 scheme) is 0.018 m, a value significantly smaller than the maximum loss differences observed in the impeller and guide vanes, suggesting that stay vane losses are less affected by clocking effects. Regarding stay vane losses, the C0 scheme consistently demonstrates the smallest time-averaged loss across all operating conditions.

Figure 16 illustrates the time-domain and frequency-domain variations of volute total pressure loss. Under stable conditions, the volute loss exhibits greater stability throughout the impeller cycle, with all schemes showing a dominant frequency at twice the blade passing frequency. Even at the 0.8Q_d condition, significant amplitudes are observed at 16fb in the frequency spectrum of all schemes. Regarding volute losses, the C8 scheme consistently demonstrates the highest time-averaged loss across all operating conditions, while the C0 scheme maintains relatively lower time-averaged losses. The differences in time-averaged losses between C8 and C0 schemes are 0.149 m, 0.036 m, and 0.061 m for 0.8Q_d, 1.0Q_d, and 1.2Q_d conditions, respectively. The variation amplitude of time-averaged volute losses with different guide vane clocking positions is significantly greater than that observed in other components. Notably, the C0 scheme consistently exhibits smaller time-averaged losses across all conditions.

The influence of guide vane clocking effects on time-averaged losses of various components is most significant at low-flow conditions (0.8Q_d) and least at the 1.0Q_d condition. For impeller and guide vane losses, all schemes show minor differences in time-averaged values under stable conditions. At 0.8Q_d condition, the C8 scheme demonstrates the smallest time-averaged loss while simultaneously enhancing loss fluctuations stability. Regarding the stationary guide vane and volute losses, the C0 scheme consistently achieves the minimum time-averaged values across all conditions, while the C8 scheme yields the maximum values. The volute loss is most significantly affected by guide vane clocking positions, resulting in poorer performance of the C8 scheme. Consequently, the C0 scheme exhibits superior hydraulic performance overall.

3.3. Analysis of Reasons for Changes in Losses

The analysis reveals that the stay vane and guide vane clocking position exert the most significant influence on volute losses. Therefore, taking the volute as the research subject, this study investigates the causes of hydraulic loss variations under different clocking schemes. Figure 17 presents the hydraulic loss distribution at the volute for the C0, C2, and C8 schemes under different flow conditions. At the 1.2Q_d condition, hydraulic losses are predominantly concentrated near the tongue, while the 0.8Q_d condition exhibits poorer distribution regularity. The 1.0Q_d condition shows negligible variations in loss distribution across different schemes. Figure 18 illustrates selected volute cross-sections for detailed analysis of loss mechanisms. Section 1 is located near the high-loss volute tongue region, with adjacent sections spaced at 90° intervals. Based on sectional loss characteristics and velocity vector distributions, it is observed that after impinging on the volute wall, symmetrical vortex structures form within the volute. The maximum hydraulic loss occurs at the interface between these vortices and the main flow. Along the flow direction in the volute, the movement of the vortex boundary away causes the hydraulic loss to gradually split into two parts. As a result, the hydraulic loss at the intermediate cross-section rapidly decreases along the flow direction. This phenomenon indicates that the hydraulic loss in the volute is directly related to the vortex structure.

The Q criterion has become the mainstream tool for identifying vortices in pumps due to its theoretical rigor, engineering applicability, and computational efficiency. It is particularly suitable for diagnosing and optimizing strong three-dimensional and transient vortex structures in rotating machinery. Therefore, this study adopts the Q criterion to study the distribution of vortices. Figure 19 shows the Q-value distribution across volute cross-sections for different schemes. The hydraulic loss along the volute flow path decreases synchronously with the reduction of vortex intensity. Additionally, due to the narrow flow passage, the vortex structure near the tongue becomes more chaotic. At 1.2Q_d condition, all volute cross-sections exhibit symmetrical vortices of similar scale, concentrating hydraulic losses in the mid-section. Therefore, the variation trend of volute losses can be determined from the loss distribution in Figure 17. From the C0 to C2 scheme, the vortex intensity gradually increases in each section, leading to higher hydraulic losses. For the 1.0Q_d condition, significant differences exist in the scale of upper and lower vortex structures, with vortex boundaries positioned higher, making it impossible to observe loss variation patterns in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. The minimal changes in vortex scale and intensity across different timing schemes result in smaller loss variations. Under 0.8Q_d conditions, due to flow field instability and reduced flow rate, the dual-vortex structure in the volute is disrupted in the C0 scheme, yielding the minimum hydraulic loss. The C8 scheme demonstrates the strongest vortex scale and intensity at Section 1 among all schemes, causing maximum hydraulic losses. This analysis reveals that stay vane and guide vane clocking positions alter volute hydraulic losses by influencing internal vortex structures. Furthermore, the C0 scheme consistently exhibits the best flow field characteristics and minimal hydraulic losses across all operating conditions.

4. Conclusions

This study conducted numerical simulations under three flow conditions to investigate the influence of the stationary blade clocking position on pump-as-turbine performance in pump mode from the perspective of external characteristics, including efficiency, head, and total pressure loss variations, as well as to determine the optimal clocking position. Based on calculations of local hydraulic loss rate parameters combined with internal flow field characteristics and unsteady loss distribution, the operating mechanism of clocking effects is explored. The main conclusions are as follows:

(1)

The clocking position of the stay vane exerts greater influence on off-design flow conditions. The maximum efficiency differences reach 0.855% and 0.805% at 0.8Q_d and 1.2Q_d conditions, respectively, while only 0.090% at the design condition. Although the impact of guide vane clocking position diminishes for off-design conditions, significant performance variations still occur at the 1.0Q_d condition. Across increasing flow rates, the maximum efficiency differences between schemes are 0.641%, 0.330%, and 0.377%, respectively. Optimal hydraulic performance across all flow conditions is achieved when the tongue is positioned at the trailing edge of guide vanes and the 1/4 chord location of stay vane passages. When the angle between the stay vane trailing edge and tongue is 0°, instability emerges in losses for the guide vane, stay vane, and volute. In addition, positioning the tongue at the mid-passage of the stationary guide vanes enhances loss fluctuation stability across all components.

(2)

The time-domain analysis of external characteristics across various flow conditions reveals that the 0.8Q_d condition represents an unstable operating regime, resulting in significant variations in component losses under different schemes. Regarding impeller and guide vane losses, all schemes demonstrate comparable time-averaged values under stable conditions. For stay vane and volute losses, the C0 scheme consistently exhibits the minimum time-averaged values across all conditions. The clocking effect proves most pronounced in the alteration of the proportion of volute losses, consequently exerting the dominant influence on pump mode performance variations.

(3)

The variation in volute hydraulic losses stems from the fact that different stay vane and guide vane clocking positions alter both the scale and intensity of the dual-vortex structure within the volute, thereby inducing significant changes in losses at the vortex interface region.