Analysis of the Energy Loss Mechanism of Pump-Turbines with Splitter Blades under Different Characteristic Heads

Abstract

The operating efficiency of high-head pump turbines is closely related to the internal hydraulic losses within the system. Conventional methods for calculating hydraulic losses based on pressure differences often lack detailed information on their distribution and specific sources. Additionally, the presence of splitter blades further complicates the hydraulic loss characteristics, necessitating further study. In this study, Reynolds-averaged Navier–Stokes (RANS) simulations were employed to analyze the performance of a pump turbine with splitter blades at three different head conditions and a guide vane opening (GVO) of 10°. The numerical simulations were validated by experimental tests using laser doppler velocimetry (LDV). Quantitative analysis of flow components and hydraulic losses was conducted using entropy production theory in combination with an examination of flow field distributions to identify the origins and features of hydraulic losses. The results indicate that higher heads are associated with lower growth rates of total hydraulic losses. In particular, the significant velocity gradients at the trailing edge of the splitter blades contribute to higher hydraulic losses. Furthermore, the hydraulic losses in the runner (RN) region are predominantly influenced by velocity gradients and not by vortices, with the flow conditions in the RN region impacting the hydraulic losses in the draft tube (DT).

1. Introduction

As the global demand for electricity continues to rise, the share of renewable energy power plants in power grids is growing rapidly. To cope with fluctuations in the power system caused by the disruption of renewable energy sources such as photovoltaics, hydroelectric power plants of different capacities are playing a crucial role in power grids [1]. Among these, pumped storage power plants, with their significant regulation capabilities, have become an important part of the grid regulation system [2]. As one of the central elements of pumped storage power plants, the pump turbine is of considerable research importance.

The performance of pump turbines in turbine mode is somewhat limited due to the need to operate in both directions. To optimize the performance of pump turbines, extensive hydraulic design studies have been conducted. Researchers typically employ numerical simulation methods [3,4,5,6,7] and model tests [8,9,10,11,12] as approaches for research on hydraulic machines. The hydraulic characteristics of pump turbines in turbine mode have been extensively investigated. Deng et al. [13] analyzed the vortex motion patterns and pressure pulsation characteristics of pump turbines in vaneless space in turbine mode using numerical simulations and experiments. Lai et al. [14,15] conducted experimental studies on the flow characteristics in the DT of a pump turbine in turbine mode and determined the connection between velocity distribution and vortex rope motion in the DT. While most researchers have focused on flow characteristics, hydraulic loss remains the key factor affecting turbine efficiency.

When studying hydraulic losses, the pressure difference method is typically used first. However, this method does not provide a specific distribution of hydraulic losses. Therefore, researchers have proposed various alternative methods to study hydraulic losses, such as the entropy production method and the local hydraulic loss method. The entropy production method, derived from thermodynamics, has been improved and widely employed in hydraulic loss studies.

In the initial decade of this century, scholars primarily focused on fundamental theoretical research. They engaged in theoretical derivations and modeling analyses to address core issues, utilizing entropy production theory to explore topics such as laminar and turbulent flow, the viscous layer of turbulent wall flows, and more [16]. Subsequently, an increasing number of researchers started employing entropy production theory as a crucial framework for investigating process mechanisms and fluid flow states in specific types of hydraulic machinery. This application aimed at optimizing performance, understanding energy dissipation in flows, and studying flow characteristics [16]. Yan et al. [17] utilized the entropy production method to investigate the hydraulic losses of a pump turbine in the S characteristic zone and found that the vortex flow in the guide vane (GV) zone and the DT contributed significantly to the hydraulic losses. Li et al. [18] applied the entropy production method to examine the hydraulic losses of a pump turbine in the hump region and concluded that rotating stall in this region was a major source of hydraulic losses. Therefore, the entropy production method is well suited for studying hydraulic losses in pump turbines. The local hydraulic loss method, established by Qin et al. [19], was specifically applied to the study of hydraulic losses in pump turbines. Currently, the entropy production method is more widely used due to its superior accuracy. Although entropy production methods have been widely used in the study of energy losses in pump turbines, research on pump turbines with splitter blades has not been conducted.

As a means of validating and supplementing numerical simulations, non-contact measurement methods are widely used in hydromechanics. Two main techniques commonly used in hydraulic machines are particle image velocimetry (PIV) and LDV. PIV is well suited for studying the evolution of flow structures [20,21], while LDV offers higher accuracy in local velocity measurements [22,23]. When the velocity of a point needs to be measured in pump turbines, the LDV technique has become the preferred experimental approach due to its superior accuracy.

This paper investigates the mechanism of hydraulic losses in pump turbines operating in turbine mode at different heads. Numerical simulations are performed and validated by LDV tests. The study utilizes the entropy production method to analyze hydraulic losses and explores the relationship between losses and heads in conjunction with flow field analysis. The findings contribute to a better understanding of the hydraulic performance of pump turbines and provide insights into improving efficiency and operation.

2. Theory of the Simulation

2.1. Turbulence Model and Entropy Production Theory

In the numerical simulations, Navier–Stokes equations based on Reynolds time averaging were used, and the SST k-ω turbulence model was used [24,25,26,27,28].

The entropy production rate (EPR) of the time-averaged Reynolds flow includes the direct entropy rate caused by the time-averaged velocity and the indirect entropy rate caused by the pulsation velocity; the EPR is shown in Equation (1). The EPR caused by direct dissipation (EPDD) is shown in Equation (2), and the EPR caused by turbulence dissipation (EPTD) is calculated using Equation (3).

$${\dot{S}}_D^{‴}={\dot{S}}_{\overline{D}}^{‴}+{\dot{S}}_{D^{\prime }}^{‴}$$

(1)

$$\begin{array}{l}{\dot{S}}_{\overline{D}}^{‴}=\frac{2{\mu }_{eff}}{T}\left[{\left(\frac{\partial {\overline{u}}_1}{\partial x_1}\right)}^2+{\left(\frac{\partial {\overline{u}}_2}{\partial x_2}\right)}^2+{\left(\frac{\partial {\overline{u}}_3}{\partial x_3}\right)}^2\right]\\ +\frac{{\mu }_{eff}}{T}\left[{\left(\frac{\partial {\overline{u}}_2}{\partial x_1}+\frac{\partial {\overline{u}}_1}{\partial x_2}\right)}^2+{\left(\frac{\partial {\overline{u}}_3}{\partial x_1}+\frac{\partial {\overline{u}}_1}{\partial x_3}\right)}^2+{\left(\frac{\partial {\overline{u}}_2}{\partial x_3}+\frac{\partial {\overline{u}}_3}{\partial x_2}\right)}^2\right]\end{array}$$

(2)

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

(3)

where ${\dot{S}}_D^{‴}$, ${\dot{S}}_{\overline{D}}^{‴}$, and ${\dot{S}}_{D^{\prime }}^{‴}$ denote EPR, EPDD, and EPTD, respectively, in m^–3·K^–1; ${\overline{u}}_1$, ${\overline{u}}_2$, and ${\overline{u}}_3$ denote the time-averaged velocity components in m/s; $u_1^{\prime }$, $u_2^{\prime }$, and $u_3^{\prime }$ denote the pulsating velocity components in m/s; T indicates temperature in Kelvin; μ_eff is the effective dynamic viscosity of fluid in Pa·s; β is an empirical constant, approximated as 0.09; k denotes the turbulent kinetic energy in m²/s²; and ω denotes the turbulent vortex frequency in s^–1.

The EPR caused by wall shear stress (EPWS) is calculated using Equation (4):

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

(4)

where ${\dot{S}}_W^{″}$ denotes EPWS, W·m⁻²·K⁻¹; ${\overrightarrow{\tau }}_w$ and ${\overrightarrow{v}}_w$ indicate the shear stress and velocity near the wall, respectively.

The TEP (sum of the integrals for each entropy production rate) can be obtained by integrating ${\dot{S}}_{\overline{D}}^{‴}$, ${\dot{S}}_{D^{\prime }}^{‴}$ and ${{\dot{S}}^{″}}_W$, respectively, over the computational domain with subsequent summation.

$$S_{pro,\overline{D}}={\displaystyle \underset{V}{\int }{\dot{S}}_{\overline{D}}^{‴}dV}$$

(5)

$$S_{pro,D^{\prime }}={\displaystyle \underset{V}{\int }{\dot{S}}_{D^{\prime }}^{‴}dV}$$

(6)

$$S_{pro,W}={\displaystyle \underset{A}{\int }\frac{{\overrightarrow{\tau }}_w\cdot {\overrightarrow{v}}_w}{T}}dA$$

(7)

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

(8)

where $S_{pro}$ denotes the TEP.

2.2. Calculation Domain and Mesh Generation

The pump turbine model is derived from real units in high-head pumped storage power plants and is created using Unigraphics NX software to achieve high-precision modeling across the entire computational domain based on design blueprints. The pump turbine model used in the numerical simulation is shown in Figure 1, and the geometric parameters of the pump turbine are shown in

Table 1. Geometric parameters of pump-turbine.

Domain	Parameter	Variable/Unit	Value
RN	Blades	Zb	5
	Splitters	Zs	5
GV	Guide vanes	ZG	16
Spiral casing (SC)	Wrap angle	Φ/(°)	360
Stay Vane (SV)	Stay vanes	ZS	16

.

The full flow channel of the pump turbine was meshed using ICEM-CFD. The final grid information is shown in

Table 2. Pump turbine grid information.

Domain	Grid Type	Number of Grid Cells	y+
RN	Hexahedral	3,011,474	<15
GV	Tetrahedral	2,988,878	<20
SV	Hexahedral	1,699,875	<30
DT	Hexahedral	1,665,874	<30
SC	Hexahedral	1,856,253	<50
Total	/	11,222,354	/

. The full computational domain grid is shown in Figure 2. The cell growth ratio was set to 1.2 at all the wall boundaries.

2.3. Grid Independence Verification

Seven grids were formed, and the numbers of grid cells were about 2,130,000, 3,150,000, 4,210,000, 6,380,000, 8,390,000, 9,560,000, and 11,220,000, respectively. A sensitivity analysis of the key parameters of the external characteristics of the unit, namely, efficiency and torque, was carried out, as shown in Figure 3. It was found that an increase in the number of grid cells could significantly improve the accuracy of the key parameters. The efficiency and torque values remain stable after a grid number greater than 8 million. To further determine the influence of the grid on the calculations, an independence analysis of the grids was performed.

To confirm grid independence in this study, the Richardson extrapolation method was utilized [29,30,31,32]. Moreover, the grid convergence index (GCI) was employed to quantitatively assess the convergence of the computational results. The GCI can be calculated using the following formula:

$$GC{I}^{21}=\frac{F_S{e}_a^{21}}{{{\displaystyle r}}_{21}^{\zeta }-1}$$

(9)

$$e_a^{21}=\left|\frac{{\phi }_1-{\phi }_2}{{\phi }_1}\right|$$

(10)

$$r_{21}=\sqrt[3]{\raisebox{1ex}{$G_1$}\!\left/ \!\raisebox{-1ex}{$G_2$}\right.}$$

(11)

$$\zeta =\frac{1}{\ln (r_{21})}\left|\ln \left|\raisebox{1ex}{${\epsilon }_{32}$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_{21}$}\right.\right|+q(\zeta )\right|$$

(12)

$$q(\zeta )=\ln (\frac{r_{21}^{\zeta }-s}{r_{32}^{\zeta }-s})$$

(13)

$$s=1\cdot \mathrm{sgn}(\raisebox{1ex}{${\epsilon }_{32}$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_{21}$}\right.)$$

(14)

where $F_S$ is the safety factor, typically 1.25; $e_a^{}$ is the relative error of the two sets of grid numerical value ${\phi }_1$ and ${\phi }_2$; $G_1$ is the grid number of G1; $r$ is grid refinement factor; $\zeta$ is the convergence accuracy; ${\epsilon }_{32}$ and ${\epsilon }_{21}$ are the difference between the two sets of numerical values of the grid.

Three groups of grid plans were selected, namely, G1: 11,220,000, G2: 8,390,000, and G3: 3,150,000, which were involved in the verification of grid accuracy with flow rate and efficiency. The GCIs of flow and efficiency involved in the grid independence verification were less than 3.0% when the grid number was 11,220,000, indicating that the grid number of G1 met the requirements of computational accuracy. Grid independence verification information is shown in

Table 3. Grid independence verification.

Parameters	φ1	φ2	φ3	FS	$\mathit{\zeta }$	GCI
Q	381.75	381.25	378.35	1.2	10.78	1.55%
Efficiency	87.3	87.2	86.9	1.2	3.89	1.35%

.

2.4. Boundary Conditions and Case Information

ANSYS-Fluent software (Ansys, Inc., Canonsburg, PA, USA) was used for the numerical simulation, and the pressure boundary was used for both the inlet and outlet of the computational domain. The inlet boundary pressure is specified using a head value, while the outlet boundary pressure is set to 0. The rotating component moves using the Moving reference frame, and the rotational speed is set to a test value. The SIMPLEC algorithm is used to solve the flow fields, the second order upwind format is used to discretize the convective terms and diffusive terms, the wall condition is set to no-slip wall, and the root mean square of the residuals is set to 10^–5 [28]. During the simulation, the interface between the rotating domain and the static domain is dealt with by means of the frozen rotor. This method is usually used for steady calculations, which reduce computational requirements by assuming that the flow field does not change significantly during one rotor revolution. The characteristic operating points during the actual operation of the power plant were selected for the study. The case information is shown in

Table 4. Simulation case selection (all data in the table is from test data are used as boundary conditions for simulation).

Case	H (m)	GVO (°)	n (rev/min)	Q (m3/s)	N (MW)	η (%)	Δh (m)
Case 1	580	10	500	36.6	180.2	86.69	77.20
Case 2	600	10	500	38.4	198.9	88.10	71.40
Case 3	640	10	500	41.3	233.9	90.30	62.08

.

3. Experimental Test and Validation

3.1. Test Rig

The model pump turbine was tested on a general hydraulic–mechanical test rig, as shown in Figure 4, and the external characteristics of the unit and the velocity distribution in the DT were obtained. The test bench parameters are shown in

Table 5. Test rig parameters.

Parameters	Value
Maximum test flow rate (m³/s)	1.5
Maximum test head (m)	150
Maximum test speed (r/min)	2500
Model RN diameter (mm)	250–500
Dynamometer maximum power (kW)	500
Rated power of pump motor (kW)	2 × 850
DT pressure (kPa)	–85 to +250
Uncertainty of efficiency measurements	≤±0.25%
Model type	Reaction turbine

.

The structure of the LDV system is shown in Figure 5a, the parameters are shown in

Table 6. Parameters of LDV test system.

Parameters	Value
Speed range	–150–1000 m/s
Measurement error	0.1%
Sample frequency	400–800 MHZ
Maximum processing frequency	175 MHz
Minimum processing frequency	300 Hz
Bits	8

, and the test site is shown in Figure 5b.

3.2. Computational Validation

In order to facilitate the comparison of the velocity distribution in the DT obtained from the numerical simulation and the experiment, the velocity in the DT must be dimensionless.

$$C_u=\frac{V_u}{n{D}_1}$$

(15)

$$C_m=\frac{V_{{}_m}}{n{D}_1}$$

(16)

where C_u denotes the tangential velocity coefficient; C_m denotes the axial velocity coefficient; V_u and V_m denote the velocity obtained by the test or numerical simulation, respectively, in m/s; n is the rotational speed in rev/min; and D₁ is the RN inlet diameter of the pump turbine in m.

The head H_M of the model test is determined from Equation (17):

$$H_M={(\frac{n_M}{n})}^2{(\frac{D_M}{D_1})}^{2}H$$

(17)

where H is the head; H_M is the model test head, m; n_M is the rotational speed of model pump-turbine, rev/min; D_M is the RN inlet diameter of model pump-turbine, m.

The external characteristics and velocity distribution of the DT of Case 1 were compared, as shown in Figure 6.

As seen in Figure 6a, the efficiency values obtained from the three cases were very close to the efficiency values obtained from experimental testing, indicating that the simulated external characteristics were accurate. As seen in Figure 6b, the distribution of the simulated axial velocity coefficient and tangential velocity coefficient in the draft tube exhibited the same trend and almost identical values as the experimental results, indicating that the simulated internal flow within the unit was accurate.

4. Results and Analysis

4.1. Total Energy Loss

In order to accurately calculate the energy loss in the pump turbine under characteristic heads, the entropy production method was utilized; it mainly consists of three terms, namely, EPDD, EPTD, and EPWS. In addition, to describe the TEP increment ratio quantitatively compared to Case 1, the growth ratio (γ) is defined as

$$\gamma =\frac{{\gamma }_n-{\gamma }_1}{{\gamma }_1}(n=2,3)$$

(18)

where γ₁, γ_2, and γ₃ represent the TEP values in Cases 1, 2, and 3, respectively.

Figure 7 illustrates the entropy production values and growth rates for different entropy terms under three operating conditions. The TEP gradually decreased from Case 1 to Case 3, indicating lower total energy losses at higher heads, consistent with the data in Table 4. Among the three entropy terms, EPTD was the dominant factor, followed by EPWS and EPDD. Specifically, the EPTD accounted for over 98% in all three cases (Figure 7a). Under the turbine condition with the high head, the flow rate of the unit was large, resulting in a large velocity gradient inside the unit, and the EPDD term was closely related to the velocity gradient, which led to a significant increase in the EPDD term in Cases 2 and 3. The efficiency of the unit was higher under the condition of a high water head, indicating that the flow regime of a high head is better. Since the EPTD of the simplified calculation was controlled by the turbulent kinetic energy, the EPTD terms of Cases 2 and 3 were significantly reduced. The large flow in the unit under the high head led to the increase of the velocity in the near-wall region. Since the EPWS item was affected by the near-wall velocity, the EPWS growth rate of Case 2 and Case 3 was significant. Additionally, the negative growth rates of –3.03% and –9.42% for Cases 2 and 3, respectively, compared to the EPTD in Case 1, indicated a decrease in intra-unit turbulence intensity with increasing flow rates (Figure 7b).

Figure 8 depicts the TEP of the five components for the three cases and their growth rates compared to Case 1. Among the five components, the DT contributed the most to the TEP, followed by the RN, GV, SV, and SC. It was observed that the energy losses increased significantly along the flow direction (Figure 8a). The SC, SV, and GV components were mainly affected by inlet boundary conditions since they are inlet components of the unit. Consequently, the growth rates of the SC, SV, and GV components were more pronounced in Case 2 and Case 3 than the TEP growth rate in Case 1. Furthermore, the TEP growth rate of the impeller in Case 3 was significantly lower than that in Case 2 (–15.74% vs. –6.46%). This indicates that the RN’s efficiency zone is at a higher head (Figure 8b).

4.2. Analysis of Flow Characteristics in Inlet Components

In turbine mode, the pressure difference between the inlet and outlet of the pump turbine served as the primary driving force for the water flow, with energy losses occurring during energy conversion. The improved flow patterns in the inlet components (SC, SV, GV) contributed to lower energy losses in these components, accounting for 5.2%, 6.2%, and 8.3% in the three cases, respectively (Figure 8a).

To facilitate the comparison of the pressure distribution of the inlet part of the pump turbine in three cases (Figure 9), the dimensionless pressure coefficient C_p can be written as

$$C_p=\frac{p-p_0}{\rho \mathrm{g}H}$$

(19)

where p₀ means the average static pressure at the inlet of the SC, pa.

As shown in Figure 9, from the inlet of the worm shell to the outlet of the GV, the pressure gradually decreased, resulting in a more evenly distributed C_p value. This indicates that the number of SV and GV was set correctly, effectively balancing the water pressure created by high heads. The presence of a distinct high C_p region at the leading edge of SV and GV indicated a higher static pressure in this region. Compared to Case 1, the region with low C_p values (dark blue) near the interface between GV and RN was larger in Case 3, indicating a faster pressure drop in the inlet component under high head conditions.

Figure 10 illustrates the streamline distribution on the horizontal surface of the inlet components. In all three cases, the streamlines in the inflow part were smooth, without significant vortex formation, which was related to the proper GV angle and proper boundary conditions. Additionally, the velocity gradually increased along the flow direction, with the most pronounced increase occurring after the GV, as the water flow section through the GV area decreased. Since the GVO remained the same at 10° in all three cases, there was no significant difference in the streamline patterns. However, comparing Case 1 to Case 3, the velocity at the outlet of GV was higher in Case 3 due to the elevated flow rate under high head conditions.

Figure 11 shows the EPR distribution on the horizontal plane of the inlet components, specifically EPDD and EPTD. In all three cases, a distinct region of high EPR was observed at the trailing edge of the GV, corresponding to a significant velocity gradient within this area. Additionally, the energy loss at the SV trailing edge was smaller than that at the GV trailing edge, while the EPR value was lowest in the SC region, which agreed with the TEP distribution of the three inlet components (Figure 8a). It is noteworthy that the increase in EPR values was most pronounced in Case 3 when compared to Case 1, indicating that the higher flow rate led to larger energy losses within the inlet components and consequently to a higher TEP growth rate for the inlet components in Case 3 (Figure 8b).

4.3. Analysis of the Flow Characteristics in RN

The RN is the key component responsible for energy conversion in a water pump turbine, and its internal flow characteristics have a direct impact on energy conversion efficiency. The pressure difference on both sides of the blade is the main driving force for RN rotation, which will greatly affect the flow state inside the unit. Figure 12 displays the C_p value distribution at different span-wise surfaces of RN in the three cases. Here, span represents the dimensionless distance from hub to shroud, indicating the position of the blade-to-blade surface. For example, span = 0 indicates the blade-to-blade surface at the hub, and span = 1 indicates the blade-to-blade surface at the shroud. It can be clearly seen that the pressure inside the RN gradually decreased from the inlet to the outlet. At span = 0.05, the Cp value of the pressure side (PS) of the rotor blade was much higher than that of the suction side (SS), indicating that the pressure difference between the PS side and the SS side of the rotor blade drives the impeller to rotate. Moreover, the pressure difference on both sides of the blades was greater than that of the splitters, indicating that the ability of the splitter to drive the rotation of the impeller is weaker than that of the blades. At span = 0.5 and 0.95, the Cp distribution in the RN was consistent with that at span = 0.05, indicating that the pressure distribution of different spans is basically the same. Under the three different cases, the distribution of Cp inside the RN was basically the same, indicating that the design of the impeller of the unit is reasonable, and the pressure distribution inside the RN is basically unchanged within the operating range of the characteristic head.

In order to allow for a clearer comparison of the impeller flow characteristics in the three cases, Figure 13 intentionally shows the relative velocity streamlines at various deployed surfaces within the RN domain. In Case 1, where span = 0.05, the speed decreased continuously during water flow from the leading edge to the trailing edge of the blade. At this point, the energy of the water was gradually converted into the rotational mechanical energy of the blade. At this point, the velocity on the PS of the rotor blade was much higher than on the SS, suggesting that the water flow on the PS side was the main driving force for the rotation of the RN. Simultaneously, the alternating distribution of blades and splitters also influenced the flow pattern, making the flow pattern inside the impeller asymmetric. Compared to the velocity on the PS side of the blades, the velocity on the PS side of the splitters was higher, while the velocity on the SS side of the splitter was lower. Starting from the energy conversion of rotating machinery, the distribution of pressure and flow velocity at the inlet and outlet sections of the impeller was relatively uniform. However, considering that the geometric length of the splitter along the flow direction was shorter than that of the blades, the force bearing area of the splitter was smaller, and the velocity gradient nearby was larger. Compared to the case with span = 0.05 in Case 1, the velocity on the PS of the splitters at span = 0.5 was slightly lower, and the velocity at the RN outlet was also slightly lower. At the same time, the relative velocity streamlines between the blades were smoother at span = 0.5, indicating a better flow pattern at the center line of the blade profile. Furthermore, at span = 0.95, the relative velocity flow direction on the SS side of the leading edge of the splitters deviated from the center line of the blade profile, and there was noticeable low-speed recirculation on the SS side of the trailing edge of the long blades. Comparing the three different cases, it was noticeable that in Case 3, the velocity difference between the two sides of the RN was smaller, with a lower velocity on the PS side and a higher velocity on the SS side. Additionally, the relative velocity streamlines between the blades in Case 3 were smoother and aligned better with the center line of the blade profile. This indicated that Case 3 had the best flow pattern, followed by Case 2, and Case 1 had the worst performance.

To investigate the relationship between vortices and energy loss within the RN, absolute helicity, defined as the absolute value of the dot product between the velocity vector and vorticity vector, was used to characterize the degree of vortex spiralization. Energy loss is represented by EPR. Figure 14 and Figure 15 show the distribution of absolute helicity and EPR at different unfolded surfaces of the RN domain. For span = 0.05 in Case 1, the leading edge of the splitter on the SS exhibited high helicity and high EPR, indicating significant vortex spiraling and energy loss in this area. At span = 0.5 in Case 1, there was high EPR and high-velocity flow at the trailing edge of the splitter, but no appreciable helicity, suggesting that the energy loss in this region was mainly due to the large velocity gradient and not the presence of vortices. In Case 1, the helicity distribution inside the impeller at span = 0.95 presented obvious asymmetry, and the leading edge of the splitter had obvious high helicity and energy loss, indicating that the high-speed flow off the centerline existed at the leading edge of the splitter. Significant helicity leads to significant energy loss. At the same time, compared with span = 0.5, the flow direction of the relative velocity near the leading edge of the SS side splitter at span = 0.95 deviated more from the centerline of the blade profile, resulting in higher helicity and EPR, thus exacerbating the impeller internal flow state asymmetry. Comparing the three cases, it was observed that the RN domain in Case 3 had the smoothest streamlines, the lowest helicity, and the lowest EPR. This contributed to the lowest TEP within the RN in Case 3. Consequently, as the head increased, the flow pattern within the RN domain improved, resulting in a reduction in overall helicity and EPR and ultimately a reduction in TEP within the RN (Figure 8).

In the process of flowing from the leading edge of the RN to the trailing edge, the water flow gradually transitioned from the state of rotating around the axis to the state of axial outflow. The swirling degree is closely related to the operating efficiency of the unit and the flow state in the RN. In order to quantitatively describe the degree of swirl in the RN under three characteristic water heads, the swirl number (S_w) is calculated as follows [33]:

$$S_w=\frac{{\displaystyle {\int }_0^R{U}_a{U}_t{r}^2\mathrm{d}r}}{R{\displaystyle {\int }_0^R{U}_a^{2}r\mathrm{d}r}}$$

(20)

where U_a is the axial velocity, U_t is tangential velocity, and R is the hydraulic radius, representing the impeller radius.

Figure 16 represents the S_w values on different horizontal planes of RN in the three cases. According to the physical structure of the RN, it could be divided into three regions: R1 is the region where the water flows in tangentially, R2 is the transition region, and R3 is the region where the water flows out axially. It can be clearly seen that the S_w value in the R1 domain was much greater than 1.0, indicating that the tangential velocity of the water flow was much greater than the axial velocity. Moreover, the S_w value dropped sharply along the axial direction, which showed that the kinetic energy of the water flow was quickly converted into the rotational mechanical energy of RN, which was also related to the greater pressure difference between the two sides of the blade in the R1 domain. In the R2 domain, the S_w value was less than 1.0, indicating that the main flow direction of the water flow was axial. At the same time, the S_w value decreased slowly along the axial direction, and it could be seen that the energy transferred by the water flow to RN was reduced. In the R3 domain, the S_w value was maintained at a low level, far below 1.0, and the velocity circulation still existed in the outlet domain. Comparing the three cases, it could be seen that on the same level, the S_w value of case 1 was the highest, followed by case 2, and that of case 3 was the least. Especially in the R3 domain, the average S_w values of the three cases were 0.23, 0.22, and 0.19, respectively. It could be seen that the velocity circulation of the water flow in Case 3 was the smallest, which was also the performance of Case 3 with less energy loss.

4.4. Analysis of the Flow Characteristics in DT

Based on the calculation results of the entropy production in Figure 7, it was observed that the TEP in the DT was larger than that in the RN domain. This was partly due to the presence of significant turbulence at the RN outlet and the larger integration volume of EPR in the DT. Since EPTD was the main contributor to the TEP in the system and was calculated indirectly from the turbulent kinetic energy, analyzing the level of turbulence in the DT could help identify the source of the energy loss. By examining the streamlines in the DT domain, as shown in Figure 17, it could be seen that there was significant tangential velocity at the inlet of the DT and that the velocity near the wall was much higher than that in the central region of the DT. Additionally, the overall flow velocity decreased from the inlet to the outlet of the DT. Figure 18, Figure 19 and Figure 20 represent the distribution of TKE, spiralization, and EPR on the vertical plane of the DT, respectively. In Case 1, there was significant spiralization near the wall in the inlet region of the DT, consistent with the tangential flow distribution. Moreover, the inlet region of the DT exhibited high TKE and EPR values, and their distributions closely matched, indicating that the high level of turbulence in the inlet region resulted in significant energy loss. Moreover, regions of high EPR in the DT exhibited relatively low spiralization, suggesting that the increase in TKE was caused by the velocity gradient rather than the presence of vortices. Comparing the three different cases, it is clear that Case 3 had significantly lower TKE and EPR values compared to Case 1 and Case 2, indicating that the improved flow properties in the RN domain reduced the turbulence level and energy loss in the DT.

5. Conclusions

In this paper, the hydraulic loss mechanism of a high-head pump turbine with splitter blades was studied using three-dimensional visualization techniques. The numerical simulation was verified by an experimental LDV test, and the velocity coefficients at the section of the draft tube were in good agreement. Using entropy generation theory combined with the flow field distribution, the hydraulic loss and internal flow state changes of the unit under three characteristic heads, such as velocity, pressure, helicity, and EPR, were compared to explore the root causes and characteristics of the hydraulic loss. The following main conclusions were drawn:

(1)

Among the three terms of entropy production, EPTD dominates, accounting for over 98% of the TEP. Within the five flow components, RN and DT play a dominant role, and TEP increases significantly along the flow direction. In all three cases, the growth rate of TEP decreases with increasing head, suggesting that the high-efficiency region of the turbine is at high-head operating conditions.

(2)

In the inlet components, the presence of a large velocity gradient at the trailing edge of the GV leads to a significant EPR. Furthermore, under high head conditions, the higher flow rate increases the velocity in the GV area, increasing the velocity gradient at the trailing edge and causing more energy loss. The average energy loss growth rate of inlet components in case 2 and case 3 are 14.87% and 42.69%, respectively.

(3)

In the RN domain, at spans of 0.05 and 0.95, the high spiralization and high EPR are observed at the leading edge of the splitter on SS where the flow direction deviates from the center line of the blade profile. This deviation, together with the high flow velocity, leads to significant spiraling and significant energy losses. At span = 0.5, the high EPR and high flow velocity are observed at the trailing edge of the splitter, with no appreciable spiralization occurring. This indicates that the energy loss in this region is mainly due to the large velocity gradient and not to the presence of vortices. In the RN domain, the S_w value continues to decrease along the axial direction, indicating that the water kinetic energy is continuously converted into the rotational mechanical energy of the RN. Especially in the R3 domain, the average S_w values of the three cases are 0.23, 0.22, and 0.19, respectively.

(4)

In Case 1, the region close to the wall in the inflow section of the DT exhibits a high tangential velocity, which leads to significant spiralization. Additionally, the central region of the inflow section with large velocity gradients generates higher TKE and EPR. Case 3 shows significantly lower TKE and EPR values compared to Case 1, indicating that the improved flow properties in the RN domain reduce turbulence and energy losses in the DT.