FSI analysis of francis-99 hydrofoil employing SBES model to adequately predict vortex shedding

Abstract

The added effects from the fluid on a structure submerged in water significantly affect its dynamic response. Since the hydraulic turbine runner is geometrically complex and involves complicated flow phenomena, the research on simple hydrofoil offers a unique opportunity to investigate added effects and mutual interaction of the elastic structure and vortical flow. For this purpose, the fluid structure interaction of Francis-99 hydrofoil was analysed using the Stress Blended Eddy Simulation (SBES). Advantage of this hybrid RANS-LES turbulence model over RANS models is shown by its enhanced ability to represent vortex shedding. The results of modal sensitivity analysis showed, that fillets of the fixed hydrofoil have negligible influence on the natural frequencies of the hydrofoil and therefore the simplified geometry was used. The modal analysis of fully fixed hydrofoil both in the air and submerged in water were carried out to investigate the added mass effect. Moreover, the hydrodynamic damping for various flow velocities was also investigated for the first bending mode. Overall results are complemented by sensitivity analysis of time step size and mesh for both structural and fluid domains. The results showed that the computed damping ratio above the lock-in and vortex shedding frequency at lock-in are largely underestimated. Therefore, the geometry with blunt trailing edge was additionally tested.

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FSI analysis of francis-99 hydrofoil employing SBES model to
adequately predict vortex shedding
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Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
IOP Publishing
doi:10.1088/1742-6596/1296/1/012002
1
FSI analysis of francis-99 hydrofoil employing SBES model to
adequately predict vortex shedding
P Čupr1, D Štefan1, V Habán1 and P Rudolf1
1V. Kaplan Department of Fluid Engineering, Faculty of Mechanical Engineering,
Brno University of Technology, Technická 2896/2, Brno, Czech Republic
cupr@fme.vutbr.cz
Abstract. The added effects from the fluid on a structure submerged in water significantly affect
its dynamic response. Since the hydraulic turbine runner is geometrically complex and involves
complicated flow phenomena, the research on simple hydrofoil offers a unique opportunity to
investigate added effects and mutual interaction of the elastic structure and vortical flow. For
this purpose, the fluid structure interaction of Francis-99 hydrofoil was analysed using the Stress
Blended Eddy Simulation (SBES). Advantage of this hybrid RANS-LES turbulence model over
RANS models is shown by its enhanced ability to represent vortex shedding. The results of
modal sensitivity analysis showed, that fillets of the fixed hydrofoil have negligible influence on
the natural frequencies of the hydrofoil and therefore the simplified geometry was used. The
modal analysis of fully fixed hydrofoil both in the air and submerged in water were carried out
to investigate the added mass effect. Moreover, the hydrodynamic damping for various flow
velocities was also investigated for the first bending mode. Overall results are complemented by
sensitivity analysis of time step size and mesh for both structural and fluid domains. The results
showed that the computed damping ratio above the lock-in and vortex shedding frequency at
lock-in are largely underestimated. Therefore, the geometry with blunt trailing edge was
additionally tested.
1. Introduction
Current trends on the energy market lead to the extension of turbine operating range, i.e. the turbine
operation under off-design conditions is more and more requested. Consequently, the turbine
components are exposed to the extreme flow conditions from almost no-load to overload experiencing
high-frequency rotor stator interaction, low-frequency vortex rope, inter-blade vortices, cavitating
structures and repeated start up and shut down, resulting in both periodic and stochastic excitation forces.
The coincidence between the periodic rotor-stator interaction or stochastic excitation and the runner
natural frequencies is unavoidable. In order to avoid some runner failure and potential fatigue problems,
the runner dynamic response investigation with focus on hydrodynamic damping is necessary [1], [2].
The dynamic response of the hydroelectric turbine structural components is strongly influenced by the
occurrence of the fluid flow and its added effects (added mass, damping and stiffness). The prediction
of added mass effect causing the natural frequency shift has been widely documented in the literature
[3], [4], [5], [6], [7]. However, the prediction of hydrodynamic damping and added stiffness both
occurring due to the flow has only recently and only partially been treated in the domain of hydraulic
turbines.

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
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doi:10.1088/1742-6596/1296/1/012002
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In recent years both the experimental and numerical hydrodynamic damping investigation has been
studied by researchers. Kimber studied the response of cantilevered beams both in the air and submerged
in water [8]. Yao investigated the influence of the trailing edge (TE) shape on the hydrodynamic
damping using a simple hydrofoil [9]. Coutu and Seeley used the piezoelectric actuators made of Macro
Fiber Composites (MFC actuators) to excite the hydrofoil and to measure its dynamic response [10],
[11]. Various techniques have been applied and developed to predict the hydrodynamic damping in the
field of numerical simulations. The full two-way fluid structure interaction approach was used in several
cases and was applied on a simple hydrofoil [12], [13] as well as on the whole turbine runner [14].
However, this approach is very time consuming and the proper analysis setting is not easy and
straightforward. Recently, the alternative methods of “Modal Work Approach”, based on the one way
coupled FSI simulation, have been presented [15], [16], [17], [18].
The hydrodynamic damping prediction in hydraulic turbines has not been totally understood yet.
Therefore, the experimental damping investigation was done at NTNU Trondheim in Norway both for
the high head Francis turbine runner and a simple hydrofoil [19], [20]. Since the hydraulic turbine runner
is geometrically complex and connected with different complicated flow phenomena, the research on a
simple hydrofoil, as a canonical case, is reasonable for study of dynamic response of elastic structure
submerged in flowing water.
This paper presents the results of hydrodynamic damping estimation in terms of one-way fluid
structure interaction using a “Modal Work Approach” applied on Francis-99 hydrofoil geometry (figure
1). Results on this geometry pointed out the difficulty of accurate prediction of vortex shedding
frequency and consequently accurate estimation of hydrodynamic damping. The main reason was linked
to the difficulty in onset of boundary layer separation on a round TE with no exactly defined separating
point [21]. Consequently, the results of vortex shedding and hydrodynamic damping are in parallel
investigated for another hydrofoil with modified geometry, where the original TE, which has one side
rounded, is replaced by a blunt TE of thickness 4.5 mm (see, figure 2) . The location of boundary layer
separation on hydrofoil with sharp blunt TE is fully determined and thus expected vortex shedding
frequency might be easily captured in CFD simulation.
Figure 1. Longitudinal cross-section through test section geometry of Francis-99 hydrofoil.
The paper is structured as follows: Section 2 describes Francis-99 hydrofoil test case, the results of
modal analysis for both hydrofoil geometries are summarized in Section 3, the Section 4 describes in
details CFD simulation and its set up, Section 5 refers about vortex shedding and results of computed
hydrodynamic damping analysis with compared with provided measurements are documented in
Section 6.
2. Francis-99 hydrofoil test case
The hydrofoil designed especially for the Francis-99 research project has a chord length of 0.25 m and
its maximal thickness is 0.012 m. After 0.15 m from the leading edge, the thickness was tapered down

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
IOP Publishing
doi:10.1088/1742-6596/1296/1/012002
3
to 4.5 mm at TE, before being chamfered and rounded on one side. The hydrofoil test section of a
squared 0.15 m × 0.15 m internal cross-section was milled from a single piece of aluminum
alloy (7075 T651) and mounted into the test rig.
Two MFCs actuators from PI Ceramic were mounted on each side of the foil at the widthwise center,
close to the trailing edge. These patches were used for hydrofoil excitation in a sinusoidal pattern phase-
separated by 180°, resulting in a bending action induced in the blade. The two main results are available:
i. Damping ratio versus discharge, ii. Natural frequency versus discharge.
The measurements were done by Bergan et al. [19]. For more details about Francis-99 workshop and
experimental results see [22].
3. Modal Analysis
Considering an oscillating hydrofoil submerged into water, its dynamic response is altered by the added
effects. This can be modelled as a single degree of freedom (1-DOF) oscillator. The hydrofoil
oscillations can be described by equation of motion (1)
    
(1)
where ms is the mass of the hydrofoil, ds is the structural damping, ks is the stiffness of the structure and
F(t) is the force acting on the hydrofoil from the fluid. The equation can be rewritten in another form (2).
    
(2)
The following added effects are present: mw is the added mass of still water, dw is damping resulting
from viscous effects, df is damping resulting from momentum exchange between the flow and the
structure due to the oscillation, kw is added stiffness due to the compressibility effects and kf is added
stiffness due to the flow.
The added mass effect mw controls the frequency of oscillations and causes the shift of natural
frequencies compared to the values measured in the air. The typical approach to investigate the added
mass effect is the acoustic modal analysis. Carrying out the modal analysis of the hydrofoil both in the
air and submerged in water, the natural frequency drop can be investigated and the added mass of water
can be calculated using equation (3), where mw is the mass of the added water, ms is the mass of the
hydrofoil, fwater and fair are the hydrofoil natural frequencies calculated in water and air respectively.
 






 




(3)
Since two different geometries were provided, i.e. the hydrofoil with and without fillets, the modal
analysis was carried out for both of them, including the test section walls (figure 1). Additionally, the
modal analysis of hydrofoil with blunt TE and without fillets was also calculated.
All three geometries were made of the same material (aluminium alloy 7075 T651). The boundary
conditions are set in respect to the experimental setup. The inlet and outlet cross-sections are fully fixed.
The computational mesh was created using 99 000 quadratic structural elements SOLID 186 (including
fillets) and 200 000 acoustic elements in case of the hydrofoil submerged in water. The material
properties are summarized in table 1. The results of modal analysis for all three geometries are presented
in the three following tables (table 2 -table 4). The first mode shape of the Francis-99 hydrofoil is shown
in the figure 2. Only the symmetric half of the mode shape is shown to provide better look into the test
section. The maximum displacement is located in the middle of the trailing edge.
The modal analysis results showed that only negligible difference is between the geometries with
and without fillets. Therefore, the first natural frequency of the hydrofoil without fillets (641.78 Hz) is

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
IOP Publishing
doi:10.1088/1742-6596/1296/1/012002
4
taken into account for further steps. This value shows a good agreement with the experimental
value (622.8 Hz; difference ca. 3%).
Table 1. Material properties of the structure.
Density
Young Modulus
Poisson Ratio
Aluminium alloy
2810 kg/m3
71700 MPa
0.33
Table 2. Results of modal analysis of Francis-99 hydrofoil without fillets.
Modal analysis of Francis-99 hydrofoil without fillets
Mode
Natural frequency
Ratio
water/air
Added mass
Added mass/hydrofoil
mass
Air
Water
(Hz)
(Hz)
(%)
(kg)
(-)
1
1711.1
641.78
37.5
162.8878
6.109
2
1791.5
1063.80
59.4
48.9596
1.836
3
1912.7
1083.40
56.6
56.4474
2.117
4
2222.8
1456.50
65.5
35.4402
1.329
5
2452.3
1469.70
59.9
47.5752
1.784
6
2645.2
1762.20
66.6
33.4184
1.253
Table 3. Results of modal analysis of Francis-99 hydrofoil with fillets.
Modal analysis of Francis-99 hydrofoil with fillets
Mode
Natural frequency
Ratio
water/air
Added mass
Added mass/hydrofoil
mass
Air
Water
(Hz)
(Hz)
(%)
(kg)
(-)
1
1743.3
672.76
38.6
152.4401
5.715
2
1793.3
868.88
48.5
86.9553
3.260
3
1968.2
1127.00
57.3
54.6826
2.050
4
2224.1
1457.10
65.5
35.4744
1.330
5
2551.5
1469.20
57.6
53.7770
2.016
6
2645.4
1744.50
65.9
34.6656
1.300

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
IOP Publishing
doi:10.1088/1742-6596/1296/1/012002
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Table 4. Results of modal analysis of hydrofoil with blunt TE without fillets.
Modal analysis of hydrofoil with blunt TE without fillets
Mode
Natural frequency
Ratio
water/air
Added mass
Added mass/hydrofoil
mass
Air
Water
(Hz)
(Hz)
(%)
(kg)
(-)
1
1702.08
637.05
37.4
163.7213
6.139
2
1791.39
1066.90
59.6
48.5202
1.819
3
1897.48
1077.50
56.8
56.0382
2.101
4
2222.60
1458.00
65.6
35.3078
1.324
5
2453.08
1471.30
60.0
47.4697
1.780
6
2645.14
1749.30
66.1
34.3115
1.286
Figure 2. The first mode shape of Francis-99 hydrofoil (symmetric half).

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
IOP Publishing
doi:10.1088/1742-6596/1296/1/012002
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4. CFD simulations
In this section the methods used for CFD analysis are summarized. Geometry simplification,
computational mesh, solver set-up and turbulence modelling are discussed in particular subsections.
4.1. Geometry
The physical geometry of hydrofoil includes fillets creating the intersection of hydrofoil and side walls.
This geometrical feature might pose difficulties for building of structured hexahedral grid thus in case
of turbine runners is often neglected without significant impact on the results [23], [24]. As shown in
Section 3, the fillets exclusion does not have any significant effect on the modal shapes and relevant
natural frequencies. After several preliminary simulations it was deduced that the fillets might be
neglected also for the estimation of hydrodynamic damping, thus further only geometry without fillets
is considered.
In order to limit the influence of inlet and outlet boundaries, the geometry for mesh has total length
of 1.56 m in streamwise direction with hydrofoil placed approximately in the middle (figure 3 a).
(a)
(b)
(c)
(d)
Figure 3. Computational domain for numerical CFD simulation (a), mesh resolution on hydrofoil
surface (b), Detail of mesh resolution near the leading edge (c), Detail of mesh resolution near the
trailing edge (d).
4.2. Mesh
Computational mesh was built using ICEM and consists of only hexahedral elements. Two different
mesh sizes were tested in order to estimate the influence on the vortex shedding frequency. In order to
employ turbulence models with low-Re near wall formulation, the near wall mesh resolution respects
the y+ < 1 for the maximal mean velocity in test-section. Two mesh sizes were tested prior to final
simulation of Francis-99 hydrofoil geometry. The initial mesh (Mesh #1) consists approximately
3.7 mil. elements and the refined mesh (Mesh #2) with approximately 1.6 times more elements, see table
5 and figure 3 b-d.

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
IOP Publishing
doi:10.1088/1742-6596/1296/1/012002
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Table 5. Mesh sizes for Francis-99 hydrofoil geometry.
Francis-99 geometry
Mesh #1
Mesh #2
Elements
3 657 600
5 976 000
Nodes
3 750 292
6 114 290
As shown in figure 4 the maximal y+ on hydrofoil surface at v = 25 m/s is y+ = 1.29 on the leading
edge (LE). The trailing edge (TE) is well resolved with y+ value around 0.75.
Figure 4. Contours of y+ on hydrofoil surface, LE view (left) and TE view (right).
The mesh for hydrofoil geometry with blunt TE (see figure 5) was created with lower resolution,
approximately equal to resolution of Mesh #1, see table 6. It was found that even using this relatively
coarse mesh the vortex shedding and consequently hydrodynamic damping were well predicted.
Table 6. Mesh size for hydrofoil geometry with blunt TE.
Geometry with blunt TE
Elements
3 221 100
Nodes
3 304 392
Figure 5. Mesh resolution downstream of hydrofoil for geometry with blunt TE.
4.3. Turbulence models

Francis 99: Fluid structure interactions in Francis turbines
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Several turbulence models were tested prior to the final simulations. The main reason was, that it was
impossible to capture the Karman vortex street using two-equation model either Standard k-ε based on
the wall function approach or SST k- based on the Low-Re approach. Consequently, the Scale-
Resolving Simulation (SRS) models were employed. Using the coarse mesh (consists of 3.66 mil.
elements), the Scale-Adaptive Simulation (SAS) model [25] had difficulties and only the Stress Blended
Eddy Simulation (SBES) model was able to easily capture unsteadiness of vortex shedding. Thus for
the further simulations only the SBES model was employed. SBES is a hybrid RANS-LES model using
the shielding/blending function to automatically switch between RANS and LES solution. Contrary to
the original Detached Eddy Simulation (DES) model family this model features a much improved
shielding function to protect RANS boundary layers. The SBES model transitions much quicker and on
much coarser grids from RANS to SRS mode in separating shear layers than classical DES [25].
4.4. Solver
For all presented CFD simulations the Ansys CFX v19.1 solver was used. The unsteady simulations
started from the result of steady-state solution for particular flow rate Q. The sensitivity study was done
in order to choose appropriate time-step size dt for chosen turbulence model and maximal Q based on
the change of vortex shedding frequency fs.
It was found that the time-step size dt plays important role on the vortex shedding frequency.
Nevertheless, even using very small time-step dt = 1e-6 s the simulated vortex shedding frequency fs CFD
underestimates experimental value by 11% at resonance fs EXP = 630 Hz for Q = 0.248 m3/s (v = 11 m/s).
All unsteady simulations were running using Second Order Backward Euler transient scheme and
High Resolution scheme for turbulence. While for k-ε, SST k-, and SAS models the High Resolution
scheme was selected for advection, the Bounded Central Difference scheme was preferred for SBES
model.
5. Vortex Shedding
The accurate simulation of vortex shedding behind the object placed in the fluid stream is the challenging
task. Especially if the object has a rounded geometry without sharp edges and thus the exact point of
boundary layer separation is unknown. In case of narrow hydrofoil with symmetrical profile the main
part of interests is the trailing edge (TE). The shape of TE might either accelerate or delay the boundary
layer separation which consequently leads to the vortex shedding in a form of von Kármán vortex street.
This flow phenomenon periodically acts on the hydrofoil structure and if the shedding frequency is in
resonance with the natural frequency of hydrofoil the strong structural load may lead to cracks. This is
very important aspect for turbomachinery design since the Francis turbine runner is composed of several
blades resembling hydrofoil-like flow. Nowadays, for turbines working in extended operating ranges
the high blade load might frequently appear if the vortex shedding frequency fits the blade natural
frequency [1].
For above mentioned reasons the accurate prediction of vortex shedding frequency is very important.
As previously mentioned we encountered difficulties to correctly capture vortex shedding frequency and
consequently the hydrodynamic damping in simulations of Francis-99 hydrofoil geometry. Thus the
additional geometry with modified TE was tested. Both hydrofoil geometries are described in following
subsections.
5.1. Francis-99 hydrofoil geometry
The Francis-99 hydrofoil geometry as shown in figure 6 has a one side of the trailing edge rounded
which resembles the so called Donaldson trailing edge [27].

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
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doi:10.1088/1742-6596/1296/1/012002
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Figure 6. Francis-99 hydrofoil geometry.
The vortex shedding behind this hydrofoil is shown in figure 7 in a form of vorticity iso-surface. For
lock-in (v = 11 m/s) the simulated vortex shedding frequency was fs = 560 Hz which is 11% lower than
the measured one.
Figure 7. Vortex shedding at v = 11 m/s visualized by isosurface of vorticity component perpendicular
to the mean flow (-1000 s-1 in blue, 1000 s-1 in red), Francis-99 hydrofoil geometry.
5.2. Hydrofoil geometry with blunt TE
As mentioned in introduction the accurate prediction of boundary layer separation is challenging task in
case of hydrofoil geometry with one side of TE rounded (as Francis-99 hydrofoil geometry). For this
purpose, the hydrofoil geometry with blunt TE is created in order to have known point of BL separation.
This yields only small modification of original Francis-99 hydrofoil geometry where the total length of
the hydrofoil remains and the rounded corner was replaced by the material to create symmetrical sharp
blunt TE, see figure 8.

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
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doi:10.1088/1742-6596/1296/1/012002
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Figure 8. Hydrofoil geometry with sharp blunt TE.
The vortex shedding behind this hydrofoil is shown in figure 9 in a form of vorticity iso-surface. For
the same velocity v = 11 m/s of Francis-99 hydrofoil lock-in the simulated shedding frequency of this
hydrofoil was fs = 495 Hz.
Figure 9. Vortex shedding at v = 11 m/s visualized by isosurface of vorticity component perpendicular
to the mean flow (-1000 s-1 in blue, 1000 s-1 in red), geometry with sharp blunt TE.
6. Hydrodynamic damping estimation
The first six mode shapes and corresponding natural frequencies were calculated using the
(acoustic) modal analysis. However, the hydrodynamic estimation presented in this paper was
performed only for the first mode shape. This enables to use simplified way of hydrodynamic damping
estimation in a form of one-way FSI.
For this purpose, the unsteady simulation with prescribed mesh motion is carried out. The hydrofoil
surface and the test section walls move according to the mode shape and its frequency. This satisfies the
assumption of hydrofoil oscillation on the first natural frequency and corresponding mode shape. The
prescribed amplitude 0.001 m was selected with 350 time steps per mode period.
Consequently, the “Modal Work Approach” is used for estimation of damping ratio  according to
the following equation
  

(4)

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
IOP Publishing
doi:10.1088/1742-6596/1296/1/012002
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where, Wm is the modal work exchanged between structure and fluid, Mw is the modal mass in water,
is modal natural angular frequency and u is the modal reference amplitude.
The Wm might be computed using equation (5).
  










(5)
where p is pressure, 


is surface normal vector (pointing into the fluid),  is wall shear stress vector, 


is mode shape velocity vector, t denotes time, A is surface area and T is period of one blade oscillation
[16], [17], [18].
Using CFX the area integral of variable Wall Power Density (W) over moving walls of prescribed
mode shape might be monitored during one period T of mode cycle. Using the time integration, the
modal work Wm is then calculated in same way as described by (5).
The results of damping ratio  estimated from CFD simulation and compared with measurements are
shown in figure 10 for both Francis-99 hydrofoil and hydrofoil with blunt TE.
Figure 10. Results of simulated damping ratio vs experimental measurements.
7. Conclusions
Presented study showed, that the numerical estimation of hydrodynamic damping on Francis-99
hydrofoil geometry might not be a straightforward and easy task. While for flow velocities below the
lock-in the damping ratio is in good agreement with measurements, the simulated results above the lock-
in are largely underestimated (around 50%). Another disagreement was found for estimation of vortex

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
IOP Publishing
doi:10.1088/1742-6596/1296/1/012002
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shedding frequency. The measured shedding frequency in lock-in (fs, EXP = 630 Hz) was underestimated
in simulation by 11%, although relatively fine mesh (6 mils. elements), small time step (dt = 1e-6 s) and
enhanced turbulence modelling (SBES model) were used.
On the other hand, interesting comparison between simulation and experiment was achieved for
geometry with blunt TE. The damping ratio for lower velocities is underestimated by around 35%, but
results for higher velocities (v > 16 m/s) follow the measurements with error less than 5%.
The interesting fact is also, that the lock-in for hydrofoil with blunt TE was moved towards higher flow
rate (somewhere between v = 16 – 17 m/s) with vortex shedding frequency around fs = 740 Hz. All this
was caused only by relatively small geometrical change of hydrofoil TE from one side rounded to the
fully blunt. For the turbine manufacturer this fact is very important since only small change of TE
geometry on the turbine blade might cause large change in vortex shedding dynamics and consequently
in added parameters influencing structural performance and lifespan.
Authors would like to acknowledge effort of the workshop organizers to provide open experimental
data, which are necessary prerequisite to finetune the computational methods and tools for further
tackling of even more complicated FSI problems connected with hydrofoil loading by collapsing
cavitation clouds and vortices [13], [28], [29].
Acknowledgement
This paper has been supported by the EU project Computer Simulations for Effective Low-Emission
Energy funded as project No. CZ.02.1.01/0.0/0.0/16_026/0008392 by Czech Republic Operational
Programme: Research, Development and Education, Priority axis 1: Strengthening capacities for high-
quality research and the collaboration.
Nomenclature
1-DOF One degree of freedom
A Area (m2)
BL Boundary Layer
CFD Computational Fluid Dynamics
DES Detached Eddy Simulation
ds,w,F Damping (N s m-1)
F Force (N)
fair Natural frequency in air (Hz)
fs Vortex shedding frequency (Hz)
FSI Fluid Structure Interaction
fwater Natural frequency in water (Hz)
k Turbulent kinetic energy (m2 s-2)
ks,w,F Stiffness (N m-1)
LES Large Eddy Simulation
ms,w Mass (kg)
p Pressure (Pa)
Q Flow rate (m3 s-1)
RANS Reynolds Averaged Navier-Stokes
Re Reynolds number
SAS Scale Adaptive Simulation
SBES Stress Blended Eddy Simulation
SRS Scale-Resolving Simulation
SST Shear Stress Transport
t time (s)
TE Trailing Edge
u Modal reference amplitude
v Flow velocity (m s-1)

Francis 99: Fluid structure interactions in Francis turbines
IOP Conf. Series: Journal of Physics: Conf. Series 1296 (2019) 012002
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Wm Modal Work (J)
x Deflection
y+ Non-dimensional wall distance
ε Dissipation of turbulent kinetic energy (m2 s-2)
 Damping ratio
τ Shear Stress (Pa)
ω Angular velocity (rad s-1)
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