Energy Balance and Local Unsteady Loss Analysis of Flows in a Low Specific Speed Model Pump-Turbine in the Positive Slope Region on the Pump Performance Curve

The positive slope on the pump performance curve of pump-turbines suggests potential operational instabilities in pump mode. Previous research has indicated that the increase of the hydraulic loss caused by sudden changes of flow patterns in pump-turbines is responsible for the positive slope, however its detailed flow mechanism is still unclear. A low specific speed model pump-turbine was numerically investigated against experiments in the present study, by applying unsteady RANS (Reynolds-Averaged Navier–Stokes equations) simulations with a v2-f turbulence model. The mechanism of occurrence of the positive slope on the pump performance curve was discussed regarding the energy balance, as this region appears when the value of ∂Pu ∂Q is larger than the critical value Pu Q . An unsteady local loss analysis, derived from the energy equation, was conducted to illustrate the contribution of local flow patterns to the loss in corresponding hydraulic components. The variation of the kinetic energy of the mean flow was taken into account for the first time so that this method can be applied to highly time dependent flow patterns, e.g., a rotating stall in the present study. The investigations on the flow patterns revealed that some guide vane channels stalled with a larger discharge coefficient than the positive slope region. Several guide vane channels near the stalled channels were stalling with minor decrease of the discharge coefficient, leading to sudden increases of the input power and the loss. When the discharge coefficient slightly decreased in further, the pump-turbine operated into the positive slope region, and the rotating stall with 3 stall cells appeared, proven by the FFT (Fast Fourier Transform) and cross-phase analysis on the pressure fluctuations.


Introduction
Pumped hydroelectric storage (PHS) is widely established for utility-scale electricity storage, utilizing relatively cheap electricity from the power grid during off-peak hours to move water from a lower reservoir to an upper one to store energy, and releasing the water from the upper reservoir to generate power at a higher price during periods of high electricity demand. This technology has been developed rapidly in the past several decades, to compensate the increasing application of the renewable energy, for its benefits of load-levelling, grid frequency regulation, and spinning reserves [1].
The earliest designs of PHS used separate pump and turbine units. Since the 1950s, the utilization of single reversible pump-turbines has become the dominant design for PHS, among which the Francis where H, H loss , ∆z, U, g, and Q represent the head of the pump-turbine, the head loss in pipes, the altitude difference between the upper and lower reservoirs, the velocity in the pipe, the gravitational acceleration and the flow rate respectively. The head loss is expressed with the flow rate Q, a loss coefficient K s and a reference section area A: A hydraulic inductance L h is introduced to express the inertia term in Equation (1): Therefore, the governing equation is simplified as Small perturbations of the flow rate (δQ) and the head (δH) have been introduced into Equation (4) to study the instability of the system, and Equation (4) becomes: Then the governing equation to the system can be obtained by ignoring the higher order terms from (Equation (4) to Equation (5)), as follows: Equation (6) can be written as: Since L h > 0, the stability condition derived from the governing equation implies: Since the value of K s Q A 2 is always positive, the occurrence of positive slopes on pump performance curves is an insufficient but necessary condition of instability.
While positive slopes on pump performance curves are often detected in pump-turbines, the theoretical pump performance curve should be monotonically decreasing according to the conservation of angular momentum with the blade-congruent flow assumption, i.e., an infinite number of congruent blades without thickness, hence all the fluid flows along the blade. The measured head is less than the theoretical head due to a number of factors, e.g., the viscosity of fluid, turbulence, and the vortex. The difference between them is defined as "hydraulic loss". The positive slope is often detected in fact due to a sudden increase of hydraulic loss in partload condition [5], as shown in Figure 2. The flow mechanisms of this phenomenon in a pumping system, including the pump-turbine, are extraordinary complex. Gülich [5] summarized them as a "sudden change" of flow patterns at the entrance or exit of the impeller, leading to the sudden increase of the hydraulic loss. In pump-turbines diverse flow pattern changes, including rotating stall [6,7], flow separation shift [8,9] and pre-rotation of inflow [6], have been reported when operating in the positive slope region. Thus, the flow pattern in the pump-turbine working in this region may be highly turbulent, unsteady and swirling.
Although various experimental studies including PIV (Particle Image Velocimetry) measurements [10], and bubble injections [11] were conducted for the purpose of studying the flow patterns in the pump-turbine, the descriptions of these flow patterns were still limited. Numerical simulation is thereby a major tool, which is less expensive, more flexible and leads to a more detailed database of the flow field. Unsteady simulation is necessary to predict flow patterns in the pump-turbine operating in the positive slope region, considering that these flow patterns are highly time-dependent (e.g., rotating stall). While LES (Large Eddy Simulation) is better to predict the flow field in theory, the corresponding requirements on the spatial discretization are too high to be acceptable in most engineering cases [7]. Hence RANS (Reynold Averaged Navier-Stokes) simulations with linear turbulence models were widely applied to study the flows in pump-turbines [12][13][14]. The results proved its capability to predict the performances, pressure fluctuations and flow patterns at design conditions, while the precision at off-design conditions still need improvement. The possible explanations for this inaccuracy of simulation at off-design conditions may lie in the unsteadiness and turbulent flow structures, which may be damped out in the RANS simulations with linear turbulence models [15][16][17]. In order to avoid this disadvantage, some research applied v 2 -f, a non-linear turbulence model, to simulate the flow field in the pump-turbine at pump mode [12,13]. Not only the performance, the pressure fluctuations and the flow patterns were also well predicted.
As discussed above, the hydraulic loss represents the difference between the measured head and the theoretical head, indicating the difference between the real flow and the ideal flow with the blade-congruent flow assumption. Although it is generally accepted that the sudden change of flow patterns leading to a sharp increase of the hydraulic loss is responsible to the positive slope [1], its detailed flow mechanism is still unclear, since it is difficult to quantitatively measure the difference described above, and to discuss the relationship between this difference and the hydraulic loss. Instead, it is easier to discuss the relationship between the flow patterns and the energy through the energy equation. For the purpose of a better understanding of the flow phenomena related to the positive slope, unsteady RANS simulations with v 2 -f turbulence model were conducted to adequately predict the flow patterns in a low-specific speed pump-turbine operating at partload conditions, near the positive slope region. The energy balance analysis was applied to discuss the mechanism of the occurrence of the positive slope. The unsteady local loss analysis based on the energy equation was also conducted to study the contribution of local flow to the loss. positive slope, unsteady RANS simulations with v 2 -f turbulence model were conducted to adequately predict the flow patterns in a low-specific speed pump-turbine operating at partload conditions, near the positive slope region. The energy balance analysis was applied to discuss the mechanism of the occurrence of the positive slope. The unsteady local loss analysis based on the energy equation was also conducted to study the contribution of local flow to the loss.

Methodology
In the present study, a low specific speed ( , where n, Q represent the rotating speed and the flow rate respectively) model pump-turbine was investigated. The geometric parameters of the pump-turbine are listed in Table 1. The pump performance curves of this pumpturbine at various guide vane openings (GVO) were experimentally examined on the DF150 Hydraulic Test Rig of Dongfang Electric Corporation (DEC), China. The details of the related experiments were stated in Ref [18]. The geometric parameters used in the numerical simulations described in the next Sections are selected according to the experimental parameters. Figure 3 demonstrates the experimental results of pump performance curves, where the relative GVO α is defined as the ratio of the individual GVO over the optimum GVO (18).   positive slope, unsteady RANS simulations with v 2 -f turbulence model were conducted to adequately predict the flow patterns in a low-specific speed pump-turbine operating at partload conditions, near the positive slope region. The energy balance analysis was applied to discuss the mechanism of the occurrence of the positive slope. The unsteady local loss analysis based on the energy equation was also conducted to study the contribution of local flow to the loss.

Methodology
In the present study, a low specific speed ( , where n, Q represent the rotating speed and the flow rate respectively) model pump-turbine was investigated. The geometric parameters of the pump-turbine are listed in Table 1. The pump performance curves of this pumpturbine at various guide vane openings (GVO) were experimentally examined on the DF150 Hydraulic Test Rig of Dongfang Electric Corporation (DEC), China. The details of the related experiments were stated in Ref [18]. The geometric parameters used in the numerical simulations described in the next Sections are selected according to the experimental parameters. Figure 3 demonstrates the experimental results of pump performance curves, where the relative GVO α is defined as the ratio of the individual GVO over the optimum GVO (18).

Methodology
In the present study, a low specific speed (n q = n √ Q H 0.75 = 27, where n, Q represent the rotating speed and the flow rate respectively) model pump-turbine was investigated. The geometric parameters of the pump-turbine are listed in Table 1. The pump performance curves of this pump-turbine at various guide vane openings (GVO) were experimentally examined on the DF150 Hydraulic Test Rig of Dongfang Electric Corporation (DEC), China. The details of the related experiments were stated in Ref [18]. The geometric parameters used in the numerical simulations described in the next Sections are selected according to the experimental parameters. Figure 3 demonstrates the experimental results of pump performance curves, where the relative GVO α is defined as the ratio of the individual GVO over the optimum GVO (18). ϕ = Q nD 3 and ψ = gH n 2 D 2 represent the discharge coefficient and the energy coefficient respectively, where n and D respectively represent the rotational speed and the diameter of the impeller. Detailed examination of each curve shows that the positive slope region exists for each GVO curve except at the optimum GVO.   The computational domain of the pump-turbine built for numerical simulations is shown in Figure 4, including its major hydraulic components, i.e., the draft tube, the impeller, the guide vanes and stay vanes, and the spiral casing. Velocity flow inlet and static pressure outlet were chosen as the boundary conditions. Unsteady numerical simulations based on RANS method were applied in the present study. The time taken by the impeller to rotate 2̊ was determined as the computational time step size. The Courant number is less than 1, leading to convergent simulations. The calculation load is also acceptable at the same time. The mesh in the impeller was set as "grid motion" with impeller rotational speed (1,100 r/min), and "no-slip" conditions were imposed on all stationary walls. The "sliding interface" was set between the rotating and stationary components to transmit the data. For the purpose of measuring the pressure fluctuations, static pressure monitoring points were positioned near the upper wall of the draft tube, the vaneless space (the gap between the impeller and the guide vane), the guide vane and the stay vanes channels in the simulation ( Figure 5). The pressure fluctuations at the locations marked in Figure 5 were also measured in the experiments. In addition, the middle spanwise hub-to-shroud surface and the guide vane symmetry plane were set in the pump-turbine to discuss the flow patterns in the impeller and the guide vane channels, respectively ( Figure 6).
A non-linear turbulence mode, i.e., v2-f turbulence model, is applied in the present study, in which no wall function is utilized. In this turbulence model, two more equations, namely the transport equation for 2 v , , and the elliptic equation for the relaxation function f , ( ) , where the turbulence time and length scales are respectively defined as max , 6 The computational domain of the pump-turbine built for numerical simulations is shown in Figure 4, including its major hydraulic components, i.e., the draft tube, the impeller, the guide vanes and stay vanes, and the spiral casing. Velocity flow inlet and static pressure outlet were chosen as the boundary conditions. Unsteady numerical simulations based on RANS method were applied in the present study. The time taken by the impeller to rotate 2 was determined as the computational time step size. The Courant number is less than 1, leading to convergent simulations. The calculation load is also acceptable at the same time. The mesh in the impeller was set as "grid motion" with impeller rotational speed (1100 r/min), and "no-slip" conditions were imposed on all stationary walls. The "sliding interface" was set between the rotating and stationary components to transmit the data. For the purpose of measuring the pressure fluctuations, static pressure monitoring points were positioned near the upper wall of the draft tube, the vaneless space (the gap between the impeller and the guide vane), the guide vane and the stay vanes channels in the simulation ( Figure 5). The pressure fluctuations at the locations marked in Figure 5 were also measured in the experiments. In addition, the middle spanwise hub-to-shroud surface and the guide vane symmetry plane were set in the pump-turbine to discuss the flow patterns in the impeller and the guide vane channels, respectively ( Figure 6).
A non-linear turbulence mode, i.e., v2-f turbulence model, is applied in the present study, in which no wall function is utilized. In this turbulence model, two more equations, namely where the turbulence time and length scales are [19,20]. The turbulence viscosity is defined as µ t = ρC u v 2 T. The coefficients are respectively selected as α = 0.6, C 1 = 1.4, C 2 = 0.3, C ε1 = 1.4, C ε2 = 1.9, C η = 70, C µ = 0.22, C L = 0.23, σ k = 1, σ ε = 1.3, and C ε1 = C ε1 1 + 0.045 k/v 2 . Although this model was originally developed for attached or mildly separated boundary layers, it could also accurately simulate flows dominated by separation [19,20]. Semi-Implicit Method for Pressure-Linked Equations Consistent (SIMPLEC) algorithm with a second-order spatial discretization and a first-order implicit transient formulation was chosen as the pressure-velocity coupling method, which is widely applied in the simulations of flows in pump-turbines.
A structured computational grid system for the whole hydrodynamic domain was developed in the commercial software the Integrated Computer Engineering and Manufacturing code for Computational Fluid Dynamics (ICEM-CFD). Local grid refinements to the boundary layers were applied in the impeller, guide vane and stay vane regions to achieve y + ≈ 1 for the first grid layer, considering that no wall function is utilized.
To determine the number of the grids for the whole computational domain, three grid systems with different number of grids were respectively built. Then η h (the hydraulic efficiency) and ψ at the Best Efficiency Point (BEP) of the optimum GVO in the three grid systems were computed, as listed in Table 2. The relative variation of both η h and ψ is larger than 5% when the number of the grid increased from grid system 1 to 2, while it is smaller from 2 to 3. Therefore, the grid system 3 with 17,000,000 number of grid is acceptable for the investigations in the present study.
Energies 2018, 11, x FOR PEER REVIEW 6 of 22 19,20]. The turbulence viscosity is defined as The coefficients are respectively selected as Although this model was originally developed for attached or mildly separated boundary layers, it could also accurately simulate flows dominated by separation [19,20]. Semi-Implicit Method for Pressure-Linked Equations Consistent (SIMPLEC) algorithm with a second-order spatial discretization and a first-order implicit transient formulation was chosen as the pressure-velocity coupling method, which is widely applied in the simulations of flows in pump-turbines.
A structured computational grid system for the whole hydrodynamic domain was developed in the commercial software the Integrated Computer Engineering and Manufacturing code for Computational Fluid Dynamics (ICEM-CFD). Local grid refinements to the boundary layers were applied in the impeller, guide vane and stay vane regions to achieve y + ≈1 for the first grid layer, considering that no wall function is utilized.
To determine the number of the grids for the whole computational domain, three grid systems with different number of grids were respectively built. Then ηh (the hydraulic efficiency) and ψ at the Best Efficiency Point (BEP) of the optimum GVO in the three grid systems were computed, as listed in Table 2. The relative variation of both ηh and ψ is larger than 5% when the number of the grid increased from grid system 1 to 2, while it is smaller from 2 to 3. Therefore, the grid system 3 with 17,000,000 number of grid is acceptable for the investigations in the present study.

Validation of the simulations
The pump performance curves of the pump-turbine at three various GVOs (α=45%, 75% and 145%) are well predicted, as shown in Figure 7, including the occurrence of the positive slope at all GVOs. The calculated fluctuations of the pressure coefficient (

Validation of the Simulations
The pump performance curves of the pump-turbine at three various GVOs (α = 45%, 75% and 145%) are well predicted, as shown in Figure 7, including the occurrence of the positive slope at all GVOs. The calculated fluctuations of the pressure coefficient (C p = p 1 2 ρu 2 2 ,where u 2 represents the relative velocity at impeller exit) at vaneless space (gv1), and the FFT (Fast Fourier Transform) results at working conditions I (ϕ = 0.294) and II (ϕ = 0.227) are illustrated in Figures 8 and 9, respectively. Pressure fluctuation frequencies were normalized against the rotational frequency f n . It can be seen that the pressure fluctuations at the blade passing frequency (BPF) and its harmonic were well predicted in the simulations. In addition, a pressure fluctuation component with a low frequency was detected in both experiments (8% f n = 1.46 Hz) and simulations (9% f n = 1.65 Hz) at condition II.

Pump performance and head loss
The pump performance curve and head loss coefficient (

Pump Performance and Head Loss
The pump performance curve and head loss coefficient (ψ L = gH L n 2 D 2 , where H L = H inlet − H outlet is the head loss in the corresponding hydraulic component. H inlet and H outlet are the head at the inlet and the outlet of this component, respectively) at α = 75% are illustrated in Figure 10. Three working conditions near the positive slope are defined as III, IV, and V from large to small discharge coefficient. It should be mentioned that condition V is the left bound of the positive slope region. The head loss coefficient in the impeller and guide vane channels are larger than in the other hydraulic components for all conditions. The total head loss coefficient increases gradually with decreasing discharge coefficient, while a sudden increase is detected when the discharge coefficient decreases from III to IV.

Conditions III and IV
The FFT results of the pressure coefficient at conditions III and IV (out of the positive slope region) are shown in Figure 11. The dominant frequency of the fluctuations is the blade passing frequency (f1=7fn=128Hz). The amplitude of this fluctuation component is strongest near the entrance of the guide vane channel, matching the conclusion in the previous study [21,22].

Condition V
The FFT results of pressure coefficient at condition V are shown in Figure 12. A low-frequency pressure fluctuation signal (f2=9%fn=1.65Hz) is detected and becomes the dominant frequency fluctuation, instead of the fluctuation at blade passing frequency. This low-frequency fluctuation, which is stronger at the vaneless space than that at other locations, may be caused by the rotating stall according to previous studies [1,13].
In the case of rotating stalls, the number of the stall cells could be determined with the analysis on the cross-phase delay of monitoring points at deferent circumferential positions, according to:

Conditions III and IV
The FFT results of the pressure coefficient at conditions III and IV (out of the positive slope region) are shown in Figure 11. The dominant frequency of the fluctuations is the blade passing frequency (f 1 = 7f n = 128 Hz). The amplitude of this fluctuation component is strongest near the entrance of the guide vane channel, matching the conclusion in the previous study [21,22].

Conditions III and IV
The FFT results of the pressure coefficient at conditions III and IV (out of the positive slope region) are shown in Figure 11. The dominant frequency of the fluctuations is the blade passing frequency (f1=7fn=128Hz). The amplitude of this fluctuation component is strongest near the entrance of the guide vane channel, matching the conclusion in the previous study [21,22].

Condition V
The FFT results of pressure coefficient at condition V are shown in Figure 12. A low-frequency pressure fluctuation signal (f2=9%fn=1.65Hz) is detected and becomes the dominant frequency fluctuation, instead of the fluctuation at blade passing frequency. This low-frequency fluctuation, which is stronger at the vaneless space than that at other locations, may be caused by the rotating stall according to previous studies [1,13].
In the case of rotating stalls, the number of the stall cells could be determined with the analysis on the cross-phase delay of monitoring points at deferent circumferential positions, according to: IV III V

Condition V
The FFT results of pressure coefficient at condition V are shown in Figure 12. A low-frequency pressure fluctuation signal (f 2 = 9% f n = 1.65 Hz) is detected and becomes the dominant frequency fluctuation, instead of the fluctuation at blade passing frequency. This low-frequency fluctuation, which is stronger at the vaneless space than that at other locations, may be caused by the rotating stall according to previous studies [1,13].
In the case of rotating stalls, the number of the stall cells could be determined with the analysis on the cross-phase delay of monitoring points at deferent circumferential positions, according to: where N is the number of the stall cells. θ represents the separation angle between two monitoring points at deferent circumferential positions, while θ RS represents the cross-phase delay of the low-frequency fluctuation caused by the rotating stall between the two points. The cross-phase delays of the low-frequency fluctuation (1.65 Hz) between several couples of monitoring points were illustrated in Table 3. It could be concluded that three rotating cells exist since the separation angles of the three couples of monitoring points are 90, while the cross-phase delays are about 270 = 90 × 3. The cross-phase delays of the three couples of points vary slightly, suggesting that the rotational speed of stall cells is not constant at different circumferential positions.   Table 3. Cross-phase delays between several couples of monitoring points at condition V.

Energy balance analysis
The mechanical energy transfers to the fluid in pump-turbines through the rotation of the impeller. Part of it becomes the mechanical energy of the fluid, while the rest is converted to the internal energy and turbulent kinetic energy due to the viscosity and turbulence. The mechanical power transferred to the fluid through the impeller is defined as the input power in Mz is the axial moment acting on the impeller and ω is the rotational speed of the impeller. The mechanical power obtained by the fluid is defined as the useful power u P gHQ ρ = .
To satisfy the energy balance, the difference of Pin and Pu is defined as the loss L i n u P P P = − .
The head H can be regarded as a function of the flow rate Q and it can be derived from the definition of the useful power that: Regarding Q＞0, it can be concluded that the necessary and sufficient condition of the occurrence of the positive slope is that the value of u P Q ∂ ∂ is larger than the critical value u P Q . When reducing the flow rate, the decrease of the input power and the increase of the torque can result in the decrease of the useful power, which may lead to the positive slope in the pump performance curve.
For the simulating results in the present study, the input power coefficient (

Energy Balance Analysis
The mechanical energy transfers to the fluid in pump-turbines through the rotation of the impeller. Part of it becomes the mechanical energy of the fluid, while the rest is converted to the internal energy and turbulent kinetic energy due to the viscosity and turbulence. The mechanical power transferred to the fluid through the impeller is defined as the input power P in = M z ω, where M z is the axial moment acting on the impeller and ω is the rotational speed of the impeller. The mechanical power obtained by the fluid is defined as the useful power P u = ρgHQ. To satisfy the energy balance, the difference of P in and P u is defined as the loss P L = P in − P u .
The head H can be regarded as a function of the flow rate Q and it can be derived from the definition of the useful power that: Regarding Q > 0, it can be concluded that the necessary and sufficient condition of the occurrence of the positive slope is that the value of ∂P u ∂Q is larger than the critical value P u Q . When reducing the flow rate, the decrease of the input power and the increase of the torque can result in the decrease of the useful power, which may lead to the positive slope in the pump performance curve. For the simulating results in the present study, the input power coefficient (λ = P in n 3 D 5 ), the useful power coefficient (λ u = P u n 3 D 5 ,) and the loss coefficient in each hydraulic component (λ L = P L n 3 D 5 ) are shown in Figure 13. λ in and λ u increase gradually with increasing discharge coefficient, with an exception of λ in at the condition III to IV. A sudden decrease of λ L at condition III to IV was also detected. λ L in the impeller and the guide vane channels are larger than that in the other hydraulic components.
The values of ∂λ u ∂ϕ and λ u ϕ at each condition are calculated as shown in Figure 14, to validate the theoretical conclusion of stability criterion above. Since only finite number of conditions can be calculated in the simulations, the value of ∂λ u ∂ϕ was computed according to the following equation: where the subscript i indicates the ith condition, while i+1 means the condition next to i with a larger discharge coefficient. Only at condition V, where the positive slope on pump performance curve was detected, ∂λ u ∂ϕ is larger than λ u ϕ , indicating that the value of ∂P u ∂Q is larger than the critical value P u Q . This result fits with the theoretical conclusion, suggesting that the substantial decrease of λ u from condition IV to V results in the positive slope in the pump performance curve. A further investigation shows that for the decrease of λ u from condition IV to V, about 80% is attributed to the decrease of the input power coefficient while the remaining 20% is caused by the increase of the loss coefficient.
For a further understanding of the sudden variation of P u , the mechanism of the decrease of the input power and the increase of the loss are discussed in Sections 4 and 5 respectively.
where the subscript i indicates the ith condition, while i+1 means the condition next to i with a larger discharge coefficient. Only at condition V, where the positive slope on pump performance curve was detected, u λ ϕ ∂ ∂ is larger than u λ ϕ , indicating that the value of u P Q ∂ ∂ is larger than the critical value u P Q . This result fits with the theoretical conclusion, suggesting that the substantial decrease of λu from condition IV to V results in the positive slope in the pump performance curve. A further investigation shows that for the decrease of λu from condition IV to V, about 80% is attributed to the decrease of the input power coefficient while the remaining 20% is caused by the increase of the loss coefficient.
For a further understanding of the sudden variation of Pu, the mechanism of the decrease of the input power and the increase of the loss are discussed in Sections 4 and 5 respectively.

Analysis of the input power
The input power Pin is equal to the product of the axial moment acting on the impeller Mz and the rotational speed ω. Since the pump-turbine is operating at a constant rotational speed in the present study, the increase of the input power indicates the increase of the axial moment, which is generated by the pressure and shear force distributed on the impeller. The two effects of pressure and shear force on the moment on the impeller were calculated, respectively. The pressure effect where the subscript i indicates the ith condition, while i+1 means the condition next to i with a larger discharge coefficient. Only at condition V, where the positive slope on pump performance curve was detected, u λ ϕ ∂ ∂ is larger than u λ ϕ , indicating that the value of u P Q ∂ ∂ is larger than the critical value u P Q . This result fits with the theoretical conclusion, suggesting that the substantial decrease of λu from condition IV to V results in the positive slope in the pump performance curve. A further investigation shows that for the decrease of λu from condition IV to V, about 80% is attributed to the decrease of the input power coefficient while the remaining 20% is caused by the increase of the loss coefficient.
For a further understanding of the sudden variation of Pu, the mechanism of the decrease of the input power and the increase of the loss are discussed in Sections 4 and 5 respectively.

Analysis of the input power
The input power Pin is equal to the product of the axial moment acting on the impeller Mz and the rotational speed ω. Since the pump-turbine is operating at a constant rotational speed in the present study, the increase of the input power indicates the increase of the axial moment, which is generated by the pressure and shear force distributed on the impeller. The two effects of pressure and shear force on the moment on the impeller were calculated, respectively. The pressure effect IV III V Figure 14. ∂λ u ∂ϕ and λ u ϕ in every condition.

Analysis of the Input Power
The input power P in is equal to the product of the axial moment acting on the impeller M z and the rotational speed ω. Since the pump-turbine is operating at a constant rotational speed in the present study, the increase of the input power indicates the increase of the axial moment, which is generated by the pressure and shear force distributed on the impeller. The two effects of pressure and shear force on the moment on the impeller were calculated, respectively. The pressure effect contributes over 98% to the moment. Therefore, the pressure distribution on the impeller was analyzed to explain the sudden increase of the input power coefficient from condition III to IV. Figure 15 shows the intersections of every blade and the middle spanwise hub-to-shroud surface. The lines on the pressure sides and the suction sides of blades are respectively marked. Then the time-averaged pressure distributions on the lines on the both sides were respectively averaged, and compared between conditions III and IV. The increase of the pressure on the pressure side near the exit of the impeller from condition III to IV leads to the variation of M z . This is responsible for the sudden change of the input power.
For a further understanding of this variation, the instantaneous pressure distributions on the pressure side of the impeller for both conditions are shown in Figure 16. It could be concluded that the region with high pressure near the exit of the impeller extended to the neighbor blade from condition III to IV, resulting in the increase of blade loading on the pressure side near the exit of the impeller. The instantaneous streamlines in the guide vane channels and pressure distributions on the pressure side of impeller are shown in Figure 17. It could be concluded that the stalled guide vane channels are responsible for the high-pressure regions on the blade. Several guide vane channels are stalling from condition III to IV, leading to the extension of the high-pressure region on the impeller. contributes over 98% to the moment. Therefore, the pressure distribution on the impeller was analyzed to explain the sudden increase of the input power coefficient from condition III to IV. Figure 15 shows the intersections of every blade and the middle spanwise hub-to-shroud surface. The lines on the pressure sides and the suction sides of blades are respectively marked. Then the time-averaged pressure distributions on the lines on the both sides were respectively averaged, and compared between conditions III and IV. The increase of the pressure on the pressure side near the exit of the impeller from condition III to IV leads to the variation of Mz. This is responsible for the sudden change of the input power.
For a further understanding of this variation, the instantaneous pressure distributions on the pressure side of the impeller for both conditions are shown in Figure 16. It could be concluded that the region with high pressure near the exit of the impeller extended to the neighbor blade from condition III to IV, resulting in the increase of blade loading on the pressure side near the exit of the impeller. The instantaneous streamlines in the guide vane channels and pressure distributions on the pressure side of impeller are shown in Figure 17. It could be concluded that the stalled guide vane channels are responsible for the high-pressure regions on the blade. Several guide vane channels are stalling from condition III to IV, leading to the extension of the high-pressure region on the impeller.  contributes over 98% to the moment. Therefore, the pressure distribution on the impeller was analyzed to explain the sudden increase of the input power coefficient from condition III to IV. Figure 15 shows the intersections of every blade and the middle spanwise hub-to-shroud surface. The lines on the pressure sides and the suction sides of blades are respectively marked. Then the time-averaged pressure distributions on the lines on the both sides were respectively averaged, and compared between conditions III and IV. The increase of the pressure on the pressure side near the exit of the impeller from condition III to IV leads to the variation of Mz. This is responsible for the sudden change of the input power.
For a further understanding of this variation, the instantaneous pressure distributions on the pressure side of the impeller for both conditions are shown in Figure 16. It could be concluded that the region with high pressure near the exit of the impeller extended to the neighbor blade from condition III to IV, resulting in the increase of blade loading on the pressure side near the exit of the impeller. The instantaneous streamlines in the guide vane channels and pressure distributions on the pressure side of impeller are shown in Figure 17. It could be concluded that the stalled guide vane channels are responsible for the high-pressure regions on the blade. Several guide vane channels are stalling from condition III to IV, leading to the extension of the high-pressure region on the impeller.

Analysis of the loss
To better understand the relationship between the flow patterns and the loss in pump-turbines, a local loss analysis based on the energy equation was conducted at conditions III, IV, and V, to study the flow mechanism of the positive slope. Compared with previous studies [23][24][25], the variation of the kinetic energy of the mean flow is taken into account for the first time so that this local loss analysis could be applied to study the highly time dependent flow patterns. This is important for the discussion in the present study because of the existence of rotating stall, a typical highly time dependent flow pattern.

Local loss analysis method
For an incompressible turbulent flow without heat transfer or temperature variation, the energy equation in stationary hydraulic components (including the draft tube, guide vane channels, stay vane channels and the spiral casing) can be simplified as follows:

Analysis of the Loss
To better understand the relationship between the flow patterns and the loss in pump-turbines, a local loss analysis based on the energy equation was conducted at conditions III, IV, and V, to study the flow mechanism of the positive slope. Compared with previous studies [23][24][25], the variation of the kinetic energy of the mean flow is taken into account for the first time so that this local loss analysis could be applied to study the highly time dependent flow patterns. This is important for the discussion in the present study because of the existence of rotating stall, a typical highly time dependent flow pattern.

Local Loss Analysis Method
For an incompressible turbulent flow without heat transfer or temperature variation, the energy equation in stationary hydraulic components (including the draft tube, guide vane channels, stay vane channels and the spiral casing) can be simplified as follows: ∂ ∂t where D ij = ∂u i ∂x j + ∂u j ∂x i , µ represents the kinematic viscosity, a and a are the time averaged value and the fluctuation of a physical quantity a, respectively. Then each term in Equation (12) is integrated over the total volume V of a hydraulic component, and Equation (12) becomes: In order to analyze the flow in the impeller, the input power should be taken into account. A source term is added to the equation as follow: By comparing the definition of P L and Equation (14), the loss in every hydraulic component can be expressed as follows: Thus, the loss in every hydraulic component is determined by five terms. Term 1 represents variation of the kinetic energy of the mean flow. Terms 2 and 3 correspond to the diffusion of the kinetic energy of the mean flow. Term 4 is the turbulent kinetic energy production, which is responsible for the transfer of the kinetic energy of the mean flow to the turbulent kinetic energy. Term 5 corresponds to the viscous dissipation.
Since RANS simulations were conducted in the present study, it should be noted that the Reynold stress ρu i u was calculated as follows: where µ t is the eddy viscosity, k represents the turbulent kinetic energy and δ ij is Kronecker delta, being equal to 1 if i = j and 0 otherwise.

Physical Phenomena Responsible for the Loss
According to the analysis in the previous discussion, the loss includes four parts: the variation of the kinetic energy of the mean flow (Term 1), the kinetic energy of the mean flow diffusion (Term 2 and 3), the turbulent kinetic energy production (Term 4), and the viscous dissipation (Term 5). Since the rotating stall is a highly time dependent flow pattern, and do not exist in conditions III and IV, it is not necessary to discuss the contribution of the variation of the kinetic energy of the mean flow at the two conditions. The contribution of each part to the loss will be investigated and the relationship with the flow patterns will be discussed. Figure 18 illustrates the compositions of the loss coefficient at conditions III and IV. The draft tube is not taken into account, since the loss in that component is low compared to the others (shown in Figure 13). It could be concluded that the turbulent kinetic energy production is much larger than the other two terms, suggesting that most loss of the energy converts to the turbulent kinetic energy. This result is supported by the fact that the flow in pump-turbines is highly turbulent. Regarding the larger losses in the impeller and guide vane channels than in the other hydraulic components, the distributions of the turbulent kinetic energy production in these two components were specifically discussed.

Condition III and IV
The instantaneous distributions of the turbulent kinetic energy dissipation in the impeller are shown in Figure 19, where the streamlines in guide vane channels are also shown. Here P t = (−ρu i u j ) ∂u i ∂x j indicates the turbulent kinetic energy production. The flow downstream the trailing edge of the blades corresponds to a high turbulent kinetic energy production since the flow near the pressure side and suction side of the blades, with different flow velocity, mix in this region. The P t near the blade upstream the stalled guide vane channels is higher than the P t near the blade upstream the unstalled guide vane channels. Figures 20 and 21 show the instantaneous distributions of P t and streamline in guide vane channels on the symmetry plane at conditions III and IV, respectively. The P t in the stalled guide vane channels is higher than in the unstalled channels. The regions between straight flows (with high flow velocity), and separated flows (with low flow velocity), correspond to high P t . The strong velocity gradient in these regions may be responsible for the large P t . is not necessary to discuss the contribution of the variation of the kinetic energy of the mean flow at the two conditions. The contribution of each part to the loss will be investigated and the relationship with the flow patterns will be discussed.
Condition III and IV Figure 18 illustrates the compositions of the loss coefficient at conditions III and IV. The draft tube is not taken into account, since the loss in that component is low compared to the others (shown in Figure 13). It could be concluded that the turbulent kinetic energy production is much larger than the other two terms, suggesting that most loss of the energy converts to the turbulent kinetic energy. This result is supported by the fact that the flow in pump-turbines is highly turbulent. Regarding the larger losses in the impeller and guide vane channels than in the other hydraulic components, the distributions of the turbulent kinetic energy production in these two components were specifically discussed.
The instantaneous distributions of the turbulent kinetic energy dissipation in the impeller are shown in Figure 19, where the streamlines in guide vane channels are also shown. Here  is not necessary to discuss the contribution of the variation of the kinetic energy of the mean flow at the two conditions. The contribution of each part to the loss will be investigated and the relationship with the flow patterns will be discussed.
Condition III and IV Figure 18 illustrates the compositions of the loss coefficient at conditions III and IV. The draft tube is not taken into account, since the loss in that component is low compared to the others (shown in Figure 13). It could be concluded that the turbulent kinetic energy production is much larger than the other two terms, suggesting that most loss of the energy converts to the turbulent kinetic energy. This result is supported by the fact that the flow in pump-turbines is highly turbulent. Regarding the larger losses in the impeller and guide vane channels than in the other hydraulic components, the distributions of the turbulent kinetic energy production in these two components were specifically discussed.
The instantaneous distributions of the turbulent kinetic energy dissipation in the impeller are shown in Figure 19, where the streamlines in guide vane channels are also shown. Here It was discussed in the previous section that there are several stalled guide vane channels at conditions III while the other several guide vane channels are stalling from condition III to IV. The P t in these channels are higher than that in the other channels, which may be responsible for the sudden increase of the loss from condition III to IV, as shown in Figure 13. It was discussed in the previous section that there are several stalled guide vane channels at conditions III while the other several guide vane channels are stalling from condition III to IV. The Pt Figure 13.

Condition V
The analysis of the pressure fluctuations at condition V reveals the existence of rotating stall, suggesting that the flow pattern at this condition is highly time dependent, and that the contribution of the variation of the kinetic energy of the mean flow should be taken into account. Figure 22 illustrates the compositions of the loss coefficient at conditions V. The contribution of the variation of mean kinetic to the loss is insignificant so the conclusion is the same: most lost is converted to the turbulent kinetic energy.
The relationship between the flow patterns and the Pt distribution should be discussed at several time instances. Four various time instances are determined as shown in Figure 23. The instantaneous distributions of streamlines, pressure and Pt in guide vane channels at these time instances are illustrated in Figure 24. The occurrence of the flow separation in Channel 1 downstream the monitoring point gv1 is detected at time 'a', where the pressure at gv1 is increasing while gv6 is decreasing, suggesting that the flow in Channel 1 is stalling. A high-pressure region in the channel neighboring the fully stalled channel is detected. The region with high Pt is located between the straight flow and the separated flow. Then the flow in Channel 1 becomes fully stalled at time 'b' and the high-pressure region circumferentially moves to Channel 1. The Pt in the region with separated flow is still high, but lower than that at time 'a' since no straight flow exists in this channel. At time 'c', Channel 1 is recovering where the pressure at gv1 is decreasing while it is increasing at gv6. The existence of both straight flow and separated flow in Channel 1 at this time instance leads to a high It was discussed in the previous section that there are several stalled guide vane channels at conditions III while the other several guide vane channels are stalling from condition III to IV. The Pt Figure 13.

Condition V
The analysis of the pressure fluctuations at condition V reveals the existence of rotating stall, suggesting that the flow pattern at this condition is highly time dependent, and that the contribution of the variation of the kinetic energy of the mean flow should be taken into account. Figure 22 illustrates the compositions of the loss coefficient at conditions V. The contribution of the variation of mean kinetic to the loss is insignificant so the conclusion is the same: most lost is converted to the turbulent kinetic energy.
The relationship between the flow patterns and the Pt distribution should be discussed at several time instances. Four various time instances are determined as shown in Figure 23. The instantaneous distributions of streamlines, pressure and Pt in guide vane channels at these time instances are illustrated in Figure 24. The occurrence of the flow separation in Channel 1 downstream the monitoring point gv1 is detected at time 'a', where the pressure at gv1 is increasing while gv6 is decreasing, suggesting that the flow in Channel 1 is stalling. A high-pressure region in the channel neighboring the fully stalled channel is detected. The region with high Pt is located between the straight flow and the separated flow. Then the flow in Channel 1 becomes fully stalled at time 'b' and the high-pressure region circumferentially moves to Channel 1. The Pt in the region with separated flow is still high, but lower than that at time 'a' since no straight flow exists in this channel. At time 'c', Channel 1 is recovering where the pressure at gv1 is decreasing while it is increasing at gv6. The existence of both straight flow and separated flow in Channel 1 at this time instance leads to a high

Condition V
The analysis of the pressure fluctuations at condition V reveals the existence of rotating stall, suggesting that the flow pattern at this condition is highly time dependent, and that the contribution of the variation of the kinetic energy of the mean flow should be taken into account. Figure 22 illustrates the compositions of the loss coefficient at conditions V. The contribution of the variation of mean kinetic to the loss is insignificant so the conclusion is the same: most lost is converted to the turbulent kinetic energy.
The relationship between the flow patterns and the P t distribution should be discussed at several time instances. Four various time instances are determined as shown in Figure 23. The instantaneous distributions of streamlines, pressure and P t in guide vane channels at these time instances are illustrated in Figure 24. The occurrence of the flow separation in Channel 1 downstream the monitoring point gv1 is detected at time 'a', where the pressure at gv1 is increasing while gv6 is decreasing, suggesting that the flow in Channel 1 is stalling. A high-pressure region in the channel neighboring the fully stalled channel is detected. The region with high P t is located between the straight flow and the separated flow. Then the flow in Channel 1 becomes fully stalled at time 'b' and the high-pressure region circumferentially moves to Channel 1. The P t in the region with separated flow is still high, but lower than that at time 'a' since no straight flow exists in this channel. At time 'c', Channel 1 is recovering where the pressure at gv1 is decreasing while it is increasing at gv6. The existence of both straight flow and separated flow in Channel 1 at this time instance leads to a high P t region. The flow in Channel 1 recovers to straight flow at time 'd' and only a minor separated region near the trailing edge of the guide vane is detected.
The instantaneous distributions of streamlines, pressure and P t in the impeller are illustrated in Figure 25. The existence of a rotating stall at condition V indicates that the flow pattern leading to the stalled guide vane channels moves circumferentially. The P t near the blade upstream the stalled guide vane channels is always higher than that upstream the unstalled guide vane channels. Pt region. The flow in Channel 1 recovers to straight flow at time 'd' and only a minor separated region near the trailing edge of the guide vane is detected. The instantaneous distributions of streamlines, pressure and Pt in the impeller are illustrated in Figure 25. The existence of a rotating stall at condition V indicates that the flow pattern leading to the stalled guide vane channels moves circumferentially. The Pt near the blade upstream the stalled guide vane channels is always higher than that upstream the unstalled guide vane channels. Pt region. The flow in Channel 1 recovers to straight flow at time 'd' and only a minor separated region near the trailing edge of the guide vane is detected. The instantaneous distributions of streamlines, pressure and Pt in the impeller are illustrated in Figure 25. The existence of a rotating stall at condition V indicates that the flow pattern leading to the stalled guide vane channels moves circumferentially. The Pt near the blade upstream the stalled guide vane channels is always higher than that upstream the unstalled guide vane channels.