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Flow-induced pulsations in Francis turbines during startup - A consequence of an intermittent energy system

Authors:

Abstract

Hydraulic turbines are increasingly responsible for regulating the electric grid, due to the rapid growth of the intermittent renewable energy resources. This involves a large increase in the number of starts and stops, which cause severe flow-induced pulsations and fluctuating forces that deteriorate the machines. Better knowledge of the evolution of the flow in the machines during transients makes it possible to avoid hazardous conditions, plan maintenance intervals, and estimate the costs of this new kind of operation. The present work provides an in-depth and comprehensive numerical study on the flow-induced pulsations and evolution of the flow field in a high-head model Francis turbine during a startup sequence. The flow simulation is carried out using the OpenFOAM open-source CFD code. A thorough frequency analysis is conducted on the fluctuating part of different pressure probes and force components, utilizing Short-Time Fourier Transform (STFT) to extract the evolution of the frequency and amplitude of pulsations. Low-frequency oscillations are detected during the startup, which are induced by the complex flow structure in the draft tube. A decomposition is performed on the draft tube pressure signals, and the variations of the synchronous (plunging) and asynchronous (rotating) modes are studied. The plunging mode is stronger at minimum and deep part load conditions, whereas the rotating mode is dominant during the presence of the Rotating Vortex Rope (RVR) at part load. The velocity field in the draft tube is validated against experimental data, and the complex flow structures formed during the startup procedure are explained using the λ2 vortex identification method.
Flow-induced pulsations in Francis turbines during startup - A
consequence of an intermittent energy system
Saeed Salehi
*
, Håkan Nilsson
Division of Fluid Dynamics, Department of Mechanics and Maritime Sciences, Chalmers University of Technology, Gothenburg, SE, 412 96, Sweden
article info
Article history:
Received 3 November 2021
Received in revised form
10 January 2022
Accepted 26 January 2022
Available online 7 February 2022
Keywords:
High head Francis turbine
Startup sequence
Flow-induced pulsations
Plunging and rotating modes
Rotating vortex rope (RVR)
OpenFOAM
abstract
Hydraulic turbines are increasingly responsible for regulating the electric grid, due to the rapid growth of
the intermittent renewable energy resources. This involves a large increase in the number of starts and
stops, which cause severe ow-induced pulsations and uctuating forces that deteriorate the machines.
Better knowledge of the evolution of the ow in the machines during transients makes it possible to
avoid hazardous conditions, plan maintenance intervals, and estimate the costs of this new kind of
operation. The present work provides an in-depth and comprehensive numerical study on the ow-
induced pulsations and evolution of the ow eld in a high-head model Francis turbine during a
startup sequence. The ow simulation is carried out using the OpenFOAM open-source CFD code. A
thorough frequency analysis is conducted on the uctuating part of different pressure probes and force
components, utilizing Short-Time Fourier Transform (STFT) to extract the evolution of the frequency and
amplitude of pulsations. Low-frequency oscillations are detected during the startup, which are induced
by the complex ow structure in the draft tube. A decomposition is performed on the draft tube pressure
signals, and the variations of the synchronous (plunging) and asynchronous (rotating) modes are studied.
The plunging mode is stronger at minimum and deep part load conditions, whereas the rotating mode is
dominant during the presence of the Rotating Vortex Rope (RVR) at part load. The velocity eld in the
draft tube is validated against experimental data, and the complex ow structures formed during the
startup procedure are explained using the
l
2
vortex identication method.
©2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
1. Introduction
The use of renewable electric energy resources has been
growing fast to respond to the increasing global electric energy
consumption. Nowadays, the inevitable intermittency of electrical
energy resources such as solar and wind power is compensated
through hydropower systems [1e3]. The hydraulic turbines are not
necessarily working at the steady Best Efciency Point (BEP) con-
dition anymore. They are being used in different transient oper-
ating sequences to stabilize the electrical grid, leading to entirely
different engineering requirements for such machines.
Transient operations usually produce complex ow structures,
such as ow separation, vortices, destructive pressure pulsations,
cavitation, etc. Frequent occurrence of such undesirable ow
structures could seriously deteriorate the turbine lifetime and
cause fatigue stresses, wear and tear on different components [4].
Currently, Francis turbines may experience over 500 start-stop
cycles per year [5], while they are usually designed to tolerate up
to 10 cycles [6,7]. Undoubtedly, the accumulated damages from
such abundant cycles degrade the machine's performance and may
lead to its failure. Hence, it is crucially important to study and
provide a profound understanding of the turbine ow eld during
transient operations such as startup.
Gagnon et al. [8] examined the inuence of startup schemes on
the fatigue-based life expectancy of a Francis turbine. It was
explained that an optimization of the scheme could improve the
turbine lifetime. Nicolle et al. [9] assessed the startup operations of
a low-head Francis turbine using a reduced CFD model. Two
different startup scenarios based on the guide vane opening
scheme were investigated. Comparisons were made with limited
experimental measurements and a general agreement was
achieved.
The impact of the guide vane opening scheme on the startup
procedure of a high-head Francis turbine has been experimentally
*Corresponding author.
E-mail addresses: saeed.salehi@chalmers.se (S. Salehi), hakan.nilsson@chalmers.
se (H. Nilsson).
Contents lists available at ScienceDirect
Renewable Energy
journal homepage: www.elsevier.com/locate/renene
https://doi.org/10.1016/j.renene.2022.01.111
0960-1481/©2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Renewable Energy 188 (2022) 1166e1183
assessed by Trivedi et al. [10]. The angular speed of the guide vanes
was for one scheme almost twice as for the other scheme. Inap-
propriate rapid rotation of the guide vanes amplied the unstead-
iness and developed undesirable pressure pulsations. Goyal et al.
[7] performed an experimental study on the same high-head
Francis turbine during startup. The startup sequence was split
into two phases, namely, phase I, to synchronize the turbine with
the generator, and phase II, to reach the steady-state condition. The
second phase was accomplished using three different guide vane
opening schemes which ended in Part Load (PL), BEP, and High Load
(HL) conditions, respectively. The Rotating Vortex Rope (RVR) fre-
quency was observed in both velocity and pressure data of the rst
scheme.
More recently, the startup of a prototype Francis turbine was
experimentally and numerically investigated [11]. Two guide vane
opening schemes, namely, conventional and reduced opening limit
schemes were studied and it was shown that the reduced scheme
decreased the fatigue damage. The draft tube vortices were shown
to have a signicantly higher impact on the dynamical stresses
compared to the interblade vortices. It was concluded that dis-
turbing the draft tube vortex could alleviate the damaging effects
on the runner during startup.
Although the experimental investigations are trustworthy re-
sources to assess the turbine ow eld during startup, they are
expensive and there are many limitations on accessibility and
measured ow details. Numerical studies provide a reliable addi-
tion to assess and understand the details of the ow eld during
turbine startup. The startup is recognized as one of the most
harmful operating conditions of hydraulic turbines [8]. Therefore,
achieving a profound understanding of the complex ow eld of a
hydraulic turbine during startup is essential to reduce the
damaging effects and improve the life expectancy of these
machines.
The present article provides a comprehensive and detailed
analysis of the transient ow eld and its pulsations during a
startup sequence of a Francis turbine. Such in-depth analyses are
crucial for a better understanding of the hazardous pulsations to be
able to ultimately reduce and avoid them. The simulation is per-
formed utilizing the OpenFOAM open-source CFD code. The varia-
tion of the pressure eld, velocity eld, and forces are carefully
assessed. One of the main focuses of the present study is to extract
the ow-induced pulsations. No investigations are found in the
literature on the draft tube pressure signal decomposition during a
startup sequence. In the current study, for the rst time, the vari-
ation of plunging and rotating modes of the uctuating pressure
during the startup operation is examined. An in-depth explanation
of the complex ow structures downstream the runner, which play
a crucial role in the generation of the pulsations, is presented. The
paper is organized as follows. The investigated test case, including
the geometrical and operational details, is introduced in Section 2.
The mathematical formulations of the assessed problem are
described in Section 3.1, while the details of the numerical frame-
work are described in Section 3. Section 4provides the numerical
results and discussions, and nally, the concluding remarks of the
paper are provided in Section 5.
2. Investigated test case
A high-head Francis turbine model is used as the investigated
test case. The Francis-99 turbine model, provided by the Francis-99
workshop series [12], is a 1:5.1 scale model of a prototype Francis
turbine [13]. The runner consists of 15 full-length and 15 splitter
blades. The prototype and model net heads are about H
prototype
z
377 m and H
model
z12 m, respectively.
Fig. 1a and b show two cross-sections of the Francis-99 model.
The axial and horizontal in-plane velocity components have been
experimentally measured at a PIV plane. The PIV plane is shown as
a red line and a gray shaded area in the z-normal and y-normal
sections, respectively. The velocity measurements are reported on
three PIV lines, two horizontal lines (Lines 1 and 2), and one axial
line (Line 3). Moreover, the static pressure is reported for three
sensor locations, namely, VL2, DT5, and DT6. In the experiments,
the draft tube pressure sensors were piezoelectric, and only
instantaneous uctuation of pressure was measured [14]. Two
additional numerical probes (RP1 and RP2) are dened in the
rotating zone (Runner) to sample the pressure eld throughout the
sequence. The numerical probes are placed in the middle of one
runner passage (in between two neighboring main and splitter
blades) at different axial positions.
The current work concerns a startup sequence that commences
from the minimum load operating condition. The guide vanes are
nearly closed with an opening angle of
a
¼0.8
and the ow rate is
Q¼0.022 m
3
/s. The guide vanes open up, rotating around their
axes, and the ow rate increases. The transient sequence ends at
the BEP condition in which
a
¼9.84
and Q¼0.199 59 m
3
/s. The
runner rotational speed remains constant at
u
¼333 rpm during
the entire transient sequence.
3. Computational framework and numerical aspects
The CFD simulation is carried out with OpenFOAM-v1912
[15,16]. The governing equations are discretized using the nite-
volume approach on a collocated mesh. The current section
briey describes the governing equations and the employed nu-
merical methods and schemes. More detailed information about
the numerical aspects of the performed CFD simulation is provided
by Salehi et al. [17], who used the same approach for a shutdown
sequence of the same case.
3.1. Mathematical formulation
A transient incompressible turbulent ow can be modelled by
the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equa-
tions, given by
vU
j
vx
j
¼0;(1)
vU
i
vtþ
vðU
i
U
j
Þ
vx
j
¼1
r
vp
vx
i
þv
vx
j
n
vU
i
vx
j
u
i
u
j
!;(2)
where
r
u
i
u
j
represents the unknown Reynolds stress tensor. The
Shear Stress Transport (SST) based Scale-Adaptive Simulation
URANS model (i.e., SST-SAS) [18,19] is here employed for the
calculation of the Reynolds stress tensor. SST-SAS is a turbulence-
resolving URANS model, used for simulations of industrial tran-
sient ows. Its formulation decreases the local eddy viscosity to
resolve the turbulent spectrum and break-up of large eddies,
providing LES-like solutions. Several research studies veried the
performance of the SST-SAS model in the simulation of hydraulic
machinery ows [11,20e25].
3.2. Discretization schemes
The second-order backward implicit scheme is employed for the
discretization of the temporal derivative terms. The time step of the
simulation is chosen as
D
t¼1.25 10
4
s, corresponding to runner
and guide vane rotations of 0.25
and 1.625 10
4
in each time
step. The average and maximum CFL numbers at the highest ow
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
116 7
rate (BEP) are 0.025 and 55. It should be noted that the CFL number
is less than 2 for 99.4% of the cells.
The convective terms in the momentum equation are dis-
cretized using the Linear-Upwind Stabilised Transport (LUST)
scheme [26], which blends the central and second-order upwind
schemes with a blending factor of 0.75. In other words, the face
values are calculated blending 75% second-order central and 25%
second-order upwind schemes, balancing accuracy and numerical
stability. The second-order upwind scheme approximates other
convective terms (i.e. in kand
u
equations).
The Laplacian terms in the transport equations are estimated
using the second-order central scheme. An explicit non-orthogonal
correction due to the high skewness of the cells at some locations is
inevitable because of the complex geometry.
3.3. Pressure-velocity coupling
The PIMPLE pressure correction algorithm is employed for the
pressure-velocity coupling. It combines two pressure correction
algorithms, namely, SIMPLE [27] and PISO [28] as outer and inner
correction loops, respectively. A maximum of 10 outer correction
loops is performed in each time step, controlled by a residual cri-
terion. At most time steps the ow solution is converged after four
outer correction loops. Each outer loop conducts two inner
correction loops. After each inner loop, one additional non-
orthogonal correction loop is performed to assure convergence of
Fig. 1. Two sections of the Francis-99 model, showing PIV plane, velocity lines, pressure sensors.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
116 8
explicit terms. It has been shown that the OpenFOAM imple-
mentation of the pressure correction algorithm is in line with the
Rhie-Chow interpolation technique [29,30].
3.4. Boundary conditions
The guide vanes open up by rotating with a constant rotational
speed of 1.3
/s. The time-variation of the guide vane opening angle
is plotted in Fig. 2a. As seen in the gure, a smooth transition is
implemented at the start and stop of the rotation (t¼2 and t¼9s)
to minimize the numerical instability caused by the sudden
movement of the guide vanes. The total time of the sequence,
t¼12 s, corresponds to 66.52 runner revolutions.
The guide vane movement is imposed through an ad-hoc
developed boundary condition that requires the guide vane rota-
tional speed as input. Therefore, the rotational speed
u
of the guide
vanes is shown in Fig. 2b.
It is assumed that the inlet volume ow rate of the turbine varies
linearly with respect to the guide vanes angle. This assumption is
according to the Francis-99 workshop series recommendation due
to inaccurate measurements of the ow rate during transient
operation [12]. Hence, a time-varying spatially uniform velocity,
according to the ow rate, is imposed at the inlet of the spiral
casing. A xed turbulence intensity (I¼7%) and viscosity ratio (
n
t
/
n
¼100) is considered for the inow condition. The inlet pressure is
extrapolated from the inside domain using a zero-gradient
assumption. All quantities at the outlet boundary are computed
using the zero-gradient condition, except the pressure which is set
by a xed value.
As previously described, there are four different mesh regions in
the simulation (spiral casing, guide vanes, runner, and draft tube).
The Cyclic Arbitrary Mesh Interface (cyclicAMI)[31,32] was uti-
lized to transfer the information between the different domains.
In order to reach a statistically stationary state at minimum load
condition, the ow is solved for 4 s ow time corresponding to over
22 runner rotations, and then the startup sequence presented in
Fig. 2 is initiated.
3.5. Dynamic mesh framework
CFD analysis of the transient operation of Francis turbines in-
cludes two types of simultaneous mesh motion, i.e, mesh defor-
mation of the guide vane domain due to the rotation of each guide
vane and solid body rotation of the runner domain. Therefore, a
Laplacian displacement mesh morphing solver is employed to
deform the guide vane domain mesh while the solid-body rotation
function handled the runner rotations. In each time step, the mesh
is updated at the beginning of the rst PIMPLE outer correction
loop. Then, the face uxes are calculated based on the face swept
volumes and relative uid velocity [33,34].
The mesh morphing is governed by a Laplace equation, given by
V,ð
G
V
d
cell
Þ¼0;(3)
where
G
is the motion diffusivity and
d
cell
is the displacement
vector of the cell centers. The Laplace equation is solved for the cell-
centered displacement (
d
cell
) and then the solution is interpolated
to get the point displacements (
d
points
). Finally, the new point lo-
cations (at time tþ
D
t) are simply computed as
x
tþ
D
t
point
¼x
t
point
þ
d
t
point
:(4)
The motion diffusivity (
G
) is obtained using a quadratic inverse
distance scheme with respect to the guide vane surfaces.
Severe mesh deformation of the guide vane region due to the
large rotation of the guide vanes during the startup sequence could
potentially result in low-quality mesh cells and consequently
deterioration of convergence and accuracy of the numerical results.
Therefore, in this study, the mesh quality parameters were moni-
tored during the mesh deformation. The guide vane region was
remeshed two times at guide vane openings of
a
¼3.47
and
a
¼6.85
to maintain an acceptable mesh quality. More informa-
tion on the numerical aspects and mesh deformation, as well as the
open-source case and codes of the current study, is provided by
Salehi and Nilsson [35], as the same case and codes are employed in
the present work.
A block-structured mesh is created for the CFD simulation. The
mesh at BEP contains a total of 16 million cells (for more infor-
mation please see our previous studies [17,35]).
3.6. Parallel processing
The scotch [36] domain decomposition approach is used to split
the computation domain and distribute roughly equal loads to the
processors while minimizing their interconnections. The job is
submitted to a Linux cluster using 320 CPU cores. The full startup
sequence consumed a computational cost of 170,000 core hours.
4. Results and discussion
This section presents the results of the transient startup
sequence of the Francis-99 model turbine.
Fig. 2. Variation of (a) guide vane angle and (b) guide vane rotational speed during the startup sequence.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
116 9
4.1. Pressure uctuations
As previously described, a number of pressure probes, namely
VL2, DT5, DT6, RP1, and RP2, were dened in the computational
domain (see Fig. 1), and the variation of the static pressure is
recorded throughout the entire startup sequence. The experimental
results of the static pressure are available for the VL2 probe, while
only the pressure uctuations were monitored at DT5 and DT6. In
transient (time-varying) turbulent ows, as in the current case, the
obtained signals (for instance, pressure) consists of two different
parts, the mean and uctuating parts ðp
0
¼pp
̄
Þ. The mean signal
changes through time due to the variation of the operating condi-
tion. Therefore, in order to extract the uctuating pressure, the
instantaneous mean should be calculated. The present study em-
ploys the Savitzky-Golay nite impulse response lter [37] for
smoothing the obtained signals and calculating instantaneous
mean and uctuations. A variable window size is chosen to capture
the uctuations more accurately. The window is much smaller at
the start and end of the transient sequence, where the variation of
the pressure level due to the change in operating condition is
sharper.
Fig. 3 shows the time-variation of the static pressure, its
instantaneous mean, and the uctuations of the static pressure
from its instantaneous mean. In general, the numerical prediction
of the VL2 pressure (Fig. 3a) sufciently matches the experimental
data, although with slightly lower values at the BEP condition at the
end of the sequence. The maximum relative error, calculated as |
p
num
p
exp
|/p
exp
100, is 4.25%. Each plot contains a zoomed view
that covers a 90
rotation of the runner in either the stationary
minimum load or BEP condition. The VL2 zoomed views show clear
smooth pressure pulsations due to the Rotor-Stator Interaction
(RSI) in the vaneless space (between the runner blades and the
guide vanes). Since the runner consists of 30 full and splitter blades,
7.5 pressure pulsations can be seen in these zoomed views. The
vaneless space static pressure uctuates around a nearly constant
mean pressure at the minimum load condition. Some low-
frequency oscillations are also visible at the minimum load condi-
tion in the VL2 pressure, which could be due to large unsteady ow
structures in the massively separated ow in the draft tube. When
the startup sequence commences at t¼2 s, the guide vanes start
opening up, and consequently, the pressure increases with the
turbine ow rate growth. The rate of the pressure increment is
initially higher and then it reduces and reaches a constant level
until the end of the sequence. The numerical results suggest an
overall pressure rise from 160 kPa (at minimum load) to 174 kPa (at
BEP). The numerical results reach a stationary condition at t¼9s
when the sequence nishes. In contrast, the experimental pressure
results show that the ow still needs some time to reach the steady
condition, due to dynamics in the experimental open-loop hy-
draulic system. Some low-frequency oscillations are also visible in
the numerical pressure results after the initiation of the sequence.
These oscillations are most likely produced by large ow structures
formed in the draft tube in the low load conditions and will be
discussed in detail later. One can see such pulsations more appar-
ently in the uctuating pressure shown in Fig. 3b. Distinct periodic
oscillatory patterns are seen between t¼4.5 s and t¼6.5 s that are
probably caused by the formation and diminish of the RVR. The
static and uctuating pressure in one of the draft tube probes (DT6)
is also shown in Fig. 3c and d. There is not a clear sign of the RSI
uctuations in the presented zoomed views at BEP. Here again,
Fig. 3. Time-variation of static pressure (a and c) and its uctuations from the instantaneous mean (b and d) for two probe locations during startup.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1170
large oscillations are visible in the draft tube pressure at low load
conditions.
There are two statistically stationary phases in the whole
simulated sequence, namely, the initial minimum load and the nal
BEP conditions. The Fast Fourier Transform (FFT) analysis technique
enables us to identify the excited frequencies and their amplitude
of the obtained signals. Therefore, FFT was applied on the uctu-
ating part of the VL2 and DT6 pressures at both stationary condi-
tions, and the results are plotted in Fig. 4. It should be noted that in
the present study, all the frequencies are normalized by the runner
rotational frequency f
n
¼5.543 Hz. The runner blade passing fre-
quency f
b
¼30f
n
(15 full-length blades and 15 splitter blades) is the
dominant frequency in the VL2 probe. The amplitude of f
b
is much
larger at BEP compared to the minimum load condition. Peaks are
visible at the harmonics of the runner passing frequency (15f
n
and
60f
n
). A low frequency of approximately 0.3f
n
is also excited at the
minimum load condition of the VL2 pressure, which could be
explained by the large separated ow region in the draft tube at
such conditions. The draft tube uctuating pressure (DT6) seems to
be most excited at the above-mentioned low frequency. A moderate
peak can also be seen at the frequency of 15f
n
, corresponding to the
full-length blade passing frequency, as only full-length blades are
elongated to the draft tube. In other words, the DT6 draft tube
probe can sense the rotation of the full-length blades much more
than the splitters.
Due to the time-varying nature of the obtained signals in tran-
sient sequences, such as turbine startup, both the excited fre-
quencies and their amplitudes change throughout the sequence.
Hence, a Short Time Fourier Transform (STFT) analysis is required
for time-frequency analysis. STFT divides the full-time domain into
small subdomains and performs the Fourier transform on each
subdomain. The time-variations of the amplitudes of different
frequencies of the VL2 and DT6 uctuating pressures are illustrated
as spectrograms in Fig. 5. The runner blade passing frequency
(f
b
¼30f
n
) is the dominant frequency of the vaneless space pressure
throughout the whole sequence (Fig. 5a). The harmonic frequencies
(i.e., 15f
n
,45f
n
,60f
n
,75f
n
, etc.) are also clearly excited. A wide range
of excited stochastic frequencies are visible in the minimum load
condition (t<2 s), indicating a complex ow eld including large
separations and vortex breakup. When the guide vanes start to
open up and the ow rate increases, such frequencies diminish
slightly after t¼2 s. The zoomed view of the VL2 spectrogram
suggests the existence of low-frequency high-amplitude oscilla-
tions during the transient sequence. The RVR phenomenon is most
likely responsible for such types of pulsations. The DT6 spectro-
gram denotes a deterministic frequency of 15f
n
, corresponding to
the passing of the full-length blades. The RVR low-frequency os-
cillations are also clearly visible here.
The time-variation of the amplitude of different excited fre-
quencies is extracted from the STFT calculations and presented in
Fig. 6. For the VL2 sensor, the runner blade passing frequency (30f
n
)
is dominant throughout the whole sequence. The amplitude is
nearly constant and slightly increases when the turbine reaches the
BEP condition. The amplitude of the RVR frequency (0.3f
n
)is
increased with the initiation of the transient sequence and then
decreases as the large RVR structures diminish when the turbine
approaches the BEP condition. On the other hand, for the DT6
probe, the amplitude of 0.3f
n
is dominant in the entire sequence,
except for the BEP condition where the large draft tube vortical
structures are washed away. A sudden rise and then decrease is
observed in the amplitude of 0.3f
n
in the middle of the sequence
due to the formation and collapse of the RVR.
Hydraulic turbine draft tube cones generally experience two
different types of pressure pulsations at low load conditions
[38,39]. The pressure signals can be decomposed into synchronous
and asynchronous modes. The synchronous mode (also known as
the plunging mode) is somehow similar to the water hammer
pressure waves which travel throughout the whole hydraulic sys-
tem. The asynchronous mode (rotating mode), produced by the
local instabilities such as the RVR, is only active in the cross-
sections. The pressure signal decomposition can be performed us-
ing the unsteady signals of two different pressure probes which are
positioned at the opposite sides of the draft tube cone with the
same height, through
p
sync
¼p
1
þp
2
2ðSynchronous component or plunging modeÞ;
p
async
¼p
1
p
2
2ðAsynchronous component or rotating modeÞ:
(5)
A few researchers have studied the draft tube pressure signal
decomposition to identify the appearance of plunging and rotating
modes at low load conditions of hydraulic turbines (e.g.,
Refs. [38e42]). However, no investigation can be found in the
literature on the decomposition of plunging and rotating modes of
a hydraulic draft tube during a startup sequence. Extracting such
modes from pressure signals can be particularly helpful for
explaining the appearance and collapse of the RVR in transient
sequences like shutdown and startup.
In the present test case, the DT5 and DT6 sensors are placed on
opposite sides (180
apart) of the conical part of the draft tube and
could be used for signal decomposition. First, the synchronous and
Fig. 4. Fast Fourier Transform of the uctuating pressure in the stationary conditions at the beginning and end of the sequence (minimum load and BEP).
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1171
asynchronous modes of the draft tube pressure are computed by
inserting DT5 and DT6 pressure signals into Eq. (5) and then the
uctuating parts of each mode are extracted. Fig. 7a presents the
time-variation of the uctuating part of both modes of the draft
tube pressure. The high-frequency pressure uctuations are mostly
captured by the synchronous mode. In other words, such uctua-
tions are in-phase for both probes and are sensed by DT5 and DT6
probes at the same time. Low-frequency oscillations are seen in
both synchronous and asynchronous modes at minimum load
conditions which decreases when the turbine approaches the BEP
condition, indicating that complex ow structures formed at deep
part load conditions have strong plunging effects. On the other
hand, the asynchronous mode does not show any clear sign of high-
frequency uctuations, and only low-frequency pulsations are
observed. The sudden rise of rotating mode after t¼4 s could be a
sign of the formation of rotating vortical structures (i.e., RVR) which
decays with further increasing turbine load after t¼6 s. After t>7s
the turbine approaches the BEP condition and the large vortical
ow structures inside the draft tube cone vanish, the pressure
uctuations predominantly contribute to the synchronous mode,
and the asynchronous mode is rather negligible at the design
condition.
In order to better understand the formation and collapse of the
RVR and its impact on the decomposed pressuremodes, a bandpass
lter with a narrow frequency range of 0.1f
n
, centered at the
fundamental frequency of RVR (0.3f
n
), was applied to the decom-
posed signals to isolate the RVR effects in the plunging and rotating
modes. As previously seen in Figs. 4 and 5, the frequency of 0.3f
n
is
Fig. 5. Spectrogram of uctuating pressure signal throughout startup sequence.
Fig. 6. Variation of uctuating pressure amplitude over time.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1172
the dominant frequency inside the draft in a wide range of low-load
conditions. The ltered signals displayed in Fig. 7b reveal that the
plunging effect is mostly the dominant mode at minimum load and
deep part load conditions (before t¼4 s), suggesting that dis-
integrated stochastic ow structures at such conditions primarily
cause axial pulsations that are sensed throughout the whole system
at the same time. Nonetheless, when the startup sequence of the
turbine initiates the rotating effects gradually increase in time
while the plunging mode weakens. The fact that the rotating mode
is dominant between t¼4.5 s and t¼6.5 s could be a clear sign of
the formation and collapse of the RVR. As expected, no large
vortical structures should exist at the BEP condition. Therefore,
both the plunging and rotating modes decay after t¼7s.
A time-dependent frequency analysis was performed on the
decomposed signals and the results are shown as spectrograms in
Fig. 8. Here again, synchronous uctuations are observed at mini-
mum load and deep part load conditions (t<4 s), while asyn-
chronous pressure pulsations can be detected between t¼4.5 s and
t¼6.5 s. More specially, the fundamental frequency of the RVR is
much more pronounced in the rotating mode than the plunging
mode during the presence of RVR (between t¼4.5 s and t¼6.5 s),
as also pointed out by Goyal et al. [42].
As explained in Section 2, two probes are dened in the rotating
domain of the runner, namely, RP1 and RP2 (see Fig. 1b) and their
pressure variation throughout the startup sequence is demon-
strated in Fig. 9. Predictably, the RP1 pressure is generally higher
than that at RP2, as it is closer to the runner inlet. Both pressure
probes exhibit a gradual rise during the transient sequence. The
RP1 pressure increases by 13.3 kPa, whereas the RP2 pressure
grows by 7.2 kPa. High-frequency RSI uctuations are visible
through the provided zoomed views. Here the probes are rotating
with the runner and therefore the pressure is expected to show a
peak whenever the probe is passing a guide vane trailing edge. The
uctuating part of the pressure shows stronger high-frequency RSI
uctuations for RP1 as it is closer to the guide vanes. Both probes
contain low-frequency oscillations which are slightly amplied
between t¼4.5 s and t¼6.5 s.
Fig. 10 plots the FFT of the uctuating pressure of the rotating
probes at the stationary conditions (minimum load and BEP). As
expected, the uctuations have a dominant frequency at the guide
vane passing frequency (f
gv
¼28f
n
) which is stronger at BEP. The rst
harmonic of this frequency (f
gv
¼56f
n
) also shows a small peak.
Additionally, some low-frequency peaks, due to the formation and
breakup of vortical ow structures are detected by the FFTanalysis at
minimum load conditions. Since the probes are rotating, the runner
rotation frequency (f
n
) and its rst few harmonics (2f
n
,3f
n
,4f
n
,etc.)
are also excited at both conditions. An STFT analysis can further
explain the variation of the amplitudes of the excited frequency
during the transient sequence. Fig. 11 presents rather similar trends
for the time-variation of the amplitudes for both rotating probes. The
guide vane passing frequency is a deterministic and dominant fre-
quency during the whole sequence. The zoomed views (Fig. 11band
d) denote that at minimum load condition, a vast range of stochastic
frequencies is excited which decay after a short while into the
Fig. 7. Time-variation of uctuating synchronous and asynchronous pressure modes. (a) Decomposed signal and (b) bandpass ltered signals.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1173
transient sequence. Although an excited low frequency is observed
during the formation of the RVR (between t¼4.5 s and t¼6.5 s), the
value of that frequency is larger than the RVR fundamental fre-
quency. When the turbine reaches the design condition, the excita-
tion of the runner rotation frequency (f
n
) and its harmonics are
clearly visible in the zoomed view of both probes.
4.2. Force pulsations
Sharp variations and oscillations of forces and moments exerted
on different parts of hydraulic turbines during transient operations
could cause serious damages and negatively affect the lifetime of
the turbine. Therefore, performing force analysis during transient
sequences like the startup is essential for mitigating such damaging
effects. Forces and moments acting on the runner surfaces (i.e., hub,
shroud, main blades, and splitters) as well as a single guide vane are
monitored during the startup sequence and the results are pre-
sented in this section. Although the runner force analysis is per-
formed on the whole runner in the present work, investigating the
uctuating forces on one individual runner blade is suggested for
future studies as it could be benecial for assessing the fatigue ef-
fects and lifetime.
Fig. 12 shows the xand zcomponents of the force acting on the
runner, as well as the runner torque (zcomponent of moment
Fig. 8. Spectrogram of uctuating part of (a) plunging and (b) rotating modes.
Fig. 9. Time-variation of static pressure (a and c) and its uctuations from the instantaneous mean (b and d) for two rotating probes (RP1 and RP2) during startup.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1174
vector). F
x
shows strong low-frequency oscillations between
t¼4.5 s and t¼6.5 s, in which the vortex rope is formed and rotates
around the turbine axis (Fig. 12a). At the BEP condition, the x-force
is uctuating around a non-zero value, indicating that the ow
distribution around the runner is not perfectly axisymmetric. The
zoomed view of the uctuating part of F
x
(Fig. 12d) denotes both
high and low-frequency oscillations at the BEP condition. The axial
force (z-force) is initially oscillating at negative (downward) values
at the minimum load condition and shortly after the commence-
ment of the transient sequence it increases and becomes positive
(upward) and continues its growth until the BEP condition
(Fig. 12b). Comparing the uctuating parts of F
z
and F
x
suggests that
the low-frequency oscillations during the formation and collapse of
the RVR are much weaker for F
z
(compared to their corresponding
instantaneous mean). In other words, the RVR mostly affects the
horizontal (radial) forces rather than the axial force. This is
compatible with the signal decomposition analysis presented in
Section 4.1. The axial forces are expected to oscillate with the
plunging mode of the RVR, while the radial forces vary with the
rotating mode. As elaborated in Fig. 7b, the rotating mode of the
RVR is the dominant mode during t¼4.5 s and t¼6.5 s, and thus
the radial force oscillations are greater. The variation of the runner
axial torque through time exhibits a smooth linear growth in ab-
solute value of the torque with turbine load increase, from 29.8 N m
at minimum load to 630.3 N m at BEP. It is also seen that the
uctuating part of the torque signal is negligible with respect to its
instantaneous mean. Fig. 12f reveals that the formation of the RVR
barely affects the uctuating torque and the maximum uctuating
Fig. 10. Fast Fourier Transform of the uctuating pressure of the rotating probes (RP1 and RP2) during the stationary conditions (minimum load and BEP).
Fig. 11. Spectrogram of uctuating pressure of rotating probes (RP1 and RP2) throughout startup sequence.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1175
torque that occurs at the minimum load condition is less than
1.5 N m .
The spectrograms in Fig. 13 exhibit STFT analysis of the runner
forces. The runner blade passing frequency is a deterministic
dominant frequency throughout the entire sequence for both the
horizontal and axial forces (Fig. 13a and c). The f
b
frequency is less
isolated in the f
z
force and it is more affected by the wide range of
stochastic frequencies. The zoomed view of the F
x
spectrogram
(Fig. 13b) shows mainly stochastic frequencies at minimum load
that vanish with the initiation of the sequence. Then the impacts of
the RVR on the horizontal forces are clearly seen as low-frequency
oscillations. However, the axial forces present a wide range of
excited low-frequencies that are not limited to the formation or the
collapse of the RVR (Fig. 13d). Here again, we can deduce that
complex ow structures at deep part load have strong plunging
effects and result in uctuations of the axial force. The runner
rotation frequency (f
n
) has an important role in the variation of the
horizontal forces at the BEP condition.
To further assess the variation of forces in the Francis-99 startup
sequence, the forces and moments acting on a single guide vane are
studied. Fig. 14 depicts the time-variation of the radial force and
torque (axial moment around the guide vane rotational axis) of the
guide vane nearest to the volute tongue. The negative torque acts as
to open the guide vane and vice versa. Both plots display smooth
variations during the transient sequence with nearly constant
ranges of RSI uctuations, which are larger for the radial force. The
F
r
is maximum at minimum load and reduces with load increase.
More importantly, during the formation and collapse of the RVR
(between t¼4.5 s and t¼6.5 s), F
r
oscillates with some low-
frequency oscillation but M
z
does not show any impact from the
RVR. The spectrogram of F
0
r
, illustrated in Fig. 15, demonstrates a
broad span of stochastic frequencies at the minimum load condi-
tion. This could be the impact of complex separated ow structures
formed behind the trailing edge of the guide vanes at minimum
load condition. Fig. 16 employs an iso-surface of
l
2
¼7500 s
2
to
reveal these structures. As expected, the 0.3f
n
frequency is
distinctly evident during the existence of the RVR in Fig. 15.
4.3. Velocity variation
The velocity eld is sampled through the entire startup
sequence along the three lines shown in Fig. 1 and the numerical
results are compared to the experimental data for validation. The
variation of the ow eld is carefully examined to understand and
explain the draft tube ow eld during the turbine startup. It
should be mentioned that in this work, the horizontal velocity (U)
represents the velocity component parallel to Line 1 and 2 (similar
but not identical to radial velocity), while the normal velocity (V)is
the velocity component normal to the PIV plane (similar but not
identical to tangential velocity).
Figs. 17e19 present the time-variation of the numerical velocity
components along the three PIV lines (previously shown in Fig. 1).
The axial and horizontal velocity components are compared to the
experimental measurements. The variable srepresents the curve
length of each line, which is normalized by its maximum in all
plots. The comparison reveals that the numerical axial velocity (W)
trend is quite similar to the experiment and thus is adequately well
predicted by the simulation. At minimum load condition, the axial
velocity direction is upward all over both Lines 1 and 2, varying
with low-frequency oscillations. This indicates a massive reversed
ow region that covers the entire extent of both lines, while the
small mass ow through the draft tube cone passes outside those
lines. Then, when the guide vanes start to open up at t¼2 s, the
reversed ow region gradually gets smaller. The low-frequency
oscillations amplify with the establishment of the RVR. After
reaching the BEP condition, the reversed ow region completely
vanishes and the ow is entirely in the downward direction. At the
design condition, the magnitude of Wincreases with the distance
Fig. 12. Variation of components of forces acting on the runner during startup.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1176
Fig. 13. Spectrogram of uctuating forces.
Fig. 14. Variation of radial force and torque acting on one guide vane.
Fig. 15. Spectrogram of uctuating part of radial force acting on one guide vane.
Fig. 16. Complex vortical structures behind guide vane trailing edges at minimum load
condition. Iso-surface of
l
2
¼7500 s
2
.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1177
to the draft tube walls, whereas it is slightly decreased at the center
(s/s
max
¼0.5) due to the runner cone wake. The uctuations around
the center are induced by the vortex shedding created behind the
runner cone. The Wcontours on Line 3, which is located at the
center of the draft tube, denote the existence of the same reversed
ow region which is diminished just before reaching the design
condition. Both the numerical and experimental data show a rather
sharp change in velocity direction around t¼8s.
Fig. 18 indicates that the horizontal velocity mainly uctuates
around zero in both the numerical and experimental data. The
uctuations are initially moderate at minimum load condition and
clearly magnify during the formation and collapse of the RVR (be-
tween t¼4.5 s and t¼6.5 s). Then, at the design conditions, both
the numerical and experimental results show near-zero Uvalues
with small uctuations.
The normal velocity component (V) was not measured in the
experimental study. Therefore, Fig. 19 displays only the numerical
results of the time-variation of the normal velocity. A strong
swirling ow exists at minimum load condition. The commence-
ment of the startup sequence and load increase temporarily re-
duces the normal velocity. However, the Vcomponent remarkably
increases and oscillates with the creation of the RVR. After the
collapse of the RVR, the Vvelocity smoothly reduces. Suddenly
before the steady BEP condition, the direction of the Vvelocity and
consequently the swirl orientation changes and a weak counter-
rotating ow exists at the design condition.
The time-variation of the velocity eld is further assessed using
two points, namely Point 1 and 2 (see Fig. 1), in Fig. 20. Both point
are placed on Line 1 at radial positions of R
Point1
¼46.5 mm and
R
Point2
¼125.80 mm, corresponding the normalized curve lengths
of s/s
max
¼0.367 and 0.038, respectively. Generally, the horizontal
velocity oscillates around a near-zero instantaneous mean value.
After t¼4 s, large oscillations are visible in the Uvelocity of Point 2,
which is closer to the draft tube wall, whereas Point 1 (close to the
center) mainly experience such large uctuation later (After t¼6 s).
This implies that the rotating vortical structures (i.e., RVR) form far
away from the draft tube center. With load increment, the vortical
structures integrate and form a more stable vortex around the
center, which results in remarkable uctuations of Uon Point 1
after t¼6 s. At the BEP condition, where normally a slender central
vortex is observed, Point 1 uctuates to some extent while Point 2 is
quite stable.
The axial velocity initially oscillates around a positive (upward)
instantaneous mean at both points in Fig. 20. The turbine load
increment gradually increases the magnitude of the axial velocity
to a stable point at BEP. The numerical and experimental data show
compatible trends. Similar to the Ucomponent, the Wcomponent
at Point 1 does not show large oscillations until t¼6 s, while at
Point 2 it exhibits extensive uctuations already after t¼4s.
To examine the swirling ow in the draft tube, the normal ve-
locity (V) is presented in Fig. 20 as well, although no experimental
data is available. Positive values of Vat Points 1 and 2 indicate a
swirling ow in the same direction as the runner rotation, and vice
versa.
Water turbines are designed such that a nearly non-swirling
ow leaves the runner at BEP. The presence of a weak swirling
ow at the design condition could help the ow to stay attached to
the draft tube walls. A residual positive (in the same direction of the
runner) swirling ow exists at partial load condition, while a higher
ow rate than design condition (high load) forms a negative
(counter-rotating) residual swirl.
As expected, at minimum load condition a considerable positive
tangential component exists, especially on Point 2 which is further
from the draft tube center, indicating a large remaining positive
Fig. 17. Time-variation of axial velocity (W) along experimental PIV lines.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1178
swirl in the draft tube. Similar to Uand W, the Vvelocity at Point 2
experiences the large oscillations of the RVR sooner than at Point 1.
When the large rotational RVR structures in the draft tube are
diminished, the Vvelocity decreases as the load increases. It settles
at an insignicant negative value, indicating a weak counter-
rotating swirl at the design condition which could be intentional
to keep the ow attached to the draft tube walls.
4.4. Flow structures in the draft tube
The formation and breakup of the vortical ow structures inside
the draft tube during the startup sequence is analyzed in this sec-
tion. Based on a previous study [17], the
l
2
-criterion is employed to
identify and visualize the vortical ow structures. It assumes a
vortex to be a region with two negative eigenvalues of the S
2
þ
U
2
tensor [43], where Sand
U
are the strain and rotation tensors, given
by
S¼1
2VUþVU
T
;
U
¼1
2VUVU
T
:(6)
Therefore, a vortex can be identied as a region with a negative
second largest eigenvalue,
l
2
. The OpenFOAM function object
Lambda2 changes the sign of the S
2
þ
U
2
tensor eigenvalues, and
thus a positive value should be used for the creation of the
l
2
iso-
surfaces.
Fig. 18. Time-variation of horizontal velocity (U) along experimental PIV lines.
Fig. 19. Time-variation of normal velocity (V) along experimental PIV lines.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1179
A video is supplied with the article for the readers to see the
time-evolution of the vortical structures during the startup oper-
ation. Fig. 21 utilizes an iso-surface of
l
2
¼750 s
2
to unveil the
evolution of the draft tube vortical structures by 12 snapshots
during the turbine startup sequence. The corresponding times (t),
guide vane opening angles (
a
), and the turbine load (normalized
ow rate Q/Q
BEP
) are denoted below each gure. The transient
sequence starts from the minimum load condition. At this condi-
tion, a massively separated ow eld with a signicant residual
positive swirl exists downstream of the runner (Fig. 21a). As a
result, large persistent vortical structures are visible upstream of
the draft tube elbow that produces low-frequency pulsations in the
ow eld and turbine forces. When the transient sequence initiates
at t¼2 s, the guide vanes start to open up and the turbine load
increases. Consequently, the growing ow rate washes down the
large vortical structures (Fig. 21b and c).
At t¼4.2 s (Fig. 21d) the aforementioned large vortices are
completely vanished and instead elongated vortical structures are
formed downstream the runner. These are formed due to the
instability of the shear layer between the swirling downward and
separated upward ow regions. The separated region is still quite
large and thus the shear layer is close to the draft tube wall. Four
distinct draft tube vortices are formed in this region. Continuing the
startup sequence, the vortical ow structures develop and expand
(Fig. 21e). Thereafter, further opening the guide vanes, the stagnant
(reversed ow) region shrinks. Accordingly, the unstable vortical
structures gradually integrate and form a large unstable coherent
structure that is helically wrapped around the stagnant region
(Fig. 21f and g). An integrated rotating vortex rope is clearly
distinguishable at time t¼6.0 s (Fig. 21g) due to KelvineHelmholtz
instability of the sharp shear layer. This is in accordance with the
results presented in Sections 4.1e4.3, where distinct low-frequency
high-amplitude oscillations were observed between t¼4.5 s and
t¼6.5 s.
The additional augmentation of the ow rate decreases the
runner residual swirl and squeezes the stagnant region. Inevitably,
the integrated central vortex becomes more stable and moves to-
ward the center of the draft tube (Fig. 21h and i). One can see small
vortices that rotate around the central axis and merge into a stable
slender vortex that is attached to the runner cone (Fig. 21j and k).
Finally, the turbine reaches the BEP condition where a small
negative (counter-rotating) swirl leaves the runner and forms a
stable and nearly stationary vortex at the center of the draft tube.
5. Conclusion
The present paper provides a detailed numerical study on the
pulsations originated by the transient ow features during the
startup of a high-head model Francis turbine. Our results contribute
to a better knowledge of the evolution of the ow in the hydraulic
turbines during startup operation that could provide the possibility
to avoid harmful conditions and have a better estimate of mainte-
nance intervals and costs.
The high-frequency pulsations generated by the blade passing
rotor-stator interaction (30f
n
) were the dominant excited frequency
in the vaneless space throughout the entire startup sequence, while
the guide vane passing frequency (28f
n
) was the dominant mode
for the pressure probe inside the runner rotating domain. Low-
frequency high-amplitude oscillations were observed in the mid-
dle of the sequence, suggesting the formation of the RVR. A signal
decomposition of the draft tube pressure indicated that the com-
plex ow structures formed at minimum and deep part load con-
ditions have strong plunging (synchronous) effects. Increasing
turbine load gave a sudden rise in rotating (asynchronous) mode
during the formation of the RVR.
Low-frequency oscillations of the RVR affect the radial forces
Fig. 20. Time-variation of horizontal (U), axial (W), and normal (V) velocity components on Points 1 and 2 compared to the experimental PIV data.
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1180
acting on the runner more than the axial force. The blade passing
frequency is less isolated in the axial force compared to the radial
component. The axial force is greatly affected by the plunging ef-
fects at the minimum load condition and thereby its STFT shows
mainly stochastic frequencies. A frequency analysis of the
uctuating radial force exerted on the guide vanes revealed a broad
span of stochastic frequencies at the minimum load condition due
to the massively separated ow eld behind the nearly closed guide
vanes.
The velocity eld in the draft tube revealed the presence of a
Fig. 21. Illustration of draft tube vortical structures using an iso-surface of
l
2
¼750 s
2
at different times corresponding to different guide vane openings (
a
).
S. Salehi and H. Nilsson Renewable Energy 188 (2022) 1166e1183
1181
large quasi-stagnant region with a large positive residual swirl that
reduces during the sequence. Large persistent vortical structures
were observed inside the draft tube at the minimum load condi-
tion. They are responsible for the low-frequency oscillations in such
conditions. Gradually increasing the turbine load results in an
integration of the unstable vortical structures and the formation of
the RVR. At BEP a slender stable vortical structure was observed
near the center of the draft tube.
CRediT authorship contribution statement
Saeed Salehi: Conceptualization, Methodology, Software,
Investigation, Writing ereview &editing. Håkan Nilsson:
Conceptualization, Supervision, Writing ereview &editing,
Funding acquisition.
Declaration of competing interest
The authors declare that they have no known competing
nancial interests or personal relationships that could have
appeared to inuence the work reported in this paper.
Acknowledgements
The current research was carried out as a part of the Swedish
Hydropower Centre - SVC. SVC is established by the Swedish En-
ergy Agency, EnergiForsk and Svenska Kraftn
at together with Luleå
University of Technology, The Royal Institute of Technology,
Chalmers University of Technology and Uppsala University, www.
svc.nu.
The computations were enabled by resources provided by the
Swedish National Infrastructure for Computing (SNIC) at NSC
partially funded by the Swedish Research Council through grant
agreement no. 2018e05 973.
The investigated test case is provided by NTNU eNorwegian
University of Science and Technology under the Francis-99 work-
shop series.
Appendix A. Supplementary data
Supplementary data to this article can be found online at
https://doi.org/10.1016/j.renene.2022.01.111.
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... Recently, Salehi et al. [41,42] presented a detailed numerical analysis of the transient flow field of the Francis-99 turbine during the shutdown and stratup sequences. Different physical aspects of the flow field, such as pressure fluctuations, force analysis, velocity field variation, draft tube vortical structures, etc., were thoroughly analyzed. ...
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