47

ＯＴＥＣ
Vol. 25（2020），47 ～ 59

The Flow Characteristics Investigation of the S-Shaped
Region for Pump-Turbine in Small Guide Vane OpeningHaiyun QIU*1, Pengcheng GUO*1, Yoichi KINOUE*2, Longgang SUN*1*1 State Key Laboratory of Eco-hydraulics in Northwest Arid Region of China,
Xi’an University of Technology, Xi'an 7, 10048, China* Institute of Ocean Energy, Saga University, 840-8502, Japan
2

Abstract
Pumped-storage power station is the most effective regulator in the current power grid. Its core components, pumpturbine, requires higher stability and flexibility. But the two unstable regions, which are the S-Shaped characteristic
region of the turbine mode and the hump area of the pump mode, become a bottleneck restricting the development ofpump turbine and its system. The two unstable regions are also one of the key issues affecting the safety of units andpower system. This paper takes a pump-turbine model of a pumped-storage power station as the research object, andconducts a numerical simulation study on the flow characteristics in the S-Shaped characteristic region of the small
guide vane opening. By analyzing the flow streamlines and velocity distributions of different span-wises in the runner,
it is found that when the pump turbine is operating at partial operating conditions, a vortex structure is formed on the
hub side of the runner, a water ring is formed on the middle span-wise, and the back flow forms on shroud side. Whenthe unstable flow structure appears in the runner, the high axial and circumferential velocity appears at the inlet of therunner and larger range of low speed areas appear in the runner passages. The inflow of the runner is obstructed. When
the water ring appears on the middle span-wise, the flow around the head of the runner is serious.
Keywords: Pumped-storage power station
structures

Pump-turbine

S-Shaped characteristics

Back flow vortex

Low frequency pressure pulsation

1. Introduction
Reversible pump-turbine units of pumped storage power stations have the advantages of rapid start-up and frequentconversion of operating conditions, can quickly respond to the needs of the power grid, and play an important role in
improving the stability of the power grid (ZUO Z G, 2016). However, as renewable energy sources such as solar andwind energy have entered the power grid in recent years, the stability of the unit has become increasingly prominent (LID Y, 2018). One of the reasons is that when the pump-turbine is operating in the turbine mode, the S-Shaped
characteristic region (SSCR) appears(ZENG W, 2016), which means that when the pump-turbine crosses the runaway
point and enters the braking zone, the slope of the characteristic curve is positive and multi-value feature is formed. This
multi-value feature can easily induce unstable problems such as start-up difficulty, large pressure fluctuation of load
rejection, and even self-excited oscillation during runaway process(XIA L S, 2017). Studies have shown that these
instability problems are related to unstable flow structures in the runner and guide vanes, such as backflow, unstablevortex structures, and rotational stall. Therefore, it is necessary to study the correlation between the flow structure andoperating stability in the SSCR.
In order to study the instability of pump turbine in SSCR, various reseachers have studied this problem by different
methods. Hasmatuchi(HASMATUCHI V,2009), and Guggenberger(GUGGENBERGER M, 2014) et al. conducted theexperimental studies on the flow field and pressure pulsation of a low-specific speed pump turbine respectively, andobserved the difference in velocity distribution and pressure pulsation in the vaneless zone as well as the dynamic change
process of unsteady flow in the SSCR. With the development of computational fluid dynamics, CFD numerical
Date of Manuscript Acceptance 2020.11.20
E-mail of corresponding author: guoyicheng@xaut.edu.cn

48

Haiyun QIU, Pengcheng GUO, Yoichi KINOUE, Longgang SUN

simulation has become one of the main methods to study the internal flow characteristics of hydraulic machinery. Sun
Longgang (Sun L G, 2019) conducted a numerical simulation study on the internal flow characteristics of the draft tube
under the partial load of a Francis model turbine, and analyzed the periodic evolution process of the vortex belt and its
induced pressure pulsation characteristics. Gou Dongming (GOU D M, 2018) et al. established a geometric model ofthe full overcurrent system of a pumped storage power station, and adopted the three-dimensional coupled turbulencecalculation method of the combined VOF two-phase flow model and single-phase flow model to conduct a numerical
study on the transition process of the runaway in the pump condition, and found that the low-frequency pressure
pulsation of the unit in the runaway condition was caused by a large number of vortices in the runner passages. Widmer(WIDMER C, 2011), Cavazzini(CAVAZZINI G,2015), and Wang Leqin (WANG L Q,2011) carried out numericalsimulation research on the internal flow field in the SSCR of a pump turbine, and concluded that the guide vane opening(GVO) affects the formation of the vortex in the vaneless region, the unstable vortex in the vaneless region, and thebackflow and rotation stall in the runner are related to the S-Shaped of the characteristic curve.
In conclusion, the flow characteristics in the SSCR are one of the current research keys of pump turbine. In order toavoid the appearance of S-Shaped region in engineering, a series of measures have been adopted, among which the early
successful method is the misaligned guide vanes (Xiao Y X, 2014). The use of the misaligned guide vanes has a certain
improvement on the stability of the pump turbine operating in the SSCR, but at the same time it also generates a strongpressure pulsation inside the unit, which brings new problems to the stable operation of the pump turbine. In order toimprove stability or eliminate the influence of the SSCR, it is necessary to study its formation mechanism. The S-Shaped
region of the pump turbine is highly correlated with the opening of the guide vane. In this paper, a model of the pumpturbine is taken as the research object, and the numerical simulation of the flow characteristics of the S-Shaped region
is carried out for the small guide vane opening.

2. Computational methodology
2.1 Pump turbine model
The numerical domain includes spiral case, stay vane, guide vane, runner and draft tube. The runner has a specificspeed(ns) is 21.82 . The ns is defined as follows:ns n Qopt(1)

3

H opt 4The inlet of the spiral case and the outlet of the draft tube were extended by 10D2m respectively to make the inflow
more in line with the actual flow and reduce the vortex intensity at the outlet of the draft tube. Under the optimal
operating conditions, the opening degree of the guide vanes is 11°, the unit discharge and the unit speed are Q11=0.4067m/s and N11=34.22r /min respectively, corresponding to the prototype head Hp=675m and the output Pp=357.5MW. The
main parameters of the pump turbine model are shown in Table 1, and the geometry of the numerical calculation domain
is shown in Figure 1.
Table1 Parameters of model pump-turbine

ns

Zr

Zg

Zs

D1m (m)D2m (m)

B0 (m)

D0 (m)n (r/min)

21.82

9

20

20

0.553

0.25198

0.03776

0.66239

1052

The flow characteristics investigation of the S-shaped region for pump-turbine
in small guide vane opening

49

Spiral Case

Draft Tube
Stay Vane
Guide Vane
Runner

Fig.1 Computational domain

2.2 Mesh
The structured hexahedral mesh is used, which is completed by ICEM-CFD software, and different parts areconnected by interfaces. Because the guide vane and the runner is stator and rotor respectively , and the upstream anddownstream flows affect each other, the flow is relatively complex in these two domains and the hydraulic loss isrelatively large. So in present, the guide vane and runner domain are the key parts for studying with more numbers ofmesh than others three parts. Meanwhile, in order to obtain the accurate flow situation near the wall surface, Special
refinements were applied at the wall surface of each overflow component, and the schematic diagram of the grid foreach component is shown in Fig2.

（a）Spiral Case（c）Runner（b）Stay Vanes and Guide Vanes（d）Draft Tube
Fig.2 Schematic diagram of mesh

50

Haiyun QIU, Pengcheng GUO, Yoichi KINOUE, Longgang SUN

In order to exclude the influence of the number of grids on the accuracy and credibility of the simulation calculation,
The grid independence checks were performed by different sizes of grid with element number from 4.18×106 to
10.89×106 at the optimal operating points. As shown in Figure 3, when the number of grids is more than 7.18×106, thechange of the head is not obvious. Considering the numerical accuracy and time cost, the selected total element numbers
is 9.5×106. The gird parameters used for the simulations are shown in Table 2.

Fig.3 Grid dependence analysis
Table.2 Number and parameters of grid elements
parameters

Spiral case

Stay vanes

Guide vanes

Runners

Draft tube

Number899954148414730825602917701

1119230

Minimum angle（°）

27.4

29

27

26

30

Minimum grid quality

0.37

0.42

0.47

0.43

0.51

2.3 Turbulence Model and Boundary Conditions
The SST k-ω turbulence model is used for simulation, which combines the advantages of the k-ε model andthe k-ω model, and can accurately predict the flow near the wall for it applies the k-ω turbulence model in theboundary layer domain and the k-ε turbulence model in the remaining regions. With the transport equation as
follows.
k
k

k
U j

Pk  * k [(   k t )
]
t
x j
x j
x j
1 k k[(    t )]  2(1  F )  2
U j  S 2   2  xi xi
t
x j
x j
x j
Where the turbulent viscosity t 
equations.

（2）

1 k
,F，Pk，β，β*，σk，σω, σω2 ,etc. are the closure coefficients of themax(1 , SF2 )The high resolution is used for advection scheme and the Second Order Backward Euler is set for transient scheme.
Time step for the simulations corresponds to two degrees of the runner rotation. The maximum number of iterations pertime-step was set to 10, and the convergence criteria of the residuals at each time-step were set to 1.0×10-3. The
simulation time of every operating point took about 15revolutions and the results of the last 5 revolutions are analyzed.
The boundary conditions were defined as follows: for the turbine operating conditions to the braking conditions,mass flow boundary was prescribed at the spiral case inlet, and static pressure was defined at the draft-tube outlet, andfor the reverse pump operating conditions, the boundary conditions for inlet and outlet were charged. The remaining
solid walls were specified as the no-slip walls.

The flow characteristics investigation of the S-shaped region for pump-turbine
in small guide vane opening51

3. Results
3.1 Comparisons between the experimental and numerical characteristic curves
When the guide vane opening is 7°, the full characteristic curve of the pump turbine is simulated at a number ofpoints in the first and fourth quadrants, and the result of experiment and simulation are compared, as shown in Figure 4.
It obvious that the steady calculation results and the experimental results have S-Shaped region. The numericalsimulation results are in good agreement with the experimental results, which demonstrates that CFD simulation settings
in the present are reasonable and reliable.

Fig.4 Comparisons between the experimental and numerical characteristic curves（a）n11-Q11 （b）n11-M11

Based on the result of the steady simulation, five operating points were chosen to carry out the unsteady simulations,which locations are shown at the Fig 5.The first operating point(OP1) is the turbine operating condition, which has large
discharge and high efficiency. The second operating point(OP2) is the turbine operating condition ,too, but is closed to
the runaway condition(OP3). The fourth operating point (𝑂𝑂𝑂𝑂4) is located in the turbine brake region, while the fifth
operating point(OP5) is the reverse operating condition. Guide vane opening is the same for all the considered operating
points. The detailed parameters for these operating points are shown in the Table 3.

Fig.5 Distributions of the operating points for the unsteady simulations

52

Haiyun QIU, Pengcheng GUO, Yoichi KINOUE, Longgang SUN

Table 3 Parameters of the five operating points

Parameters

OperatingH（m）

n11（r/min）

Q11

Operating Points

Conditions

OP1

Turbine21.17732.76

0.25213

OP2

Turbine18.50140.89

0.17858

OP3

Runaway

17.40542.17

0.10912

OP4

Turbine Braking

17.99541.47

0.06346

OP5

Reverse Pump

18.94440.89-0.06306（m³/s）

3.2 Analysis on the internal flow of runner channels
Runner is the energy conversion part of the pump-turbine, and the internal flow state is closely related to the external
characteristics of the pump turbine. In Fig 6, the distributions of the non-dimensional radial velocity component Ur andcircumferential velocity component Ut for the different conditions are presented, respectively. The velocity is madenon-dimensional by using the runner rotational velocity U=n×π×Dm/60. Positive Ur means inflow and negative onemeans outflow. Positive Ut means that the circumferential velocity is in line with the rotation direction of the runner,and negative Ut means the opposite.
In Fig 6, from the hub to the shroud, Ur is almost positive for OP1 with the features of small in the middle and large
at both ends. Except for the reverse pump operating condition(OP5), the Ur of the upper runner channel near the hubside increases as well as the high velocity area increases with the change of operating conditions. While for the lower
part of runner channel near the shroud side, negative Ur appears which means there is back flow in shroud side. The
negative Ur area increases with conditions change, suggests that increasing back flow region. The variation for Ur of
OP5 is much smaller than that of other operating conditions, and the velocity coefficient is relatively small. The positive
radial velocity area is near the shroud side, but it is almost negative on the hub side and the middle flow surface due tothe flow direction is opposite to that of the turbine.
The distribution of the circumferential velocity coefficient is quite different from the radial velocity coefficient. The
circumferential velocity coefficients of all operating conditions are positive. For OP2 to OP4, when a higher radial
velocity appears at the hub side, the circumferential velocity also appears a higher value. However, there is little
difference in the circumferential velocity distribution at the shroud side. When it operates at OP5, the circumferential
velocity is smaller than the other four operating conditions, and the distribution is more uniform.

Fig 6 Radial velocity and circumferential velocity distribution at runner inlet

The velocity coefficient distribution at the runner inlet shows that the flow structure is different at different positionsin the runner. In order to further understand the change of the flow structure in the runner, the velocity vector, meridional
velocity and circumferential velocity of the three span-wise locations in the runner were analyzed under various
operating conditions. The meridional velocity and the circumferential velocity are respectively expressed by

The flow characteristics investigation of the S-shaped region for pump-turbine
in small guide vane opening

53

corresponding velocity coefficient, where the meridional velocity coefficient Um is the ratio of the meridional velocity
Vm to the runner rotational velocity.
The velocity vectors together with the flow streamlines and the mean distributions of the non-dimensional
meridional velocity Um and circumferential velocity Ut at 0.1, 0.5, 0.9 span-wise for the five operating points of different
span-wise locations are presented in Figures 7, 8, and 9, respectively. As shown in Fig 7, on the 0.1span-wise location,
the velocity vectors together with the flow streamlines and velocity distribution of OP1 are uniform, and the flow in therunner is smooth. When operating at OP2, vortex structures appear on the pressure side of leading edge of runner blades,
and there are full vortex structures formed in several channels ,while in OP3 and OP4, full vortex structures formed in
all runner channels. When operating in OP5, there has no vortex structure in runner inlet, but the inflow of the guidevane outlet and the outflow of the runner inlet converge at the leading edge of the runner blade, bypass the head ofblades and flow into the next channel. As a result, an obvious water ring is formed, and the flow separation is severe aswell as the small vortices filled in the channels.
The meridional velocity distribution shows that when the vortex structure appears in the runner, a ring-shaped highmeridional velocity region is formed before the runner inlet, which is gradually expanded to the entire flow channel withthe change of the operating points, indicating that the flow capacity of the runner was decreasing. The inlet of the runnerexhibits a higher circumferential velocity, and the negative circumferential velocity region gradually appears on pressure
side of blades, which is consistent with the position of the vortex, indicating that the direction of rotation for the vortex
is opposite to the direction of rotation for the runner. The high meridional velocity region of the runner inlet almost
disappears when it operates at the OP5, the low meridional velocity region expands and almost fill the entire runnerchannel. And the high meridional velocity region is scattered distribution within the runner channel, wheel that lowvelocity region at the exit zone into a high-velocity region. This is because when the pump-turbine is operating at OP5,
the inlet and outlet of the runner are opposite to that of turbine operating points. The negative circumferential velocity
region in the runner is also increased compared to the other four operating points, mainly distributed in two places, one
is the band-shaped area which begins from the head of the suction side of the blade and extend into the runner; anotheris at the trailing edge of the blade and close to the suction side.

（a）OP1（b）OP2

54

Haiyun QIU, Pengcheng GUO, Yoichi KINOUE, Longgang SUN（c）OP3（d）OP4（e）OP5
Fig.7 The velocity vectors together with the flow streamlines and the mean distributions of the non-dimensional meridional
velocity Um and circumferential velocity Ut at 0.1 span-wise of the five operating points

On the 0.5 span-wise locations, the velocity vectors together with the flow streamlines and velocity distribution ofOP1 do not change much, and the velocity distribution shows that the meridional velocity is greatly affected by rotorstator interference. When operating at off-design condition, the structure is changed, as shown in figure 8, a ring-shaped
flow appears in the vaneless area before the runner inlet from OP2. The ring-shaped flow is divided into two parts when
flowing through the runner inlet, one part of the fluid flows downstream along the runner channel, and the other partflows counterclockwise in the circumferential direction to the next runner blade head and bypasses the head to enter the
next runner channel, so a cross flow is formed at the runner inlet t. The existence of the flow around makes the local
high meridional velocity and circumferential velocity area exist at the head of runner. Incomplete vortex structuresappear in part of the runner channels, and the scattered low meridional velocity region is distributed in the runner, whichindicates that the flow in the runner is disturbed. With the change of operating conditions, the density of the ring-shaped
flow in the bladeless area before the runner is enhanced to form a complete water ring. The cross flow at the inlet of therunner and the vortex structure on the pressure side become obvious. In terms of velocity distribution, the low meridional
velocity area within the runner and in the front of the runner area is increased. When operating at OP4, the runnerchannel is almost completely covered by the low meridional velocity, indicating that the runner flow is greatly hindered.
On the pressure side of the runner blade, there is a negative circumferential velocity corresponding to the vortex position.

The flow characteristics investigation of the S-shaped region for pump-turbine
in small guide vane opening

55

Affected by the cross flow, nine distinct high circumferential velocity zones are formed at the head of the runner. While
entering the reverse pump condition, the water ring in the bladeless area before the runner disappears, but there is stillturbulence and a significant meridional velocity zone at the head of the runner. There are a large number of small vortices
at the runner outlet, which cause the high meridional velocity region at the runner outlet. The circumferential velocity
distribution on middle span is similar to that on the hub side that a negative meridional velocity band is formed from thepressure side of the runner head into the runner ,and the high meridional velocity region is formed at the outlet of runner.
Compared with the flow on the hub side, when running under off-design conditions, the ring-shaped flow appearing
on the middle spin-wise of the runner gradually forms the water ring and the cross flow, and the vortex structure
gradually appears on the pressure side of the runner inlet. Because the water ring and cross flow exist in the inlet of
runner, the head turbulence of the runner is aggravated, and the local high circumferential velocity area is formed at therunner inlet. While when the vortex structure is gradually formatting, the flow capacity of the runner of is decreasing.
The low meridional velocity area at the inlet and outlet of the runner and the area inside the runner are increased,
indicating that the flow is blocked. The blocking water ring on the middle spin-wise of the runner disappeared at OP5,
but the turbulence at the head of the runner was still relatively strong.

（a）OP1（b）OP2（c）OP3

56

Haiyun QIU, Pengcheng GUO, Yoichi KINOUE, Longgang SUN（d）OP4（e）OP5
Fig.8 The velocity vectors together with the flow streamlines and the mean distributions of the non-dimensional
meridional velocity Um and circumferential velocity Ut at 0.5 span-wise of the five operating points

As shown in Fig 9, on the 0.9 span-wise location, the flow state of the OP1 on the hub side is basically the same asthat on the shroud, indicating that the flow in the runner is basically uniform and symmetrical at OP1. Compared withthe flow on the 0.5 span-wise location, no water ring is formed in the vaneless area before the blade at the shroud side,
but there is still a circulation with the greatly weakened strength. Besides the circulation is deflected upstream to forma backflow in the vaneless area. The distribution of the radial velocity coefficient of the runner inlet in the direction of
the blade height above shows that under OP2 to OP4 conditions, the radial velocity coefficient at the shroud side isnegative, which shows that the backflow occurs at the shroud side. It can be seen that the local high circumferential
speed of the vaneless area before the runner is caused by backflow. The vortex structure formed on the hub side of therunner is no longer obvious on the shroud side, but there is still flow separation on the pressure side of the runnerblade. Different from the other four operating conditions, the reverse pump forms complete vortex structures at theoutlet of runner. Affected by the vortex structure, from the hub to the shroud side, the low meridional velocity area at
the runner exit disappears, gradually forming a larger range of high meridional velocity, while the circumferential
velocity distribution does not change much. It shows that the area where the runner blades transfer energy to the fluidas the pump works gradually decreases from the shroud to the hub side.

（a）OP1

The flow characteristics investigation of the S-shaped region for pump-turbine
in small guide vane opening

57（b）OP2（c）OP3（d）OP4（e）OP5
Fig.9 The velocity vectors together with the flow streamlines and the mean distributions of the non-dimensional
meridional velocity Um and circumferential velocity Ut at 0.9 span-wise of the five operating points

The above analysis shows that the vortex structure forms at the hub side of the runner and basically disappears at
the shroud side of the runner with the change of operating points from turbine mode to the braking mode. In order tofurther illustrate the change in the flow structure of the runner inlet along the direction of the blade height, a three-58

Haiyun QIU, Pengcheng GUO, Yoichi KINOUE, Longgang SUN

dimensional streamline diagram of a runner channel under OP3 operating conditions is analyzed, as shown in Fig 10.
The inflow on the hub side bends downward at the inlet of the runner near the pressure surface and is divided into twoparts of fluid. One part forms a vortex structure inside the runner. After the vortex structure is developed inside therunner, it almost disappears at the shroud side and eventually flows downstream with the mainstream. Another part offlow has a tendency to flow upstream during the flow to the shroud side, so that a part of the fluid bypasses the runnerhead and enters the vaneless area at the middle flow surface, forming a high-speed water retaining ring and a cross-flow
structure, which hinders the inflow, reduces the inflow, and has a great influence on the speed distribution of the runner
inlet. And there is also a part of the fluid flows to the shroud side and forms the backflow structure. On the external
characteristic curve, OP2 and OP5 are in the operating region with negative slope, while OP3 and OP4 are in the region
with positive slope. When there is an unstable flow structure in the runner, the inflow of the runner is blocked. As the
speed increases, the flow rate decreases, the vortex structure, the water retaining ring structure, and the backflow
structure of the runner inlet form a complete unstable structure, which further blocks the flow, causing the pump turbine
to be unable to maintain operation at low discharge with higher speeds. Therefore, the rotation speed is reduced, and the
S-Shaped region appears after the runaway point on the characteristic curve.

Fig.10 Three-dimensional schematic diagram of flow structure changes at the inlet of the runner

4. Conclusions
By analyzing the flow structure in the runner, it is found that, except for OP5 condition, vortexes occur at the hubside of the runner with the change of operating condition, a water retaining ring appears at the middle flow surface,and a backflow occurs at the shroud side. When the unstable structure appears in the runner, the high velocity area isformed at the head of the runner blade, and a large range of low velocity area appears in the runner passages. When
operating at OP5, the water retaining ring appears at the hub side, the vortex structure appears at the runner outlet,
which is strongest near the shroud side. Besides the flow separation in the runner is serious, and both the meridional
velocity and circumferential velocity are less than that at other operating points, but the range of the low velocity areais the largest. The existence of unstable flow structure in the runner reduces the flow capacity of the runner, making
the pump turbine unable to maintain operation at low discharge with higher speeds. As a result, the runner's speed
decreases, and the characteristic curve shows an S-Shaped region before and after the runaway condition.

Acknowledgment
This work was supported by the National Natural Science Foundation of China (Grant No. (51839010), the Key
Research and Development Program of Shaanxi Province (Grant No. 2017ZDXM-GY-081), the Youth Innovation Teamof Shaanxi Universities (Grant No. 2020-29), and the cooperative Research Program of IOES (No.18A07).

ReferencesZUO Z G, FAN H G, LIU S H, et al. S-shaped characteristics on the performance curves of pump-turbines in turbine
mode – A review. Renewable and Sustainable Energy Reviews, Vol.60, (2016), pp.836-851.

The flow characteristics investigation of the S-shaped region for pump-turbine
in small guide vane opening

59

LI D Y, WANG H J, QIN Y L, et al., Numerical simulation of hysteresis characteristic in the hump region of a pumpturbine model. Renewable Energy, Vol.115, (2018), pp.433-447.
ZENG W, YANG J, YANG W. Instability Analysis of Pumped Storage Stations Under No-Load Conditions Usinga Parameter-Varying Model. Renewable Energy, Vol.90, (2016), pp.420–429.
XIA L S, CHENG Y G, YANG Z Y. Evolutions of pressure fluctuations and runner loads during runaway processesof a pump-turbine. Journal of Fluids Engineering, Vol.139, No.9, (2017).
HASMATUCHI V. Experimental investigation of a pump-turbine at off-design operating conditions. Brno:
International Meeting of the workgroup on Cavitation and Dynamic problems in Hydraulic Machinery and Systems,(2009).
GUGGENBERGER M, SENN F, SCHIFFER J. Experimental investigation of the turbine instability of a pumpturbine during synchronization. IOP Conference Series: Earth and Environmental Science, Vol.22, No.3, 2014.
Sun L G, Guo P C, LUO X F. Investigation on synchronous and asynchronous characteristics of pressure fluctuations
towards processing vortex rope in Francis Turbine draft tube. Chinese Journal of Agricultural Machinery, Vol.50, No.9,(2019), pp.122-129.
GOU D M, GUO P C, LUO X Q. 3-D combined simulation of power-off runaway transient process of pumped
storage power station under pump condition. Hydrodynamics Research and Progress, Vol.33, No.1, 2018, pp.28-39.
WIDMER C, STAUBLI T. LEDERGERBER N. Unstable Characteristics and Rotating Stall in Turbine Brake
Operation of Pump-Turbines. Vol.133, No.4, (2011).
CAVAZZINI G, COVI A, PAVESI G. Analysis of the Unstable Behavior of a Pump-Turbine in Turbine Mode:
Fluid-Dynamical and Spectral Characterization of the S-shape Characteristic. Vol.138, No.2, (2015).
WANG L Q, YIN J L, JIAO L. Numerical investigation on the “S” characteristics of a reduced pump turbine model.
Science China Technological Sciences, Vol.54, No.5, (2011), pp.1259-1266.
Xiao Y X, Wang Z W, Zhang J, et al. Numerical predictions of pressure pulses in a Francis pump turbine withmisaligned guide vanes. Journal of Hydrodynamics, Vol.24, No.2, 2014, pp.250-256.