dynamic evolutions between the draft tube pressure
TRANSCRIPT
404
International Journal of Fluid Machinery and Systems DOI: http://dx.doi.org/10.5293/IJFMS.2017.10.4.404
Vol. 10, No. 4, October-December 2017 ISSN (Online): 1882-9554
Original Paper
Dynamic evolutions between the draft tube pressure pulsations and
vortex ropes of a Francis turbine during runaway
Xiaoxi Zhang1, Qiuhua Chen2 and Jie Liao1
1School of Environmental Science and Engineering, Xiamen University of Technology
No.600 Ligong Road, Xiamen, 361024, PR China, [email protected], [email protected] 2 School of Civil Engineering and Architecture, Xiamen University of Technology No.600 Ligong Road,
Xiamen, 361024, PR China, [email protected]
Abstract
To analyze the dynamic evolutions between the draft tube pressure pulsations and vortex ropes of a Francis turbine, the
runaway transient process of a hydropower system is simulated by coupling a one-dimensional model of the water
conveyance system and a three-dimensional model of the Francis turbine. The results show that the annular-distributed
pressure pattern at the entrance of the draft tube breaks and induces small vortex ropes, which then merge into an eccentric-
distributed helical one with the transient operating point moving away from the rating region. In this process, low frequency
pressure pulsations form and continue to strengthen. When the operating point moves to the runaway point, the vortex ropes
keep dividing and merging irregularly, causing random-like pressure pulsations.
Keywords: Francis turbine, runaway, 1D-3D coupling, numerical simulation, pressure pulsations, flow patterns.
1. Introduction
Francis turbines are the most common hydro-turbine in use in hydropower engineering. When they operate in off-rating
conditions, undesirable low-frequency and high-amplitude pressure pulsations are produced, which may bring output fluctuations,
turbine vibrations and even hydraulic oscillations in the hydropower system[1]. These types of pressure pulsations are more
significant at the runner exit and the sources are the rotating vortex ropes in the draft tube[2].
Draft tube vortex ropes have always been attracting attentions from both the research community and the industry. This
phenomenon is closely related to the swirl ratio and radial velocity distribution of the water flow near the runner exit, where the
latest contributions of Bergan C. et al.[3] , Jošt D. et al.[4] , Sefan D. et al.[5] and Sonin V. et al.[6] have to be mentioned.
Additionally, studies about the mechanism of vortex ropes and their influences on the characteristics of pressure pulsations in the
past decades can be found in the reviews of Nishi M. et al.[7] and Dörfler P. et al.[8]. In recent years, dynamic characteristics of
the draft tube vortex ropes are becoming a focus in this field of research. Müller A. et al.[9] revealed that approximately 5–10 %
of the water flow near the draft tube cone passes inside the vortex rope during the process of its formation and collapse by model
test; this flow pattern will induce the discharge variation in the draft tube and a mass excitation source in the hydropower system.
Zuo Z. et al.[10] simulated the fluid flow in a Francis turbine in different operating conditions by three-dimensional
Computational Fluid Dynamics (CFD). The results illustrated that the vortex rope tends to break up into smaller scale vortex
filaments in partial load condition. Through a model test in the same condition, Foroutan H. et al.[11] found that the developing
process of the vortex rope produces oscillatory forces on the draft tube and the most dangerous region is the inner side of the
elbow. Favrel A. et al.[2] studied the relations between the vortex and pressure pulsations in different discharges of a Francis
turbine in cavitation-free conditions by particle image velocimetry. They found that the vortex trajectory broadens with the
discharge decreases but retracts when the discharge drop to a certain value. During this process, the precession frequency and the
characteristics of the pressure pulsations vary accordingly. Afterwards, they conducted the same measurements in cavitation
conditions and found that the vortex trajectory retracting with the cavitation volume growing. This phenomenon may reduce the
intensity of the pressure pulsations caused by the vortex precession[12]. The numerical simulations adopted by Gavrilov A. A. et
al.[13] demonstrated flow patterns with upward main flow and multiple downward recirculation zones near the draft tube cone, all
bringing about high energy dissipation and efficiency reduction. Skripkin S. G. et al.[14] indicated that vortex rings generate and
detach from the vortex rope periodically by model tests. In this process, vortex rings appear about every 3-5 revolutions of the
vortex rope and induce sharp pressure shocks near the draft tube wall. In summary, the number, shape, trajectory and rotationally
speed of vortex ropes vary dynamically, even in the static operating conditions, which influence the turbine pressure and discharge
Received July 13 2017; accepted for publication July 30 2017: Review conducted by Shogo Nakamura. (Paper number O17052J)
Corresponding author : Qiuhua Chen, [email protected]
405
significantly.
However, all above researches were conducted in static operating conditions rather than transient processes. During transients
in hydropower systems, the water-hammer phenomenon occurs in whole flow passages and the hydraulic inertia plays an
important role in the dynamic characteristics of hydro-turbines [15]. In this circumstance, pressure pulsations are more intense
than those under the static operating conditions[16]. More importantly, intensive pulsations on the water-hammer pressure leads to
ultra-high amplitudes in absolute pressure, which may induce fatal accidents such as turbine-generator unit lifting-up or water
column separation in the draft tube.
Although the characteristics of pressure pulsations during transients are not clear yet, they may also relate to the draft tube
vortex ropes according to the previous researches in static operating conditions. The purpose of this paper is to investigate the
relations between the draft tube pressure pulsations and vortex ropes during runaway in a prototype hydropower system. Firstly, a
numerical hydraulic model, which couples a one-dimensional model of the water conveyance system and a three-dimensional
model of the Francis turbine, is built. Then, the runaway transient process in the system is simulated and the results are validated
by the traditional one-dimensional numerical approach. Finally, the developments of the vortex ropes and characteristics of the
transient pressure pulsations are analyzed.
2. Numerical methods and validation
2.1 Numerical methods
The hydraulic model of a hydropower system, including two sections of penstocks, a Francis turbine and a section of tailrace
tunnel, is built numerically by the 1D-3D coupling approach for its advantage in both accuracy and efficiency[17]. In this
approach, the gradually varied water flow in the penstocks and tailrace tunnel is reduced to one-dimensional (1D) flow, while the
rapidly varied water flow in the turbine is modelled directly by the three-dimensional method. The whole computational domain is
shown in Fig. 1.
P2
1D zone
3D zone 1D zoneP1
T2T1P3
1 2
PA Bt
t+Δt1D MOC Grid
3D FVM Grid
Turbine
Fig. 1 The 1D-3D coupling hydraulic model of a hydropower system
Flow in the 3D part is governed by the unsteady Navier-Stocks equations and solved by the Finite Volume Method (FVM) in
the commercial code Ansys Fluent 13.0. The 3D computational domain is divided into four parts as follows: the spiral casing, the
stay and guide vanes, the runner as well as the draft tube. As shown in Fig. 1, two extension sections (P3 and T1) are added at the
beginning of the spiral casing and the end of the draft tube in order to force the gradually varied flow at the coupling boundaries.
For spatial discretization, about 3600,000 unstructured tetrahedral and prism cells are employed to the flow passage, and the
validation of this grid on simulating the hydraulic characteristics of the Francis turbine in both the steady and load rejected states
can be found in reference[18]. A time step corresponding to 1° of runner revolution and the convergence criterions of 10-5 are
selected for all equations. Besides, the rotating speed of the turbine during runaway is evaluated by the hydraulic torque acting on
the runner by Eq. (1).
d9.55
d
n M
t J (1)
Flow in the 1D part is described by the continuity and momentum equations of transient-flows in closed conduits, eq. (3) and
(4), and solved by the Method of Characteristics (MOC)[19]. The MOC code is written in C language and compiled in Ansys
Fluent 13.0 with the help of User Defined Functions (UDFs). The 1D computational domain is divided into three parts, includes
the upstream horizontal penstock (P1), the upstream inclined penstock (P2) and the tailrace tunnel (T2), details of each parts are
listed in Table 1. Uniform grids are adopted to discrete the three parts and the grid size, which is equal to 1m, is decided according
to the Courant stability condition. Finally, for synchronous computation, the time step size is set as the same as that of the 3D
model. 2
0H a Q
t gA s
(2)
02
fQ QQ HgA
t s DA
(3)
Table 1 Main parameters of the piping system
section L (m) sectional shapes D (m) f (-) a (m/s)
P1 334 Round 5.625 0.012 1000
P2 189 Round 5.625 0.012 1000
T2 234 D-shape 11.180 0.012 1000
406
Flow variables between the 1D and 3D domains are exchanged on the interface by establishing mathematical relationships
between the MOC and FVM grids. The grids near the interface are set as shown in Fig. 1, and it should be note that gradually
varied flow patterns must be kept in section 1-2 of the FVM grid. The fundamental coupling procedure is described in Fig. 2 and
executed in Ansys Fluent 13.0 by the UDFs. More details and validations of this approach can be found in reference[17].
Calculate the discharge and head at node P at the
next time step by combining C+ and C- equations:
Start
Calculate the flow variables in 1D part by MOC
Obtain the discharge and head at the 1D boundary,
node B, by using the 3D flow variables :
End
=t t + t
Input the initial flow variables in 3D part
Y
N
t<=total time?
+
P P
-
P P
: ( )2
: ( )2
t t t t t t t t
A A A A
t t t t t t t t
B B B B
gA f t gAC Q Q H Q Q H
a DA a
gA f t gAC Q Q H Q Q H
a DA a
Calculate the flow variables in 3D part by CFD
B 2 2
B 2 2 2
= ( ) ( )
= ( ) ( ) / ( ( ))
t t
t t
Q u i A i
H p i A i g A i
Obtain the total pressure at the 3D boundary,
section 1, by using the 1D flow variables :2
1 P P 2( )= g[ ( / ( )) / 2 ]t t tp i H Q A i g
Fig. 2 Flow chart of the coupling procedure (Subscripts indicate the node index and superscripts indicate the time step .)
2.2 Validation
During the runaway process, the rotating speed of the turbine increases for the load has been rejected, while the discharge
decreases due to choking caused by overspeed. As a result, flow in the turbine becomes more and more disordered, inducing flow
separations and vortex. In order to simulate the flows dominated by separations accurately, two turbulence models, RNG
(Renormalization Group) k-ε[20] and SAS (Scale Adapted Simulation)[21], are adopted and compared. It is important to note that
the numerical results of 1D MOC are presented here for validating the 1D-3D simulations, for the measured data on the prototype
turbine of this hydropower system are not available. This treatment is valid because the 1D approach, in which the turbine
performance are evaluated based on the steady-state model tests data, has been proved to be reliable by plenty of engineering
practices [19]. The simulated rotating speed and discharge are compared in Fig. 3. It can be seen that both of the two models can
give accurate results near the rating operating region. But with the rotating speed increase and the discharge decrease, the results
of SAS model are more accordant with the data from the 1D MOC than those of the RNG model, especially in discharge.
Therefore, the SAS model are more accurate in modelling the flow patterns during runaway and the simulated results are analyzed
in the following parts.
0 2 4 6 8 10140
160
180
200
220
240
260
280
n (
rad/m
in)
t (s)
1D
1D-3D (RNG k-e)
1D-3D (SAS)150
170
190
210
230
Q
Q (
m3/s
)
n
Fig. 3 Comparisions of the numerical results between the 1D and 1D-3D coupling approaches
407
3. Results and discussion
3.1. Pressure pulsations
The properties of pressure along the draft tube wall vary greatly during the runaway process. For comparison, the simulated
pressure–time signals are normalized by the average pressure of the corresponding point and the working head of the turbine in
steady-state condition using eq. (4), as shown in Fig. 4. It is obviously that the data at the entrance of the draft tube pulsate more
intensely both in frequency and amplitude than those in other positions. Therefore, pressure signals in P1 and P2, which are
symmetrical points at the draft tube entrance, are further compared and analyzed, as shown in Fig. 5. Furthermore, the time-
frequency properties are analyzed by the MATLAB-implemented wavelet transform, for the reason that pressure pulsations during
runaway are typical non-stationary signals, and the results are shown in Fig. 6. Note that all of the frequencies are divided by the
rotating frequency, fn, which is regard as the reference frequency when the turbine is operating.
0( ) / /sh p p g H (4)
Pressure pulsations in P1 and P2 can be divided into three stages according to the characteristics of the frequency and
amplitude. In the region near the rating point, approximately 0~2s, pressure variations between P1 and P2 are almost coincident
with each other, and the amplitudes are smallest during the runaway. Besides, the pulsations show significant high frequency
properties and the dominant frequency is around 16fn. Therefore the pressure pulsations in the first stage is induced by the rotor
and stator interactions (RSI) between the runner and guide vanes, for the reason that 16 is the number of the guide vanes. In the
second stage, approximately 2-8s, the amplitudes increase significantly with the transient operating point moves to the partial load
region. In this process, low frequency components, which is below fn, emerge and increase to the dominant components gradually.
Meanwhile, the pressure variations between P1 and P2 begin to diverge and develop to two low frequency oscillations which
differ by half cycle exactly. When the transient operating point approaches the runaway point, although the dominant frequency
still locates in the low frequency region, all levels of frequency components are enhanced and large enough to influence the
characteristics of the pulsations. In this stage, the pressure pulsations in both P1 and P2 show spectrums with wider range of
frequency components, which indicates obvious irregular pulsations. In summary, during runaway, the pressure pulsations at the
entrance of the draft tube demonstrate high frequency and low amplitude properties at first, then the low-frequency components
are enhanced and the amplitude increases, and finally tend to random-like pulsations in the region near the runaway point.
0 3 6 9 12-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
h (
-)
T (s)0 3 6 9 12
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
h (
-)
T (s)
0 3 6 9 12-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
h (
-)
T (s)
0 3 6 9 12-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
h (
-)
T (s)
0 3 6 9 12-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
h (
-)
T (s)
0 3 6 9 12-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
h (
-)
T (s)
P1 P2
P3 P4
P5
P6
P1
P3
P5
P4
P6
P2
Fig. 4 Pressure pulsations in different positions along the draft tube wall
0 2 4 6 8 10 12-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
10.0 10.2 10.4-0.1
0.0
0.1
0.2
h (
-)
T (s)
P1
P2
Rotating speed
160
180
200
220
240
260
1.0 1.1 1.2
-0.03
0.00
0.03
n (rp
m)
Fig. 5 Comparisions of the pressure signals between P1 and P2
408
0.2
0.1
00 2 4 6 8 10 12
20
15
10
5
f /
f n(-
)
h (-)
0.15
0.05
T (s)
Rotating frequency
Runaway point
(a) P1
0.2
0.1
00 2 4 6 8 10 12
20
15
10
5
f /
f n(-
)
h (-)
0.15
0.05
T (s)
Rotating frequency
Runaway point
(b) P2
Fig. 6 Spectrogram of the transient frequency variaion at P1 and P2
3.2. Flow patterns
The variation of pressure distribution in the draft tube can also be divided into three stages with the change of the operating
conditions. Fig. 7 shows the contours of the relative pressure, which is define as eq. (5), at the entrance of the draft tube during the
runaway process. It can be seen that near the rating point, pressure distributes as a series of concentric annulus, which indicates
axisymmetric flow patterns, see Fig. 7 (a). With the operating point deviating from the rating point, pressure in the center of the
draft tube decreases rapidly and the low pressure region brakes up into small pieces, see Fig. 7 (b) and (c). These small low
pressure regions are almost symmetrical to the center and rotate with the runner synchronously. In the second stage, the pressure
values and distances among these regions change with the transient operating point moves to the partial load region, so the
axisymmetric flow pattern is damaged and the low pressure regions trend to merge, see Fig. 7 (d)-(f). Then there forms an
eccentric-distributed pressure pattern which rotates with the runner asynchronously. Finally, when the operating point approaches
to the runaway point, the low pressure regions divide into small pieces and then remerge together, repeating this process over and
over again, see Fig. 7 (g)-(o). Besides, the low pressure regions appear randomly and their number varies discontinuously in this
process.
0( / ) / /i i i ih p p A A g H (5)
In the above three stages, vortex ropes in the draft tube develop accordingly. Fig. 8 displays the iso-surface which represents
points of a constant pressure (h is equal to 1.2). It is obvious that each low-pressure region at the entrance of the draft tube
corresponds to a vortex rope. Initially, the pressure in the draft tube is axisymmetric, and there is no obvious vortex rope. With the
breaking up of the annular-distributed pressure pattern, there generates several small vortex ropes, which distribute around the
runner hub evenly and rotates with the runner synchronously. In the second stage, these small vortex ropes merge gradually into a
large helical one, attaching to the side of the hub and rotating with the runner asynchronously, see Fig. 8 (a) and (b). In the third
stage, the vortex ropes divide and merge so frequently that they share more or less common parts in the top or bottom. Besides,
the shapes of the vortex ropes are not as regular as the previous two stages, see Fig. 8 (c)-(e). Therefore, during runaway, the
annular-distributed pressure pattern breaks up and induces small vortex ropes at first. With the transient operating point moves to
the partial load region, these small vortex ropes merge into an eccentric-distributed helical one gradually. Finally, when the
operating point moves to the runaway point, the vortex ropes keep dividing and merging irregularly.
0.2
-0.2
0
0.1
-0.1
h (-)
(a) T=0.15s (b) T=1.10s (c) T=1.85s (d) T=2.20s (e) T=2.65s
409
(f) T=3.84s (g) T=5.45s (h) T=6.60s (i) T=7.40s (j) T=8.20s
(k) T=8.75s (l) T=9.40s (m) T=10.05s (n) T=10.90s (o) T=12.20s
Fig. 7 Pressure distribution at the entrance of the draft tube during runaway
(a) T=2.65s (b) T=5.45s (c) T=8.20s (d) T=10.05s (e) T=10.90s
Fig. 8 Evolutions of the vortex ropes in the draft tube during runaway
3.3. Discussion
During transients, the relations between draft tube flow patterns and pressure pulsations are similar to those in steady-state
conditions in some specific operating regions. For example, near the rating point, pressure distributes symmetrically along the
radius direction at the entrance of the draft tube. So high frequency and low amplitude pressure pulsations, which are caused by
RSI between the runner and guide vanes, are the dominant phenomenon. Besides, in partial load region, helical vortex ropes
generate and bring low frequency and high amplitude pulsations. These results are coincidence with those in the previous studies.
However, this study also demonstrates the dynamic evolutions of flow patterns and pressure pulsations between different
operating regions. First of all, the discrete vortex ropes, which caused by the breaking of the annular-distributed pressure pattern,
induce the original low frequency pressure pulsations. Then, low frequency components are enhanced with the merging and
growing of these small vortex ropes. Finally, in the region near the runaway point, the irregular developments of the vortex ropes
not only widen the frequency band, but also bring random-like pulsations in low frequency components. In summary, the high
frequency components, which relate to the RSI, remain the same, but the low frequency components, which are induced by the
vortex ropes, continue to strengthen during runaway.
Low frequency and high amplitude pressure pulsations in the draft tube are one of the most dangerous phenomenon in
hydropower systems, so it is important for hydropower stations to implement control measures. This study illustrates that the
variation of these pressure pulsations relate to not only the operating regions, but also the dynamic evolutions of vortex ropes.
Moreover, the flow evolutions are affected by both the dynamic trajectory and the moving speed of the transient operating point.
Considering that these factors are determined by the acceleration or deceleration characteristics of water flow and the turbine
runner, so it may be possible to control the pressure pulsations in transients by changing the turbine inertia, the guide vanes
closure rule or even the piping system scheme of the hydropower station on the premise of safety.
4. Conclusions
In this paper, the runaway transient process in a hydropower system is simulated by the 1D-3D coupling approach, and the
dynamic evolutions between the draft tube pressure pulsations and vortex ropes of a Francis turbine are analyzed. The main
findings are as follows:
(1) Pressure pulsations of different frequency vary independently during runaway. The properties of high frequency components
remain the same, while the low frequency components continue to strengthen at first, and then tend to random-like
pulsations near the runaway point.
410
(2) Flow patterns in the draft tube evolve continuously during runaway. The annular-distributed pressure pattern breaks and
induces small vortex ropes at first, which then merge into an eccentric-distributed helical one gradually. Finally, when the
operating point moves to the runaway point, the vortex ropes keep dividing and merging irregularly.
(3) The development of low frequency and high amplitude pressure pulsations corresponding to the dynamic evolutions of
vortex ropes, so it may be possible to reduce these pulsations by controlling the trajectories or speed of the transient
operating point in transients, on the premise of safety.
To further complete this study, pressure pulsations in the real hydropower station should be measured for validating the above
results. Besides, only the qualitative characteristics of a certain runaway process are discussed here, further studies should be
focused on the quantitative rules by analyzing various types of transients.
Acknowledgments
This work is financially supported by the Xiamen Science and Technology Planning Project of China (3502Z20150051), the
Education Department of Fujian Province (JA15384, JAT170410) and the Scientific Research Foundation of Xiamen University
of Technology (YKJ15042R, YKJ14025R).
Nomenclature
A Cross-sectional area [m2] a Acoustic speed [m/s]
C+ The positive characteristic equation [-] C- The negative characteristic equation [-]
D Equivalent diameter [m] f Darcy-Weisbach friction factor [-]
g Gravity [m/s2] H Head [m]
H0 Working head of the turbine [m] h Relative pressure head [-]
i Node index [-] J Rotational inertia [kgm2]
L Flow passage length [m] M Hydraulic torque [Nm]
n Angular rotating speed [rpm] p Pressure [Pa]
sp Average pressure in the steady condition [Pa] Q Discharge [m3/s]
s Coordinate along a pipe [m] T Time [s]
t Time [s] Δt Time step size [s]
u Velocity [m/s] ρ Water density [kg/m3]
References [1] KC, A., Lee, Y. H., and Thapa, B., 2016, “CFD study on prediction of vortex shedding in draft tube of Francis turbine and
vortex control techniques,” Renewable Energy, Vol. 86, pp. 1406-1421. [2] Favrel, A., Müller, A., Landry, C., Yamamoto, K., and Avellan, F., 2015, “Study of the vortex-induced pressure excitation
source in a Francis turbine draft tube by particle image velocimetry,” Experiments in Fluids, Vol. 56, No. 12, pp. 215. [3] Bergan, C., Amiri, K., Cervantes, M. J., and Dahlhaug, O. G., 2015, “Preliminary Measurements of the Radial Velocity in the
Francis-99 Draft Tube Cone,” Journal of Physics: Conference Series, Vol. 579, No. 1, pp. 012014. [4] Jost, D., Skerlavaj, A., Morgut, M., Meznar, P., and Nobile, E., 2015, “Numerical simulation of flow in a high head Francis
turbine with prediction of efficiency, rotor stator interaction and vortex structures in the draft tube,” Journal of Physics:
Conference Series, Vol. 579, No. 1, pp. 012006. [5] Stefan, D., and Rudolf, P., 2015, “Proper Orthogonal Decomposition of Pressure Fields in a Draft Tube Cone of the Francis
(Tokke) Turbine Model,” Journal of Physics: Conference Series, Vol. 579, No. 1, pp. 012002. [6] Sonin, V., Ustimenko, A., Kuibin, P., Litvinov, I., and Shtork, S., 2016, “Study of the velocity distribution influence upon the
pressure pulsations in draft tube model of hydro-turbine,” IOP Conference Series: Earth and Environmental Science, Vol. 49, No.
8, pp. 82020. [7] Nishi, M., and Liu, S., 2013, “An Outlook on the Draft-Tube-Surge Study,” International Journal of Fluid Machinery and
Systems, Vol. 6, No. 1, pp. 33-48. [8] Dörfler, P., Sick, M., and Coutu, A., 2013, Flow-Induced Pulsation and Vibration in Hydroelectric Machinery, Springer
London, London, UK. [9] Müller, A., Dreyer, M., Andreini, N., and Avellan, F., 2013, “Draft tube discharge fluctuation during self-sustained pressure
surge: fluorescent particle image velocimetry in two-phase flow,” Experiments in Fluids, Vol. 54, No. 4, pp. 1514. [10] Zuo, Z., Liu, S., Liu, D., and Qin, D., 2014, “Numerical predictions and stability analysis of cavitating draft tube vor tices at
high head in a model Francis turbine,” Science China Technological Sciences, Vol. 57, No. 11, pp. 2106-2114. [11] Foroutan, H., and Yavuzkurt, S., 2015, “Unsteady Numerical Simulation of Flow in Draft Tube of a Hydroturbine Operating
Under Various Conditions Using a Partially Averaged Navier–Stokes Model,” Journal of Fluids Engineering-Transactions of the
Asme, Vol. 137, No. 6, pp. 61101. [12] Favrel, A., Müller, A., Landry, C., Yamamoto, K., and Avellan, F., 2016, “LDV survey of cavitation and resonance effect on
the precessing vortex rope dynamics in the draft tube of Francis turbines,” Experiments in Fluids, Vol. 57, No. 11, pp.168. . [13] Gavrilov, A. A., Sentyabov, A. V., Dekterev, A. A., and Hanjalić, K., 2017, “Vortical structures and pressure pulsations in
draft tube of a Francis-99 turbine at part load: RANS and hybrid RANS/LES analysis,” International Journal of Heat and Fluid
Flow, Vol. 63, pp. 158-171. [14] Skripkin, S. G., Tsoy, M. A., Kuibin, P. A., and Shtork, S. I., 2017, “Study of pressure shock caused by a vortex ring
separated from a vortex rope in a draft tube model,” Journal of Fluids Engineering-Transactions of the Asme, Vol. 139, No. 8, pp.
411
081103. [15] Zhang, X., Cheng, Y., Xia, L., Yang, J., and Qian, Z., 2016, “Looping Dynamic Characteristics of a Pump-Turbine in the S-
shaped Region During Runaway,” Journal of Fluids Engineering-Transactions of the Asme, Vol. 138, No. 9, p. 91102. [16] Trivedi, C., Cervantes, M. J., Dahlhaug, O. G., and Gandhi, B. K., 2015, “Experimental Investigation of a High Head Francis
Turbine During Spin-No-Load Operation,” Journal of Fluids Engineering-Transactions of the Asme, Vol. 137, No. 6, p. 61106. [17] Zhang, X., and Cheng, Y., 2012, “Simulation of hydraulic transients in hydropower systems using the 1-D-3-D coupling
approach,” Journal of Hydrodynamics, Vol. 24, No. 4, pp. 595-604. [18] Zhang, X., Cheng, Y., Yang, J., Xia, L., and Lai, X., 2014, “Simulation of the load rejection transient process of a francis
turbine by using a 1-D-3-D coupling approach,” Journal of Hydrodynamics, Vol. 26, No. 5, pp. 715-724. [19] Chaudhry, M. H., 2014, Applied Hydraulic Transients, Third Edition, Third Edition, Springer New York, New York, USA. [20] Yakhot, V., and Orszag, S. A., 1986, “Renormalization group analysis of turbulence. I. Basic theory,” Journal of Scientific
Computing, Vol. 1, No. 1, pp. 3-51. [21] Menter, F. R., and Egorov, Y., 2010, “The Scale-Adaptive Simulation Method for Unsteady Turbulent Flow Predictions. Part
1: Theory and Model Description,” Flow, Turbulence and Combustion, Vol. 85, No. 1, pp. 113-138.