Research on the Energy Distribution of Hump Characteristics Under Pump Mode in a Pumped Storage Unit Based on Entropy Generation Theory

Abstract

To alleviate the pressure on grid regulation and ensure grid safety, pumped storage power stations need to frequently start and stop and change operating conditions, leading to the pump-turbine easily entering the hump characteristic zone, causing flow oscillation within the unit and significant changes in its input power, resulting in increased vibration and grid connection failure. The spatial distribution of energy losses and the hydrodynamic flow features within the hump zone of a pump-turbine under pumped storage operation are the focus of the study. The SST k-ω turbulence model is applied in CFD simulations of the pump-turbine within this work, focusing on the unstable operating range of the positive slope, with model testing providing experimental support. The model test method combines numerical simulation with experimental verification. The LEPR method is used to quantitatively investigate the unstable phenomenon in the hump zone, and the distribution law of energy loss is discussed. The results show that, at operating points in the hump zone, up to 72–86% of the energy dissipation is attributed to the runner, the guide vane passage, and the double vane row assembly within the guide vane system. The flow separation in the runner’s bladeless area evolves into a vortex group, leading to an increase in runner energy loss. With decreasing flow rate, the impact and separation of the water flow intensify the energy dissipation. The high-speed gradient change and dynamic–static interference in the bladeless area cause high energy loss in the double vane row area, and energy loss mainly occurs near the bottom ring. In the hump operation zone, the interaction between adverse flows such as vortices and recirculation and the passage walls directly drive the sharp rise in energy dissipation.

1. Introduction

With accelerated advancement in the global energy transition towards green and low-carbon energy sources, the global energy structure is making a rapid shift towards low-carbonization, pumped storage, as the most mature and scalable energy storage technology for power systems, as it plays an irreplaceable and crucial role in peak shaving, frequency, and voltage regulation, and as a backup power source. The pump-turbine, as the core equipment of pumped storage power stations, directly affects the economic efficiency, stability, and reliability of the entire pumped storage unit with its operational performance. However, during operation of the pump-turbine under pump mode, the presence of hump characteristics has always been a significant factor restricting its safe and stable operation. Since hump characteristics were discovered in 1998 [1], many researchers have conducted studies on the cambered area of pump-turbines using physical experiments and numerical simulations. Li et al. [2] investigated a pumped storage pump-turbine, analyzing flow characteristics in the camber region under pump conditions. The results showed that increased flow angles induce flow separation and vortices on the pressure surface. In double-blade configurations, low-velocity zones surround pressure and suction surfaces, forming vortices that block flow paths, increase energy loss, reduce hydraulic efficiency, and trigger hump region formation. Jia J. [3] examined hump characteristic variations in pump-turbines relative to guide vane outlet angles. Their findings indicate that, as the outlet angle increases, the hump intensity rises, and onset occurs earlier; conversely, smaller angles yield higher head due to flow restriction. Wang H. [4] and Zhang et al. [5] investigated a specific pump-turbine via computational simulations. Their findings revealed recirculation flow at the runner inlet, elevating energy dissipation and ultimately precipitating the hump characteristics in the system. Wang et al. [6] characterized internal flow field and pressure pulsations within the vaneless space of a pump-turbine model under turbine operating conditions, employing an integrated approach combining numerical simulation and model testing. Zhao et al. [7] analyzed the formation mechanism of hump regions in low-head pumped-turbines via numerical and experimental approaches. Their findings identified backflow at the blade leading edge and angular vortices at the runner outlet as the primary flow instabilities responsible for hump region development. Yang et al. [8] investigated pressure pulsations in the pump-turbine hump region via DES simulations, identifying intense vortex activity and pressure fluctuations as the primary drivers of these pulsations. Zhang Z. [9] employed the SBES turbulence model to simulate pump-turbine performance. During hump region operation under low-flow conditions, the energy dissipation percentage in the adjustable guide vanes markedly escalated and intensified as the flow rate diminished. This further demonstrates that heightened hydraulic losses in the adjustable guide vanes result from pronounced flow separation on their suction surfaces, which generates substantial vortex formations. Li et al. [10] investigated camber characteristics in a pump-turbine via combined experimental and numerical methods. The results revealed that, upon entering the hump region, circumferentially-aligned vortex clusters emerged in the series cascade. Their intensity and spatial extent varied with flow rate, with vortices at the series stage position proven to cause hump characteristics. Wu et al. [11] conducted a model test based on a pump-turbine of a certain power station. They analyzed the head, power, and pressure pulsation in the zero-flow zone of the pump under different opening degrees of the movable guide vanes. They concluded that the head at the zero-flow condition of the pump would show a significant hump phenomenon as the opening degree of the movable guide vanes increased. Pan et al. [12] investigated pump-turbines using the DES turbulence model, identifying hydraulic losses as the primary driver of head instability and showing that flow pattern differences were the source of the losses; they proposed that, by appropriately adjusting the guide vane opening, the flow state could be improved, and hydraulic losses could be reduced. Yang et al. [13] identified that vortical structures and flow separation at the centrifugal impeller inlet and draft tube outlet are the primary contributors to hydraulic energy loss. According to Denton J D [14], entropy generation serves as a robust indicator of energy losses within hydraulic systems. While Moore J [15] established the initial regional entropy generation framework using RANS turbulence models, its validity depends on the equilibrium assumption between turbulent dissipation and production rates, imposing inherent theoretical limitations. To overcome the shortcomings of Moore’s entropy generation model, Kramer-Bewa [16] formulated a pioneering localized entropy generation approach, which directly performs Reynolds averaging on the entropy generation rate without the need to introduce any assumptions. Based on this, Herwig and Kock [17] developed a systematic methodology for quantifying local entropy generation rates. Through implementation of this equation system, local entropy generation can be directly computed during the post-processing phase of computational fluid dynamics simulations, eliminating the requirement for solving supplementary differential or transport equations. Li et al. [18,19] applied entropy generation theory to identify pump mode hump instability and hysteresis to backflow at the runner inlet rear shroud and flow separation across the double-row cascade. Zhou et al. [20] executed an in-depth examination of flow-induced losses in hydraulic components across the hump region of pump-turbines, leveraging entropy generation principles. The results established that head deterioration in hump operation stems from pronounced hydraulic loss escalation. The theory’s capacity to quantify irreversible energy dissipation during flow processes furnishes a transformative perspective for interpreting the hydraulic signature of the hump region.

Prior research has chiefly examined pump-turbine hump characteristic genesis and evolution via internal flow mechanisms, neglecting core energy dissipation principles. Standard hydraulic loss evaluation in units utilizes pressure difference techniques for energy performance assessment through aggregate efficiency calculations. This method, based on energy conservation, determines component energy losses from inlet–outlet energy differentials. However, it cannot resolve internal energy loss distributions within flow components. Entropy generation theory permits accurate quantification of irreversible energy dissipation during fluid flow, offering a novel paradigm for hump region hydraulic analysis. This work investigates the hump region of a pumped storage pump-turbine. Applying entropy generation theory, we quantitatively analyze instability phenomena in this zone. Through characterizing energy dissipation distribution, this study establishes a theoretical basis for pump-turbine optimization, enhancing operational stability and efficiency.

2. Mathematical Model

2.1. Governing Equation

The internal flow characteristics of a pump-turbine are fundamentally governed by the laws of mass conservation, momentum conservation, and energy conservation. In the context of numerical simulation, these three laws correspond to the continuity equation, the Navier–Stokes equations, and the energy equation, respectively. For the present study, the working fluid is assumed to be incompressible water at ambient temperature, and heat transfer effects are neglected during operation. However, energy losses inevitably occur in mechanical processes due to viscous dissipation and friction. Therefore, the total energy equation is considered to account for these effects, and the internal flow within the pump-turbine is predominantly described by the continuity equation, the Navier–Stokes equations, and the total energy equation. The equations are as follows:

$${\displaystyle \frac{𝜕\left(\rho U_i\right)}{𝜕x_i}}=0$$

(1)

$${\displaystyle \frac{𝜕\left(\rho U_i{U}_j\right)}{𝜕x_i}}=-{\displaystyle \frac{𝜕P}{𝜕x_i}}+{\displaystyle \frac{𝜕}{𝜕x_j}}\left[{\mu }_e+({\displaystyle \frac{𝜕U_i}{𝜕x_j}}+{\displaystyle \frac{𝜕U_j}{𝜕x_i}})\right]$$

(2)

$${\displaystyle \frac{𝜕}{𝜕t}}\left(\rho h_{tot}\right)-{\displaystyle \frac{𝜕p}{𝜕t}}+{\displaystyle \frac{𝜕}{𝜕x_j}}\left(\rho u_j{h}_{tot}\right)={\displaystyle \frac{𝜕}{𝜕x_j}}\left({\lambda }_t{\displaystyle \frac{𝜕T}{{𝜕}_{x_j}}}-\overline{u_{j{h}_{sta}}}\right)+{\displaystyle \frac{𝜕}{𝜕x_j}}\left[u_j(2\mu \overline{S_{ij}}-\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right]$$

(3)

where $\rho$ denotes liquid density, $U$ and $P$ represent time-averaged velocity and pressure, respectively, ${\mu }_e$ is effective viscosity, $h_{tot}$ represents total enthalpy, $\overline{S_{ij}}$ denotes the strain rate tensor, ${\lambda }_t$ is the effective thermal conductivity, and $h_{sta}$ represents static enthalpy.

2.2. Turbulence Model

The RNG k-ε, Realizable k-ε, and SST k-ω turbulence models are predominantly employed in pump-turbine simulations. Among these, the RNG k-ε model is characterized by its high Reynolds number formulation, which assumes fully developed turbulence and negligible molecular viscosity effects. However, in near-wall regions where molecular viscosity dominates, the RNG k-ε model incurs limitations for investigating hydraulic stability in pump-turbines. The Realizable k-ε turbulence model incorporates an additional formulation to account for turbulent viscosity, which enhances its performance in rotational flows, strong adverse pressure gradient boundary flows, flow separation, and secondary flows. Nevertheless, similar to other k-ε models, it exhibits a tendency to delay the prediction of flow separation compared to experimental observations. This characteristic may result in an overly optimistic assessment of the flow regime for smooth surface separation. In contrast, the SST k-ω turbulence model combines the advantages of both the k-ε and k-ω models by employing a wall-function-based blending function. The model integrates low-Re k-ω for boundary-adjacent regions and transitions to high-Re k-ε in the core flow domain. This approach enables more accurate predictions across a broader range of flows, especially for separation induced by adverse pressure gradients. The SST k-ω model demonstrates excellent stability and reasonable accuracy in the numerical simulation of rotating machinery [21,22]. Therefore, this study employs the SST k-ω model for turbulence closure. The governing equations are given by

$${\displaystyle \frac{𝜕\left(\rho k\right)}{𝜕t}}+{\displaystyle \frac{𝜕\left(\rho k{u}_i\right)}{𝜕x_i}}=\widetilde{P_k}-{\beta }^{\ast }\rho \omega k+{\displaystyle \frac{𝜕}{𝜕x_i}}\left(\left(\mu +{\sigma }_k{\mu }_t\right){\displaystyle \frac{𝜕k}{𝜕x_i}}\right)$$

(4)

$${\displaystyle \frac{𝜕\left(\rho \omega \right)}{𝜕t}}+{\displaystyle \frac{𝜕\left(\rho \omega u_i\right)}{𝜕x_i}}=\alpha \rho S^2-\beta \rho {\omega }^2+{\displaystyle \frac{𝜕}{𝜕x_i}}\left(\left(\mu +{\sigma }_{\omega 1}{\mu }_t\right){\displaystyle \frac{𝜕k}{𝜕x_j}}\right)+2(1-F_1)\rho {\sigma }_{\omega 2}{\displaystyle \frac{1}{\omega }}{\displaystyle \frac{𝜕k}{𝜕x_i}}{\displaystyle \frac{𝜕\omega }{𝜕x_i}}$$

(5)

$${\mu }_t={\displaystyle \frac{{\rho \alpha }_{1}k}{\mathrm{m}\mathrm{a}\mathrm{x}\left({\alpha }_1\omega S{F}_2\right)}}$$

(6)

where $P_k$ denotes the production term of turbulent kinetic energy, $S$ is the invariant measure of the strain rate, $\mu$ is the molecular viscosity, $F_1,{ F}_2$ is the blending function, ${\mu }_t$ is turbulent eddy viscosity, and ${\beta }^{\ast }, \beta$, ${\sigma }_k,$ ${\sigma }_{\omega 1}$, ${\sigma }_{\omega 2}$, and ${\alpha }_1$ are constants of the turbulence model [23]. Here, ${\beta }^{\ast }$ = 0.09, ${\sigma }_k$ = 0.85, ${\sigma }_{\omega 1}=0.5,$ ${\sigma }_{\omega 2}$ = 0.856, and ${\alpha }_1=0.31$.

2.3. Entropy Generation Theory

The report from Gong [24] was the first study to innovatively introduce the entropy generation method to evaluate hydraulic losses of pumps and turbines, integrating fluid mechanics and thermodynamics. The advantage of this method lies in its ability to quantitatively assess the energy dissipation of pumps and turbines and determine the location of energy dissipation. Based on the RANS method, entropy generation originates from two primary sources: energy dissipation and heat transfer. Within turbulent flows, the entropy generation attributed to energy dissipation further bifurcates into viscous dissipation (induced by mean velocity) and turbulent dissipation (resulting from turbulent pulsations).

Total entropy generation in Reynolds time-averaged flow arises from dual sources: direct contribution driven by time-averaged velocity and indirect contribution attributable to velocity pulsations. It can be calculated using the following formula:

$${\dot{S}}_{\overline{\mathrm{D}}}^{‴}={\displaystyle \frac{\mu }{T}}\left[{\left({\displaystyle \frac{𝜕\overline{u}}{𝜕y}}+{\displaystyle \frac{𝜕\overline{v}}{𝜕x}}\right)}^2+{\left({\displaystyle \frac{𝜕\overline{u}}{𝜕z}}+{\displaystyle \frac{𝜕\overline{w}}{𝜕x}}\right)}^2+{\left({\displaystyle \frac{𝜕\overline{w}}{𝜕y}}+{\displaystyle \frac{𝜕\overline{v}}{𝜕z}}\right)}^2\right]+{\displaystyle \frac{2\mu }{T}}\left[{\left({\displaystyle \frac{𝜕\overline{u}}{𝜕x}}\right)}^2{\left({\displaystyle \frac{𝜕\overline{v}}{𝜕y}}\right)}^2+{\left({\displaystyle \frac{𝜕\overline{w}}{𝜕z}}\right)}^2\right]$$

(7)

$$\begin{gathered} {\dot{S}}_{D^{\prime }}^{‴}={\displaystyle \frac{{\mu }_{eff}}{T}}\left[{\left({\displaystyle \frac{𝜕u^{\prime }}{𝜕y}}+{\displaystyle \frac{𝜕v^{\prime }}{𝜕x}}\right)}^2+{\left({\displaystyle \frac{𝜕u^{\prime }}{𝜕z}}+{\displaystyle \frac{𝜕w^{\prime }}{𝜕x}}\right)}^2+{\left({\displaystyle \frac{𝜕w^{\prime }}{𝜕y}}+{\displaystyle \frac{𝜕v^{\prime }}{𝜕z}}\right)}^2\right] \\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+{\displaystyle \frac{2{\mu }_{eff}}{T}}\left[{\left({\displaystyle \frac{𝜕u^{\prime }}{𝜕x}}\right)}^2{+\left({\displaystyle \frac{𝜕v^{\prime }}{𝜕y}}\right)}^2+{\left({\displaystyle \frac{𝜕w^{\prime }}{𝜕z}}\right)}^2\right] \end{gathered}$$

(8)

$${\mu }_{eff}=\mu +{\mu }_t$$

(9)

where $\overline{u}$, $\overline{v}$, and $\overline{w}$ are the Reynolds-averaged velocities; $u^{\prime }$, $v^{\prime }$, and $w^{\prime }$ are the pulsating velocities; T is the temperature; ${\mu }_{eff}$ represents the effective viscosity; and $\mu$ is the molecular viscosity.

The Reynolds time-averaged approach inherently precludes direct resolution of fluctuating velocity components, consequently rendering the associated entropy production unsolvable. According to the entropy generation theory proposed by Kock et al. [17], the SST k-ω framework enables approximation of entropy production due to velocity fluctuations via the ω-specific dissipation rate.

$${\dot{S}}_{D^{\prime }}^{‴}=\beta {\displaystyle \frac{\rho \omega k}{T}}$$

(10)

where $\beta$ is the empirical coefficient, taking the value of 0.09; $k$ denotes the turbulent kinetic energy; and $\omega$ represents the frequency of turbulent eddies.

Furthermore, the entropy generation rate has a strong wall effect within fluid machinery, resulting in entropy generation. An adaptive high-precision wall function was proposed by Duan et al. [25] to quantify entropy generation in the near-wall region. The mathematical expression for wall entropy generation rate is given by

$${\dot{S}}_w^{″}={\displaystyle \frac{\overrightarrow{\tau }\cdot \overrightarrow{v}}{T}}$$

(11)

where $\overrightarrow{\tau }$ is the wall shear stress; while $\overrightarrow{v}$ is the velocity near the wall surface.

Total entropy generation can be quantified through domain integration of the entropy production rates induced by mean flow and turbulent fluctuations, with subsequent aggregation of the integrated components. The expression is as follows:

$$S_{pro,\overline{D}}={\int }_v{\dot{S}}_{\overline{D}}^{‴}\mathrm{d}V$$

(12)

$$S_{pro,D^{\prime }}={\int }_v{\dot{S}}_{D^{\prime }}^{‴}\mathrm{d}V$$

(13)

$$S_{pro,W}={\int }_A{\displaystyle \frac{\overrightarrow{\tau }\cdot \overrightarrow{v}}{T}}\mathrm{d}A$$

(14)

$$S_{pro}=S_{pro,\overline{D}}+S_{pro,D^{\prime }}+S_{pro,W}$$

(15)

where $S_{pro}$ is the total entropy generation; $S_{pro,\overline{D}}$ quantifies mean flow dissipation contribution; $S_{pro,D^{\prime }}$ represents turbulent fluctuation dissipation; ${\mathrm{a}\mathrm{n}\mathrm{d} S}_{pro,W}$ characterizes wall shear dissipation.

3. Model Setup and Validation

3.1. Numerical Model Setup

This study focuses on experimental investigations conducted on a pump-turbines operating within the scaled-down test platform for pumped storage units at the Pumped Storage Engineering Technology Research Center, Zhejiang University of Water Resources and Electric Power. Concurrently, three-dimensional geometric models of all pump-turbine components were developed using NX UG 12 software. Each component pump-turbine model was assembled to construct the entire fluid computational domain. The computational domain comprises five components, namely the spiral casing, stay vanes, guide vanes, runner, and draft tube. The complete computational domain is illustrated in Figure 1, and the key geometric parameters of each component are listed in

Table 1. Geometric parameters of the pump-turbine.

Variable	Symbol	Value
Design flow rate	$Q_d$	0.12 [m3/s]
Design pump head	$H_d$	7.84 [m]
Design rotational speed	$n_d$	400 [r/min]
Count of runner blades	$Z_b$	12
Count of stay vanes	$Z_v$	20
The inlet runner at hub	$D_h$	300.1 [mm]
Runner outlet diameter	$D_{iout}$	574 [mm]

.

In this study, the three-dimensional numerical model was computed utilizing CFD software ANSYS CFX 2022 R2 on a high-performance workstation equipped with two 32-core AMD EPYC processors, 128 GB of RAM, and a 64-bit operating system.

3.2. Boundary Conditions Setting

Based on the model test setup, the inlet and outlet boundary conditions for the numerical simulation were defined as follows: for the pump operating mode, the draft tube inlet was designated as the inlet boundary of the entire computational domain, specified as a mass flow inlet. The spiral casing outlet served as the outlet boundary, configured as an outlet average static pressure boundary with its value set to zero. All walls were treated as no-slip walls, with wall functions automatically assigned. To account for the rotational motion of the runner component, the Multiple Reference Frame (MRF) model was employed. The runner domain was defined as a rotating zone, while all stationary components were defined as stationary zones. Data transfer between components was facilitated through interfaces. Interfaces between rotating and stationary zones were defined as rotating-stationary type, while interfaces between stationary zones were defined as stationary-stationary type. All interfaces were handled using the General Grid Interface (GCI) method. For steady simulations, the interface type for stationary zone interfaces was set to None, whereas the rotating-stationary interfaces utilized the Frozen Rotor method. For unsteady simulations, a steady-state solution was first obtained by iterating 1000 steps until the velocity residual converged to 10⁻⁴ using the SST k-ω turbulence model. This steady-state solution was then used as the initial condition for the subsequent unsteady simulation. The transient simulation employed a carefully calibrated temporal resolution of 180 time steps per complete runner revolution (360°), equating to a 2° angular advancement per computational step. For the 400 rpm operating condition under investigation, this corresponds to a physical time step size of $∆t$ = 0.000833 s, as determined by the relationship $∆t$ = 60/(400 × 180). This temporal discretization ensures proper resolution of the blade passing frequency (80 Hz for 12 blades) and its harmonics, while maintaining numerical stability. The simulation was conducted for 50 full runner revolutions to achieve fully developed periodic flow conditions [26,27].

To accurately predict the internal flow characteristics and energy distribution within the hump region of the pump-turbine under pump mode, the simulation results from the final three rotational cycles were selected for analysis.

3.3. Grid Independence

In this study, the entire flow passage geometric model domain of the pump-turbine is defined as the computational domain. ANSYS ICEM 2022 R2 is employed to generate hexahedral mesh for this domain, offering high precision and computational efficiency for all components. Given that the numerical model is scaled from a prototype pump-turbine with large dimensions, the Reynolds number reaches extremely high levels (approximately 10⁷) under fully developed turbulent flow conditions. To achieve a target Y+ value near 1 while maintaining grid quality would require an impractically large number of grid cells. Therefore, during numerical simulation, wall treatment is set to automatic mode, allowing the solver to automatically adjust the Y+ values for all component walls within the applicable range of the SST k-ω turbulence model [28,29,30,31].

Grid independence verification was performed to ensure computational accuracy and precision by evaluating eight sets of grids with varying dimensions. Under guide vane opening conditions of 11°, validation across the entire computational domain was conducted at the pump operating point of maximum efficiency, with head coefficient and efficiency selected as evaluation criteria. As shown in Figure 2, the grid independence verification results indicate that, when the grid count increases from 2,368,511 to 5,857,496, the head and efficiency exhibit significant variations. Further increasing the grid count to 9,227,089 reduces the computational errors of head and efficiency to within 4%. While high-quality grids yield more reliable numerical results, higher grid counts demand greater computational resources and longer simulation cycles. Therefore, given the precision requirements, the model with 5,857,496 grids was selected for subsequent numerical simulations. The computational grid is presented in Figure 3. The quality metric refers to the minimum orthogonal quality of the grids, with a value range of 0–1. Values closer to 1 indicate better mesh orthogonality. In this study, all computational domains achieved orthogonal quality values above 0.3. Detailed parameters for whole grid component are listed in

Table 2. Grid information of each component in the calculation domain.

Domain	Coarse		Medium		Fine
	Grids	Quality	Grids	Quality	Grids	Quality
Volute	632,967	0.35	994,290	0.43	2,282,624	0.47
Runner	1,473,216	0.30	1,795,213	0.33	2,032,896	0.37
Guide vanes	287,280	0.42	642,180	0.45	1,347,758	0.48
Stay vanes	508,329	0.44	1,461,133	0.53	1,848,867	0.53
Draft tube	579,478	0.48	964,680	0.50	1,714,944	0.51
Total	3,481,270		5,857,496		9,227,089
Computational Time	48 min		50 min		55 min

.

In Figure 2, the head coefficient ψ is defined as

$$\psi ={\displaystyle \frac{2gH}{u_1^2}}={\displaystyle \frac{2gH}{{\omega }^2{R}_1^2}}$$

(16)

where $H$ is the head; and $u_1$ is the circumferential velocity at the runner outlet.

3.4. Test Verification

Model tests of the pump-turbine were conducted on two actual pump-turbine test units at the Pump-Turbine Test Facility of the Water Storage and Hydropower Engineering Technology Research Center of Zhejiang University of Water Resources and Electric Power, as shown in Figure 4. The comprehensive efficiency test error of the actual test bench machines was no more than ±0.25%. The external characteristics under an open angle of 11° of the movable guide vanes were compared with the results from the numerical simulation. The comparison situation is shown in Figure 5. The head and efficiency values derived from the numerical simulation exhibited excellent agreement with the experimental measurements, demonstrating minimal deviation at the optimal operating point. The numerical simulation error of the head was less than 5% at each flow condition point; the maximum error of the efficiency was 5.3%, occurring at the minimum flow condition point, and the errors at the other condition points were all within 5%. The results indicate that the numerical simulation results have good accuracy.

Table 3. Measurement equipment parameters.

Parameter	Equipment	Type	Precision
Flow velocity	Electromagnetic Flowmeter	WP-EMF-A	±0.2%
Head	pressure sensor	WIDEPLUS-8	±0.15%
Pressure pulsation	pressure sensor	WIDEPLUS-8	±0.15%

summarizes the measurement instruments and their specific parameters.

4. Results and Discussion

4.1. Analysis of External Characteristic Test Results

Experimental analysis of the pump-turbine’s external characteristics under different guide vane openings in pump mode yields hump characteristic curves for various openings. As shown in Figure 6, distinct hump characteristics appear under small flow conditions across all guide vane openings. Notably, the hump occurrence positions vary as flow increases. With decreasing guide vane openings, the pump head in the low-flow region progressively rises, the slope of the head curve becomes steeper, and the hump phenomenon intensifies.

When operating in the hump region, pump-turbines exhibit unstable phenomena such as noise and vibration. Therefore, it is essential to investigate the internal flow characteristics of pump-turbines. However, model tests are subject to errors and make it difficult to observe internal flow phenomena, rendering experimental data insufficient to reveal the physical mechanism underlying hump-induced instability. Subsequently, CFD numerical simulation results will be utilized to provide further explanation for the internal flow patterns responsible for the hump phenomenon observed in the experiments.

4.2. Entropy Generation Evolution Characteristics

In order to facilitate quantitative assessment of the local entropy generation rate, a dimensionless quantity, the energy dissipation coefficient $\epsilon$, is defined, and its expression is as follows:

$$\epsilon =S_{pro}{\displaystyle \frac{Q_b}{P_b}}\sqrt{{\displaystyle \frac{H}{g}}}$$

(17)

where $P_b$ and $H$ are the shaft power and pump head, respectively, at $Q_b$.

The energy losses of the draft tube, runner, movable guide vanes, and fixed guide vanes, as well as the volute components under different flow conditions, are shown in Figure 7. From Figure 7, it can be seen that the energy losses occurring in the runner and the double-row cascade dominate, accounting for 72% to 86% of the total energy loss. With increasing flow rate, energy loss exhibits abrupt variations at hump region operating conditions. From the larger flow point to the optimal operating point (φ = 0.86–1), energy loss in the double-row cascade tends to rise progressively, contrasting sharply with the substantial reduction observed in the runner. The draft tube has a larger energy loss under small flow conditions and is relatively sensitive to flow changes. As the flow increases, the energy loss of the draft tube continuously decreases and reaches the minimum at the optimal efficiency flow point. On the contrary, the energy loss of the volute changes significantly under large flow conditions and is small. with relatively gentle change, under small flow conditions.

Comprehensive energy loss analysis in turbomachinery necessitates quantifying each flow component’s entropy generation contribution to the irreversibility of the total system.

Figure 8 shows the distribution of the proportion of entropy generation for each component. From Figure 8, it can be seen that, at the hump region curve operating conditions, the entropy generation losses of the runner and the guide vanes account for a relatively large proportion. In the draft tube, the entropy generation loss decreases with the increase in flow rate, while in the volute, the trend is opposite. Within the flow coefficient range of φ = 0.43–0.86, the proportion of entropy generation losses of the runner and the double-row cascade does not change much. Within the range of φ = 0.86–1, the entropy generation loss of the runner decreases with the increase in flow rate, and the total proportion of entropy generation losses of the double-row cascade gradually increases, from 45% to 66%.

Entropy generation analysis can compare the magnitudes of energy losses of various components under the same operating conditions, but it cannot determine the specific locations where the energy losses occur. In the future, the pump-turbine hump region will be studied by combining the specific distribution of energy dissipation.

4.3. Correlation Between Entropy Generation Loss and Internal Flow Distribution

4.3.1. Analysis Between Runner Energy Loss and Internal Flow Characteristics

In the runner region, a two-dimensional plane is obtained by unfolding along the spanwise direction. Span denotes the dimensionless span value along the flow plane perpendicular to the mainstream. The cross-sectional positions at different blade heights of the runner are clearly indicated in Figure 9. Three spans (Span 0.1, Span 0.5, Span 0.9) are selected to represent three stream surfaces: near the front shroud, the middle region, and near the back shroud of the runner.

Entropy generation can serve as a quantitative method to describe energy losses in systems or individual components, with LEPR (Local Entropy Production Rate) capable of representing energy losses and dissipation areas in the vaneless space and runner. Compared to conventional methods for evaluating hydraulic losses, entropy generation rate enables intuitive visualization and pinpointing of locations with significant energy dissipation within components.

To reveal the correlation between runner energy loss and internal flow pattern variations under different hump region conditions, this study analyzes the entropy generation and velocity streamline distributions on different surfaces of the runner. By examining the entropy generation distribution at the blade inlet, the energy loss within the runner flow passages is evaluated. Figure 7 illustrates the local entropy generation rate and streamline distribution at different blade spans under hump region operating conditions.

As shown in Figure 10, at a flow coefficient φ = 1, the flow pattern within the runner passages is favorable across all span-wise surfaces. No significant flow separation occurs on the blade surfaces, and the velocity distribution at the runner outlet is uniform. No backflow is observed at the runner inlet, resulting in low entropy generation rates in this region. However, localized high entropy generation rates still exist near the blade leading edges. This is attributed to the small angle between the incoming flow direction and the blade setting angle, which weakens flow separation. When the flow coefficient decreases to φ = 0.86, a single vortex structure initially present at the blade leading edge migrates toward the runner back shroud, forming multiple vortex clusters. These vortex clusters distributed along the blade pressure side result in high local entropy generation rates. The runner flow passages evolve into separated vortices, partially obstructing the flow channels. Backflow exchanges momentum with the outgoing flow from the runner outlet, causing significant hydraulic losses in the vaneless space and leading to unstable flow conditions. In the flow coefficient range of φ = 0.21–0.54, as flow rate increases, the attack angle between the runner blade setting angle and inlet flow angle decreases, mitigating flow separation phenomena. The high local entropy generation rate zones within the flow passages diminish. Corresponding streamline diagrams reveal that this is attributed to the weakening of backflow at the blade inlet, accompanied by an increased velocity gradient around the backflow region and the disappearance of associated unstable low-speed turbulence.

4.3.2. Analysis Between Dual-Blade-Row Cascade Energy Loss and Internal Flow Characteristics

Figure 11 and Figure 12, respectively, show the LEPR and streamline distribution at different spans of the dual-blade-row cascade. In Figure 8, it is observed that, under various flow coefficients, the LEPR values are notably higher in the regions of the guide vanes and portions of the stay vanes. In Figure 8, when φ = 0.21–0.54, the area of energy loss distribution exceeds that of other operating conditions. At φ = 0.21, the maximum LEPR intensity occurs predominantly at the guide vane region of Span 0.1. The indicates that energy losses are primarily concentrated adjacent to the guide vane crown, with high LEPR mainly distributed at the guide vane leading edge and trailing edge. As flow rate increases, when φ = 0.43, entropy loss begins to shift from the vaneless region between guide vanes to the flow passages of stay vanes. Entropy distribution concentrates primarily in the vaneless space between guide vanes and stay vanes at Span 0.5. Meanwhile, high LEPR values are localized at the trailing edges of guide vanes at Span 0.1 and Span 0.9. It is primarily attributed to vortices within the guide vane passages and high-speed circulation induced by runner rotation. When φ = 0.54, entropy generation concentrates within the stay vane passages. It is primarily caused by diversion of flow at the runner outlet by the guide vane leading edges. Due to an excessive attack angle formed between the flow direction and the vane setting angle, part of the flow forms vortices on the pressure side of the stay vanes, while another part collides with low-velocity flow near the vane walls, creating high velocity gradients that generate energy losses. This process thus contributes to the formation of hump characteristics. At the higher flow coefficient φ = 0.86, the local entropy loss regions concentrate primarily on the pressure surface of the guide vanes, with a reduction in local entropy generation intensity. The high-entropy zone near the bottom ring diminishes in size. Within the guide vane area, high local entropy generation is mainly localized at the leading edge portions. It is attributed to the increased flow velocity from the runner outlet, which impinges on the guide vanes and generates localized high entropy generation rates. When flow coefficient φ = 1, entropy generation concentrates primarily within the fixed guide vane passages. Vortical structures in these passages elevate convective intensity and velocity gradients in adjacent regions, inducing substantial hydraulic losses. Concurrently, elevated entropy zones appear across multiple spanwise sections at the tongue regions, with peak entropy intensity occurring at Span 0.1. This phenomenon arises from the maximal attack angle between the tongue’s setting angle and flow direction, triggering flow separation and backflow at the tongue area across different span surfaces. Consequently, significant energy dissipation occurs in these regions.

4.3.3. Analysis Between Draft Tube Energy Loss and Internal Flow Characteristics

Figure 13 illustrates the entropy generation rate and streamline distribution across three axial planes (A-A, B-B, C-C) within the draft tube’s conical diffuser section. These cross sections reveal the spatial variation and distribution characteristics of entropy generation. From the streamline diagram, it can be seen that, at the optimal operating point and the point with larger flow rate, due to relatively stable flow, the streamlines are relatively smooth, and no vortex structure appears. The entropy generation distribution diagram reveals the absence of high entropy generation zones within the core flow region of the draft tube. Across the axial cross sections (A-A, B-B, C-C), the transverse entropy generation rate exhibits non-uniform distribution yet maintains an overall circular pattern. During the propagation process from downstream, the entropy generation rate gradually decreases. At the optimal operating condition with φ = 1 and the flow point, the flow state in the tail water chamber is stable, and there is no vortex structure. As the flow rate decreases, when φ = 0.54, non-uniform streamline distribution is observed at the draft tube outlet, with primary energy losses concentrated in the conical diffuser section. Analysis of flow simulation results reveals that secondary flows form under the influence of centrifugal forces and wall viscous effects within this section. Notably, the flow rate exhibits an inverse relationship with circulation scale: smaller flow rates correspond to larger circulation patterns. When the flow rate decreases to φ = 0.21, circulation emerges near the wall of the conical diffuser outlet. The circulation center shifts closer to the central axis of the diffuser, and the inception point of circulation advances toward the elbow section, forming high hydraulic loss zones along the circumferential wall surfaces. Under low-flow conditions, reduced flow velocity promotes wall separation in both the conical diffuser and elbow sections, generating large-scale vortices and backflow that induce significant entropy generation losses.

4.3.4. Analysis of Entropy Generation Distribution on Runner Blade Wall Surfaces

As the sole rotating component in the pump-turbine, the runner exhibits highly complex fluid dynamics due to its blade geometrical distortions. During operation, energy dissipation from water–wall interactions across hydraulic components contributes to hydraulic losses. As fluid sequentially interacts with guide vanes and runner blades, wall entropy generation on the runner blades was systematically investigated, with the results illustrated in Figure 14. As shown in Figure 14, the wall entropy generation areas on the runner blades are primarily distributed at the inlet and outlet edges, gradually decreasing toward the blade’s central region. Within the flow coefficient range of φ = 0.21–0.53, significant entropy generation is observed at the blade trailing edge outlets, with the maximum entropy generation occurring at the minimum flow point. As the flow rate increases, the entropy generation intensity progressively diminishes. The high entropy generation intensity at the runner blade trailing edge outlets is mainly attributed to the impact phenomena that occur when water flows through the runner at the blade leading edge, resulting in relatively high entropy generation losses. Within the flow coefficient range of φ = 0.86~1, increasing flow rates lead to heightened entropy generation intensity on the pressure surface of fixed guide vanes, with high-entropy regions predominantly concentrated near the trailing edge. This phenomenon results from both the formation of larger attack angles between partial flow streams and the cascade blade and rotor–stator interactions between the runner and movable guide vanes, which induce flow impingement against the pressure surface and trailing edge regions.

5. Conclusions

Based on the above research, the following conclusions can be drawn:

(1) When operating in the hump region, entropy generation in the main flow zone predominantly originates from adverse flow phenomena, including flow separation, backflow, and vortices—within the runner and double-row blade passages. These components account for a significant proportion of energy losses, specifically 72–86% of total entropy generation.

(2) For operating conditions below the optimal efficiency point (φ = 1), energy loss distribution along spanwise directions from the inlet to the outlet exhibits significant variations within the runner and double-row blade passages. As the flow rate further decreases, flow separation on the runner suction surface evolves into vortex clusters that fill the runner flow passages, obstructing the flow. Vortices developing near the runner rear hub extend into the draft tube, resulting in increased energy losses in the draft tube.

(3) Vortices in the runner inlet passage disturb the internal flow pattern, causing non-uniform velocity distribution at the runner outlet and increasing energy losses in the vaneless space and guide vanes. As the flow rate decreases, the velocity triangle at the guide vane inlet undergoes significant changes, inducing flow separation at the leading edge of guide vanes. Under hump region operating conditions, most of the guide vane passages become blocked, trapping water in the vaneless space to rotate with the runner direction. Due to rotor–stator interactions, backflow and high velocity gradients occur in the vaneless space, resulting in substantial energy losses.