1 3
Exp Fluids (2015) 56:215
DOI 10.1007/s00348-015-2085-5
RESEARCH ARTICLE
Study of the vortex‑induced pressure excitation source
in a Francis turbine draft tube by particle image velocimetry
A. Favrel
1
· A. Müller
1
· C. Landry
1
· K. Yamamoto
1
· F. Avellan
1
Received: 4 August 2015 / Accepted: 9 November 2015 / Published online: 26 November 2015
© Springer-Verlag Berlin Heidelberg 2015
changes and loses its coherence, leading to a drastic reduc-
tion in the intensity of the induced excitation source.
1 Introduction
Beyond a critical degree of swirl, rotating flows experience
a hydrodynamic phenomenon, commonly called vortex
breakdown. It is characterized by a sudden change in the
flow structure with the formation of a stagnant point and
a recirculation zone (Lucca-Negro and O’Doherty 2001).
Under certain conditions, the development of a vortical
structure precessing around the zone of recirculation may
be observed (Gupta et al. 1984). This phenomenon, referred
to as the precessing vortex core (PVC), is encountered in a
wide range of engineering applications, leading to the pro-
duction of an abundant literature reporting experimental
and theoretical investigations (see Escudier 1987 and Syred
2006 for a review).
In the case of Francis turbines operating with a discharge
lower than the nominal discharge (part-load condition), the
vortex breakdown arises immediately at the runner outlet
and produces a precessing helical vortex core in the draft
tube. Its rotational frequency lies between 0.2 and 0.4 times
the runner frequency (Nishi et al. 1982). The vortex preces-
sion in the draft tube acts as an excitation source for the
hydromechanical system, inducing the propagation of pres-
sure fluctuations. In case of resonance, pressure pulsations
of high amplitude occur in the draft tube, which may put
at risk the stability of the power plant (Rheingans 1940).
As such off-design operations are increasingly required for
the large-scale integration of intermittent energy sources
into the grid, such as solar and wind, an accurate assess-
ment of the stability of hydropower plants over a broad
operating range is essential. This led to the development
Abstract Francis turbines operating at part-load experi-
ence the development of a precessing cavitation vortex
rope at the runner outlet, which acts as an excitation source
for the hydraulic system. In case of resonance, the result-
ing pressure pulsations seriously compromise the stabil-
ity of the machine and of the electrical grid to which it is
connected. As such off-design conditions are increasingly
required for the integration of unsteady renewable energy
sources into the existing power system, an accurate assess-
ment of the hydropower plant stability is crucial. However,
the physical mechanisms driving this excitation source
remain largely unclear. It is for instance essential to estab-
lish the link between the draft tube flow characteristics and
the intensity of the excitation source. In this study, a two-
component particle image velocimetry system is used to
investigate the flow field at the runner outlet of a reduced-
scale physical model of a Francis turbine. The discharge
value is varied from 55 to 81 % of the value at the best effi-
ciency point. A particular set-up is designed to guarantee a
proper optical access across the complex geometry of the
draft tube elbow. Based on phase-averaged velocity fields,
the evolution of the vortex parameters with the discharge,
such as the trajectory and the circulation, is determined
for the first time. It is shown that the rise in the excitation
source intensity is induced by an enlargement of the vor-
tex trajectory and a simultaneous increase in the precession
frequency, as well as the vortex circulation. Below a cer-
tain value of discharge, the structure of the vortex abruptly
* A. Favrel
arthur.favrel@epfl.ch
1
Laboratory for Hydraulic Machines, Ecole Polytechnique
Fédérale de Lausanne, Av. de Cour 33 bis, 1007 Lausanne,
Switzerland
Exp Fluids (2015) 56:215
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215 Page 2 of 15
of one-dimensional flow models, where the contributions
of Dörfler (1982), Couston and Philibert (1998), and more
recently Alligné et al. (2014) have to be cited. Generally,
the cavitation draft tube flow is modelled by a cavitation
compliance parameter according to the definition intro-
duced by Brennen and Acosta (1976), while the excitation
source is represented by a source term in the momentum
equation.
These models however require experimental validation.
Particle image velocimetry is commonly used for the inves-
tigation of flow instabilities in hydraulic machines, from
pump-turbines (Guggenberger et al. 2014; Widmer et al.
2011) to Francis turbines. In case of the latter, the axial
velocity field in a vertical section of the draft tube cone was
recently measured both at part-load (Iliescu et al. 2008;
Kirschner et al. 2012) and at full-load conditions (Müller
et al. 2013). From an analytical point of view, Susan-Resiga
et al. (2006) modelled the mean swirling flow at the run-
ner outlet by three elementary vortices, whose parameters
are identified on the basis of time-averaged velocity pro-
files obtained experimentally. Kuibin et al. (2013) derived
an analytical model recovering the time-averaged velocity
profiles and the precession frequency. However, despite the
large quantity of research dealing with the part-load issue
(see Nishi and Liu 2013 for a review), the exact physical
mechanisms driving the excitation source remain unclear
and sparsely documented. A better understanding of this
aspect is an important step towards an accurate prediction
of pressure fluctuations and their transposition from the
model to the prototype scale, which represents the ultimate
challenge.
The present work aims at investigating the influ-
ence of the discharge on the vortex rope parameters and
structure, and at highlighting how the intensity of the
excitation source is linked to the vortex rope dynamics.
The tangential flow field is investigated at the outlet of
a reduced-scale physical model of a Francis turbine by
means of a two-component PIV system. Measurements
are taken in two different horizontal cross sections of the
draft tube cone. The unsteady wall pressure is measured
in order to describe the evolution of both the excitation
frequency and the amplitude of pressure pulsations with
the discharge. The measurements are taken in cavita-
tion-free conditions in order to exclude the influence of
parameters related to the presence of a cavitation volume.
The latter drastically reduces the natural frequency of the
system (Landry et al. 2014), which results in the risk of
resonance (Favrel et al. 2014). Moreover, the presence of
cavitation in the vortex core is prone to modify the vortex
dynamics and consequently to influence the excitation
source.
The test case is presented in Sect. 3, together with the
evolution of both the excitation frequency and the ampli-
tude of the pressure pulsations with the discharge, leading
to the identification of three different flow regimes. The
set-up for the PIV measurements is presented in Sect. 4,
along with the methodology for calculating the evolution of
the velocity fields over one precession cycle. The identifi-
cation of both the vortex core limits and the vortex centre
trajectory is also included. The behaviour of the precess-
ing vortex core parameters as a function of the discharge is
presented in Sect. 5. The used methodology and the physi-
cal mechanisms inducing the excitation source are finally
discussed in Sect. 6.
2 Background
If the pressure level in the draft tube is decreased, the
cavitation development is promoted and the vortex rope
appears. A visualization is presented in Fig. 1a for the
present test case. The periodical precession of this vortex
induces the propagation of pressure fluctuations in the
hydraulic system at the precession frequency. In the draft
tube cone, these pressure fluctuations can be decom-
posed into two distinct components, the convective one
and the synchronous one (Nishi et al. 1982). The convec-
tive component is a local pressure fluctuation induced
by the rotation of the pressure field with the vortex. It
is directly related to the local convection of the precess-
ing vortex core in the draft tube cone. The synchronous
component has an equal phase and amplitude for pres-
sure sensors located in the same cross section of the cone
and is able to propagate into the entire hydraulic system.
Dörfler (1982) identified this component as the result of
the excitation source induced by the vortex precession in
the draft tube elbow. In cavitation conditions, the pres-
ence of a gaseous volume causes a significant drop of
the wave speed in the draft tube and therefore of the sys-
tem’s eigenfrequencies. When the first eigenfrequency
approaches the precession frequency with a growing
cavitation volume, the synchronous pressure component
becomes dominant and the system can enter into reso-
nance, as illustrated in Fig. 1b.
However, in cavitation-free conditions, it is assumed that
the first eigenfrequency of the system is sufficiently high
to avoid any amplification of the synchronous component.
Therefore, the evolution of both the excitation source inten-
sity and the excitation frequency with the discharge value
can be quantified in cavitation-free conditions by measur-
ing pressure fluctuations directly in the feeding pipe, where
they are purely synchronous.
Exp Fluids (2015) 56:215
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3 Test case
3.1 Reduced‑scale model
The test case is a 1:16 reduced-scale physical model of a Fran-
cis turbine with a specific speed of
ν = 0.27
. It is installed on
a closed-loop test rig of the EPFL Laboratory for Hydraulic
Machines (see Fig. 2). The facility includes two axial double-
volute pumps generating the specified head. The value of the
discharge is adjusted through the guide vanes opening of the
turbine model. The cone of the draft tube is made of Plexi-
glas, providing an optical access for the laser sheet of the PIV
system. Flush-mounted piezo-resistive pressure transducers
are installed to measure the absolute wall pressure at different
locations of the test rig. Two pairs of sensors are installed in
two different cross sections of the draft tube cone. The pres-
sure sensors located in the same cross section are separated
with an angle equal to
π
. In addition, one sensor is installed in
the feeding pipe to evaluate the amplitude of the synchronous
pressure component independently of the convective compo-
nent observed in the draft tube cone.
3.2 Operating conditions
Operating conditions in hydraulic machines are character-
ized by two non-dimensional parameters, the speed factor
n
ED
and the discharge factor
Q
ED
which are defined accord-
ing to IEC Standards (1999) by:
where n is the runner frequency, Q is the discharge, D is the
runner outlet diameter, and E is the specific hydraulic energy.
In the present investigation, the speed factor is kept constant
at the rated value of
n
ED
= 0.288
, which corresponds to the
value at the best efficiency point (BEP). The runner frequency
n and the hydraulic specific energy
E = gH
are kept constant
at the values
n = 13.33 Hz
and
E = 263 J kg
−1
, respectively.
Pressure measurements are first taken in a wide range of
operating conditions in cavitation-free conditions by modify-
ing step by step the discharge from 50 to 100 % of the dis-
charge value
Q
0
at the BEP, in order to investigate the influ-
ence of the discharge on both the precession frequency and
the amplitude of the synchronous pressure pulsations.
3.3 Pressure fluctuations related to different flow
regimes
In cavitation-free conditions, pressure oscillations at a dis-
tinct frequency equal to about 0.25 times the runner fre-
quency are observed for discharge values lower than 85 %
of the value at the BEP. This indicates the development of a
precessing vortex rope in the draft tube. For discharge val-
ues between 50 and
85 %
of the value
Q
0
, the precession
frequency
f
PVC
is determined by computing the auto-spec-
tral density function of the pressure signal measured in the
feeding pipe. Its evolution with the discharge is presented
in Fig. 3a. The corresponding amplitude
G
xx
(f
PVC
)
of the
auto-spectra is also determined, and its evolution with the
discharge is plotted in Fig. 3b.
The results presented in Fig. 3 suggest that three different
flow regimes occur in the range of the investigated operating
(1)
n
ED
=
nD
√
E
(2)
Q
ED
=
Q
D
2
√
E
Vortex
Precession
Flow
direction
(a)
0 5 10 15
−0.1
0
0.1
0.2
n × t
(-)
(-)
C
p
(b)
Fig. 1 Visualization of the cavitation precessing vortex rope together
with two pressure signals measured in the same cross section of the
draft tube cone in resonance conditions. a Visualization of the cavita-
tion vortex rope. b Wall pressure signals measured in the cone
Exp Fluids (2015) 56:215
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215 Page 4 of 15
conditions. Within the first regime, from
Q = 78
to 85 % of
the value
Q
0
, the precession frequency remains quasi-con-
stant and equal to about 0.26 times the runner frequency. The
amplitude of the synchronous pressure pulsations at this fre-
quency increases as the discharge decreases but remains low,
indicating a weak hydroacoustic excitation. From 62 to 78 %
of the BEP, the precession frequency increases linearly from
0.26 to 0.34 times the runner frequency as the discharge
decreases. In parallel, the amplitude of the pressure fluctua-
tions significantly increases and reaches its maximum for a
discharge value equal to
Q = 0.65 × Q
0
, which indicates a
significant rise in the excitation source intensity in this seg-
ment. Below a discharge value of
Q = 0.62 × Q
0
, the results
suggest that the dynamics of the precessing vortex rope are
drastically modified. The precession frequency does not fol-
low a particular law anymore, and the amplitude of the pres-
sure fluctuations swiftly decreases, which indicates a signifi-
cant reduction in the excitation source intensity.
A comparison between the cross spectrum of two pres-
sure signals measured in the same cross section of the cone
at two different discharge values,
Q = 0.64 × Q
0
(Regime
2) and
Q = 0.55 × Q
0
(Regime 3), is presented in Fig. 4.
The cross spectrum obtained at
Q = 0.64 × Q
0
presents a
very distinguishable peak at a low frequency, correspond-
ing to the precession frequency. This type of spectrum is
obtained for all discharge values between 62 and 85 % of
the value at the BEP. The spectrum at
Q = 0.55 × Q
0
dis-
plays a similar qualitative shape, while being shifted to
the right on the frequency axis and to the top regarding
the mean amplitude level. The energy at the precession
Feeding pipe
Flow
Direction
Laser
Pressure Sensors
Vertical cut-view
of the optical access
CCD Camera
Fig. 2 Reduced-scale physical model of the Francis turbine installed on EPFL test rig PF3, with the set-up for the PIV measurements and a ver-
tical cut-view of the camera support frame and the optical access
0.5 0.6 0.7 0.8 0.9
0
Q / Q
0
(-)
×10
−3
1
2
3
4
G
xx
(f
PVC
)
0.5 0.6 0.7 0.8 0.9
0.2
0.25
0.3
0.35
0.4
Regime 3
Regime 2
Regime 1
f
PVC
/ n
Q / Q
0
(-)
(a)
(b)
(-)
(Hz
-1
)
Fig. 3 a Precession frequency
f
PVC
made non-dimensional by the
runner frequency n and b auto-spectral amplitude at
f
PVC
for a pres-
sure signal measured in the feeding pipe as a function of the dis-
charge. The pressure signal and the discharge are made dimensionless
by the turbine-specific head and the discharge value
Q
0
, respectively
Exp Fluids (2015) 56:215
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Page 5 of 15 215
frequency of 0.35 times the runner frequency is however
spread over a broader frequency range, indicating a loss of
periodicity in the pressure signal and therefore in the flow
structure. At the sight of these observations, it is not possi-
ble anymore to extract a well-defined periodical behaviour
in the velocity fields within regime 3.
4 Acquisition and analysis of PIV data
4.1 Experimental set‑up
The flow field is investigated in the draft tube cone by
means of PIV performed in two horizontal cross sections,
situated
0.39 × D
and
1.02 × D
downstream the runner
outlet (Fig. 5), D being the exit diameter of the runner.
The diameters of the measurement sections are, respec-
tively, equal to
1.05 × D
and
1.14 × D
. The light sheet of
2-mm thickness is generated by a double-pulsed Nd:YAG
laser and a cylindrical lens. The time interval between
two consecutive pulses is adjusted at each operating point
to limit the out-of-plane particle displacement to 1/10 of
the laser sheet thickness (75 μs for a discharge equal to
Q = 0.320 m
3
s
−1
, corresponding to an average axial veloc-
ity
Cm = 2.57 m s
−1
in the Section 2). A waterbox is used
to minimize the optical deformation of the laser sheet
induced by the inclined wall of the cone.
The images are recorded using a CCD camera placed
perpendicular to the laser sheet at the bottom of the draft
tube elbow. The characteristics of both the laser and the
camera are given in Table 1. A curved Plexiglas window fit-
ting the shape of the draft tube is installed, providing an
optical access for the camera. The camera is aligned with
the coordinate system of the test rig (see Fig. 5) for a direct
measurement of the velocity components Cx and Cy in this
system. Due to the strong optical distortion introduced by
the singular optical access, particular attention is paid to
the calibration procedure. For each measurement section,
a calibration image is taken with a circular dotted target
covering the measurement section. The camera is fixed on
a metal frame attached to the draft tube elbow, in order to
preserve the relative position of the camera when the draft
0 0.1 0.2 0.3 0.4 0.5
10
−8
10
−6
10
−4
10
−2
f / n
(-)
|G
xy
|
f
PVC
(a)
0 0.1 0.2 0.3 0.4 0.5
f / n
(-)
0
θ
xy
(rad)
f
PVC
Q = 0.64 × Q
0
Q = 0.55 × Q
0
(b)
(Hz
-1
)
Fig. 4 a Amplitude and b phase of the cross-spectral density func-
tion of two pressure signals for the discharge values
Q
=
0.64
×
Q
0
(dotted line) and
Q
=
0.55
×
Q
0
(solid line). The frequency and the
pressure signals are made dimensionless by the runner frequency
and the turbine-specific head, respectively. The pressure sensors are
located in the same cross section of the cone and are separated by an
angle equal to
π
1.02 × D
0.39 × D
section 1
section 2
D
y
z
x
CCD camera
Plexiglass
Window
Fig. 5 Vertical cut-view of the reduced-scale model, with the stream-
wise position of the PIV measurement sections. The used coordinate
system is indicated by the unit vectors x, y, and z
