PAR
Steven Roth
Suisse
2012
A ma princesse, Jelena
If fate doesn’t make you laugh,
then you don’t get the joke.
On dit qu’un travail de thèse est avant tout une aventure personnelle. Or, en préambule de
ce document, il m’importe de démentir cette affirmation en exprimant ma profonde grat
itude à celles et ceux qui ont, de quelque façon que ce soit, contribué à l’accomplissement
et la réussite de ma thèse. Face à l’immensité de l’océan, ces personnes ont su apporter, à
temps, leur soutien moral, leurs connaissances techniques ou leur vaste savoir scientifique,
afin que le navire sur lequel je me trouvais ne sombre pas et arrive à bon port.
En premier lieu, je ne pourrais manquer de remercier le Prof. François Avellan qui,
en ses qualités de directeur de thèse, a su orchestrer le bon déroulement de cette belle
aventure. Je lui suis reconnaissant de m’avoir proposé de réaliser une thèse de doctorat
au sein de son laboratoire et de m’y avoir vivement encouragé. Il a su ensuite maintenir
le cap de ce petit navire tantôt pris dans la “pétole”, tantôt malmené par la tempête.
En second lieu, mes remerciements vont directement au Dr. Mohamed Farhat, chef
de pont et codirecteur de thèse, qui a, généreusement, consacré une bonne partie de
son précieux temps à illuminer la crête des vagues et à éviter les scélérates. Les longues
discussions, néanmoins très intéressantes, tant à la cafétéria qu’à la table à cartes, m’ont
été d’une grande aide.
Ce travail de recherche a été mené dans le cadre du projet Eureka Hydrodyna II dont
les partenaires sont ALSTOM Hydro, ANDRITZ Hydro, VOITH Hydro et UPCCDIF.
Je remercie toutes les personnes impliquées dans ce projet qui ont su faire preuve d’intérêt
durant les nombreuses réunions techniques et qui ont réussi à extraire le meilleur de moi
même tout au long de ces années de thèse. Mes remerciements s’adressent également à la
Commission pour la Technologie et l’Innovation, CTI, et à Swisselectric Research qui ont
également supporté le projet financièrement. Je remercie aussi les membres du jury, les
Professeurs Gmür et Egusquiza ainsi que le Dr. Hübner, pour les questions intéressantes
soulevées lors de la défense de thèse privée.
J’en viens maintenant aux équipiers sans lesquels les manoeuvres sur le pont ne se
seraient, tout simplement, pas aussi bien déroulées. Vlad, dont la rigueur et la générosité
n’ont cessé de me grandir, m’a tendu la main à plusieurs reprises même dans les déferlantes
les plus impressionnantes. El Francisco a toujours été présent lors des manoeuvres les
plus délicates, et y a gracieusement apporté ses connaissances techniques et scientifiques
étendues ainsi que son sourire des plus rayonnants. J’ai eu aussi énormément de plaisir
à travailler avec Maxime, qui a réussi à partager son grand savoir technique et pratique,
ainsi qu’à garder son calme lorsqu’il entendait pour la énième fois : “Euh, tu peux changer
la neuf avec la dix... stp...”. Je garde encore le bon souvenir, parmi tant d’autres, d’un
doux réveil à la lueur de l’aube méridienne, au son d’une guitare désaccordée et aussi
sèche que moi! Priorité ou pas : “De l’eau !!!”
Je ne peux m’arrêter là dans le témoignage de ma reconnaissance pour toutes celles
et ceux qui ont fait de ce beau voyage intellectuel un souvenir inoubliable. Je continue
avec l’équipe du bureau d’étude. Je remercie ainsi Philippe F., Alain, Vincent, Pierre B.
et Philippe C. qui ont toujours été disponibles au cours de cette longue traversée. Du
côté du groupe “J’aime”, je tiens à remercier tout particulièrement Georges avec qui les
échanges sur le canal 16 ont été très sympathiques et enrichissants. Viennent ensuite
Ran, Lillie, Davide, Ambrosio, Matteo et celui que j’aimerais particulièrement montrer
du doigt, Sébastien B. Ma gratitude revient aussi à HenriPascal qui nous a laissé assez
d’eau pour manoeuvrer correctement malgré la marée et le peu de place à quai.
Il me faut également remercier Pierre M. qui, dans son rôle de supérieur direct, a su
maintenir la bonne humeur dans notre petite équipe. Ensuite, comment oublier l’équipe
des mécaniciens qui, aux côtés de Maxime, a toujours été prête à mettre les mains dans le
cambouis : Louis, Christian S., David, Jérôme, Mattias, Victor, JeanDaniel et Raymond.
Aussi, ma reconnaissance va directement à Isabelle qui a su démêler les noeuds admin
istratifs les plus complexes tout au long du voyage. Finalement, je ne saurais oublier
d’exprimer ma gratitude à Shadije pour son travail et sa gentille disponibilité durant les
premières années de thèse.
Aussi, “au moindre coup de Trafalgar, c’est l’amitié qui prenait le quart”. Une joyeuse
équipe de fous furieux a peuplé les cales du navire tout au long de la traversée. Et ces
aventureslà pourraient faire l’objet d’une thèse à ellesseules, dont la plupart des chapitres
serait frappée par la censure. Tout comme la lune orchestre la marée, le vaetvient des
doctorants influence cycliquement l’ambiance au sein de cette belle équipe. J’ai regretté
le départ de certains et me suis réjoui de l’arrivée d’autres... Je souhaite une longue et
belle vie à cette équipe sans cesse renouvelée.
Un, deux, trois... quinze... c’est parti ! J’ai juste croisé Bob, Yes pur !, en prenant
place à bord du bateau alors que lui en débarquait, mais on s’est souvent croisé en ville
par la suite... Merci à toi pour tous ces bons moments et pour tes conseils scientifiques
toujours avisés !
Merci aussi à toi Philippe A. pour cette belle amitié dont la charpente résiste à tout
coup de vent ! Tu n’as cessé de m’encourager à parvenir sans trop de dommage à la fin
de ma thèse. Que se soit la tête dans les enceintes du Mad, autour d’un bon petit plat
ou lors de réunions inter minables, je garde également un excellent souvenir de nos fous
rires.
Aussi, je ne peux oublier l’ami Ruchon qui a, un jour, osé me dépasser en windsurf
au large d’Hermance... Tcheu c’taguenet, i’pensait que j’pétouillais avec mon cassoton,
ou bien... Il a ensuite fallu qu’on se fasse un surfin’ road trip au Portugal pour nouer
définitivement notre amitié ! Bon vent à toi !
Comment oublier de saluer l’amitié qui me lie au Couple, avec un grand “C”, du labo,
j’ai nommé Séb et Cilecé ! Merci Cilecé de m’avoir mis dans le bain du couplage fluide
structure et de m’avoir convaincu de faire une thèse... J’ai aussi apprécié ta compagnie
dans le meilleur bureau du LMH ! Séb, j’ai bien aimé cette douce rivalité entre voileux et
aileux ! Merci à toi pour les nombreux bras de fer. La prochaine fois, je t’aurai !
Une passion commune a tissé nos liens d’amitié, Olivier B. Je suis heureux d’avoir pu
compléter en ta compagnie les derniers miles manquants ! Oh dear, les fameux bords tirés
au large de Guernesey resteront longtemps gravés dans ma mémoire !
Ensuite, viennent les moussaillons Pacot, Martino, Marc et Andres avec qui je partage
une grande amitié depuis les études au Poly. Merci à toi, Paquito, d’avoir réussi à me
ramener sain et sauf à la maison après cette partie de cachecache à vélo... Je garde
aussi un très bon souvenir de nos mois de colocation dans le Dix Vingt ! Martino, je me
souviendrai encore longtemps de nos longues discussions dans lesquelles nous avons su
refaire le monde à notre sauce, puisque c’était mieux avant, et oui... Marc, j’ai apprécié
ton sympathique flegme hispanofribourgeois qui défie même jusqu’à la gravité! Andres,
tu m’as été d’un énorme soutien moral dans les moments les plus durs de la rédaction
et je t’en remercie infiniment. J’ai beaucoup apprécié ton humilité et ta bonne humeur
quotidienne qui font de toi un ami hors pair.
Parmi les personnes qui ont pris place à bord en cours de route, je ne veux pas oublier
le petit Matthieu, qui a, malgré lui, réussi à me faire bien rire... Il a souvent été le seul
à comprendre mon humour ainsi que le sujet de ma thèse, mais je peine encore à cerner
son humour... et le sujet de sa thèse... Mais ça va venir, 120 secondes ne suffisent pas...
Manquent encore, à la pelle : Christian L., Christian V., Arthur, Ebrahim et Arturo, à
qui je souhaite un bon voyage. Et ceux qui ont débarqué plus tôt, à la louche : Ali, Amir,
Marco, Martin et Stefan. Et finalement ceux qui n’étaient que de passage : Joao, Marcelo,
Fatine, Felipe, Filippo, Florine et Philippe K. Merci tout spécialement à toi Danail d’avoir
pris la peine d’éclairer mon chemin lorsque j’étais dans le doute.
Aussi, je ne pourrai omettre de saluer mes potes du bout du lac qui m’ont soutenu par
leur amitié de longue date ! Fred, en particulier, a su être présent quand il le fallait le plus
et me faire oublier, par nos nombreux délires, que je me trouvais dans une belle galère !
Merci aussi à mes amis de Lausanne qui ont toujours pris des nouvelles de l’avancement
de mon travail et ont su me faire voir d’autres horizons.
Ensuite, je souhaite remercier toute ma famille, et particulièrement mes parents. Je
leur suis infiniment reconnaissant de m’avoir donné l’éducation, la motivation et les
moyens qui m’ont permis d’arriver là où je me trouve. Merci de m’avoir offert toutes
ces années qui n’auraient tout simplement pas été aussi belles sans vous. Je ne peux
oublier de remercier plus spécialement ma maman qui a eu le courage de lire ma thèse et
d’en améliorer la qualité rédactionnelle.
Finalement, je ne pourrais conclure ces quelques lignes sans remercier de tout mon
coeur ma belle princesse, Jelena. Le vent a fait converger nos chemins pour mon plus
grand bonheur. L’amour que tu me portes m’a été d’un soutien sans égal durant ma thèse.
Tu as réussi à me motiver jusqu’au bout, à me relever quand tout parraissait perdu et à
me prendre par la main sur le chemin de ton coeur. Nous embarquons ensemble, pour le
restant de nos jours, sur un petit bateau construit de nos propres mains. Et ce bateau
est le plus beau qui soit.
Steven Roth
Résumé
Hydraulic pumpturbines are subject to a high periodic excitation due to the RotorStator
Interaction, RSI. Basically, the RSI is caused by the impeller blade passage in the wake of
the guide vanes in generating mode, or upstream from the guide vanes in pumping mode.
Therefore, the structural parts, notably the guide vanes, suffer from high cycle fatigue
strength.
The dynamic behavior of the guide vanes is influenced by the surrounding flow. Ad
ditional inertia and dissipation strongly affect the structural vibrations; the added mass
and the hydrodynamic damping being of the same order of magnitude as the structural
mass and damping. In addition, should the entire guide vane cascade be considered, the
neighboring guide vanes are influencing each other through the fluid medium. Their eigen
frequencies as well as the vibration amplitudes close to resonance may, thus, be strongly
modified.
A poor assessment of their dynamic behavior during the design stage may lead to
premature failures due to RSI in the early stage of commissioning. So far, researchers have
studied the RSI phenomenon, but have not established an analytical description. They
have also investigated the added mass, especially the one acting on vibrating runner
blades. However, few studies are related to the hydrodynamic damping in hydraulic
machines. Moreover, to the author’s knowledge, researchers have not yet considered
neither the influence of the guide vane vibrations on the pressure fluctuations arising
from the RSI nor the coupling between the guide vanes.
Therefore, the present experimental work considers the response of the guide vanes in
a pumpturbine reduced scale model to the RSI excitation. The pumpturbine is operated
at the Best Efficiency operating Point, BEP, in turbine mode. The guide vane cascade
consists of a complex mechanical system featuring many degrees of freedom. The study
aims to show that the cascade may be viewed as a 2nd order mechanical system.
The impulse response of immersed guide vanes is enabled with the use of a spark plug
flush mounted in the bottom ring in a guide vane channel. This type of measurements is
successfully undertaken in water, model at rest, and model in operation.
Keeping the operating conditions of the BEP constant, the impeller rotation frequency
is then swept and the guide vanes are therefore excited by the RSI over a wide frequency
range. The combination of zb impeller blades with zo guide vanes makes apparent many
different rotating diametrical pressure modes. The guide vanes respond up to the RSI 5th
harmonic, but are mostly excited at the frequencies corresponding to the RSI fundamental
f = zb n and the 2nd harmonic f = 2zb n.
The amplitude of the fluctuating bending displacement and torsion angle of the guide
vanes is strongly varying across the impeller frequency range. The ranges of the 1st and
the 5th RSI harmonic frequency contain the frequency of the 1st bending eigenmode and
the 1st torsion eigenmode, respectively. The pressure fluctuations close to the vibrating
guide vanes are strongly varying and may even decrease by 50% at resonance. Therefore,
a transfer of energy between the vibrating structure and the flow pressure should occur.
The influence of an adjacent guide vane on the vibrations of a guide vane is found
to vary significantly between its position on the pressure side and suction side of the
latter. Regarding the guide vane bending vibrations, the hydrodynamic force acting on a
guide vane induced by its neighboring guide vane on the pressure side is up to 10 times
higher than the force induced by its suction side neighbor. As for the guide vane torsion
vibrations, the hydrodynamic torque acting on a guide vane induced by its neighboring
guide vane on the pressure side is up to 5 times higher than the force induced by its
suction side neighbor.
The hydrodynamic damping coefficient and the added mass corresponding to the vi
brations of the adjacent guide vanes are successfully identified and an influence matrix is
built. These two terms are shown to depend strongly on the relative amplitude of their
vibrations, the absolute flow velocity and the phase shift between their vibration signals.
Taking into account the periodicity condition, the influence matrix is built in order
to predict the dynamics of the entire guide vane cascade. Four and six different eigen
modes are investigated for the case of bending and torsion motions, respectively. The
eigenvalue real part of each bending eigenmode remains positive on the investigated im
peller frequency range, that is the mechanical system is stable. On the other hand, the
eigenvalue real part of the torsion eigenmode which is the most likely to be excited by the
RSI becomes negative. This means that the mechanical system is unstable and premature
failures of the guide vanes are expected. Finally, two different ways to prevent damage
to the guide vanes excited at the RSI 5th harmonic frequency are proposed. On the one
hand, it is shown that by increasing the structural damping constant by a factor 2, the
mechanical system becomes stable. On the other hand, the modification of the shape of
the cascade eigenmode is achieved by mistuning the cascade, such that its shape does
no longer match the shape of the RSI pressure mode. This way, even if the mechanical
system remains unstable, the risk of damaging the guide vanes is reduced.
Keywords: Fluidstructure coupling, Hydraulic Machines, PumpTurbines, Rotor
Stator Interaction, Guide vane cascade, wicket gate cascade.
Contents
I Introduction 1
1 Problem overview 3
1.1 Thesis document organization . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Hydraulic pumpturbines 5
2.1 Hydropower generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Pumpedstorage power plants . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Pumpturbine technology . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Pumpturbine components . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 RotorStator Interaction phenomenon . . . . . . . . . . . . . . . . . . . . 13
2.4.1 Physical principles . . . . . . . . . . . . . . . . . . . . . . . . . . 14
II Investigation methodology 27
4 Investigated pumpturbine 29
4.1 Pumpturbine characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Test facility 37
6 Measuring apparatus 39
6.1 Impulse excitation system . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.1.1 In air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.1.2 In water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.2 Structural vibration measurement . . . . . . . . . . . . . . . . . . . . . . 40
6.3 Flow pressure measurement . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.4 Measuring chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8 Impulse response 63
8.1 Guide vanes in place, dewatered model . . . . . . . . . . . . . . . . . . . 63
8.1.1 Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.1.2 Structural damping . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.2 Guide vanes in place, still water . . . . . . . . . . . . . . . . . . . . . . . 68
8.2.1 Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8.2.2 Hydrodynamic damping . . . . . . . . . . . . . . . . . . . . . . . 70
8.3 Guide vanes in place, model in operation . . . . . . . . . . . . . . . . . . 72
8.3.1 Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8.3.2 Hydrodynamic damping . . . . . . . . . . . . . . . . . . . . . . . 75
14 Perspectives 137
Appendices 141
A Signal processing 141
A.1 Random data spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . 141
References 167
Latin
L̂18n
ps Transfer function between pressure (pres. side) and the force due to the RSI [m2 ]
L̂18n
ss Transfer function between pressure (suct. side) and the force due to the RSI [m2 ]
N Impeller rotation frequency [rpm]
th
Oi Denomination of the i guide vane
P Pumpturbine power [W]
Q Pumpturbine volumetric discharge [m · s1 ]
3
R Radius [m]
R3 Radius at a position between stay and guide vanes [m]
Ti Total torsion torque acting on the ith guide vane [N · m]
f
Ti,j Fluctuating torsion torque due to Oj vibrations acting on Oi [N · m]
Ti0 Fluctuating torsion torque due to change of incidence angle of Oi [N · m]
TiRSI Fluctuating torsion torque due to the RSI acting on Oi [N · m]
T Measuring period [s]
U Peripheral impeller velocity [m · s1 ]
V Fluid control volume [m3 ]
W Relative flow velocity [m · s1 ]
W Narrowest width of the guide vane channel [m]
Wh Hamming window function []
Z Elevation [m]
c0 Speed of sound [m · s1 ]
e Unit vector []
f Frequency [Hz]
f0 Eigenfrequency [Hz]
g Gravitational acceleration [m · s2 ]
gi Denomination of the ith pressure sensor in the rotorstator gap
gHX Fluid specific energy at the section X [J · kg1 ]
gHrX÷Y Fluid specific energy losses between the sections X and Y [J · kg1 ]
h (t) Sinusoidal exponentially decreasing fitting curve []
h0 Yintercept of the function h (t) []
st
k− 1 nodal diameter number []
k+ 2nd nodal diameter number []
l length [m]
m,m0 Integers for RSI harmonics []
n Impeller rotation frequency [Hz]
n− 1st diametrical pressure mode rotating frequency [Hz]
Greek
Subscripts
Superscripts
Dimensionless Numbers
Cmax · Dh
Re Reynolds number Re = []
ν
p
cp Pressure factor cp = []
ρ·E
y
cy Displacement factor cy = []
δ
α · L2
cα Rotation factor cα = []
δ
Q
ϕ1̄e Discharge coefficient ϕ1̄e = π2 3
[]
4
D1̄e n
E
ψ1̄e Energy coefficient ψ1̄e = π2 2 2
[]
4
D1̄e n
2πf L2
κ Reduced frequency κ= []
Cref
ϕ0.5
1̄e
ν Specific speed ν= 0.75
[]
ψ1̄e
Acronyms
Introduction
Problem overview
Hydraulic pumpturbines are subject to high periodic loading due to the RotorStator
Interaction, RSI. Basically, the RSI is caused by the impeller blade passage in the wake of
the guide vanes in generating mode, or upstream from the guide vanes in pumping mode.
Therefore, the structural parts, notably the guide vanes, suffer from high cycle fatigue
strength.
The dynamic behavior of the guide vanes is influenced by the surrounding flow. Ad
ditional inertia and dissipation strongly affect the structural vibrations; the added mass
and the hydrodynamic damping being of the same order of magnitude as the structural
mass and damping. In addition, should the entire guide vane cascade be considered, the
neighboring guide vanes are influencing each other through the fluid medium. Their eigen
frequencies as well as the vibration amplitudes close to resonance may, thus, be strongly
modified.
A poor assessment of the guide vane dynamics during the design stage may lead to
premature failures due to RSI in the early stage of commissioning. So far, researchers have
studied the RSI phenomenon, but have not established an analytical description. They
have also investigated the added mass, especially the one acting on vibrating runner
blades. But, few studies are related to the hydrodynamic damping in hydraulic machines.
Moreover, to the author’s knowledge, researchers have not yet considered neither the
influence of the guide vane vibrations on the pressure fluctuations arising from the RSI
nor the coupling between the guide vanes.
Therefore, the present experimental work considers the response of the guide vanes in
a pumpturbine reduced scale model to the RSI excitation. The pumpturbine is operated
at the Best Efficiency operating Point, BEP, in turbine mode. The guide vane cascade
consists of a complex mechanical system featuring many degrees of freedom. The study
aims to show that the cascade may be viewed as a 2nd order mechanical system. The
fluidstructure coupling parameters are then identified and are shown to depend strongly
on the flow velocity and the vibration phase and amplitude. From the measurements
on two isolated guide vanes, the response of the entire guide vane cascade to the RSI
excitation may be predicted. Finally, solutions are proposed to prevent damage to the
guide vanes.
Hydraulic pumpturbines
1973 2009
0.6% 3.3%
21.0% 16.2%
38.3% 40.6%
3.3%
13.4%
12.1%
21.4% 5.1%
24.7%
Figure 2.1: Distribution of electrical production by energy source for the years 1973 and
2009 [43].
Alternative sources of electricity generation are diverse. Nuclear fission and fusion con
stitute the alternative nonrenewable sources. Solar, biomass, wind, hydropower, ocean
thermal gradients, ocean tides and ocean waves are the available renewable sources. With
the necessary current tendency to reduce the industrial impacts on the environment, these
alternative sources, renewable and nonrenewable, are likely to be more widely used in
the future. In 2009, they are already sharing 8% more of the world electrical production
than in 1973, while the electrical energy generation has rised from 6115 TWh to 20055
TWh [43].
With the growth of the world population and the global warming consequences, the
water [1] and energy supply will constitute great challenges in the 21st century [76]. More
over, water and energy present a mutual necessity [39] in the sense that the access to water
needs energy and water is needed for energy production. In energy production, water is
indirectly employed for cooling, wasting, extracting and conducting heat. But, water may
also be directly used for electrical energy production, in socalled hydropower installations.
Nowadays, hydropower is being utilized in over 160 countries. The net installed ca
pacity has reached 980 GW at the end of 2009. Hydropower largely contributes to the
global energy mix by providing 17% of the world’s estimated installed electrical capacity
and 72% of the estimated renewable energysourced capacity at the end 2010 [44]. The
International Hydropower Association, IHA, estimates that hydropower plants commis
sioning will grow at an average rate of 3% per year [44], whilst the electricity consumption
is forecasted to grow at an average rate of 2.3% until 2035 [81].
The worldwide installed hydropower capacity and hydropower plants still under con
struction at beginning2008 are shown by region in Figure 2.2 [87]. Europe has the
greatest number of installed capacity, followed by Eastern Asia and Middle East which
notably have the greatest number of hydropower plants under construction. In emerging
350
[GW] P Under construction
Installed capacity
300
250
200
150
100
50
0
ric
a
ric
a
ric
a ia st pe
Af e e an Ea ro
Am Am Oc
e
dle Eu
d
rth uth an
d Mi
No So sia
d
an
nA sia
ter nA
es r
W ste
Ea
economies, like in Asia and Latin America, hydropower is currently strongly developing.
As an example, the Chinese government has set a 300 GW hydropower capacity target for
2020, and China currently has a sufficient number of projects under construction to meet
the target. Pumpedstorage plants are notably required among the many hydroelectric
projects in China, in order to meet the future peak electricity demand [84]. In low income
countries, as in Africa, the ”translating of [recognized substantial hydropower potential]
into commissioned projects still encompasses many difficulties” [41]; 90% of Africa’s cur
rent hydropower plants being operational in only eight countries. In developing countries,
energy supply security is currently one of the main issues to guarantee the growth of the
economy [85]. A correct transfer of technology is the key component to meet the economic
target by ensuring a technical independence. Finally, OECD economies in North America
and Europe focus on the rehabilitation of existing plants [55], [15], [82], the development
of new technologies [59], [65] and the construction of pumpedstorage power plants, such
as Germany’s Goldisthal pumpedstorage scheme [14].
Switzerland has about 2% of the world installed hydropower capacity. Figure 2.3
shows the distribution of the electrical production in 2010. 56.5 % of electrical power is
5.4%
24.2%
38.1%
32.3%
NaS battery Large energy and power density High cost Pertinent solution Pertinent solution
Operating security
MetalAir battery Large energy density Hard recharge  Pertinent solution
Liion battery Large energy and power density High cost Pertinent solution Nonmature solution
Good efficiency Special recharge cycle
NiCd battery Large energy and power density Pertinent solution Conceivable solution
Efficiency
PbAcid battery Low cost Life span Pertinent solution Pertinent solution
Flywheel High power Small energy density Pertinent solution Pertinent solution
Magnetic storage SMES High power Small energy density Pertinent solution
High cost
Super capacitance Life span Small energy density Pertinent solution Nonmature solution
Good efficiency
ge
ge
ra
ra
to
sto
ss
[MW] P ge
30
1h
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Compressed
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7
SMES
Hydropower
1
Inertia wheel
Redox
Super
capaci
tance
0.1 Energy
battery
Power
battery E
Portable
battery 0.1 1 10 [MWh]
Figure 2.4: Utilization range of storage sources [75]: instantaneously available power P
versus the stored energy E.
advantage of over 100 years old experience. Moreover, pumpedstorage powerplants in
ject money into the economy, since electricity tends to be generated when its price is high
and the water, to be pumped when the price is low.
In 2009, more than 127 GW of pumpedstorage power plants were operating through
out the world [66]. A growth rate of 60% over the next four years was expected.
Thanks to its geographical location and its topology, Switzerland plays a key role in the
European electrical network regulation. In Table 2.2, the exports and imports of electrical
energy in 2010 and during winter 2009/2010 are given [62]. The balance between exports
and imports is close to zero which illustrates the important position of Switzerland in
the European electrical energy business. The revenues from electricity exports are 5064
millions Swiss francs (7.65 cts./kWh) and the expense for electricity imports are 3736
millions Swiss francs (5.60 cts./kWh).
Table 2.2: Electrical energy exports and imports of Switzerland in 2010 and 2009 [62].
The daily electricity production in Switzerland is illustrated in Figure 2.5 for four dates
in 2010, whereas the daily electrical consumption is shown in Figure 2.6. The production is
classified by types of power plants. Because of technological requirements, all powerplants
except hydropower plants are generating a constant power all day long. Only hydropower
may ensure the balance between production and consumption. In summer, when the
electrical consumption is low, the extra energy is used to feed the storage basin; an
important part being also exported at an economically profitable price.
After the Fukushima Daiichi nuclear power plant accident in Japan due to the earth
quake and tsunami in March 2011, Switzerland decided, less than two months later, to
abandon plans to build new nuclear reactors; the operation of the last one being sus
pended in 2034. This decision will lead to a turnaround of the electrical supply strategy
by switching from a centralized to a decentralized and irregular electrical production [63].
The pumpedstorage power plants precisely meet the requirements to ensure a safe and
high quality supply of electricity. At least 3 huge pumpedstorage plants are already
planned to be commissioned up to 2015 increasing the hydropower capacity by 2 GW, [5]
and [6].
12000
10000
8000
6000
4000
2000
0
0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24
Storage hydropower plants Runofriver hydropower plants Nuclear power plants Other power plants
Figure 2.5: Daily electrical production of four days in 2010 in Switzerland [61].
12000
10000
8000
6000
4000
2000
0
0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24
Consumption exept pumping Comsumption for storage pumping Export balance Import balance
Figure 2.6: Daily electrical consumption of four days in 2010 in Switzerland [61].
turbines may be used [53]. Both the pump and the turbine are designed to be operated
at the most favorable operating conditions, making them still competitive [6].
Owing to the high costs of separate machines and the improved pump design, reversible
pumpturbines are often used. One impeller/runner, rotating in one direction for pumping
and in the other direction for generating electricity, is coupled to the motorgenerator.
The machine may be Francis (radial), Deriaz (diagonal) or bulb (axial) type, whether it
is used for application in high, medium or low head ranges, respectively [53]. Reversible
pumpturbines may be single or multistage designs [70].
However, due to the advance in research leading to a better design in pump mode and
because of the presence of adjustable guide vanes enabling more flexible operating con
ditions, singlestage reversible Francistype pumpturbines are nowadays mainly chosen
[54]; the power regulation being, therefore, more efficient. Moreover, the pumpedstorage
plants currently tend to be located where a very high head is available in order to save
the capital costs per unit of stored energy and to reduce the size of the reservoirs and the
powerhouse [78]. Moreover, a pumpedstorage plant does not produce energy, but rather
“transfers the energy from times of low demand to peak demand periods”[60]. High rate
of utilization induces frequent start and stop which reduces the reliability of the utilities
in comparison with conventional generating hydraulic machines.
In addition, pumpedstorage is currently a dominant ancillary services provider. More
than a simple peak power supplier, a pumpedstorage plant offers indeed services “to
improve system reliability, such as frequency control, voltage regulation, and reserve
operation”[79]. In the 21st century, to keep the pumpedstorage competitive against
other generating sources, focus must be put, in particular, on quicker mode changes, the
reduction of capital costs, the improvement of part load efficiency and the extensive use
of the adjustable speed technology.
Spiral casing
Impeller
Pump
mode
Turbine
mode
Stay vanes
Guide vanes
Draft tube
where the − sign is used in generating mode and the + sign in pumping mode; gHX being
the specific energy available at the fluid section X and gHrX÷Y , the specific energy losses
from the fluid section X to Y .
The hydraulic power Ph corresponds to the power available between the sections I and
¯
I and is obtained as follows:
Ph = ρQE , (2.2)
ρ and Q being the water density and flow discharge at the pumpturbine section I, re
spectively.
The Figure 2.9 shows the mechanical power balance in the impeller in both generating
and pumping modes. In the same way as the hydraulic power, the transformed power Pt
is defined as:
Pt = ρQt Et , (2.3)
where Qt and Et are the transferred discharge and specific energy, respectively. The
difference between Pt and Ph is the volumetric and friction losses between the sections I
and 1 and the respective low pressure sections I¯ and 1̄.
By definition, Q > 0 in generating mode and Q < 0 in pumping mode. According to
eq. 2.3, the power sign follows the same sign convention. The power P supplied to or by
the electrical machine, depending on whether one is generating or pumping, is given by:
B
upper reservoir
A
generatin
pumping
g I B
A lower reservoir
I
pumpturbine
η being the total efficiency, ηe , the energetic efficiency, ηh , the volumetric efficiency, ηrm ,
the mechanical efficiency in the impeller expressing the losses by disc friction and ηm , the
mechanical efficiency in the shaft expressing the power losses in the bearings.
Finally, the velocity triangles at the low and high pressure sides of the machine in
generating mode, on the external stream line e, are drawn in Figure 2.10. The Euler
equation expresses the transferred specific energy Et as a function of the absolute flow
velocity C and the peripheral impeller velocity U at the low and high pressure sides of
the machine, as follows:
In the present study, the RSI is the source of the guide vane vibrations. By addressing
the physical principles of this phenomenon, one is interested to determine the excitation
function of the mechanical system constituted by the guide vane cascade.
Generating mode
Qt Pt
Pumping mode
1
Qt
Pt
Figure 2.9: Mechanical power balance in the impeller in both generating and pumping
modes.
1e
1e
the guide vanes is spatially perturbed due to the periodical flow velocity defects in the
potential wake of the guide vanes [47]. The potential effects depend strongly on rotor
stator gap thickness. In axial gas turbines, Dring et al. [29] state that “the potential flow
over a row of airfoils can cause unsteadiness in both the upstream and downstream rows
if the axial gap between them is less than approximately the airfoil chord”.
Second, the fluid viscosity reinforces the flow velocity defect in the guide vane wake [48].
The resulting potential and viscous wakes interaction causes the relative flow velocity W1
at the impeller blade leading edge to fluctuate in time due to the periodic spatial variations
of the absolute flow velocity C1 , the impeller peripheral velocity U1 staying constant, see
Figure 2.11. The frequency at which W1 fluctuates is f = mzo n, m being an integer and
zo , the guide vanes number. The shape of the guide vanes wake, the opening angle and the
rotorstator gap thickness are responsible for the number of harmonics m present in the
fluctuations. The fluctuations of W1 , both in terms of magnitude and direction, induce
oscillations of the pressure field around the impeller blades. As a result, the impeller
blades are undergoing a fluctuating lift force. It is rather difficult to distinguish between
potential wake and viscous wake effects, but Kemp and Sears [48] note that, “the unsteady
forces arising from passage through viscous wakes are of about the same size as those due
to [potential wakes]”. Arndt et al. [3] adds that, if the rotorstator clearance is small,
both wake and potential mechanisms initiate RSI.
Third, the possible vortex shedding, whose physics and generation process on an iso
lated hydrofoil is well described by Ausoni [4] and Zobeiri [91], at the guide vane trailing
edge may play a role in the RSI as well. If flow conditions and guide vane geometry are
favorable to vortex shedding, the impeller inflow velocity W1 may be influenced.
Finally, vibrating blades may have an effect on RSI. Collard and Cizmas [21] numer
n n
∆θ= 2π n ∆t
U1 W1
U1 W1
C1
C1
Figure 2.11: Influence of guide vane wake flow defect on the impeller inlet velocity triangle.
ically study the effects of vibrating rotor blades in an axial gas turbine. The efficiency
is found to decrease by approximately 1.8% compared to the turbine with rigid blades.
Giesing [35] proposes a general method for determining the unsteady incompressible flow
around one or more vibrating bodies placed in small amplitude gust fields. Kahl [45]
studied the effects of mistuning and coupling in a turbomachinery bladings. Beretta [11]
proves that the effect of rotor blade vibrations may be linearly superposed to rotor blades
gust response in an axial gas turbomachine. Rottmeier [72] highlight the dependence of
the gust phase on the pressure field around rotor blades.
The physics resulting from all these root phenomena is very complex. Research is
incidentally devoted to simplified numerical methods to take into account the RSI [20].
Moreover, these root phenomena produce other effects, such as guide vane vibrations [58],
impeller vibrations [23], shaft vibrations [38] and spinning pressure modes in the stator.
In pumpturbines, BlancCoquand and Lavigne [13] note that the fluctuating pressure
field due to the RSI is much more complex in the rotorstator clearance than away from
this zone. Many rotating pressure modes are actually present close to the rotorstator
clearance, but only the most energetic pressure modes remain in the spiral casing.
To the author’s knowledge, in the case of radial hydraulic machines, the RSI physics
has never been analytically approached. Tanaka [77], Franke et al. [34] and Dubas [30]
address the problem with elementary fluid flow principles and construct the RSI pressure
mode shapes based on requirements to satisfy consistency and the fluid physics. The
pressure consists of a modulation of the stator and rotor pressure fields which are firstly
decomposed in Fourier series. This approach makes apparent the RSI spinning modes,
but apparently fails to analytically predict the predominance of one spinning mode on
another; the predominance being only qualitatively appreciated based on the fact that a
mode with few diametrical nodes is more energetic. Nevertheless, numerous RSI studies
are based on this simplified approach, [92] and [33]. In addition to the complexity of the
root phenomena, the lack of data, [16] and [25], may be the reason for the absence of
analytical approach to explain RSI physics. In the following lines, this approach to the
RSI is given.
The stationary ps (θ) and rotating pr (θr ) pressure fields are decomposed in Fourier
series as follows:
∞
X
ps (θ) = ps,m cos (mzo θ + φm ) (2.6)
m=0
∞
X
pr (θr ) = pr,m0 cos (m0 zb θr + φm0 ) (2.7)
m0 =0
m and m0 being integers, θ, the angular position in the stator, θr , the angular position
in the rotor, ps,m , the amplitude of the mth component of the stationary pressure Fourier
0
series and pr,m0 , the amplitude of the m th component of the rotating pressure Fourier
series.
The resulting pressure field p (θ) may be seen as a modulation which may be expressed
as the product of the two pressure fields:
The combination of given integers m and m0 makes apparent two rotating pressure
modes featuring k+ and k− diametrical nodes. As mentioned above, this approach fails
to predict which of these two modes predominates and, therefore, does not agree with
the experiments in terms of relative amplitude. Nevertheless, it successfully predicts the
number of diametrical nodes that would feature a pressure mode oscillating at a given
frequency.
Another interesting approach to RSI has been developed by Blake [12], which succeeds
in predicting the mode that predominates for a combination of given integers m and m0 .
Since it is based on the acoustic fluid theory, some assumptions are not in accordance
with the potential flow theory. Even though the results may be controversial, the author
has adapted the theory to the case of a radial hydraulic pumpturbine. The approach
may be found in Appendix E.
• Body oscillators: either a rigid structure or structural part, that is elastically sup
ported so that it can perform linear or angular movements, or a structure or struc
tural part that is elastic in itself so that it can perform flexural movements.
• Fluid oscillators: a passive mass of fluid that can undergo oscillations usually gov
erned either by fluid compressibility or by gravity.
According to this approach, in the case of the guide vane cascade, the zo guide vanes
consist of zo rigid body oscillators. The flow compressibility and the gravity effect are
neglected, so that there is no fluid oscillator. The guide vanes are extraneously excited
by the RotorStator Interaction. Moreover, the vibrations of the guide vanes consist of a
movementinduced excitation.
where I S denotes the bending structural inertia, C S the bending structural damping, K S
the bending structural stiffness and Fi the external force acting on the guide vane Oi .
The torsion angle αi of the guide vane Oi is governed by the expression:
where J S denotes the torsion structural inertia, DS the torsion structural damping, LS
the torsion structural stiffness and Ti the external torque acting on the guide vane Oi .
The fluctuating force contribution Fi0 linearly dependent of the angle of incidence is
zero when the flow velocity is zero.
The lift force Fi results from the pressure field and from wall shear stresses on the
guide vane Oi surface as follows [8]:
Z Z
Fi (t) = − (p(t) [I] · n) · ei,2 dAi + (τ (t) · n) · ei,2 dAi , (3.4)
Ai Ai
p (t) = pa (t) − patm being the gauge pressure obtained by subtracting the atmospheric
pressure patm from the absolute pressure pa , Ai the guide vane Oi surface, τ the shear
stress tensor, [I] the identity matrix, n the vector normal to the guide vane surface and
ei,2 the vector perpendicular to the chord profile, see Figure 3.1.
For high Reynolds number, the viscosity effects may be neglected. Moreover, by
ignoring any turbulence effect that might occur, one obtains:
Z
Fi (t) ≈ − (p(t) [I] · n) · ei,2 dAi (3.5)
Ai
For the case without moving guide vanes, eq. 3.11 yields:
p̄ + pRSI (xi,k , t)
c¯p + cRSI
p =
ρ·E
1 1
= 1 − [g (Z (xi,k ) − ZB̄ ) + ∇Φ̄ (xi,k ) 2
E 2
∂ΦRSI (xi,k , t)
+ + ∇Φ̄ (xi,k ) · ∇ΦRSI (xi,k , t)] (3.12)
∂t
Replacing eq. 3.12 in eq. 3.11, one obtains:
∂Φf (xi,k , t)
1
cp = c¯p + cRSI
p − f
+ ∇Φ̄ (xi,k ) · ∇Φ (xi,k , t)
E ∂t
= c¯p + cRSI
p + cfp (3.13)
4Φ = 0 , in V (3.14)
4Φf = 0 (3.15)
For small displacements, this potential has to satisfy the following Neumann condition
on the boundary made by the guide vane surface Aj , see Figure 3.1:
∂Φf (xj,k , t)
= ẏj (t) ej,2 · nj,k + (α̇j (t) ej,3 × xj,k ) · nj,k , for xj,k ∈ Aj (3.16)
∂n
ej,2 V
ej,1
Aj+1 nj
Aj
Aj+zo
Aj+2
Aj+zo 1
Aj+3
Aj+zo2
As shown in eq. C.16 in Appendix C, the potential at any position xi,k in the fluid
volume V may be expressed as a function of the boundary condition:
zo
f
X ∂Φf (xj,k , t)
Φ (xi,k , t) = Λj (xi,k )
j
∂n
zo
X zo
X
= Λyj (xi,k ) · ẏj (t) + Λαj (xi,k ) · α̇j (t) (3.17)
j j
From eqs. 3.3, 3.6 and 3.18, the force acting on the guide vane Oi may be expressed
as:
Fi (t) = F̄i + FiRSI (t)
− Kif,α · αi (t)
Xzo
f f,α
− Ii,j · ÿj (t) + Ii,j · α̈j (t)
j
z
X o
f f,α
− Ci,j · ẏj (t) + Ci,j · α̇j (t) (3.19)
j
Kif,α being the fluid torsion stiffness acting on the guide vane Oi ; Ii,j
f,α
, the added mass
f
on the guide vane Oi due to the torsion motion of the guide vane Oj ; Ii,j , the added
f,α
mass on the guide vane Oi due to the bending motion of the guide vane Oj ; Ci,j , the
hydrodynamic damping constant on the guide vane Oi due to the torsion motion of the
f
guide vane Oj and Ci,j , the hydrodynamic damping constant on the guide vane Oi due to
the bending motion of the guide vane Oj .
For the torque, the same procedure may be followed and the hydrodynamic torque is
expressed as:
Ti (t) = T̄i + T RSI (t)
− Lf,α
i · αi (t)
Xz o
f f,α
− Ji,j · ÿj (t) + Ji,j · α̈j (t)
j
zo
X
f f,α
− Di,j · ẏj (t) + Di,j · α̇j (t) (3.20)
j
where Lf,α
i
f,α
is the fluid torsion stiffness acting on the guide vane Oi ; Ji,j the added mass
f
on the guide vane Oi due to the torsion motion of the guide vane Oj ; Ji,j the added
f,α
mass on the guide vane Oi due to the bending motion of the guide vane Oj ; Di,j the
hydrodynamic damping constant on the guide vane Oi due to the torsion motion of the
f
guide vane Oj ; Di,j the hydrodynamic damping constant on the guide vane Oi due to the
bending motion of the guide vane Oj .
Using eqs. 3.1, 3.2, 3.19 and 3.20 together, the dynamics of the guide vane cascade is
governed by the following set of equations:
(3.22)
(3.23)
K1f,α
K1S 0 . . . 0 ... ... 0
0 KS . . . 0
2 0 K2f,α ... 0
.. .. . . .. .. .. .. ..
.
. . . . . . .
0 . . . . . . KzS 0 ... ... Kzf,α
[K] = o
0 . . . . . . 0 LS + Lf,α
o (3.24)
1 1 ... ... 0
0
0 . . . 0 0 LS
2 + Lf,α
2 ... 0
. . . . . . .. ..
. . . . . .
. . . . . . . .
0 ... ... 0 0 ... . . . LSzo + Lfzo
Münch et al. [56] found that a single hydrofoil immersed in a flow and exhibiting
torsion motion may be seen as a 2nd order mechanical system. The hydrodynamic loading
is modeled as a combination of inertia, damping and stiffness effects. In eq. 3.21, these
terms are modeled by the diagonal terms of the matrices [I], [C] and [K].
The experimental investigation in the present document is devoted to the identification
of the nondiagonal terms of the matrices. Therefore, it aims to show that the guide vane
cascade behaves as a 2nd order mechanical system. Moreover, as reported in [49], the
nondiagonal terms depend on the vibration phase, the distance between the vibrating
structure and the amplitude of the vibrations. In addition to these parameters, Basak
and Raman [7] found that the Reynolds number has also an influence on the coupling
between two neighboring vibrating structures. The present study intends to highlight
the dependency of these parameters in the case of the guide vane cascade. Faced to the
complexity of the entire cascade, the mechanical system is first reduced to a simple 2
DOF system by considering only two neighboring guide vanes. Then, we will show that
this approach allows to analyze the dynamics of the entire cascade. Finally, solutions are
proposed to prevent damage to the guide vanes.
The Caughey condition is necessary and sufficient for the mechanical system to feature
2zo real modes. This condition is expressed:
If the Caughey condition is not satisfied, the modes are complex and the formulation
of the problem may be established using the Duncan transformation [28]. Therefore, the
eq. 3.25 may be written as follows:
− [I] [0]
[G] = (3.29)
[0] [K]
Applying the change of variables z = [B] q, the system of equations in eq. 3.27 may
be decoupled:
where [B] is the change of base matrix and where [H o ] = [B]T [H] [B] and [Go ] =
[B]T [G] [B] are diagonal matrices.
Multiplying by [H o ]−1 , the eq. yields to:
q̇ + [H o ]−1 [Go ] q = q̇ + [∆] q = 0 (3.31)
where [∆] is the diagonal matrix containing the eigenvalues of the matrix [H o ]−1 [Go ].
The system regroups 4zo independent equations:
q̇p + δp q = 0 , p = 1, 2, ...4zo (3.32)
Each eigenmode has not only different amplitude βlp but also different phase φlp . The
eigenshape must therefore be defined in the phase space. For dissipative motion with
complex modes, the solution finally yields:
2·zo
X
y= βip Yp e−λp t cos (ωp t − ψip − φp ), for i = 2zo + 1, 2zo + 2...4zo (3.39)
p
where φp and Yp are defined by the initial conditions y (0) and ẏ (0).
Taking into account the periodicity condition of the guide vane cascade, the phase
shift between two adjacent components of the eigenvector is expressed as:
2π
∆φp = φl+2,p − φl,p = ·p (3.40)
zo
Investigation methodology
Investigated pumpturbine
Characteristics Value
Specific speed ν 0.17
Number of impeller blades zb 9
Number of guide vanes zo 20
Outer impeller diameter D1e 527 mm
Impeller diameter at low pressure side D1̄e 250 mm
Height of the distributor channel Bo 36 mm
Narrowest width of guide vane channel W 25 mm
7.0
y1e
0.88
[]
0.92
6.5
0.94
0.96
0.98
6.0
12°
5.5
BEP
34°
5.0
18°
4.5
4.0
j1e
0.8
4
Figure 4.2: Relative efficiency η/ηmax hill chart in generating mode for opening angles αo
ranging from 12◦ to 34◦ .
The Best Efficiency operating Point, BEP, is reached at 18◦ opening angle and corre
sponds to a discharge coefficient ϕ1̄e = 0.36 and an energy coefficient ψ1̄e = 5.3. These
coefficients are defined in [42] as follows:
Q
ϕ1̄e = π2 3
= 0.36 (4.1)
4
D1̄e n
E
ψ1̄e = π2 2 2
= 5.3 (4.2)
4
D1̄e n
Q being the discharge and E the specific energy of the machine.
The specific speed ν is expressed from these two coefficients as follows:
ϕ0.5
ν= = 0.17 (4.3)
ψ 0.75
The guide vanes O10 and O11 , see Figure 4.3, are modified, their stems being more
flexible than those of the usual guide vanes. The material is similar (CuSn12) for all the
O10
O11
D1e= 250 mm
D1e= 527 mm
guide vanes. The assembly of the two types of guide vanes in the pumpturbine model is
shown in Figure 4.4. The drawings of the modified guide vane featuring a flexible stem
and the usual guide vane having a stiff stem are given in Figures 4.5 and 4.6, respectively.
As may be observed, the span width of the modified guide vanes is 0.2 mm smaller.
Therefore, we avoid any contact of the modified guide vanes with the upper or lower
flange, and we ensure reliable measurement of bending and torsion motions. Moreover,
the modified guide vanes are loosened from their lever and held in place with stiff stainless
Clamp
Actuating ring
Impeller blade
Guide vane
Stay vane
A
B
Guide vane
Stay vane
A
Figure 4.4: Assembly of the two types of guide vanes in the pumpturbine model.
Figure 4.5: Working drawing of the modified guide vane featuring a flexible stem.
Figure 4.6: Working drawing of the usual guide vane having a stiff stem.
Figure 4.7: Clamping system to hold the modified guide vane in place.
Test facility
The pumpturbine reduced scale model is placed in the EPFL PF2 test rig, see Figure
5.1, featuring a maximum specific energy E = 1.25 · 103 J·kg1 , a maximum discharge
Q = 1.4 m3 ·s1 ; the rotation speed being limited to N = 2.5 · 103 rpm. The test rig
fulfills the IEC standards 60193 [42] and reaches 0.2 % accuracy for the shaft torque, the
discharge and the differential pressure measurements. The closed loop circuit is made up
of one circulating pump, driven by a 1000 kW power electric motor, a valve controlling
the head losses, an air vessel for controlling the Net Positive Suction Head, NPSH. The
pumpturbine reduced scale model is coupled to a 300 kW electrical generator.
Sdp being the output voltage signal of the sensor and adp the calibration coefficient
obtained with a linear regression on the measured values.
0.4
0 0.50 0
0.3
0.1
Sdp SQ
−1.0 0 −1.0
6 8 10 [V] 12 6 8 10 12 [V] 14
Figure 5.2: Calibration of the Rosemount differential pressure sensors for specific energy
E measurement (left) and calibration of the electromagnetic flow meter (right)
Qref − aQ · SQ
0Q = (5.2)
Qref
SQ being the output voltage signal of the flow meter and aQ the calibration coefficient
obtained with a linear regression on the measured values.
Finally, the impeller rotation speed N is measured with the help of a tachometer:
the HEIDENHAIN ERA 1801AK.10039000 counter provides 9000 impulses per impeller
revolution to the HEIDENHAIN IBV 600 electronic converters.
Measuring apparatus
6.1.2 In water
To create an impulse excitation in water, a nonintrusive system is used, see [67]. A
BOSCH Super W7DC spark plug, see Figure 6.1 is flush mounted on the wall of the
bottom flange in a guide vane channel at the location SP, see Figure 6.2.
Figure 6.1: BOSCH Super W7DC spark plug used as a nonintrusive system to get the
impulse excitation in water
The discharge of 1.55 µF capacitor under a 4.3 kV voltage supply in a very short time
(∆t < 25 µm) generates a discharge energy of 14.3 J producing a rapid increase of the
water temperature. The explosive growth of a vapor bubble produces a strong pressure
wave traveling towards the hydrofoil at the speed of sound. The hydrofoil is, thereby,
impulsively excited on a wide band frequency range. Several collapses and rebounds of
the bubble occur, but the first pressure wave is the strongest.
SP
The visualization of the spark generated bubble is obtained with an ultra high speed
PHOTRON FASTCAM SA1.1 video system at 8’000 frames per second with a resolution
of 832x608 pixels. The growing and then collapsing vapor bubble can be observed in
Figure 6.3
Figure 6.3: Spark generated bubble visualization at zero flow velocity (time in [ms])
Torsion: +
T2
T1
Bending strain gage
5V Wheatstone bridge
T4
T3

 +
SOi,α
Figure 6.4: Strain gage Wheatstone bridges for torsion and bending monitoring.
F
y
2.3°
α
T
Figure 6.5: Direction of positive force and torque.
The force and torque calibration diagrams are drawn in Figure 6.6.
The calibration curves for the guide vanes O10 and O11 are given in Figures 6.7 and
6.8, for force and torque, respectively.
In these figures, the output voltage values of the two Wheatstone bridges, SOi ,y and
SOi ,α , corresponding to the reference forces Fref and torques Tref applied to the guide
vanes are plotted. The absolute measurement uncertainty, Fref and Tref are also given
a) b)
F F F F
Figure 6.6: Force a) and Torque b) calibration diagrams (lever arm units: [mm]).
50 0.5 50 0.5
0 0 0 0
−50 −0.5 −50 −0.5
εF11 [N]
−150 −1.5 −150 −1.5
SO10 ,y SO11 ,y
−250 −2.5 −250 −2.5
−0.4 0 0.4 0.8 [V] 1.2 −1.2 −0.8 −0.4 0 0.4 [V] 0.8
Figure 6.7: Bending force calibration for the guide vane O10 (left) and O11 (right).
where the sensitivity coefficients aOi , bOi , cOi and dOi are obtained with a multivariate
linear regression.
On the one hand, the output voltage signals SO10 ,α and SO11 ,α are 283 and 117 times
less sensitive than SO10 ,y and SO11 ,y , respectively, when applying a bending force. On the
30
SO10 ,α 3 30 SO11 ,α 0.3
[N m] Tref εT10 [N m] [N m] Tref
20 2 20 0.2
10 1 10 0.1
0 0 0 0
0.2 [V] 0.4 0.6 0.8 1 −0.3 −0.2 −0.1 0 [V] 0.1
Figure 6.8: Bending force calibration for the guide vane O10 (left) and O11 (right).
other hand, the output voltage signals SO10 ,y and SO11 ,y are 4 and 11 times less sensitive
than SO10 ,α and SO11 ,α , respectively, when applying a torsion torque.
Using the expression Fi = K s · yi and Ti = Ls · αi , K s and Ls being the structural
bending stiffness and torsion stiffness, respectively, the bending displacement and torsion
angle may be linked to the Wheatstone bridge output signals:
!
aOi bOi ∗
aOi b∗Oi
yi K s K s SOi ,y SOi ,y
= cOi dOi = ∗ (6.4)
αi s s
SOi ,α cOi d∗Oi SOi ,α
L L
a b c d
where a∗Oi = KOsi , b∗Oi = KOsi , c∗Oi = KOsi and d∗Oi = KOsi are the sensitivity coefficients for
bending displacement and torsion angle. These coefficients may be used for displacement
and angle monitoring during dynamic tests, since the Wheatstone bridge output voltage
is proportional to the structural strain, which is proportional to the bending deflection
and torsion angle.
The characteristics of the guide vanes equipped with strain gages are given in Table
6.2.
Table 6.2: Characteristics of the guide vanes O10 and O11 equipped with strain gages.
O10 O11
Static force range ±150 N ±150 N
Static force abs. uncertainty 0 ÷ 2.3 N 0 ÷ 1.4 N
Static torque range ±33 N·m ±33 N·m
Static torque abs. uncertainty 0 ÷ 1.5 N·m −1.1 ÷ 1.1 N·m
Sensitivity coefficient aOi 195.5 N·V−1 178.7 N·V−1
Sensitivity coefficient bOi 41.7 N·V−1 15.7 N·V−1
Sensitivity coefficient cOi 0.1 N·m·V−1 0.1 N·m·V−1
Sensitivity coefficient dOi 21.9 N·m·V−1 16.3 N·m·V−1
Static displ. range ±0.117 mm ±0.117 mm
Static displ. abs. uncertainty 0 ÷ 2 · 10−3 mm 0 ÷ 1 · 10−3 mm
Static angle range ±9.1 · 10−3 rad ±9.1 · 10−3 rad
Static angle abs. uncertainty 0 ÷ 0.4 · 10−3 rad −0.3 · 10−3 ÷ 0.3 · 10−3 rad
Sensitivity coefficient a∗Oi 1.53·10−4 mm·V−1 1.4·10−4 mm·V−1
Sensitivity coefficient b∗Oi 0.33·10−4 mm·V−1 0.12·10−4 mm·V−1
Sensitivity coefficient c∗Oi 0.28·10−4 rad·V−1 0.28·10−4 rad·V−1
Sensitivity coefficient d∗Oi 60.66·10−4 rad·V−1 45.15·10−4 rad·V−1
Bandwidth < 25 kHz < 25 kHz
The pressure sensors are statically calibrated up to 0.5 MPa absolute pressure. The
same conditioning electronics as for measurement is used, in order to include the condi
tioning effects in the sensitivity coefficients. The sensors are placed in a vessel in which
the pressure, measured with a high precision reference sensor, may be modified. Ten
pressure values ranging from 0.1 to 0.5 MPa are tested by increasing and decreasing the
pressure. The calibration sensitivity coefficients are identified using a linear regression.
The absolute measurement uncertainty is defined as follows:
psi = pref − asi · Ssi (6.5)
where asi is the pressure sensor si sensitivity coefficient and Ssi is the output voltage.
The calibration curve of the pressure sensor s10 is given, as an example, in Figure 6.10.
The characteristics of the pressure sensors are given in Table 6.3.
Value
Range 0.5 MPa
Abs. uncertainty ±0.5 kPa
Sensitivity 70 kPa·V1
Bandwidth < 25 kHz
O10
O11
s10
s11
g10 g9
g8
g11 g7
g12
g6
g13
g5
g14
X
g4
g15
g3
g16
g2
g17
g1
g18 g20
g19
Figure 6.9: Locations of the flush mounted pressure sensors on the head cover.
SS
10
εp,S10
0.5 0.5
0.2
0.3
0.1
0.0
0.2
−0.1
−0.2
S ps
0.1
10
−0.3
−2 −1 0 1 2 3 4 [V] 5
and pressure fluctuation signals are simultaneously recorded using NI PXI digitizers with
24 bits A/D resolution at 5 kHz sampling frequency over 15.36 s, keeping the operating
conditions of the pumpturbine constant.
Storage
Guide vanes Instrumented
displacement y guide vanes
NI PXI
Flow pressure Absolute
p pressure sensors
Chapter 6. Measuring apparatus
Electromagnetic
Discharge
Impeller
rotating speed n Tachometer
Table 7.1: RSI pressure mode in the stator corresponding to the first m and m0 combi
nations, for a pumpturbine featuring zb = 9 impeller blades and zo = 20 guide vanes:
Numbers of diametrical nodes k− and k+ ; dimensionless frequency f /n of the modes
monitored in the stator; phase shift ∆φk between two pressure signals monitored at two
angular positions distant by ∆θ = 2π
zo
; dimensionless rotating frequencies, n− /n and n+ /n;
Bessel function values, Jk− and Jk+ , in the acoustic wave propagation approach.
These rotating pressure modes are experimentally highlighted in the following lines.
One focuses on the pressure fluctuations in the rotorstator gap at the BEP, see Figure
4.2. The specific energy of the pumpturbine is set to E = 435 J·kg1 and the impeller
rotation speed at 16.3 Hz. The 20 guide vanes feature a stiff stem in order to get rid of
any fluidstructure coupling.
The pressure fluctuations are monitored using the 20 pressure sensors located in the
rotorstator gap, namely gi (1 < i < 20), during 2.56 s. The sampling frequency is 51.2
kHz.
In Figure 7.2, the phase average of the pressure fluctuations monitored by the two
adjacent pressure sensors, g10 and g11 , is plotted. The phase average is obtained over 42
impeller revolutions representing 2.56 s.
The two pressure signals are phase shifted by 3.5 rad. The frequencies f = 9n and
f = 18n predominate in both signals.
The FFT is applied to signals which are 214 samples long. A hamming window is
used. The spectra are averaged over 8 records. In Figure 7.3, the waterfall diagram of
the amplitude spectra of the pressure factor fluctuations is plotted against the angular
position θ of the pressure sensors in the rotorstator vaneless gap. Most of the spectral
density energy is concentrated at the frequencies f = mzb n revealing the RSI modes.
n
f = 9n
+ k= 9
 n= n
+ 
 +
n
+ 
 +
+ 
f = 18n
 + n k= 2
+ n= 9n
 +   +
n
n f = 27n
 + k= 7 + 
+  n= 3.86n
 n +
+ 
 + f = 36n
n k= 4
+ +  n= 9n

 +
 n +
n +
f = 45n 
k= 5
 + n= 9n
 +
+ 
n
 +
+ 
 +
Figure 7.1: Shape of the five most predominant RSI rotating pressure modes for the case
of zb = 9 impeller blades and zo = 20 guide vanes.
0.03
[] c'p g10 g11
0.02
0.01
−0.01
t n
−0.02
0 0.2 0.4 0.6 0.8 [] 1
Figure 7.2: Phase average of the pressure factor fluctuations monitored by the sensors g10
and g11 over 42 impeller revolutions.
[] c'^p 
θ
[°]
f/n
[]
Figure 7.3: Waterfall diagram of the amplitude spectra of the pressure factor fluctuations
against the angular position θ of the pressure sensors in the rotorstator vaneless gap,
namely gi (1 < i < 20).
In Figure 7.4, the spectra amplitude value of pressure fluctuations ĉ0p  at the frequen
cies f = mzb n corresponding to the RSI fundamental and the harmonics are represented
against the angular position θ in the rotorstator gap. The standard deviation indicating
the variation from the mean value is represented by intervals.
The low standard deviations highlight the reliability of the measurements. The pres
sure fluctuations amplitude at the RSI 2nd harmonic is slightly higher than the one at the
fundamental. At the 5th harmonic f = 45n, the pressure fluctuations amplitude is more
than 8 times inferior to the amplitude at the 2nd harmonic frequency f = 18n.
0.005
^ 
[] c'
p
0.004
f=18n
f=9n
0.003
f=27n
0.002
0.001
f=36n
f=45n
0
θ
0 90 180 270 [°] 360
Figure 7.4: Values of the amplitude spectra of the pressure factor fluctuations ĉ0p  at the
frequencies f = mzb n corresponding to the RSI fundamental and the first four harmonics,
against the angular position θ in the rotorstator gap.
: impeller revolution
: impeller rotation
direction
Figure 7.5: Qualitative circumferential evolution of the pressure fluctuations in the rotor
1
stator gap during 9n s at a frame rate F P S = 10 · 9n.
rotating at the same frequency as the impeller n− = n, see Figure 7.6. As already
mentioned above, in the literature, some authors mention the presence of a 11nodal
diameter mode at this frequency. It is certainly a confusion, since the 9nodal diameter
mode clearly predominates in this study. Moreover, the combination of zb impeller blades
with zo guide vanes induces a spatial modulation which might erroneously be confused
with a 2nodal diameter mode. Finally, the small number of monitoring points compared
to the number of diametrical nodes may make erroneously believe that the mode is rotating
in the opposite direction to the impeller.
At the RSI 2nd harmonic, the circumferential shape of the pressure mode is a 2nodal
diameter mode rotating at n− = 9n in the opposite direction to the impeller, see Figure
7.7, as forecasted by the theory. This mode may already be observed in the pressure
fluctuations circumferential evolution without filtering, see Figure 7.5.
At the RSI 3rd harmonic, the circumferential shape of the pressure mode is a 7nodal
diameter mode rotating at a rotating frequency slightly inferior to n− = 4n, as predicted,
see Figure 7.8.
At the RSI 4th harmonic, the pressure mode features 4nodal diameters and rotates
at n− = −9n in the opposite direction to the impeller, see Figure 7.9.
At the RSI 5th harmonic frequency, the circumferential shape of the pressure mode
features 5nodal diameters rotating at the frequency n− = 9n in the same direction as the
Figure 7.6: Qualitative circumferential evolution of the pressure fluctuations at the RSI
fundamental f = 9n in the rotorstator gap, during 21 of impeller revolution at a frame
rate F P S = 10 · 2n.
Figure 7.7: Qualitative circumferential evolution of the pressure fluctuations at the RSI
1
2nd harmonic f = 18n in the rotorstator gap, during 18 of impeller revolution at a frame
rate F P S = 10 · 18n.
Figure 7.8: Qualitative circumferential evolution of the pressure fluctuations at the RSI
1
3rd harmonic f = 27n in the rotorstator gap, during 2·3.86 of impeller revolution at a
frame rate F P S = 10 · (2 · 3.86n).
Figure 7.9: Qualitative circumferential evolution of the pressure fluctuations at the RSI
1
4th harmonic f = 36n in the rotorstator gap, during 18 of impeller revolution at a frame
rate F P S = 10 · 18n.
Figure 7.10: Qualitative circumferential evolution of the pressure fluctuations at the RSI
1
5th harmonic f = 45n in the rotorstator gap, during 18 of impeller revolution at a frame
rate F P S = 10 · 18n.
Impulse response
The results reported in this chapter are related to the impulse response in air, in water,
model at rest, and model in operation. They are all the more relevant since the study of
the guide vane cascade forced response, see Chapter 9, is based on them.
The modal analysis in air aims to detect the eigenmodes of the guide vanes and to
compare them to the theoretical eigenmodes. The eigenfrequencies, the eigenshape and
the structural damping related to the mode are studied.
The modal analysis in water, model at rest, intends to detect the eigenmodes of the
guide vanes immersed in still water and, therefore, to identify the added mass as well as
the hydrodynamic damping.
The impulse response of the guide vanes placed in the operating machine highlight the
interaction between the vibrating guide vanes. It is the necessary preliminary approach
to the guide vanes forced response study reported in the Chapter 9.
8.1.1 Eigenfrequencies
In Figure 8.1, the first 0.25 s of typical normalized bending displacement and torsion angle
impulse response signals of the two guide vanes, O10 and O11 , are plotted.
1.0 1.0
y α
[] ymax O10 O11 []
αmax O10 O11
0.5 0.5
0 0
−0.5 −0.5
t t
−1.0 −1.0
0 0.05 0.1 0.15 0.2 [s] 0.25 0 0.05 0.1 0.15 0.2 [s] 0.25
Figure 8.1: Impulse response in air: time signal of normalized bending displacement (left)
and torsion angle (right).
In Figure 8.1, one may observe that the torsion signal is damped more than the bending
signal. A single frequency predominates in the bending signal, whereas the torsion signal
contains more frequencies.
In Figure 8.2, the amplitude spectra of the normalized bending displacement and
torsion angle of the two guide vanes, O10 and O11 , are plotted.
0.3 0.03
^ ^
[]  ^y  O10 O11 []
α
^  O10 O11
ymax αmax
0.2 0.02
0.1 0.01
f f
0 0
0 200 400 600 800 [Hz] 1000 0 200 400 600 800 [Hz] 1000
Figure 8.2: Impulse response in air: amplitude spectra of the guide vane O10 and O11
normalized bending displacement (left) and of the normalized torsion angle (right).
In Appendix B.4.1, the first bending and the first torsion eigenfrequencies are analyt
ically estimated using the RayleighRitz method, RR. The 1st bending eigenfrequency is
R−R R−R
found to be f0,y = 306 Hz and the 1st torsion eigenfrequency is estimated at f0,α = 645
Hz. Two main frequencies, around 300 Hz and 740 Hz, are present in the bending and
torsion signals, see Figure 8.2. In the following lines, it is shown that these two frequencies
correspond to those of the eigenmodes analytically estimated.
Both analytical and experimental eigenfrequencies are reported in Table 8.1. The
inertia, I S and J S , related to the first bending mode and to the first torsion mode,
respectively, are also given. They are deduced from the stiffness constant, K S and LS ,
analytically determined in Appendix B.3 as follows:
KS
IS = (8.1)
(2πf0,y )2
LS
JS = (8.2)
(2πf0,α )2
Table 8.1: First bending and first torsion eigenfrequencies of the guide vane O10 and
S S
O11 placed in air, f0,y and f0,α ; relative difference f0,y
S and f0,α
S from the theoretical
S S
eigenfrequencies; corresponding structural mass/inertia, I and J ; and relative difference
from theoretical values, I S and J S .
The f0,y
S absolute value is less than 3% for both guide vanes and, therefore it may
easily be concluded that the 302 Hz and 297 Hz frequencies correspond to the first bending
eigenfrequency of the guide vanes O10 and O11 , respectively.
The f0,α
S values are comprised between 14% and 15%. Nevertheless, one may state
that the 740 Hz and 736 Hz frequencies are those of the first torsion eigenmode of the
guide vanes O10 and O11 , respectively, for three main reasons. First, no other frequency
predominates in the corresponding frequency range. Second, the torsion signal is much
more responsive at this frequency than the bending signal. Finally, the identification of the
analytical torsion eigenfrequency is based on the balance between the deformation and the
kinetic energies, see Appendix B.4.1. For simplicity reason, these energies are determined
for a pure torsion motion. In the real case, the eccentricity of the hydrofoil center of
gravity from the stem neutral axis should make the deformation energy slightly higher
and, at the same time, should increase the theoretical value of the torsion eigenfrequency.
Concerning the impulse response spectra plotted in Figure 8.2, the amplitude of the
peaks must be put into perspective. One should keep in mind that the Fourier transform
is applied to an exponentially decreasing oscillating signal. The bending mode is much
more energetic than the torsion mode; the former being even present in the torsion signal.
The torsion mode is much more damped and, therefore, only the bending one remains on a
vast range of the torsion signal length, making its amplitude high in the averaged spectra.
Finally, on both angle and displacement spectra, a smaller peak appears at f = 264 Hz
and f = 270 Hz for the guide vane O10 and O11 , respectively. We suppose the presence of
the frequency of the bending eigenmode in the direction perpendicular to the ydirection.
The torsion strain gage bridge is indeed more sensitive to this motion type. The peak
hardly appears in the bending displacement signal; the bending strain gage bridge being
theoretically not sensitive to this type of motion.
The frequency response magnitude and phase of the Butterworth filters for the cases of
bending and torsion modes are plotted in Figure 8.3. It may be observed that the filters
may alter the time signals in terms of phase if the cutoff frequencies are not equally
positioned on both sides of the concerned frequency. Nevertheless, in terms of amplitude,
the signals are not modified so long as the cutoff frequencies are sufficiently far from each
other and, therefore, the filter has no repercussion on the damping identification accuracy.
Then, by applying the Hilbert transform [68], the envelope of the filtered signal is
obtained. Finally, an exponential fitting of the envelope is applied for estimating the
damping value. The fitting curve h (t) is expressed as:
In Figure 8.4, the identification procedure of structural damping of the bending mode
and of the torsion mode is illustrated for the case of the guide vane O10 . The original
signal, the filtered signal, the envelope and the fitting curve are plotted.
0
10 0
^ ^
Arg(B)
[] B [rad] bending torsion
−1
10 π
−2
10 2π
−3
10 3π
−4
f f
10 4π
0 200 400 600 800 [Hz] 1000 0 200 400 600 800 [Hz] 1000
Figure 8.3: Butterworth filter frequency response magnitude and phase for the bending
and the torsion modes.
The damping constants, C S and DS and damping factors, hSy and hSα are defined as
functions of the damping coefficients as follows:
C S = 2I S λSy (8.4)
λSy
hSy = S
(8.5)
2πf0,y
DS = 2J S λSα (8.6)
λSα
hSα = S
(8.7)
2πf0,α
(8.8)
The damping values are averaged over 16 records. The mean values of the damping
constants and factors are listed in Table 8.3. The standard deviations are also indicated.
Table 8.3: Structural damping constants and factors of the guide vanes O10 and O11 .
O10 O11
damping factor hSy (0.40 ± 0.01) % (0.51 ± 0.05) %
damping constant C S (5.00 ± 0.12) kg · s−1 (6.57 ± 0.44) kg · s−1
damping factor hSα (1.50 ± 0.07) % (1.46 ± 0.09) %
damping constant DS (0.024 ± 0.001) kg · m2 · s−1 (0.024 ± 0.002) kg · m2 · s−1
The two guide vanes O10 and O11 feature relatively similar damping constant values.
The damping of the guide vane O11 bending mode is nevertheless 30% higher than the
guide vane O10 , whereas the damping of the guide vane O11 torsion mode is 3% lower
1.0 1.0
y10 original signal
α10
[] y10 max [] original signal
filtered signal
α10max
filtered signal
envelope envelope
0.5 fitting curve 0.5 fitting curve
0 0
−0.5 −0.5
t t
−1.0 −1.0
0 0.05 0.1 0.15 0.2 [s] 0.25 0 0.05 0.1 0.15 0.2 [s] 0.25
Figure 8.4: Identification procedure of structural damping of the bending mode (left) and
of the torsion mode (right) for the case of the guide vane O10 in air.
than the guide vane O10 . The low standard deviation values indicate fair measurements
and reliable damping results.
8.2.1 Eigenfrequencies
In Figure 8.5, the first 0.25 s of typical normalized bending displacement and torsion angle
signals, for the two guide vanes, O10 and O11 , placed in water, model at rest, are plotted.
The bending and torsion signals are damped more than in air. Moreover, the guide vane
O10 presents a higher damping than the guide vane O11 . One may also remark the lower
frequency of the vibrations, at least in the bending displacement signal.
In Figure 8.6, the amplitude spectra of the normalized bending displacement and of
the normalized torsion angle are plotted. In the torsion angle amplitude spectra, the am
plitude at the bending eigenfrequency is lower than the one observed in the corresponding
amplitude spectra in air, because of the higher damping of the vibrations in water, model
at rest, and because of the added mass.
The Table 8.4 lists the eigenfrequencies of the guide vanes O10 and O11 placed in still
water, the added mass/inertia for the bending and torsion modes and the ratio of added
1.0 1.0
y α
[]
ymax O10 O11 []
αmax O10 O11
0.5 0.5
0 0
−0.5 −0.5
t t
−1.0 −1.0
0 0.05 0.1 0.15 0.2 [s] 0.25 0 0.05 0.1 0.15 0.2 [s] 0.25
Figure 8.5: Impulse response in water, model at rest: time signal of normalized bending
displacement (left) and torsion angle (right).
0.1 0.05
^ ^
[]  ^y  O10 O11 []
α
^  O10 O11
ymax αmax
0.08 0.04
0.06 0.03
0.04 0.02
0.02 0.01
f f
0 0
0 200 400 600 800 [Hz] 1000 0 200 400 600 800 [Hz] 1000
Figure 8.6: Impulse response in still water: magnitude of amplitude spectra of the guide
vane O10 and O11 normalized bending displacement (left) and of the normalized torsion
angle (right).
to structural mass/inertia. The fluid mass motion induced by the vibrations of the guide
vanes reduces the value of the eigenfrequencies. The added mass/inertia, I f and J f , is
deduced from the structural inertia, I S and J S , and from the eigenfrequencies in air and
in water, model at rest. The eigenfrequencies in air may be estimated as follows:
r
S 1 KS
f0,y = (8.9)
2π r I S
S 1 LS
f0,α = (8.10)
2π J S
Table 8.4: Guide vanes first bending and first torsion eigenfrequencies in water, model
f f
at rest, f0,y and f0,α , added mass/inertia, I f and J f , for the bending and torsion modes,
respectively, and ratio of structural inertia to added mass/inertia.
O10 O11
f
f0,y 231 Hz 233 Hz
f
I 0.23 kg 0.21 kg
I f /I S 69.7 % 61.8 %
f
f0,α 626 Hz 630 Hz
J f,α 6.95 · 10−5 kg · m2 6.45 · 10−5 kg · m2
J f,α /J S 39.7 % 36.4 %
The fluid stiffness in still water is zero, see Section 3.3, and using eqs. 8.9 to 8.12, one
obtains the expression for the added mass I f and the added inertia J f,α :
!2
S
f0,y
If = IS · f − 1 (8.13)
f0,y
!2
S
f 0,α
J f,α = J S · f − 1 (8.14)
f0,α
For the bending and the torsion modes, the added mass and the added inertia are
significant and, thus, strongly affect the eigenfrequencies. The influence is nevertheless
higher for the bending mode than for the torsion mode. For the bending mode, the added
mass represents 69.7% and 61.8% of the structural mass for the guide vanes O10 and O11 ,
respectively. For the torsion mode, the added inertia represents 39.7% and 36.4% of the
structural inertia for the guide vanes O10 and O11 , respectively.
may be modified in terms of phase if the cutoff frequencies are not positioned equally on
both sides of the concerned frequency. No alteration in terms of amplitude should occur
so long as the cutoff frequencies are sufficiently far from each other. Therefore, the filters
have no repercussion on the damping identification accuracy.
0
10 0
^ ^
Arg(B)
[] B [rad] bending torsion
−1
10 π
−2
10 2π
−3
10 3π
−4 f f
10 4π
0 200 400 600 800 [Hz] 1000 0 200 400 600 800 [Hz] 1000
Figure 8.7: Butterworth filter frequency response magnitude and phase for the bending
and torsion modes in still water.
Then, the similar procedure as in air is followed for estimating the damping of the
guide vane in water, model at rest.
In Figure 8.8, the bending and torsion eigenmodes damping identification procedure
is illustrated for the case in water, model at rest. The original signal, the filtered signal,
the envelope and the fitting curve are plotted for the guide vane O10 .
The damping values are averaged over 8 records for the two guide vanes. The damping
constant and factor mean values are listed in Table 8.6. The standard deviations are
also indicated. The hydrodynamic damping constants, C f and Df,α , are determined
by subtracting the structural damping constants, C S and DS , from the total damping
constants, C = 2Iλy and D = 2Jλα , by assuming the linearity of damping with the
frequency. Therefore, the hydrodynamic damping may directly be expressed as function
of the inertia terms, the total damping coefficients, λy and λα and the structural damping
constants, C S and DS , as follows:
C f = C − C S = 2Iλy − C S = 2 I S + I f · λy − C S
(8.15)
f,α S S S f S
D = D − D = 2Jλα − D = 2 J + J · λα − D (8.16)
1.0 1.0
y10 α10
[] original signal [] original signal
y10  max α10 max
filtered signal filtered signal
envelope envelope
0.5 fitting curve 0.5 fitting curve
0 0
−0.5 −0.5
t t
−1.0 −1.0
0 0.05 0.1 0.15 0.2 [s] 0.25 0 0.05 0.1 0.15 0.2 [s] 0.25
Figure 8.8: Damping identification procedure of the bending mode (left) and the torsion
mode (right) for the case of the guide vane O10 placed in still water.
Table 8.6: Damping constant and factor values for the guide vanes O10 and O11 in still
water.
O10 O11
total damping factor hy (1.79 ± 0.03) % (1.67 ± 0.22) %
total damping constant C (29.2 ± 0.4) kg · s−1 (26.8 ± 3.6) kg · s−1
hydro. damp. const. C f (24.2 ± 0.5) kg · s−1 (20.2 ± 4.0) kg · s−1
total damping factor hα (1.31 ± 0.23) % (1.15 ± 0.06) %
total damping constant D (0.025 ± 0.004) kg · m2 · s−1 (0.022 ± 0.001) kg · m2 · s−1
hydro. damp. const. Df,α (0.001 ± 0.005) kg · m2 · s−1 (−0.002 ± 0.003) kg · m2 · s−1
In Table 8.6, it may be observed that the damping factor of the bending motion
is higher than the one of the torsion motion, whereas it was the inverse in air. The
two guide vanes feature relatively similar damping values. The low standard deviation
indicates reliable results. On the one hand, for the torsion mode, the hydrodynamic
damping constant value is found to be negligible, which agree with the potential flow
approach. At zero mean flow velocity, only the fluid inertia terms remain in eq. 3.20; the
dissipative terms vanishing. On the other hand, for the bending mode, the hydrodynamic
damping constant value is 4.8 and 3.0 times higher than the structural damping constant
value for the guide vanes O10 and O11 , respectively. In the absence of mean flow velocity,
the energy dissipation is certainly due to the viscosity of the fluid.
operating conditions have been chosen in order not to have neither the RSI fundamen
tal frequency nor the harmonics close to the guide vanes eigenfrequencies. Therefore,
keeping the machine operating at the BEP at 18◦ opening angle, the specific energy is
set to E = 98 J·kg1 and the rotation frequency to n = 7.4 Hz. The discharge is thus
Q = 0.108 m3 ·s1 . The flow turbulence noise must be dealt with. One will see in the
following sections that it is still present in the filtered signals. On the other hand, the
excitation procedure which consists of the rapid growth of a vapor bubble produced by
an immersed spark plug, providing a strong shock wave in the fluid, see Section 6.1.2, is
limited by the operating head. Above the specific energy value E = 98 J·kg1 , the static
pressure in the stator is too high for the shock wave to be energetic enough in order to
excite the guide vanes. Sometimes, the vapor bubble is not even formed. The signals are
recorded at 20 kHz. The Fast Fourier Transforms are applied to 214 samplelong signals,
which correspond to 0.819 s, leading to 1.22 Hz frequency resolution. The spectra are
averaged over 8 records for the two guide vanes.
Cref,u
α0
Cref,m
Cref
R3
Figure 8.9: Definition of the reference flow velocity Cref at the radius R3 .
1
The reduced frequency, comparing the oscillating period f
with the advection time
L
Cref
, is defined as follows [56]:
2πf L/2 πf L
κ= = (8.17)
Cref Cref
where L is the chord length and Cref , a reference flow velocity defined as follows, see
Figure 8.9:
Cref,m Q Q
Cref = = =
sin (αo ) A · sin (αo ) 2πR3 Bo sin (αo )
0.108
= ◦
= 4.8m · s−1 (8.18)
2π · 0.325 · 0.036 · sin (18 )
A being the surface crossed by the flow and Bo , the guide vane channel width.
The reduced frequency takes the values κy = 15 and κα = 41 for the bending and the
torsion modes, respectively, which show the relevance of the unsteady effects.
8.3.1 Eigenfrequencies
In Figure 8.10, the first 0.25 s of typical normalized bending displacement and torsion
angle signals, for the two guide vanes, O10 and O11 , placed in the operating machine,
are plotted. One may distinguish the response to a permanent excitation in the bending
displacement time signals after the impulse response is damped. The guide vane O10 is
damped less rapidly than O11 ; the response of the former to the permanent excitation
featuring an even higher amplitude. In the torsion angle time signals, the permanent
excitation is negligible compared to the impulse response.
1.0 1.0
y α
[]
ymax O10 O11 []
αmax O10 O11
0.5 0.5
0 0
−0.5 −0.5
t t
−1.0 −1.0
0 0.1 0.2 0.3 0.4 [s] 0.5 0 0.1 0.2 0.3 0.4 [s] 0.5
Figure 8.10: Guide vane impulse response, model in operation: time signal of normalized
bending displacement (left) and torsion angle (right).
In Figure 8.11, the averaged amplitude spectra of the normalized bending displacement
and of the normalized torsion angle are plotted. The permanent excitation observed in
the time signals is clearly observed at 67 Hz, 134 Hz and 200 Hz with peaks representing
the response to the fundamental, the 2nd and the 3rd harmonics of the RSI, respectively.
In the torsion angle amplitude spectra, the peaks at the two eigenfrequencies are much
higher than the one related to the RSI fundamental. That is, the guide vanes are excited
by the RSI more in terms of bending than in terms of torsion.
The Table 8.7 lists the guide vanes O10 and O11 eigenfrequencies in the model in
operation, the added mass/inertia for the bending and torsion modes and the ratio of
added to structural mass/inertia. The added mass and inertia are found using eqs. 8.13
and 8.14.
The eigenfrequencies are slightly higher when operating the machine than those in
water, model at rest. The bending eigenfrequency of the guide vane O10 is 1.3% higher
than the value in still water, whereas the torsion eigenfrequency is 2.4 % higher. In the
bending eigenfrequencies range, the guide vane O11 responds to two frequencies: 226 Hz
and 236 Hz, see Figure 8.11. The guide vane O11 torsion eigenfrequency is 1.9 % higher
than in still water. According to Conca et al. [22] and Brennen [17], the added mass and
inertia should not vary between the cases in water, model at rest, and model in operation.
These terms should actually be independent of the flow velocity. Therefore, the added
0.10 0.05
^ O10 ^
[]  ^y  O11
[]
α
^  O10 O11
ymax αmax
0.08 0.04
0.06 0.03
0.04 0.02
0.02 0.01
f f
0 0
0 200 400 600 800 [Hz] 1000 0 200 400 600 800 [Hz] 1000
Figure 8.11: Guide vanes impulse response, model in operation: amplitude spectra of
the guide vane O10 normalized bending displacement (left) and of the normalized torsion
angle (right).
Table 8.7: Guide vanes first bending and first torsion eigenfrequencies in the model in
operation, f0,y and f0,α , added mass/inertia, I f and J f , for the bending and torsion
modes, respectively, and ratio of added to structural mass/inertia.
O10 O11
f
f0,y 234 Hz 226/236 Hz
f
I 0.22 kg 0.25/0.20 kg
I f /I S 66.7 % 73.5 %/58.8 %
f
f0,α 641 Hz 642 Hz
J f,α 5.82 · 10−5 kg · m2 5.56 · 10−5 kg · m2
J f,α /J S 33.2 % 31.4 %
mass and inertia observed here must be brought by the vibrations of the neighboring guide
vanes. The Chapter 9 is incidentally devoted to the identification of the contributions of
the neighboring vibrating guide vane.
Table 8.8: characteristics of the bandpass filters in the case of the machine being operated.
torsion eigenmodes is illustrated for the guide vanes O10 and O11 , respectively. The
original signal, the filtered signal, the envelope and the fitting cure are plotted.
1.0 1.0
y10 α10
[] y10  max original signal [] original signal
filtered signal
α10 max filtered signal
envelope envelope
0.5 fitting curve 0.5 fitting curve
0 0
−0.5 −0.5
t t
−1.0 −1.0
0 0.1 0.2 0.3 0.4 [s] 0.5 0 0.1 0.2 0.3 0.4 [s] 0.5
Figure 8.12: Guide vane O10 damping identification procedure, model in operation.
1.0 1.0
y11 original signal α11
[] [] original signal
y11  max filtered signal
α11 max filtered signal
envelope envelope
0.5 fitting curve 0.5 fitting curve
0 0
−0.5 −0.5
t t
−1.0 −1.0
0 0.1 0.2 0.3 0.4 [s] 0.5 0 0.1 0.2 0.3 0.4 [s] 0.5
Figure 8.13: Guide vane O11 damping identification procedure, model in operation.
The damping coefficient values is averaged over 8 records for the guide vanes O11 and
O10 . The damping constants and factors mean values are listed in Table 8.9. The standard
deviation is also indicated.
Table 8.9: Damping constant and factor values for the guide vanes O10 and O11 in the
machine being operated.
O10 O11
total damping factor hy (1.6 ± 0.2) % (3.0 ± 0.2) %
total damping constant C (25.1 ± 3.1) kg · s−1 (49.4 ± 3.7) kg · s−1
hydro. damp. const. C f (20.1 ± 3.2) kg · s−1 (42.8 ± 4.1) kg · s−1
total damping factor hα (1.5 ± 0.0) % (1.0 ± 0.0) %
total damping constant D (0.030 ± 0.001) kg · m2 · s−1 (0.020 ± 0.000) kg · m2 · s−1
hydro. damp. const. Df,α (0.006 ± 0.002) kg · m2 · s−1 (−0.004 ± 0.002) kg · m2 · s−1
As may be qualitatively seen in Figures 8.12 and 8.13, the filtered bending signal
still contains turbulent noise. The hydrodynamic damping values are therefore underes
timated.
As a conclusion, one may expect a similar behavior of the two guide vanes between the
case in still water and the case in the machine in operation. But the results prove that
the guide vanes respond differently in the pumpturbine in operation. At these reduced
frequency values, κy = 15 and κα = 41, the dissipation due to the fluid advection should
be close to zero and the total dissipation should be similar to the one measured in water,
model at rest, [71]. On the one hand, it may be observed that the guide vane O10 features
a lower hydrodynamic bending damping constant C f = 20.1 ± 3.2 kg · s−1 than in still
water C f = 24.2 ± 0.5 kg · s−1 and, on the other hand, it may be seen that the guide vane
O11 presents a much higher hydrodynamic bending damping constant C f = 42.8 ± 4.1
kg · s−1 than in still water C f = 20.2 ± 4.0 kg · s−1 , see Table 8.9. By all evidence, the
guide vanes are mutually interacting. With the study of the guide vane forced response
reported in the next chapters, we will quantify the added mass and inertia as well as the
hydrodynamic damping brought by the vibrations of the neighboring guide vanes.
This chapter is devoted to the study of the forced response of the two modified guide
vanes, O10 and O11 , to the excitation due to RSI. The testing conditions are first given.
Then, the pressure vibrations of the guide vanes are analyzed in detail as well as the
pressure fluctuations in the proximity of the vibrating guide vanes.
In Figure 9.1, the experimentally measured values of the specific energy Eexp and the
discharge Qexp that maintain the operating conditions at the BEP are plotted against the
impeller rotational frequency n. Eexp ranges from 223 J·kg1 to 372 J·kg1 and Qexp , from
0.162 m3 ·s1 to 0.211 m3 ·s1 . In addition, the figure gives the values of the specific energy
E and discharge Q, if the expected value of ϕ1̄e and ψ1̄e have been reached. These values
are obtained as follows:
π2 2 2
E= · DIe
¯ · ψIe
¯ ·n (9.1)
2
π2 3
Q= · DIe ¯ ·n
¯ · ϕIe (9.2)
4
The relative differences ϕ1̄e and ψ1̄e are defined as:
ϕ1̄e − ϕ1̄e,exp
ϕ1̄e = (9.3)
ϕ1̄e
ψ1̄e − ψ1̄e,exp
ψ1̄e = (9.4)
ψ1̄e
Qexp Eexp
with ϕ1̄e,exp = π2 3
and ψ1̄e,exp = π2 2 2
4
D1̄e n 4
D1̄e n
With the relative differences ϕ1̄e and ψ1̄e , the offset between the experimental and
expected values of the coefficients, which is less than 1% for all investigated impeller
rotational frequencies n, is also illustrated in Figure 9.1.
380 0.22
E E exp εψ 1.0 Q Qexp εϕ 1.0
1
[J kg ] E 1e 3 1
[m s ] Q εϕ [%]
εψ [%]
1e
1e
1e
0.18
260 0.5 0.5
n 1.0 n 1.0
220 0.16
12 13 14 [Hz] 15 12 13 14 [Hz] 15
Figure 9.1: Experimental and expected values of specific energy, Eexp and E, and dis
charge, Qexp and Q, versus impeller rotational frequency n to ensure BEP operating
conditions.
The Reynolds number Re related to the flow in the guide vane channel is defined as
follows:
Cmax · Dh Q · Dh
Re = = (9.5)
ν zo · (W · Bo ) · ν
where Cmax is the maximum absolute flow velocity in the guide vane channel, W, the
narrowest width of the guide vane channel, Bo , the channel height, ν, the kinematic
viscosity and Dh , the hydraulic diameter, [74], defined as:
2 · W · Bo
Dh = (9.6)
W + Bo
The Reynolds number is comprised between 2.7 · 105 and 3.5 · 105 . The high Reynolds
number indicates the dominance of the inertial effects on the viscous effects. Moreover, at
BEP, the turbulence effects may be neglected and, therefore, the potential flow approach
given in Section 3.3 is appropriate.
Finally, the reduced frequency defined in eq. 8.17 takes the constant values κy =9.3
and κα =23.1 over the entire frequency range corresponding to the RSI 2nd harmonic and
over the frequency range related to the RSI 5th harmonic, respectively. In other words,
this means that the unsteady effects are important.
αi · L2
cα i = (9.8)
W
L being the guide vane chord length.
The RMSvalues of the displacement and angle factors, c̃yi and c̃αi , of the guide vanes
O10 and O11 are plotted on a logarithmic scale against the impeller rotating frequency n
in Figure 9.2.
The guide vane O10 experiences a maximum RMSvalue c̃y10 at n = 12.9 Hz. At this
impeller frequency, the RSI 2nd harmonic value is close to the 1st bending eigenfrequency,
231 Hz, in water, model at rest, see Section 8.2.1. As for the guide vane O11 , its maximum
RMSvalue c̃y11 is reached at n = 12.5 Hz. The corresponding RSI 2nd harmonic is 225 Hz.
Therefore, this guide vane does not preferably respond at the 1st bending eigenfrequency
in water, model at rest, but at a 3% lower frequency.
The RMSvalue of guide vane O11 torsion angle c̃α11 presents two maxima at n =
12.7 Hz and n = 14.6 Hz. The first maximum is linked to the bending mode excited
by the RSI 2nd harmonic. The second maximum is reached at an impeller frequency
whose corresponding RSI 5th harmonic is 660 Hz, being close to the 1st torsion eigenmode
frequency in water, model at rest, 626 Hz. The RMSvalue of the guide vane O10 torsion
c̃α10 presents a maximum at n = 13.0 Hz and a minimum at n = 14.6 Hz. The first
5
x10
50
40
30
20
10
n
5
12 12.5 13 13.5 14 14.5 15 15.5
[Hz]
Figure 9.2: Displacement and angle factors RMSvalues against the impeller frequency.
maximum is linked to the bending mode excited by the RSI 2nd harmonic. The minimum
may be caused by the guide vane resonance close to its 1st torsion eigenmode frequency.
The RMSvalues of the pressure factor, c̃p , monitored with the sensors s10 , s11 , g10 ,
g11 and g15 are plotted against the impeller rotating frequency n in Figure 9.3.
3
x10
c~p
17.5
[] s10 g10 s11 g11 g15
15.0
12.5
10.0
7.5
n
5.0
12 12.5 13 13.5 14 14.5 15 15.5
[Hz]
Figure 9.3: Pressure factor RMSvalues against the impeller frequency.
The pressure sensor g15 experiences a fairly constant RMSvalue around c̃p = 0.010.
The pressure factor fluctuations at this angular position are therefore not influenced by
the impeller frequency sweep, while the pressure factor fluctuations monitored by the
sensors g10 and g11 , located close to the flexible guide vanes O10 and O11 , respectively,
are strongly influenced. After a maximum reached at n = 12.5 Hz, these RMSvalues
decrease. The relative difference between the maximum and the minimum RMSvalues
for the pressure sensors g10 and g11 is 50% and 30%, respectively. Thereby, a transfer of
energy occurs locally from the flow to the vibrating guide vanes, O10 and O11 .
The pressure sensors, s10 and s11 , placed in the guide vane channel are also subject
to the influence of the vibrations, since their RMSvalues vary according to the impeller
frequency. The pressure factor fluctuations monitored by these sensors feature a higher
amplitude close to the resonance of the guide vanes. The increase of the fluctuations is
nevertheless higher for the sensor s11 positioned between the two vibrating guide vanes.
The waterfall diagrams of the power spectral density of the pressure factor monitored
by the pressure sensors g10 , g11 , g15 , s10 and s11 are plotted in Figures 9.4, 9.5 and 9.6,
respectively, against the impeller frequency n. The frequency is normalized with the
impeller frequency n to highlight the RSI fundamental frequency and its harmonics with
the multiples of the impeller blade number nf = mzb . The power spectral densities are
averaged over 8 blocks of 0.32 s, windowed with a Hamming function, each overlapping
by 50%.
2 2
4 c'^ pg10 4 c'^ pg11
[] []
6 6
8 8
10 10
12 12
n n
[Hz] f/n [Hz] f/n
[] []
Figure 9.4: Waterfall diagrams of power spectral density of the pressure fluctuation factor
monitored by the pressure sensors g10 (left) and g11 (right) against the impeller frequency
n.
4 c'^ pg  2
15
[]
6
8
10
12
n
[Hz] f/n
[]
Figure 9.5: Waterfall diagrams of power spectral density of the pressure fluctuation factor
monitored by the pressure sensors g15 , against the impeller frequency n.
8 8
10 10
12 12
n n
[Hz] f/n [Hz] f/n
[] []
Figure 9.6: Waterfall diagrams of power spectral density of the pressure fluctuation factor
monitored by the pressure sensors s10 (left) and s11 (right) against the impeller frequency
n.
sensors monitor a higher amplitude at two fixed frequencies for all the impeller frequencies
investigated. These frequencies correspond to the guide vane eigenfrequencies. Therefore,
the guide vanes reaching resonance are strongly modifying the pressure fluctuations, when
the eigenfrequencies corresponds to either the 2nd or the 5th RSI harmonics.
The waterfall diagrams of the power spectral density of the displacement factor fluc
tuations for the guide vanes O10 and O11 are plotted in Figure 9.7, against the impeller
frequency n.
The waterfall diagrams of the power spectral density of the fluctuating angle factor
10 10
12 12
14 14
16 16
n n
[Hz] f/n [Hz] f/n
[] []
Figure 9.7: Waterfall diagrams of the power spectral density of the guide vanes O10 and
O11 displacement factor fluctuations, against the impeller frequency n.
for the guide vanes O10 and O11 are plotted in Figure 9.8, against the impeller frequency
n.
8 c'^α10 2 8 c'^α112
[] []
10 10
12 12
14 14
16 16
n n f/n
[Hz] f/n [Hz]
[] []
Figure 9.8: Waterfall diagrams of the power spectral density amplitude of the guide vanes
O10 and O11 angle factor fluctuations, against the impeller frequency n.
Due to RSI excitation, most of the spectral energy is concentrated at RSI modes
frequencies f = mzb n. The response of the guide vanes to the two main RSI modes, f = 9n
and f = 18n, is nevertheless the highest. On the one hand, the 1st bending eigenmode is
clearly observed in Figure 9.7, its frequency crossing the RSI 2nd harmonic frequency at
n = 12.9 Hz. When the RSI 2nd harmonic frequency approaches the eigenmode frequency,
the response of the guide vanes greatly varies. The response of the guide vanes to the RSI
2nd harmonic depends, thus, on the impeller frequency n, because of their 1st bending
eigenmode. On the other hand, the 1st torsion eigenmode may clearly be identified in
Figure 9.8. The eigenmode frequency crosses the RSI 5th harmonic frequency and the
response of the guide vanes is severely perturbed.
One may state that the two eigenmodes appear in both the bending displacement and
the torsion angle waterfall diagrams. Nevertheless, the bending eigenmode predominates
in the displacement waterfall diagrams, whereas the torsion eigenmode predominates in
the torsion angle waterfall diagrams.
−1
10
[] c'^ p  s10 g10 s11 g11 g15
−2
10
−3
10
−4
10
10
−5 9n
105 110 115 120 125 130 135 140
[Hz]
Figure 9.9: Averaged Fourier transform magnitude of pressure factor fluctuations, mon
itored by the pressure sensors g10 , s10 , g11 , s11 , g15 , at the RSI fundamental frequency
f = 9n, for the investigated impeller frequency range.
The magnitude of the averaged Fourier transform of the pressure factor fluctuations
monitored by the pressure sensors s10 , s11 , g10 , g11 and g15 at the RSI fundamental and the
2nd to 5th harmonic frequencies is plotted against the corresponding harmonic frequency
in Figure 9.9 to 9.13, respectively, for the investigated impeller frequency range.
The magnitude of the pressure factor fluctuations monitored by each of the five sensors
does not vary with the impeller frequency at the RSI fundamental, see Figure 9.9. It stays
constant around ĉ0p  = 0.007 for the whole investigated impeller frequency range.
The magnitude of the pressure factor fluctuations ĉ0pg15 , monitored close to stiff guide
vanes, at the RSI 2nd harmonic frequency remains constant around 3.8 · 10−3 for the inves
tigated impeller frequency range, whereas the others present variations, see Figure 9.10.
The magnitude of the pressure factor fluctuations monitored by the sensor ĉ0pg11  features
a maximum to minimum ratio of 5, while the ratio reaches 12 for the pressure sensor ĉ0pg10 .
Concerning the pressure sensors positioned in the guide vane channels, the magnitude of
the pressure factor fluctuations monitored by the sensor ĉ0ps11  features a maximum to
minimum ratio of 4.5, while the ratio reaches 1.5 for the pressure sensor ĉ0ps10 . The pres
sure sensors monitor very different pressure fluctuation along the investigated frequency
range.
−1
10
[] c'
^
p s10 g10 s11 g11 g15
−2
10
−3
10
−4
10
10
−5 18n
210 220 230 240 250 260 270 280
[Hz]
Figure 9.10: Averaged Fourier transform magnitude of pressure factor fluctuations, mon
itored by the pressure sensors g10 , s10 , g11 , s11 , g15 , at the RSI 2nd harmonic frequency
f = 18n, for the investigated impeller frequency range.
At the frequency corresponding to the RSI 3rd harmonic, the three pressure sensors
positioned in the rotorstator gap do not vary according to the investigated impeller
frequency, see Figure 9.11. Moreover, the pressure sensor s11 monitors lower fluctuations
at the frequencies f = 27n ranging from 338 to 380 Hz, whereas the pressure sensor s10
seems not to be subject to the vibrations of the guide vanes, its value staying constant
along the impeller frequency range investigated. A surplus of energy is dissipated in a
unknown manner in the flow at the position of the sensor s11 . Since no variation of the
vibrations is monitored at the RSI 3rd harmonic frequency, see Figure 9.16, one knows
that this decrease of pressure fluctuations does not have any influence on the vibrating
motion of the guide vanes.
At the frequencies f = 36n corresponding to the RSI 4th harmonic, the pressure
fluctuations do not undergo any variation along the whole impeller frequency range, see
Figure 9.12. The pressure sensors in the rotorstator gap monitor fluctuations 10 times
stronger than the pressure sensors in the guide vane channel.
The magnitude of the pressure factor fluctuations recorded at the frequencies f = 45n
corresponding to the RSI 5th harmonic in the rotorstator gap does not vary according to
the investigated impeller frequency n, see Figure 9.13. Nevertheless, the pressure sensors
in the guide vane channel monitor variations for frequencies ranging from 625 to 683 Hz.
The magnitude of the pressure fluctuations factor monitored by the sensor ĉ0ps11  features
a maximum to minimum ratio of 12, while the ratio reaches 4.5 for the pressure sensor
ĉ0ps10 .
The magnitude of the averaged Fourier transform of the fluctuating displacement and
torsion angle factor, for the guide vanes O10 and O11 at the RSI fundamental and the 2nd to
5th harmonic frequencies is plotted against the corresponding frequencies on the left hand
side of Figure 9.14 to 9.18, respectively, for the investigated impeller frequency range.
−1
10
[] c'
^
p s10 g10 s11 g11 g15
−2
10
−3
10
−4
10
10
−5 27n
320 340 360 380 400 420
[Hz]
Figure 9.11: Averaged Fourier transform magnitude of pressure factor fluctuations, mon
itored by the pressure sensors g10 , s10 , g11 , s11 , g15 , at the RSI 3rd harmonic frequency
f = 27n, for the investigated impeller frequency range.
−1
10
[] c'
^
p s10 g10 s11 g11 g15
−2
10
−3
10
−4
10
10
−5 36n
420 440 460 480 500 520 540 560
[Hz]
Figure 9.12: Averaged Fourier transform magnitude of pressure factor fluctuations, mon
itored by the pressure sensors g10 , s10 , g11 , s11 , g15 , at the RSI 4th harmonic frequency
f = 36n, for the investigated impeller frequency range.
The phase shifts between the vibrating motions of the two guide vanes ∆φy10 ,y11 (f ) =
φy10 (f ) − φy11 (f ) and ∆φα10 ,α11 (f ) = φα10 (f ) − φα11 (f ) at the corresponding frequencies
f = mzb n are given on the right hand side of the five figures.
For small amplitudes, the torsion angle factor definition, see eq. 9.8, may be seen as
−1
10
[] c'
^
p s10 g10 s11 g11 g15
−2
10
−3
10
−4
10
10
−5 45n
550 600 650 700
[Hz]
Figure 9.13: Averaged Fourier transform magnitude of pressure factor fluctuations, mon
itored by the pressure sensors g10 , s10 , g11 , s11 , g15 , at the RSI 5th harmonic frequency
f = 45n, for the investigated impeller frequency range.
−2
10 3π/2
[] c'
^
y y10 y11 α10 α11 [rad] ∆φ ∆φ y10 ,y11 ∆φα10 ,α11
10
−3 ^
c'
α
5π/4
−4
10
π
−5
10
∆φk=9
−6
10 3π/4
−7
10
9n π/2
9n
105 110 115 120 125 130 135 140 105 110 115 120 125 130 135 140
[Hz] [Hz]
Figure 9.14: Averaged Fourier transform magnitude of displacement and torsion angle
factors (left) and phase shift between the displacement signals and between the angle
signals (right) at the RSI fundamental frequency, f = 9n, for guide vanes O10 and O11 ,
for the investigated impeller frequency range.
αi · L2 y tip
cα i = ≈ i (9.9)
δ δ
where yitip is the local displacement of the guide vane leading or trailing edge in the
direction perpendicular to the chord. In this way, the two factors, cyi and cαi , may easily
−2
10 π/8
[] c'
^
y y10 y11 α10 α11 [rad] ∆φ ∆φ y10 ,y11 ∆φα10 ,α11
10
−3
^
c'
α
0
−4
10
−π/8
−5
10
∆φ k=2
−6
10 −π/4
−7
10
18n −3π/8
18n
210 220 230 240 250 260 270 280 210 220 230 240 250 260 270 280
[Hz] [Hz]
Figure 9.15: Averaged Fourier transform magnitude of displacement and torsion angle
factors (left) and phase shift between the displacement signals and between the torsion
angle signals (right) at the RSI 2nd harmonic, f = 18n, for guide vanes O10 and O11 , for
the investigated impeller frequency range.
At the RSI 2nd harmonic frequencies, see Figure 9.15, the bending displacement factor
of the guide vane O10 and O11 even exceeds 10 times the torsion angle factor. The
eigenfrequency of the guide vanes found when placed in water, model at rest, is included
in the frequency range constituted by the RSI 2nd harmonic during the impeller frequency
sweep. Nevertheless, as already pointed out with the RMSvalues observation, the guide
vanes O10 and O11 do not preferably respond at the same frequency, see Figure 9.15,
as would be presumed by the modal analyzes in water, model at rest. Moreover, when
the guide vane O10 is at resonance, the amplitude of the vibrations of the guide vane
O11 clearly seem to be attenuated. The phase shift between the pressure monitored at
the location of the guide vanes O10 and O11 imposed by the RSI mode at f = 18n
is ∆φk=−2 = −0.63 rad. Therefore, ∆φy10 ,y11 and ∆φα10 ,α11 are close to this value at
f = 18nmin : ∆φy10 ,y11 (18nmin ) = −0.47 rad (∆φ = 25%) and ∆φα10 ,α11 (18nmin ) = −0.30
rad (∆φ = 52%). The high value of the relative differences ∆φ is due to the proximity
of the bending eigenmode. When increasing the impeller rotating frequency, the phase
shifts are strongly modified across structural resonance. The maximum phase shift value
is reached at 230 Hz.
−2
10 π
[] c'
^
y y10 y11 α10 α11 [rad] ∆φ ∆φ y10 ,y11 ∆φα10 ,α11
10
−3 ^
c'
α
7π/8
−4
10
3π/4
−5
10
∆φ k=7
−6
10 5π/8
−7
10
27n π/2
27n
320 340 360 380 400 420 320 340 360 380 400 420
[Hz] [Hz]
Figure 9.16: Averaged Fourier transform magnitude of displacement and torsion angle
factors (left) and phase shift between the displacement signals and between the torsion
angle signals (right) at the RSI 3rd harmonic, f = 27n, for guide vanes O10 and O11 , for
the investigated impeller frequency range.
At the RSI 3rd harmonic frequencies, see Figure 9.16, the bending displacement factor
of the guide vanes O10 and O11 is of the same order of magnitude as the torsion angle
factor, see Figure 9.16. Nevertheless, through the frequency range, one may observe
that the displacement factor increases, whereas the torsion angle factor decreases, by a
factor 1.5. The reason is that, by increasing the impeller frequency, the RSI 3rd harmonic
frequency moves away from the 1st bending eigenfrequency, 230 Hz, decreasing the bending
displacement, whereas it comes closer to the 1st torsion eigenfrequency, 630 Hz, increasing
the torsion angle. As to the phase shift between the displacement and torsion motions, it
is close to the phase shift imposed by the RSI: ∆φk=7 = 2.20 rad.
At the RSI 4th harmonic frequencies, see Figure 9.17, the torsion angle factor value
triples by increasing the excitation frequency from 36nmin to 36nmax , whereas the displace
ment factor presents a relatively constant amplitude throughout the frequency range, see
Figure 9.17. Nevertheless, between 450 and 460 Hz, one may observe a local decrease of
the displacement factors. One presumes the influence of the 1st higher harmonic of the
1st bending eigenfrequency. It is interesting to see an attenuation of the vibrations at
this eigenfrequency harmonic, whereas the vibrations are severely amplified at the prox
imity of the bending eigenfrequency, see Figure 9.15. The phase shift between the torsion
motions remains constant at ∆φk=−4 = −1.26 rad, whereas the phase shift between the
bending motions greatly varies due to the influence of the 1st higher harmonic of the 1st
bending eigenfrequency already mentioned above.
Finally, at the RSI 5th harmonic frequencies, see Figure 9.18, the maximum value of the
torsion angle factor of the guide vanes O10 and O11 is 50 times higher than the respective
displacement factor value because of the proximity of the 1st torsion eigenfrequency. The
−2
10 9π/4
[] c'
^
y y10 y11 α10 α11 [rad] ∆φ ∆φ y10 ,y11 ∆φα10 ,α11
10
−3 ^
c'α 2π
−4
10
7π/4
−5 ∆φ k=4
10
3π/2
−6
10
5π/4
−7
10
36n π 36n
420 440 460 480 500 520 540 560 420 440 460 480 500 520 540 560
[Hz] [Hz]
Figure 9.17: Averaged Fourier transform magnitude of displacement and torsion angle
factors (left) and phase shift between the displacements and between the torsion angle
signals (right) at the RSI 4th harmonic, f = 36n, for guide vanes O10 and O11 , for the
investigated impeller frequency range.
−2
10 π
c'
^
y y10 y11 α10 α11 [rad] ∆φα10 ,α11
10
−3 ^
c'α
3π/4
−4
10
−5
π/2 ∆φ k=5
10
−6
10 π/4
−7
10
45n 0
45n
550 600 650 [Hz] 700 550 600 650 [Hz] 700
Figure 9.18: Averaged Fourier transform magnitude of displacement and torsion angle
factors (left) and phase shift between the displacement signals and between the torsion
angle signals (right) at the RSI 5th harmonic, f = 45n, for guide vanes O10 and O11 , for
the investigated impeller frequency range.
guide vane O11 responds preferably at 657 Hz, whereas the amplitude of the guide vane
O10 vibrations at this frequency is strongly attenuated. The latter responds preferably
at 646 Hz and 666 Hz. The amplitude of the guide vane O11 vibrations at the frequency
f = 45n is amplified by a factor 42 in the impeller frequency range investigated, whereas
the factor is 17 for the guide vane O10 . The phase shift values between the torsion motions
present a relatively high standard deviation. Nonetheless, one may observe that, away
from resonance, the phase shift is close to ∆φk=5 = 1.57 rad. Throughout the resonance,
the phase shift greatly varies to get closer to zero near 45n = 650 Hz. At 657 Hz, the
phase shift is 2 rad and, at higher frequencies, the phase shift returns to 1.57 rad. The
phase shift between the bending displacement is not possible to get in a reliable manner,
because of the too low displacement values at these frequencies.
The waterfall diagrams of the coherence functions between the pressure factor fluc
tuations and vibrations of the guide vanes against the normalized frequency f /n are
presented for the whole range of impeller frequencies n in Figures 9.19 to 9.23. The
coherence functions are averaged over 8 blocks of 0.32 s, each overlapping by 50%.
2 2
Γcy ,cp Γcα ,cp
O g11 15 O g11 15
[] []
n n
[Hz] f/n [Hz] f/n
[] []
Figure 9.19: Waterfall diagrams of the coherence functions between the pressure factor
monitored by the sensor g15 and the guide vane O11 displacement factor (left) and between
the pressure factor monitored by the sensor g15 and the guide vane O11 torsion angle factor
(right) against the impeller frequency n.
In these figures, the RSI modes clearly appear, while the correlation is poor in between.
The coherence functions value tends to 1 at RSI modes frequencies f = mzb n, for 1 ≤
m ≤ 3. Because of the low response of the strain gages and the pressure sensors to higher
RSI harmonics, the coherence functions do not exactly reach 1 at these frequencies. At the
4th harmonic, the value of the coherence functions is the worst when the harmonic value
comes closer to twice the value of the bending eigenfrequency. In Figure 9.17, one has
already mentioned an influence of the 1st higher harmonic of this guide vane eigenmode.
At the 5th harmonic, the coherence functions take low values when the harmonic frequency
comes closer to the torsion eigenfrequencies.
Therefore, linear relations between these pressure, displacement and torsion angle
factors might be built at RSI modes frequencies. Nevertheless, the accuracy of these
linear relations will be higher for the harmonic up to the third one. The following chapter
is devoted to the identification of the hydrodynamic parameters, namely the added mass
and the hydrodynamic damping brought by the vibrations of the neighboring guide vanes.
These parameters are deduced from influence matrices which is nothing else but a linear
relations between a force and a displacement.
2 2
Γcy ,cp Γcα ,cp
O g10 15 O g10 15
[] []
n n
[Hz] f/n [Hz] f/n
[] []
Figure 9.20: Waterfall diagrams of the coherence functions between the pressure factor
monitored by the sensor g15 and the guide vane O10 displacement factor (left) and between
the pressure factor monitored by the sensor g15 and the guide vane O10 torsion angle factor
(right) against the impeller frequency n.
2 2
Γcy ,c y Γcα ,cα
O O10 11 O O
10 11
[] []
n n
[Hz] f/n [Hz] f/n
[] []
Figure 9.21: Waterfall diagrams of the coherence functions between the displacement
factors of the guide vane O10 and O11 (left) and between the torsion angle factors of the
guide vane O10 and O11 (right) against the impeller frequency n.
2 2
Γcy ,cα Γcy ,cα
O 11O 10 O 10O 11
[] []
n n
[Hz] f/n [Hz] f/n
[] []
Figure 9.22: Waterfall diagrams of the coherence functions between the displacement
factor of the guide vane O11 and the torsion angle factor of the guide vane O10 (left) and
between the displacement factor of the guide vane O10 and the torsion angle factor of the
guide vane O11 (right) against the impeller frequency n.
2 2
Γcy ,cα Γcy ,cα
O 10O 10 O 11O 11
[] []
n n
[Hz] f/n [Hz] f/n
[] []
Figure 9.23: Waterfall diagrams of the coherence functions between the displacement
factor of the guide vane O10 and the torsion angle factor of the guide vane O10 (left) and
between the displacement factor of the guide vane O11 and the torsion angle factor of the
guide vane O11 (right) against the impeller frequency n.
After having analyzed the dynamics of the guide vanes and the pressure fluctuations at
the frequency of the RSI modes in Chapter 9, the aim of the present chapter is to identify
the
f hydrodynamic
f parameters
to determine the unknown components of the matrices
f
I , C and K in eq. 3.21. The methodology is exposed prior to the analysis
of the bending and torsion modes. In Appendix D, the procedure used to identify the
hydrodynamic parameters is summarized.
The structural stiffness matrix [K s ] is known from the analytical estimation given in
Appendix B.3 and is expressed as:
From Table 8.1, the structural mass/inertia matrix may be filled up:
0.33 kg 0 0 0
0 0.34 kg 0 0
[I s ] = −4 2
(10.6)
0 0 1.75 · 10 kg · m 0
−4 2
0 0 0 1.77 · 10 kg · m
5.00 kg · s−1
0 0 0
0 6.57 kg · s−1 0 0
[C s ] = 2 −1
(10.7)
0 0 0.024 kg · m · s 0
2 −1
0 0 0 0.024 kg · m · s
By substituting these matrices in the first equation of 10.4 and since we know the
displacement/angle vector ŷ from the forced response measurements, one may obtain the
total dynamic force/torque vector F̂ applied to the guide vanes O10 and O11 . The Fourier
transform is applied to eight 214 samplelong signals y (t) overlapping by 50%, each of the
eight sequences being windowed with a Hamming function. The Fourier transforms are
not averaged to keep phase information.
At this stage of the identification procedure, the force/torque vector F̂ may be placed
in
fthe
second
equation
of eq. 10.4 to identify the unknown components of the matrices
I , Cf , Kf .
The fluid mass and inertia matrix in eq. 10.4 is expressed as:
f f f,α f,α
I10,10 I10,11 I10,10 I10,11
f f f,α f,α
f I11,10 I11,11 I11,10 I11,11
I = f f f,α f,α
(10.8)
J10,10 J10,11 J10,10 J10,11
f f f,α f,α
J11,10 J11,11 J11,10 J11,11
where the boxed components are known from Table 8.4 given in Section 8.2.1.
The hydrodynamic damping matrix in eq. 10.4 may be written as:
Cf f
C10,11 f,α
C10,10 f,α
C10,11
10,10
f f f,α f,α
f C11,10 C11,11 C11,10 C11,11
C = f f f,α f,α
(10.9)
D10,10 D10,11 D10,10 D10,11
f f f,α f,α
D11,10 D11,11 D11,10 D11,11
where the boxed components are known from Table 8.6 given in Section 8.2.2.
The fluid stiffness matrix in eq. 10.4 is expressed as follows:
f f,α
K10,10 0 K10,10 0
f f,α
f 0 K11,11 0 K11,11
K = f (10.10)
L10,10 0 Lf,α 0
10,10
f f,α
0 L11,11 0 L11,11
the fluctuating force and torque due to RSI, F̂RSI . In the two following sections, the
methodology for dealing with this underdetermined
f system
f of fequations,
and to finally
identify the missing components of the matrices I , C , K , is consecutively given
for the 1st bending and torsion eigenmodes.
At the frequencies of the RSI fundamental and the other harmonics, the displacement
factor does not exceed the torsion angle factor in such a way to reduce the vector ŷ
to this simple formulation. Therefore, one may only focus on the identification of the
hydrodynamic parameters for the vibrating behavior at the RSI 2nd harmonic frequency
f = 18n.
The eq. 10.4 is reduced to a system of 2 equations:
T
−ω 2 I S + K S + iω C S (ŷ10 (ω) ŷ11 (ω))T = F̂10 (ω) F̂11 (ω)
(10.12)
F̂ (ω) F̂ (ω) T = F̂RSI (ω) − Ĥ f (ŷ (ω) ŷ (ω))T
h i
10 11 10 11
where the influence matrix H f is expressed as follows:
f f
h
f
i Ĥ10,10 Ĥ10,11
Ĥ = f f
Ĥ11,10 Ĥ11,11
f f f f f
−ω 2 I10,10 −ω 2 I10,11
+ iωC10,10 + K10,10 + iωC10,11
= f f f f f (10.13)
−ω 2 I11,10 + iωC11,10 −ω 2 I11,11 + iωC11,11 + K11,11
f f
For high ω, the stiffness terms, K10,10 and K11,11 , being negligible in comparison to the
f
−3
inertia terms, disappear. Münch et al. [56] show that when ω → ∞, IK f ω 2 ≈ 10
h i 1.
As a consequence, only the 2 nondiagonal components of the influence matrix Ĥ f and
the fluctuating force due to RSI, F̂RSI , remain unknown.
To approximate the force contributions due to the RSI excitation, F̂RSI , two assump
tions are successively made. First, it is assumed that the guide vane O11 bending dis
placement amplitude, ŷ11 , is not high enough to induce any influence on the guide vane
O10 at the RSI 1st harmonic frequency corresponding to the lowest impeller frequency of
the investigated range fmin = 18nmin . Therefore, the force spectral value acting on O10
due to the RSI at fmin = 18nmin may be written from the 2nd equation of Eq. (10.23):
RSI
F̂10 (fmin ) = F̂10 (fmin ) + − (ωmin )2 I10,10
f f
+ i (ωmin ) C10,10 · ŷ10 (fmin ) (10.14)
The force due to the RSI excitation may be linearly linked to the pressure monitored
in the guide vane channel on its pressure side as follows:
(
RSI
F̂10 (f = 18n) = L̂s,10 · p̂RSI
s10 (f = 18n)
(10.16)
F̂11 (f = 18n) = L̂s,11 · p̂RSI
RSI
s11 (f = 18n)
where L̂s,10 and L̂s,11 are complex numbers which are supposed to be constant on the
whole impeller frequency range. To determine the force due to the RSI excitation in the
whole impeller frequency range, these constants as well as the pressure contribution from
the RSI must be known and the following procedure is followed. At the location of the
pressure sensors s10 and s11 , the guide vane vibrations influence the measurement and the
contribution due to RSI may not be isolated. Therefore, the only pressure sensor which is
not influenced by the guide vane vibrations is the sensor g15 . During measurements where
all the guide vanes of the cascade feature stiff stem, avoiding in this way the contribution
from the guide vane vibrations, the pressure monitored by the sensor g15 is supposed to
be linearly linked to the pressure monitored by the sensor s10 and s11 with the complex
constants Lsg,10 and Lsg,11 , respectively.
In the case with the two modified guide vanes, the forces due to the RSI may then be
known from the pressure monitored by the sensor g15 as follows:
F̂ RSI (f = 18n) = L̂s,10 · p̂g (f = 18n) = L̂10 · p̂g (f = 18n)
10 L̂sg,10 15 15
L̂
(10.17)
RSI s,11
F̂11 (f = 18n) =
L̂
· p̂ g 15 (f = 18n) = L̂11 · p̂ g 15 (f = 18n)
sg,11
Finally, the complex numbers L̂10 and L̂11 are derived from the system of equations
RSI
in Eq. (10.17) at the frequency f = fmin , at which the forces due to the RSI, F̂10 (fmin )
RSI
and F̂11 (fmin ) are already known from Eqs. (10.14) and (10.15). Assuming that the
complex numbers remain constant in the whole impeller frequency range, the forces due
to the RSI excitation at all these frequencies may be obtained using Eq. (10.17).
f f
The hydrodynamic parameters, namely the added mass terms, I10,11 and I11,10 and the
f f
hydrodynamic damping terms, C10,11 and C11,10 , may then be obtained from eq. 10.13 as
follows:
f
< Ĥ10,11
f
I10,11 (f = 18n) = − 2
(10.18)
ω18n
f
< Ĥ11,10
f
I11,10 (f = 18n) = − 2
(10.19)
ω18n
f
= Ĥ10,11
f
C10,11 (f = 18n) = (10.20)
ω18n
f
= Ĥ11,10
f
C11,10 (f = 18n) = (10.21)
ω18n
¡¡ At the frequencies of the RSI fundamental and the other harmonics, the torsion angle
factor does not exceed the displacement factor in such a way to reduce the vector ŷ
to this simple formulation. Therefore, one may only focus on the identification of the
hydrodynamic parameters for the vibrating behavior at the RSI 5th harmonic frequency
f = 45n.
h i
f
Ĝ being an influence matrix expressed as follows:
f
Ĝ10,10 Ĝf10,11
h i
f
Ĝ = (10.24)
Ĝf11,10 Ĝf11,11
2 f,α f,α
+ Lf,α f,α
−ω 2 J10,11 f,α
−ω J10,10 + iωD10,10 10,10 + iωD10,11
= f,α f,α f,α f,α
−ω 2 J11,10 + iωD11,10 −ω 2 J11,11 + iωD11,11 + Lf,α
11,11
As mentioned in Section 10.1.1, for high ω, the stiffness terms Lf,α f,α
10,10 and L11,11 are
neglected compared to the inertia terms.
h i As a consequence, only the 2 nondiagonal
components of the influence matrix Ĝf and the fluctuating torque due to RSI, T̂RSI i ,
remain unknown.
To approximate the torque contribution due to the RSI, T̂RSI , one firstly assumes the
guide vane O11 torsion angle α̂11 being not high enough to induce any influence on the
guide vane O10 at the RSI 5th harmonic frequency corresponding to the lowest impeller
frequency of the investigated range fmin = 45nmin . Therefore, the torque acting on O10
due to the RSI at fmin = 45nmin may be written from the 2nd equation of eq. 10.23:
RSI
T̂10 (fmin ) = T̂10 (fmin ) + − (ωmin )2 J10,10
f,α f,α
+ i (ωmin ) D10,10 · α̂10 (fmin ) (10.25)
The same procedure as for the bending case is followed to determine the torque due to
the RSI excitation in the whole impeller frequency range. Onceh this
i torque contribution
known, the nondiagonal components of the influence matrix Ĝf may be obtained from
eq. 10.23:
RSI
T̂10 − T̂10
Ĝf10,11 = (10.27)
α̂11
RSI
T̂11 − T̂11
Ĝf11,10 = (10.28)
α̂10
f,α f,α
The hydrodynamic parameters, namely the added inertia terms, J10,11 and J11,10 , and
f,α f,α
the hydrodynamic damping terms, D10,11 and D11,10 , may then be obtained from eq. 10.25
as follows:
< Ĝf10,11
f,α
J10,11 (f = 45n) = − 2
(10.29)
ω45n
< Ĝf11,10
f,α
J11,10 (f = 45n) = − 2
(10.30)
ω45n
= Ĝf10,11
f,α
D10,11 (f = 45n) = (10.31)
ω45n
= Ĝf11,10
f,α
D11,10 (f = 45n) = (10.32)
ω45n
1.0
[] I i,jf /Ii S (i,j)=(10,11)
(i,j)=(10,10)
(i,j)=(11,10)
(i,j)=(11,11)
0.8
0.6
0.4
0.2
0
18n
210 220 230 240 250 260 270 280
[Hz]
Figure 10.1: Added mass on the guide vane O10 by the vibrations of the guide vane O11 ,
f f
I10,11 ; added mass on the guide vane O11 by the vibrations of the guide vane O10 , I11,10 ;
f f
added masses I10,10 and I11,11 ; each being normalized by the corresponding structural mass
S S
I10 or I11 , at the RSI 2nd harmonic frequency f = 18n for the whole impeller frequency
range investigated.
We may observe high variations of the hydrodynamic damping due to the vibrations
of the neighboring guide vanes throughout the investigated impeller frequency range. The
f
hydrodynamic damping constant C10,11 is from 0 up to 11.7 times higher than the struc
S f
tural damping constant C10 and the hydrodynamic damping constant C11,10 represents
S
between 0 and 461% of the structural damping constant C11 . As already mentioned, the
f f
damping constants C10,10 and C11,11 are assumed constant throughout the investigated
S S
impeller frequency range and correspond to 510% of C10 and 330% of C11 , respectively.
The amplitude of the total force F̂10  acting on the guide vane O10 ; the contribution
RSI f
from the RSI, F̂10 ; the contribution from O10 vibrations, F̂10,10 ; and the contribution
f
from O11 vibrations, F̂10,11 ; at the frequency corresponding to the RSI 2nd harmonic
RSI
frequency are normalized by F̂10 and plotted in Figure 10.3. The standard deviations
are also indicated by intervals.
The amplitude of the total force acting on the guide vane O11 , F̂11 ; the contribution
RSI f
from the RSI, F̂11 ; the contribution from O11 vibrations, F̂11,11 ; and the contribution
f
from O10 vibrations, F̂11,10 ; at the frequency corresponding to the RSI 2nd harmonic
RSI
frequency are normalized by F̂11 and plotted in Figure 10.4. The standard deviations
are also indicated by intervals.
On the one hand, it may be seen that the force acting on the guide vane O10 due to
the guide vane O11 vibrations is between 14 and 100 times lower than the force due to
the guide vane O10 vibrations. On the other hand, the force acting on the guide vane O11
due to the guide vane O10 vibrations is between 2 and 10 times lower than the force due
16
(i,j)=(10,11) (i,j)=(11,10)
[] Ci,jf /Ci S (i,j)=(10,10) (i,j)=(11,11)
12
0
18n
210 220 230 240 250 260 270 280
[Hz]
Figure 10.2: Hydrodynamic damping constant acting on the guide vane O10 due to the
f
vibrations of the guide vane O11 , C10,11 ; hydrodynamic damping constant acting on the
f
guide vane O11 due to the vibrations of the guide vane O10 , C11,10 ; hydrodynamic damp
f f
ing constants C10,10 and C11,11 ; each being normalized by the corresponding structural
S S
damping constant C10 or C11 , at the RSI 2nd harmonic frequency f = 18n for the whole
impeller frequency range investigated.
f
to the guide vane O11 vibrations. The force amplitude F̂11,10  is between 2 and 10 times
f
higher than the force amplitude F̂10,11  and, consequently, the assumption leading to the
eq. 10.14 is verified. Finally, whereas the amplitude of the force due to the adjacent guide
vane vibrations remain inferior to the amplitude of the force due to RSI, the amplitude
of the force acting on a guide vane due to its vibrations may exceed the amplitude of the
force due to the RSI by a factor 7 at resonance.
In Figure 10.5, the phase shift at the RSI 2nd harmonic frequency f = 18n between
the displacement of the guide vanes O10 and O11 and the external force due to the RSI
acting on the guide vane O10 and O11 , respectively, is plotted against the corresponding
frequency. The standard deviations are also indicated by intervals.
One may see that, throughout the guide vane resonance, the phase shift at the RSI
nd
2 harmonic frequency f = 18n between the displacement and the external force due to
the RSI greatly varies. Theoretically, a resonance modifies the phase shift by −π. Here,
the resonance already occurs at a lower frequency than fmin = 18nmin and, therefore, on
the investigated impeller rotating frequency range, the phase shift varies by 2.8 rad and
2.4 rad for the guide vanes O10 and O11 , respectively. Moreover, between 18n = 230 Hz
and 18n = 240 Hz, the phase shift between the displacement of the guide vane O11 and
the external force due to the RSI locally varies due to the influence of the vibrations of
the guide vane O10 .
In Figure 10.6, the ratio of the Fourier transform magnitude of the guide vane O10
[]
^ ^
F/ F10 
RSI F^10 / F^10  F^10 /F^10 
RSI RSI RSI
10
1 F^10,10
f
/ F^10  F^10,11
RSI f
/ F^10 RSI
0
10
−1
10
10
−2 18n
210 220 230 240 250 260 270 280
[Hz]
RSI
Figure 10.3: Fourier transform magnitude of normalized total force F̂10 /F̂10  acting
nd
on the guide vane O10 at the 2 harmonic frequency and each of its contributions: the
RSI RSI
normalized contribution from RSI, F̂10 /F̂10 ; the normalized contribution of the own
f
vibrations, F̂10,10 ; and the normalized contribution from the vibrations of the guide vane
f
O11 , F̂10,11 .
bending displacement to the Fourier transform magnitude of the guide vane O11 bending
displacement is plotted at the RSI 2nd harmonic frequency f = 18n, for the investigated
impeller frequency range. At resonance, the guide vane O10 vibrates with an amplitude 3
times higher than the guide vane O11 , whereas far from resonance, the ratio is 1.
As mentioned in Section 3.3, regarding the Figures 9.15 and 10.6, one may assume
f
that at least three parameters are able to play a role in the added mass terms, I10,11 and
f f f
I11,10 , and in the hydrodynamic damping terms, C10,11 and C11,10 : the phase shift between
the displacement of the guide vanes O10 and O11 , ∆φy10 ,y11 , the relative amplitude of the
vibrations, ŷ 10 
ŷ11 
, and the reference flow velocity Cref . The added mass and hydrodynamic
terms are approached by the following expressions:
∗f −1 ŷ10 
I10,11 = 0.5 kg · s · m · Cref + 1.4 kg · · sin (∆φy10 ,y11 − 0.5) − 2.1 kg · 10−2
ŷ11 
(10.33)
∗f −1 ŷ11 
I11,10 = 2.75 kg · s · m · Cref + 3.25 kg · · sin (∆φy10 ,y11 + 1.7) − 20 kg · 10−2
ŷ10 
(10.34)
∗f ŷ10 
C10,11 = 25 kg · m−1 · Cref + 22 kg · s−1 · · sin (∆φy10 ,y11 + 1) − 185 kg · s−1 (10.35)
ŷ11 
[] ^ ^
F/ F11 
RSI
F^11 / F^11  F^11 /F^11 
RSI RSI RSI
10
1
F^11,11
f
/ F^11  F^11,10
RSI f
/ F^11  RSI
0
10
−1
10
10
−2 18n
210 220 230 240 250 260 270 280
[Hz]
RSI
Figure 10.4: Fourier transform magnitude of normalized total force F̂11 /F11  acting
nd
on the guide vane O11 at the 2 harmonic frequency and each of its contributions: the
RSI RSI
normalized contribution from RSI, F̂11 /F̂11 ; the normalized contribution of the own
f
vibrations, F̂11,11 ; and the normalized contribution from the vibrations of the guide vane
f
O10 , F̂11,10 .
0
[rad] ∆φ ∆φ y10 ,F10RSI ∆φ y11 ,F11RSI
−π/4
−π/2
−3π/4
−π 18n
210 220 230 240 250 260 270 280
[Hz]
Figure 10.5: Phase shift between the displacement and the force at the RSI 2nd harmonic
frequency f = 18n.
∗f ŷ11 
C11,10 = 5 kg · m−1 · Cref + 50 kg · s−1 · · sin (∆φy10 ,y11 + 0.4) − 21 kg · s−1 (10.36)
ŷ10 
4
[] y^10
y^11
3
0
18n
210 220 230 240 250 260 270 280
[Hz]
Figure 10.6: Ratio of the Fourier transform magnitude of the guide vane O10 bending
displacement to the Fourier transform magnitude of the guide vane O11 bending displace
ment at the RSI 2nd harmonic frequency f = 18n, for the investigated impeller frequency
range.
The eqs. 10.33 to 10.36 are assumed to be valid for the investigated ranges, namely
ŷ10 
−0.7 ≤ ∆φy10 ,y11 ≤ 0.3 and 1 ≤ ŷ11 
≤ 3.
f
In Figure 10.7, the measured values, I11,10 , of the added mass on the guide vane O11
due to the vibrations of the guide vane O10 at the frequency f = 18n, are compared with
∗f S
the approximated values I11,10 . Both are normalized with the structural mass I11 and
plotted against the corresponding frequency f = 18n. The standard deviations are also
indicated by intervals.
f
In Figure 10.8, the measured values, I10,11 , of the added mass on the guide vane O10
due to the vibrations of the guide vane O11 at the frequency f = 18n, are compared to
∗f S
the approximated values I10,11 . Both are normalized with the structural mass I10 and
plotted against the corresponding frequency. The standard deviations are also indicated
by intervals.
f
In Figure 10.9, the measured values, C11,10 , of the hydrodynamic damping constant
acting on the guide vane O11 due to the vibrations of the guide vane O10 at the frequency
∗f
f = 18n, are compared to the approximated values C11,10 . Both are normalized with the
S
structural damping constant C11 and plotted against the corresponding frequency. The
standard deviations are also indicated by intervals.
f
In Figure 10.10, the measured values, C10,11 , of the hydrodynamic damping constant
acting on the guide vane O10 due to the vibrations of the guide vane O11 at the frequency
∗f
f = 18n, are compared to the approximated values C10,11 . Both are normalized with the
S
structural damping constant C10 and plotted against the corresponding frequency. The
standard deviations are also indicated by intervals.
0.3
[] Ii,jf /Ii S I11,10
*f
/I11S f
I11,10 /I11S
0.2
0.1
0
18n
210 220 230 240 250 260 270 280
[Hz]
f ∗f
Figure 10.7: Measured values, I11,10 , and approximated values, I11,10 , of the added mass
on the guide vane O11 due to the vibrations of the guide vane O10 at the RSI 2nd harmonic
S
f = 18n, normalized with the structural mass I11 , against the corresponding frequency.
0.08
[] Ii,jf /Ii S I10,11
*f
/I10S f
I10,11 /I10S
0.04
−0.04
18n
210 220 230 240 250 260 270 280
[Hz]
f ∗f
Figure 10.8: Measured values, I10,11 , and approximated values, I10,11 , of the added mass
on the guide vane O10 due to the vibrations of the guide vane O11 at the RSI 2nd harmonic
S
f = 18n, normalized with the structural mass J10 , against the corresponding frequency.
The approximated added mass and the hydrodynamic damping constants fit the mea
sured values adequately. For the bending case, the guide vane cascade is therefore shown
to behave as a 2nd order mechanical system whose parameters are dependent of the flow
6
[] Ci,jf /Ci S C11,10
*f
/C11S f
C11,10 /C11S
18n
210 220 230 240 250 260 270 280
[Hz]
f ∗f
Figure 10.9: Measured values, C11,10 , and approximated values, C11,10 , of the hydrody
namic damping constant acting on the guide vane O11 due to the vibrations of the guide
vane O10 at the RSI 2nd harmonic f = 18n, normalized with the structural damping
S
constant C11 , against the corresponding frequency.
20
[] Ci,jf /Ci S C10,11
*f
/C10S f
C10,11 /C10S
15
10
0
18n
210 220 230 240 250 260 270 280
[Hz]
f ∗f
Figure 10.10: Measured values, C10,11 , and approximated values, C10,11 , of the hydrody
namic damping constant acting on the guide vane O10 due to the vibrations of the guide
vane O11 at the RSI 2nd harmonic f = 18n, normalized with the structural damping
S
constant C10 , against the corresponding frequency.
velocity, the vibration phase shift between two adjacent guide vanes and the relative
amplitude of their vibrations.
1.0
[] Ji,jf,α/Ji S (i,j)=(10,11)
(i,j)=(10,10)
(i,j)=(11,10)
(i,j)=(11,11)
0.5
−0.5
−1.0
−1.5
45n
550 600 650 [Hz] 700
Figure 10.11: Added inertia on the guide vane O10 by the vibrations of the guide vane O11 ,
f,α f,α
J10,11 ; added inertia on the guide vane O11 by the vibrations of the guide vane O10 , J11,10 ;
f,α f,α
added inertia J10,10 and J11,11 ; each being normalized by the corresponding structural
S S
inertia J10 or J11 , at the RSI 5th harmonic frequency f = 45n for the whole impeller
frequency range investigated.
f,α
The added inertia on the guide vane O11 by the vibrations of the guide vane O10 , J10,11 ,
varies slightly according to the excitation frequency f = 45n. It represents between 5%
S
and 5% of the structural mass of the guide vane O10 , J10 . The added inertia on the
f,α f,α
guide vane O10 by the vibrations of the guide vane O11 , J11,10 , varies more than J10,11 and
S
fluctuates between 45% and 30% of the structural mass of the guide vane O11 , J11 . As
f,α f,α
mentioned in Section 8.2.1, the added mass J10,10 and J11,11 represent 39.7% and 36.4%
of the structural mass of the guide vane O10 and O11 , respectively.
The hydrodynamic damping constant acting on the guide vane O10 due to the vi
f,α
brations of the guide vane O11 , D10,11 ; the hydrodynamic damping constant acting on
f,α
the guide vane O11 due to the vibrations of the guide vane O10 , D11,10 ; the hydrody
f,α f,α
namic damping constant D10,10 and D11,11 ; each being normalized by the corresponding
S S
structural damping constant D10 or D11 , are plotted in Figure 10.12 against the RSI 5th
harmonic frequency f = 45n for the whole impeller frequency range investigated. The
standard deviations are also indicated by intervals.
40
[] Di,jf,α/Di S (i,j)=(10,11) (i,j)=(11,10)
30 (i,j)=(10,10) (i,j)=(11,11)
20
10
−10
−20
−30
45n
550 600 650 [Hz] 700
Figure 10.12: Hydrodynamic damping constant acting on the guide vane O10 due to the
f,α
vibrations of the guide vane O11 , D10,11 ; hydrodynamic damping constant acting on the
f,α
guide vane O11 due to the vibrations of the guide vane O10 , D11,10 ; hydrodynamic damp
f,α f,α
ing constants D10,10 and D11,11 ; each being normalized by the corresponding structural
S S
damping constant D10 or D11 , at the RSI 5th harmonic frequency f = 45n for the whole
impeller frequency range investigated.
10
2 T^10 / T^10  T^10 / T^10 
RSI RSI RSI
^ ^
[] T/ T10 
RSI
T^10,10
f
/ T^10  T^10,11
RSI f
/ T^10  RSI
1
10
0
10
−1
10
10
−2 45n
550 600 650 [Hz] 700
RSI
Figure 10.13: Fourier transform magnitude of normalized total torque T̂10 /T̂10  acting
th
on the guide vane O10 at the 5 harmonic frequency and each of its contributions: the
RSI RSI
normalized contribution from RSI, T̂10 /T̂10 ; the normalized contribution of the own
f
vibrations, T̂10,10 ; and the normalized contribution from the vibrations of the guide vane
f
O11 , T̂10,11 .
The amplitude of the total torque T̂11  acting on the guide vane O11 ; the contribution
RSI f
from the RSI, T̂11 ; the contribution from O11 vibrations, T̂11,11 ; and the contribution
f
from O10 vibrations, T̂11,10 ; at the frequency corresponding to the RSI 5th harmonic
RSI
frequency are normalized by T̂11 and plotted in Figure 10.14. The standard deviations
are also indicated by intervals.
It may be seen that the torque acting on the guide vane O10 due to the guide vane O11
vibrations is 5 times lower than the torque due to the guide vane O10 vibrations. Similarly,
the torque acting on the guide vane O11 due to the guide vane O10 vibrations is 5 times
f
lower than the torque due to the guide vane O11 vibrations. The torque amplitude T̂11,10 
f
is 5 times higher than the torque amplitude T̂10,11  and, consequently, the assumption
leading to eq. 10.25 is verified. Finally, the amplitude of the torque due to the adjacent
guide vane vibrations may exceed the amplitude of the torque due to RSI by a factor 10
at resonance.
In Figure 10.15, the phase shift at the RSI 5th harmonic frequency f = 45n between
the torsion angle of the guide vanes O10 and O11 and the external torque due to the RSI
acting on the guide vane O10 and O11 , respectively, is plotted against the corresponding
frequency. The standard deviations are also indicated by intervals.
The phase shift between the torsion angle and the external torque varies by −π
throughout the resonance, namely from 625 Hz to 683 Hz. Between 45n = 650 Hz
and 45n = 660 Hz, the phase shift between the torsion angle of the guide vane O10 and
the external torque due to the RSI locally varies due to the influence of the vibrations of
10
2 T^11 / T^11  T^11 / T^11 
RSI RSI RSI
^ T^ 
[] T/ RSI
11 T^11,11
f
/ T^11  T^11,10
RSI f
/ T^11  RSI
1
10
0
10
−1
10
10
−2 45n
550 600 650 [Hz] 700
RSI
Figure 10.14: Fourier transform magnitude of normalized total torque T̂11 /T̂11  acting
th
on the guide vane O11 at the 5 harmonic frequency and each of its contributions: the
RSI RSI
normalized contribution from RSI, T̂11 /T̂11 ; the normalized contribution of the own
f
vibrations, T̂11,11 ; and the normalized contribution from the vibrations of the guide vane
f
O10 , T̂11,10 .
π/2
[rad] ∆φ ∆φ α10 ,T10RSI ∆φα11 ,T11RSI
−π/2
−π
−3π/2 45n
550 600 650 [Hz] 700
Figure 10.15: Phase shift between the torsion angle and the torque at the RSI 5th harmonic
frequency f = 45n.
In Figure 10.16, the ratio of the Fourier transform magnitude of the guide vane O10
torsion angle to the Fourier transform magnitude of the guide vane O11 torsion angle at
the RSI 5th harmonic frequency f = 45n, for the investigated impeller frequency range.
At resonance, the guide vane O10 vibrates with an amplitude 9 times lower than the guide
vane O11 , whereas, far from resonance, the ratio tends to 1.
2.0
[] α^ 10 
α^ 11 
1.5
1.0
0.5
0
45n
550 600 650 [Hz] 700
Figure 10.16: Ratio of the Fourier transform magnitude of the guide vane O10 torsion
angle to the Fourier transform magnitude of the guide vane O11 torsion angle at the RSI
5th harmonic frequency f = 45n, for the investigated impeller frequency range.
As for the bending motion case, three parameters are able to play a role in the added
f,α f,α f,α f,α
inertia, J10,11 and J11,10 , and in the hydrodynamic damping, D10,11 and D11,10 : the phase
shift between the torsion angle of the guide vanes O10 and O11 , ∆φα10 ,α11 , the relative
amplitude of the vibrations, α̂α̂11
10 

, and the reference flow velocity Cref . These terms are
approached by the following expressions:
∗f 2 α̂10 
J10,11 = 1.5 kg · m · · sin (∆φy10 ,y11 − 1.5) + 0.10 kg · m · 10−5
2
(10.37)
α̂11 
∗f α̂11 
J11,10 = 1 kg · m · s · Cref 2
+ 1.5 kg · m · · sin (∆φy10 ,y11 − 1.0) − 9 kg · m ·10−5
2
α̂10 
(10.38)
∗f α̂10 
D10,11 = 0.03 kg·m·Cref +0.08 kg·m2 ·s−1 · ·sin (∆φy10 ,y11 )−0.27 kg·m2 ·s−1 (10.39)
α̂11 
∗f α̂11 
D11,10 = −0.06 kg·m·Cref +0.08 kg·m2 ·s−1 · ·sin (∆φy10 ,y11 − 3.1)+0.54 kg·m2 ·s−1
α̂10 
(10.40)
f,α
In Figure 10.17, the measured values, J11,10 , of the added mass on the guide vane O11
due to the vibrations of the guide vane O10 at the frequency f = 45n, are compared to
∗f,α S
the approximated values J11,10 . Both are normalized with the structural inertia J11 and
plotted against the corresponding frequency. The standard deviations are also indicated
by intervals.
1.0
[] Ji,jf,α/JiS J11,10
*f,α
/J11S f,α
J11,10 /J11S
0.5
−0.5
−1.0
−1.5
45n
550 600 650 [Hz] 700
f ∗f
Figure 10.17: Measured values, J11,10 , and approximated values, J11,10 , of the added inertia
on the guide vane O11 due to the vibrations of the guide vane O10 at the RSI 5th harmonic
S
f = 45n, normalized with the structural inertia J11 , against the corresponding frequency.
f,α
In Figure 10.18, the measured values, J10,11 , of the added mass on the guide vane O10
due to the vibrations of the guide vane O11 at the frequency f = 45n, are compared to
∗f,α S
the approximated values J10,11 . Both are normalized with the structural inertia J10 and
plotted against the corresponding frequency. The standard deviations are also indicated
by intervals.
0.4
[] Ji,jf,α/JiS J10,11
*f,α
/J10S f,α
J10,11 /J10S
0.2
−0.2
−0.4
45n
550 600 650 [Hz] 700
f ∗f
Figure 10.18: Measured values, J10,11 , and approximated values, J10,11 , of the added inertia
on the guide vane O10 due to the vibrations of the guide vane O11 at the RSI 5th harmonic
S
f = 45n, normalized with the structural inertia J10 , against the corresponding frequency.
f,α
In Figure 10.19, the measured values, D11,10 , of the hydrodynamic damping constant
acting on the guide vane O11 due to the vibrations of the guide vane O10 at the frequency
∗f,α
f = 45n, are compared to the approximated values D11,10 . Both are normalized with
S
the structural inertia D11 and plotted against the corresponding frequency. The standard
deviations are also indicated by intervals.
f,α
In Figure 10.20, the measured values, D10,11 , of the hydrodynamic damping constant
acting on the guide vane O10 due to the vibrations of the guide vane O11 at the frequency
∗f,α
f = 45n, are compared to the approximated values D10,11 . Both are normalized with
S
the structural inertia D10 and plotted against the corresponding frequency. The standard
deviations are also indicated by intervals.
The approximated added inertia and the hydrodynamic damping constants fit the
measured values adequately. For the torsion case, the guide vane cascade is therefore
shown to behave as a 2nd order mechanical system whose parameters are dependent of
the flow velocity, the vibration phase shift between two adjacent guide vanes and the
relative amplitude of their vibrations.
40
[] Di,jf,α/DiS D11,10
*f,α
/D11S f,α
D11,10 /D11S
20
−20
45n
550 600 650 [Hz] 700
f ∗f
Figure 10.19: Measured values, D11,10 , and approximated values, D11,10 , of the hydro
dynamic damping constant acting on the guide vane O11 due to the vibrations of the
guide vane O10 at the RSI 5th harmonic f = 45n, normalized with the structural damping
S
constant D11 , against the corresponding frequency.
20
[] Di,jf,α/DiS D10,11
*f,α
/D10S f,α
D10,11 /D10S
10
−10
45n
550 600 650 [Hz] 700
f ∗f
Figure 10.20: Measured values, D10,11 , and approximated values, D10,11 , of the hydro
dynamic damping constant acting on the guide vane O10 due to the vibrations of the
guide vane O11 at the RSI 5th harmonic f = 45n, normalized with the structural damping
S
constant D10 , against the corresponding frequency.
Eigenvalue problem
The aim of this chapter is to consider the eigenvalue problem linked to the bending motion
and the torsion motion. The associated eigenvalues being complex, the eigenvectors are
complex as well and defined in the phase space. The eigenfrequencies determined in this
chapter are compared with the frequencies at which the guide vanes respond preferably
in the tests.
The Caughey condition is satisfied if [δC] = 0. The four elements of the relative
residual [δC] matrix at the RSI 2nd harmonic frequency are given in Figure 11.1. The
standard deviations are indicated by intervals.
In Figure 11.1, it may clearly be observed that the Caughey condition is not satisfied.
The two diagonal terms of the [δC] matrix present a value of 10% around the bending
eigenfrequency value. At higher frequencies, the matrix components δC10,10 and δC11,11
are rising up to 0.4 and 0.3, respectively. One has to mention the small values of the
standard deviation, except for frequencies where the denominator in eq. 11.1 is close to
zero and for frequencies close to f = 18nmax .
Since the Caughey condition is not satisfied, the eigenfrequencies of the mechanical
system are found by following the procedure given in Section 3.4. The eigenfrequencies of
the 2 DOF mechanical system are plotted in Figure 11.2 against the excitation frequency
corresponding to the RSI 2nd harmonic frequency f = 18n, for the investigated impeller
frequency range.
In Figure 11.2, one may see that the eigenfrequencies vary depending on the impeller
rotation frequency. As the mechanical system has two DOF, two eigenfrequencies exist
for each impeller rotating frequency. Nevertheless, the resonance of the guide vanes is
reached when the linear curve f0,y,i crosses one or the other eigenfrequency curves. As
1.0
[] δC i,j δC 10,10 δC10,11
δC11,10 δC11,11
0.5
−0.5
18n
−1.0
210 220 230 240 250 260 270 280
[Hz]
Figure 11.1: Component of the relative residual matrix [δC] measuring the satisfaction
of the Caughey condition for the case of bending motion, against the RSI 2nd harmonic
excitation frequency f = 18n.
250
[Hz] f0,y,i f0,y,1 f0,y,2
245
8n
i 1
0f ,y, =
240
235
230
225
220
18n
210 220 230 240 250 260 270 280
[Hz]
Figure 11.2: Eigenfrequencies of the 2 DOF system, against the RSI 2nd harmonic exci
tation frequency f = 18n.
may be observed in the figure, in the present case, resonance occurs at 228 and 234 Hz,
which corroborate the results, see Figure 9.15: the guide vane O10 reaches resonance at
234 Hz, whereas the amplitude of the vibrations of the guide vane O11 at this frequency
is attenuated. The latter responds preferably at 228 Hz.
250
[Hz] f0,α,i f0,α,1 f0,α,2
245
n
,i =18
240
0f ,α
235
230
225
220
18n
210 220 230 240 250 260 270 280
[Hz]
δCi,j
2.0
[] δC10,10 δC10,11
1.5
δC11,10 δC11,11
1.0
0.5
−0.5
−1.0
−1.5
−2.0
45n
550 600 650 [Hz] 700
Figure 11.4: Component of the relative residual matrix [δD] measuring the satisfaction
of the Caughey condition for the case of torsion motion, against the RSI 5th harmonic
excitation frequency f = 45n.
750
[Hz] f0,α,i f0,α,1 f0,α,2
45n
700
=
f0,a,i
650
600
550
500
45n
550 600 650 [Hz] 700
Figure 11.5: Eigenfrequencies of the 2 DOFs system against the RSI 5th harmonic exci
tation frequency f = 45n.
the torsion angle factor fluctuations of the two guide vanes do not show any resonance at
610 Hz.
in Figure 11.6 against the excitation frequency corresponding to the RSI 5th harmonic
frequency f = 45n. According to Figure 11.6, the resonance of the corresponding conser
vative mechanical system should be reached between 625 and 650 Hz. The shift of the
eigenfrequencies due to dissipation is greater for the torsion than for the bending mode.
750
[Hz] f0,α,i f0,α,1 f0,α,2
700
n
=45
f0,a,i
650
600
550
500
45n
550 600 650 [Hz] 700
This chapter aims to predict the behavior of the entire cascade if all the zo = 20 guide
vanes feature a flexible stem, from the measurements on two instrumented guide vanes in
the case where the others present a stiff stem. The structural inertia/mass, damping and
stiffness of each guide vane are supposed to be identical.
The zo = 20 eigenvectors are identified following the procedure exposed in Section 3.4.
We restrict the investigation to the p values (0 ≤ p ≤ 3) corresponding to the measured
phase shifts, since the eqs. 10.33 to 10.36 are valid in the investigated range of the phase
shift between two adjacent guide vanes. The relative amplitude ŷŷ10 11
is 1. In Figure 12.1,
the real part of the first four displacement eigenvectors Bp for 0 ≤ p ≤ 3 are normalized
with their maximum value and plotted.
According to eq. 3.40, each p value features a particular phase shift ∆φ = 2π
zo
·p
between two adjacent guide vanes of the cascade. Assuming a permanent response, one
may compute the resulting displacement as follows:
2 −1 RSI
ŷ (18n) = −ω18n [I] + [K] + ı · ω18n [C] F̂ (18n) (12.2)
In Figure 12.2, the eigenfrequency and the eigenvalue real part of the bending eigen
modes 0 ≤ p ≤ 3 at the RSI 2nd harmonic frequency f = 18n are plotted against the
corresponding frequency.
Re(B p)
max(Bp) p=0
2
[]
1
0
1
p=1
0
−1
p=2
1
0
−1
p=3
1
0
Oi
−1
0 5 10 15 [] 20
Figure 12.1: Real part of the first four displacement eigenvectors Bp for 0 ≤ p ≤ 3
normalized with their maximum value.
230 150
[Hz] f0,y,p [s1 ] λ0,y,p p=0 p=1 p=2 p=3
225
100
220
18n
f0,y =
50
215
210
18n 0
18n
210 220 230 240 250 260 270 280 210 220 230 240 250 260 270 280
[Hz] [Hz]
Figure 12.2: Eigenfrequency (left) and eigenvalue real part (right) of the bending modes
0 ≤ p ≤ 3, against the RSI 2nd harmonic excitation frequency f = 18n.
The frequency at which the guide vanes respond preferably varies from 221 Hz to
222 Hz according to the different eigenmodes, which is between 2% and 3% lower than
the frequency at which the guide vanes respond preferably in the tests made with two
adjacent flexible guide vanes. The real eigenvalue of each eigenmode is positive, that is
the mechanical system is stable. Since the RSI pressure mode at this frequency features
a phase shift ∆φp=2 = 2π20
· 2, all the evidence suggests that the eigenmode p = 2 is most
likely.
In Figure 12.3, the amplitude of the displacement factor of a guide vane in the cascade
at the RSI 2nd harmonic frequency f = 18n is plotted for the mode p = 2. It may be
observed that when considering the entire guide vane flexible, the vibration amplitude is
twice lower than when considering only two flexible guide vanes, the others being stiff.
−3
x10
2.0
1.0
0.5
0.2 18n
210 220 230 240 250 260 270 280
[Hz]
Figure 12.3: Measured values of displacement factor amplitude of the guide vanes O10
and O11 during the measurements and predicted values of displacement factor amplitude
of a guide vane in the cascade if each one features a flexible stem and identical structural
mass, damping and stiffness, at the RSI 2nd harmonic frequency f = 18n, for the mode
p = 2.
The zo = 20 torsion angle eigenvectors are identified following the procedure exposed
in Section 3.4. As for the case of the bending eigenmodes, one restricts the investigation
to the p values corresponding to the measured phase shifts, since the eqs. 10.37 to 10.40
are valid in the investigated range of the phase shift between two adjacent guide vanes.
The relative amplitude ŷŷ10
11
is 1. In Figure 12.4, the real part of the first six torsion angle
eigenvectors for 0 ≤ p ≤ 5 are normalized with their maximum value are plotted.
Re(B p)
max(Bp) p=0
2
[]
1
0
1
p=1
0
−1
p=2
1
0
−1
p=3
1
0
−1
p=4
1
0
−1
p=5
1
0
Oi
−1
0 5 10 15 [] 20
Figure 12.4: Real part of the first six torsion angle eigenvectors for 0 ≤ p ≤ 5 normalized
with their maximum value.
In Figure 12.5, the eigenfrequency of the torsion eigenmodes 0 ≤ p ≤ 5 at the RSI 5th
harmonic frequency f = 45n are plotted against the corresponding frequency.
In Figure 12.6, the real part of the eigenvalues corresponding to the torsion eigen
modes 0 ≤ p ≤ 5 at the RSI 5th harmonic frequency f = 45n are plotted against the
corresponding frequency.
720 720
[Hz] f0,α,p p=0 p=1 p=2 [Hz] f0,α,p p=3 p=4 p=5
700 700
680 680
660 660
45n
45n
,p =
,p =
f0,α
640 640
f0,α
620
45n 620
45n
550 600 650 [Hz] 700 550 600 650 [Hz] 700
Figure 12.5: Eigenfrequency of the guide vane cascade, corresponding to the modes
0 ≤ p ≤ 2 (left) and the modes 3 ≤ p ≤ 5 (right) against the excitation frequency
corresponding to the RSI 5th harmonic frequency f = 45n.
300 300
[s1 ] λ0,α,p p=0 p=1 p=2 [s1 ] λ0,α,p p=3 p=4 p=5
200 200
100 100
0 0
unstable unstable
−100 −100
45n 45n
550 600 650 [Hz] 700 550 600 650 [Hz] 700
Figure 12.6: Eigenvalue real part of the guide vane cascade, corresponding to the modes
0 ≤ p ≤ 2 (left) and the modes 3 ≤ p ≤ 5 (right) against the excitation frequency
corresponding to the RSI 5th harmonic frequency f = 45n.
The frequency at which the guide vanes respond preferably varies between 650 Hz and
695 Hz, see Figure 12.5, which is slightly different from the frequency at which the guide
vanes respond preferably in the experiments: up to 8% frequency shift. The real part
of the eigenvalues is strongly varying depending on the mode p. Since the RSI pressure
mode at this frequency features a phase shift ∆φp=2 = 2π 20
· 5, all the evidence suggests
that the eigenmode p = 5 is most likely. Above 660 Hz, the real eigenvalue of this mode
becomes negative, that is the mechanical system is unstable. If all the guide vanes feature
flexible stem, premature failures of the guide vanes are expected if the pumpturbine is
operated at BEP above n = 14.6 Hz.
In the following lines, we propose two different ways to prevent damage to the guide
vanes. On the one hand, by multiplying the structural damping constant by a factor of
2, the mechanical system becomes stable. In Figure 12.7, the real part of the modified
guide vane cascade eigenvalues is plotted for the eigenmode p = 5. It may be seen that
the eigenfrequencies are not shifted due to the higher damping. Moreover, the real part
remains positive in the whole impeller frequency range, which means that the mechanical
system is stable.
720 300
[Hz] f0,α,p=5=5 [s1 ] λ0,α,p
250
700
200
680
150
660
45n
100
5=
,p=
f0,α
640
50
620
45n 0
45n
550 600 650 [Hz] 700 550 600 650 [Hz] 700
Figure 12.7: Eigenfrequencies (left) and eigenvalue real part (right) of the eigenmode
p = 5, against the excitation frequency corresponding to the RSI 5th harmonic frequency
f = 45n. The guide vanes feature a structural damping constant twice higher than the
initial one.
On the other hand, the instability leads to high vibration amplitude when the shape of
the excitation, that is the RSI pressure mode, is similar to the eigenmode of the cascade.
If all the guide vanes feature similar structural parameters, the shape of the RSI pressure
mode is identical to the shape of the cascade eigenmode. By multiplying by a factor
0.8 the inertia of 10 guide vanes and placing them alternately between the guide vanes
featuring the initial inertia, the mechanical system is mistuned. In such a case, the phase
shift between the vibrations of the guide vanes Oi and Oi+1 is different from the one
between the vibrations of the guide vanes Oi+1 and Oi+2 . The shape of the structural
eigenmode is thereby changed, both in terms of amplitude and phase, and does not match
anymore the shape of the pressure mode.
Considering the eigenmode presenting 5 diametrical nodes, the phase shifts between
the eigenvector components corresponding to the guide vanes Oi and Oi+1 as well as
between the eigenvector components corresponding to the guide vanes Oi+1 and Oi+2 for
the mistuned cascade are plotted against the RSI 5th harmonic frequency f = 45n in
Figure 12.8.
[] ∆φ Oi+1  Oi
π Oi+2  Oi+1
π/2
0
45n
550 600 650 [Hz] 700
Figure 12.8: Phase shifts between the eigenvector components corresponding to the guide
vanes Oi and Oi+1 as well as between the eigenvector components corresponding to the
guide vanes Oi+1 and Oi+2 , against the excitation frequency corresponding to the RSI 5th
harmonic frequency f = 45n.
1 Oi+1
[] Im(Bp=5 )
Oi Oi+2
0
tuned cascade
mistuned cascade
−1
Re(Bp=5 )
−1 0 [] 1
Figure 12.9: Complex visualization of the eigenvectors components associated to the guide
vanes Oi , Oi+1 and Oi+2 corresponding to the tuned cascade and the mistuned cascade.
In Figure 12.10, the eigenfrequencies and the real part of the mistuned guide vane
cascade eigenvalues are plotted for the eigenmode with 5 diametrical nodes. By mistuning
the cascade, the eigenvalues are increased and the range of negative eigenvalue real part
is shifted to slightly higher excitation frequencies compared to the tuned cascade.
780 300
[Hz] f0,α,p [s1 ] λ0,α,p
740 200
700 100
5n
=4
660 0
,p
f 0,α
unstable
620 −100
45n 45n
550 600 650 [Hz] 700 550 600 650 [Hz] 700
Figure 12.10: Eigenfrequencies and real part of the mistuned guide vane cascade eigen
values for the eigenmode with 5 diametrical nodes, against the excitation frequency cor
responding to the RSI 5th harmonic frequency f = 45n.
Conclusions
Experimental investigations of the guide vane cascade dynamic response to the excitation
due to the RotorStator Interaction in a low specific speed pumpturbine reduced scale
model are reported. The investigated pumpturbine features 9 impeller blades and 20
guide vanes and is operated at the Best Efficiency operating Point at 18◦ opening angle.
The bending and torsion vibrations of the guide vanes are studied. The influence of
the adjacent guide vane vibrations are pointed out. A methodology is given to reliably
identify the hydrodynamic parameters of the mechanical system, which is shown to be of
the 2nd order. The entire guide vane cascade dynamic response is studied based on the
measurements on two guide vanes equipped with strain gages and three pressure sensors
adequately placed in the stator.
The impulse response of immersed guide vanes is obtained using a spark plug flush
mounted on the bottom flange in a guide vane channel. This type of measurements are
successfully undertaken in water, model at rest, and model in operation.
Keeping the operating conditions of the Best Efficiency Point constant, the impeller
rotation frequency is swept and the guide vanes are therefore excited by the RotorStator
Interaction, RSI, over a wide frequency range. The combination of zb impeller blades
with zo guide vanes makes apparent many different spinning diametrical pressure modes.
Nevertheless, the guide vanes are mainly excited at the frequencies f = zb n and f = 2zb n,
but also respond up to the RSI 5th harmonic.
The amplitude of the fluctuating bending displacement and torsion angle of the guide
vanes is strongly varying across the impeller frequency range. The ranges of the 2nd and
the 5th RSI harmonic frequency contain the frequency of the 1st bending eigenmode and
the 1st torsion eigenmode, respectively. The pressure fluctuations close to the vibrating
guide vanes are strongly varying and may even decrease by 50% at resonance. Therefore,
a transfer of energy between the vibrating structure and the flow pressure should occur.
The influence of an adjacent guide vane on the vibrations of a guide vane is found
to vary significantly between its position on the pressure side and suction side of the
latter. Regarding the guide vane bending vibrations, the hydrodynamic force acting on a
guide vane induced by its neighboring guide vane on the pressure side is up to 10 times
higher than the force induced by its suction side neighbor. As for the guide vane torsion
vibrations, the hydrodynamic torque acting on a guide vane induced by its neighboring
guide vane on the pressure side is up to 5 times higher than the force induced by its
suction side neighbor.
The hydrodynamic damping coefficient and the added mass corresponding to the vi
brations of the adjacent guide vanes are successfully identified and an influence matrix is
built. These two terms are shown to depend strongly on the relative amplitude of their
vibrations, the absolute flow velocity and the phase shift between their vibration signals.
Taking into account the periodic condition, the influence matrix is built in order to
predict the dynamics of the entire guide vane cascade. Four and six different eigenmodes
are investigated for the case of bending and torsion motions, respectively. On the one
hand, the bending modes feature eigenfrequencies varying from 221 Hz to 222 Hz, which
is between 2% and 3% lower than the frequency at which the guide vanes respond prefer
ably in the tests made with two adjacent flexible guide vanes. The eigenvalue real part of
each eigenmode remains positive on the investigated impeller frequency range, that is the
mechanical system is stable. On the other hand, the torsion modes feature eigenfrequen
cies varying from 650 Hz to 695 Hz. Above 660 Hz, the eigenvalue real part of the mode
which is the most likely to be excited by the RSI becomes negative. This means that the
mechanical system is unstable and premature failures of the guide vanes are expected if
the pumpturbine is operated at BEP above n = 14.6 Hz.
Finally, two different ways to prevent damage to the guide vanes excited at the RSI
th
5 harmonic frequency are proposed. On the one hand, it is shown that by increasing
the structural damping constant by a factor 2, the mechanical system becomes stable.
On the other hand, the modification of the shape of the cascade eigenmode is achieved
by mistuning the cascade, such that its shape does no longer match the shape of the RSI
pressure mode. This way, even if the mechanical system remains unstable, the risk of
damaging the guide vanes is reduced.
Perspectives
The present document does not pretend to clarify all the pending issues of fluidstructure
coupling occurring in hydraulic machines. Incidentally, it gives rise to several questions
whose answer would contribute greatly to the further development of these machines. In
addition to opening the way for further experiments, the present study proposes complete
results of a case study which may constitute a benchmark for the validation of numerical
tools.
The study intends to predict the vibrating behavior of the entire cascade featuring similar
flexible guide vanes from the local measurements on two isolated flexible guide vanes. It
would be interesting to place flexible guide vanes in the entire cascade to validate the
forecasted results. Nonetheless, the difficulty lies in manufacturing guide vanes featuring
exactly similar structural properties.
At the end of the present document, two solutions to prevent guide vane damage are
shortly investigated. Mistuning the cascade is one of the solutions, and we show that this
way, we are able to shift the shape of the structural eigenmode and, thus to reduce the
risk of damaging the guide vanes. This technique is already used in gas turbine, [45] and
[10]. It needs further attention: by optimizing the type of mistuning, a reduction of the
guide vane vibration amplitude should be possible.
After having shown the relevance of the vibration phase, amplitude and the flow velocity
on the coupling terms, it is necessary to investigate all these parameters in a simpler
case. For instance, two vibrating plates immersed in a uniform flow would allow a deeper
investigation of the mentioned parameters. Moreover, the dependence of the distance
between the vibrating structures could easily be analyzed.
Pressure fluctuations
At guide vane resonance, a transfer of energy occurs from the flow to the vibrating struc
ture. Therefore, the pressure fluctuations in the rotorstator gap decrease. By reaching
the resonance of the guide vanes, the periodic loading due to the RSI acting on the impeller
blades might be minimized and, as a consequence, the risk of crack appearance on the
impeller might be lowered. A more detailed study of the relation between the vibrations
of the guide vanes and the decrease of the pressure fluctuations might be undertaken.
Signal processing
The values of the signals {xk,i } are associated with equally spaced time tm = m∆t in
such a way that
xk (m) = xk (tm ) (A.2)
The discrete Fourier transform components related to the j th segment are expressed
as:
Mj −1
X
x̂k (fi ) = ∆t · (xk,j (mj ) − xk,j ) Wh (mj ) e−i2πfi mj (A.4)
mj =0
The averaged magnitude of the discrete Fourier transform of the signal xk (t) is ex
pressed as:
nd
1 X
x̂k (fi ) = x̂k,j (fi ) (A.7)
nd j=1
The power spectral density estimate of the signal xk (t) is defined as follows:
nd
2 1 X
x̂k  (fi ) = x̂k,j (fi )2 (A.8)
nd j=1
The cross power spectral density estimate between the signals xk (t) and xl (t) is defined
as:
nd
2 1 X
x̂k x̂l  (fi ) = x̂∗ (fi ) · x̂l,j (fi ) (A.9)
nd j=1 k,j
The coherence function between the signals xk (t) and xl (t) is defined as follows [9]:
In the present document, for readability reason, the ¯ sign for averaged magnitude of
Fourier Transform, power spectral density and cross power spectral denstity is omitted.
l
x
F
βF
y
Figure B.1: Embedded beam deflection yF and rotation βF due to a concentrated force
F at its free end.
The effect of a concentrated torque T applied at its free end, see Figure B.2, provides
the following elastic line equation:
T 2
yT (x) = x (B.3)
2EIz
Tl
βT = (B.4)
EIz
l
x
T
βT
y
Figure B.2: Embedded beam deflection yT and rotation βT due to a concentrated torque
T at its free end.
T α T
Figure B.3: Embedded beam deflection due to concentrated torque around the neutral
axis.
T
α (x) = x (B.5)
GIp
4 4
π (Dext − Dint )
Ip = (B.7)
32
1 2 3 4
Figure B.4: Guide vane partitioning, hydrofoil concentrated mass and embedding location.
Table B.1: Guide vane dimensions, Dext , Dint and l, moment of inertia, Iz and Ip , and
mass per unit length µ.
Denoting the segment number by i and considering small rotations at extremities, the
total deflection y obtained when applying a concentrated force F at the free extremity of
Using eqs. B.1 to B.4, the guide vane bending stiffness K s is finally recovered by
dividing the force F by the deflection y as follows:
F
Ks = = 1.28 · 106 N·m1 (B.9)
y
The total angular deflection α due to a torque T around the neutral axis may be
expressed as the sum of the angular deflection αi of each of the four segments as follows:
4
X
α= αi (B.10)
i=1
1 2 3
Figure B.5: Usual guide vane stem partitioning, hydrofoil lumped mass and embedding
location.
Table B.2: Usual guide vane dimensions, Dext , Dint and l, moment of inertia, Iz and Ip ,
and mass per unit length µ.
s
In the same manner as for the modified guide vanes, the bending Kusual and torsion
s
Lusual stiffness yields:
s
Kusual = 2.94 · 106 N·m1 (B.12)
The bending stiffness of the usual guide vanes is 2.3 higher than the one of the modified
guide vanes and the torsion stiffness of the usual guide vanes is 2.15 higher than the one
of the modified guide vanes.
with
i−1
X
yi,tot (x) = (yF,j (Lj ) + yT,j (Lj ) + (βF,j−1 + βT,j−1 ) · Lj )
j=1
+ (βF,i−1 + βT,i−1 ) · x + yF,i (x) + yT,i (x) (B.15)
R−R
The first torsion eigenfrequency f0,α is evaluated as follows:
s P4
1
R−R 1 2
T i=1 αi
f0,α = 1 = 645Hz (B.16)
2π 2
· J∗
with
!2 Z
4 Z Li i−1 D1,ext 4
X X D
J∗ = ρ αi (x) + αj (Lj ) π dD dx
i=1 0 j=1 D1,int 16
4
!2
X
+ αi (Li ) · Jm (B.17)
i=1
R−R
f0,α,usual = 945Hz (B.19)
The bending eigenfrequency of the usual guide vanes is 1.34 higher than those of the
modified guide vanes and the torsion eigenfrequency of the usual guide vanes is 1.47 higher
than those of the modified guide vanes.
Ls
Js = = 2.20 · 10−4 kg · m2 (B.21)
R−R 2
2πf0,α
Lsusual
s
Jusual = 2 = 2.20 · 10−4 kg · m2 (B.23)
R−R
2πf0,α,usual
s
On the one hand, the structural mass of the usual guide vanes Iusual is 6.4% lower
than the one of the modified guide vanes. On the other hand, the structural inertia of
s
the usual guide vanes Jusual is similar as the one of the modified guide vanes.
4Φ = 0 , in V (C.1)
4Θ (x − x0 ) = δ (x − x0 ) (C.3)
According to the Green’s function properties, one may express the potential at any
position in V as a function of the conditions imposed on the boundary A from eq. C.5:
∂Φ (x0 )
Z Z
1 0 ∂ ln r
Φ (x) = Φ (x ) dA − ln r dA , if x ∈ V and x0 ∈ A (C.6)
2π A ∂n A ∂n
∂Φ (x0 )
Z Z
Φ (x) 1 0 ∂ ln r
= Φ (x ) dA − ln r dA , if x, x0 ∈ A (C.7)
2 2π A ∂n A ∂n
We use the notation of the boundary elements method to express the relation between
the potential and the boundary conditions, in which the Einstein summation convention
is used. In this method, the boundary A is first discretized and then a collocation method
is applied, see Figure C.1.
nk
xj x j+1
xk x k+1
with
1
Υ0kj = Υkj − δkj (C.9)
2
and where Θij and Υij are defined as:
Z xj+1
1
Θkj = ln rdx (C.10)
2π xj
Z xj+1
1
Υkj = ∇x ln r · ndx , with r = xk − x (C.11)
2π xj
From eq. C.8, the potential at each position xj of the boundary A may be found as
follows:
0 −1 ∂Φ
Φj = Υkj Θkj (C.12)
∂n j
Writing the eq. C.6 in a discretized formulation, the potential in the whole domain V
is expressed:
∂Φ
Φi = Ξij Φj − Πij (C.13)
∂n j
where
Z
1
Ξij = ln rdx (C.14)
2π ∂Ω
and
Z
1
Πij = ∇x ln r · ndx , with r = xi − x (C.15)
2π ∂Ω
Finally, one may express the potential Φ at any position in V by replacing eq. C.12
in eq. C.13:
0 −1
∂Φ
∂Φ
Φi = Ξij Υkj Θkj − Πij = Λij · (C.16)
∂n j ∂n j
This coefficient finally indicates the influence of the guide vane Oi−J on the guide vane
Oi , since the potential flow Φi at the position xi is directly related to the pressure at the
wall of the guide vane Oi , see eq. 3.18 and, consequently, to the force applied on the guide
vane Oi .
In Figure C.3, the influence coefficient Λi,i−J of the guide vane Oi−J vibrations on the
potential flow Φi at the position xi is plotted for the first five neighboring guide vanes.
y y
L xi Oi1 xj+2 Oi2 xj+2
Oi
L xk+1 xk+1
2 xj+1 xj+1
αO =18° xk xk
xj xj
2πR2 2πR2
zO zO
Figure C.2: Simplified guide vane cascade represented linearly.
The influence of the guide vanes Oi−2 and Oi−5 on the guide vane Oi is found to
be 3.7 and 12.3 times lower than the influence of the guide vane Oi−1 . In the reality,
one certainly observe a lower influence of the guide vanes Oi−J (J ≥ 2), because of the
intermediate guide vanes. In this approach, the intermediate guide vanes are neglected
for the sake of simplicity.
1.0
Λi,OiJ
[]
Λi,Oi1
0.8
0.6
0.4
0.2
0
Oi1 Oi2 Oi3 Oi4 Oi5
Figure C.3: Influence coefficient Λi,i−J of the guide vane Oi−J vibrations on the potential
flow Φi at the position xi , normalized by Λi,i−1 for the first five neighboring guide vanes.
with
and the matrices whose components are built from the potential flow theory
2) The guide vanes O10 and O11 having a flexible stem, the others being stiff, the system is reduced to:
1
2
with ^ ^ ^ ^ ^ is known from the measurements
and ^ ^ ^ ^ ^ is unknown.
8) The outlined diagonal terms are known from the modal analyzes in water, model at rest.
9) Focus on the response to the frequency f=18n at which the torsion response is negligible compared to the bending:
2
where and
10) y^11 is assumed not strong enough to have any influence at the lowest value of the impeller frequency shift, and,
therefore, the force due to RSI acting on O10 may be known:
11) The force due to RSI acting on O11 is known by taking into account the phase shift of pressure mode k=2:
12) The force due to RSI on the entire impeller frequency range investigated is known by solving:
where and are assumed constant over the frequency range. The pressure terms are then linearly linked
to the pressure monitored by the sensor which is not influenced by the guide vanes vibrations.
13) The non diagonal components of the influence matrix are known by using 2 :
15) Focusing on the response to the frequency f=45n at which the bending response is negligible compared to
the torsion, and repeating the steps from 10) to 13) accordingly, the corresponding hydrodynamic parameters are
known as follows:
The spinning pressure modes resulting from RSI have been theoretically approached in
axial gas turbomachines. Blake [12] summarizes the different studies on spinning pressure
modes made in the past. Basically, the fluid in the stator is considered as an acoustic
medium. The rotating and fluctuating forces exerted on rotor blades due to RSI induce
pressure fluctuations in the stator. The combination of the stator and rotor blade numbers
makes apparent different pressure modes. Lowson [51] theoretically analyzes the noise
of an axial compressor and preliminary agreements with experiments are demonstrated.
Kaji and Okazaki [46] investigate the effects of the Mach number, the rotorstator axial
clearance, the blade spacing ratio and the drag coefficient of upstream airfoils on the
sound generated by RSI.
The theoretical approach applied to axial gas turbomachines is adapted here to the
case of a radial hydraulic pumpturbine. The water is taken as a compressible fluid.
Recently, Yan et al. [89], [88], have launched both compressible and incompressible simu
lations for understanding the RSI phenomenon in hydraulic machines. The results of the
compressible simulation corroborate the experimental results. The effect of compressibil
ity has thereby a significant influence on the RSI. The influence level is strongly linked
to the length scale. The higher the length scale, the stronger the influence. The incom
pressibility assumption often made in hydraulic machines means that the acoustic waves
travel at an infinite speed. The pressure fluctuations in the rotorstator gap may also be
instantaneously observed in the spiral casing.
The Figure E.1 shows the rotating impeller where each blade consists of a source S of
acoustic wave, which is assumed to be located at the blade leading edge. The stay and
guide vanes are removed and, therefore, the sound is assumed to propagate rectilinearly
from the rotating source of sound positioned at x (t) to an observer placed in the acoustic
medium at the fixed location x. The waves are propagating in the plane (x1 , x2 ) normal
to the impeller rotation axis. According to Blake [12], the pressure p (x, t) coming from
the source S is expressed as a function of the force Fi exerted by the blades on the fluid
in the direction i:
0 1 ∂ Fi (t − rS /co )S
[p (x, t)]S = − (E.1)
4π ∂xi rS
x2
F
FR
n
Fθ x(t)
rS x
R1 S r
θS
θ
x1
where the Einstein summation convention is used and co denotes the speed of sound and
rS , the sourceobserver vector which may be approximated as follows:
q
rS (t) = r2 + R12 − 2R1 rcos (θ − θS (t))
≈ r − R1 cos (θ − θS (t)) (E.2)
where θS (t) denotes the angular position of the sound source S and is expressed as follows:
2π
θS (t) = θ0 + S − 2πn · t (E.3)
zb
zb being the number of impeller blades and n the impeller rotation frequency.
b −1
zX
1 1 (∂Fi (t − rS /co ))S ∂t ∂rS 1 ∂rS
= − − 2 Fi (t − rS /co )
s=0
4π rS ∂t ∂rS ∂xi rS ∂xi
zXb −1
1 (∂Fi (t − rS /co ))S ∂rS
≈
s=0
4π · co · rS ∂t ∂xi
zXb −1
1 (∂Fi (t − rS /co ))S ∂ (r − R1 cos (θ − θS ))
≈
s=0
4π · co · rS ∂t ∂xi
zXb −1
1 (∂Fi (t − rS /co ))S ri
= · (E.4)
s=0
4π · co · rS ∂t r
The fluctuating force is dependent on the impeller inflow velocity field. Respecting
the spatial periodicity condition and taking into account the number of guide vanes zo in
the cascade causing the impeller inflow velocity field to periodically present defects in the
wake of the guide vanes, the fluctuating force may be written as follows:
zXb −1 X
∞
ri ri i(mzo θS (t)+ko rS )
(Fi (t − rS /co )) · = F̂i s,m · e (E.5)
r s=0 m=−∞
r
The rather complex summation in eq. E.6 is simplified using the following relation:
b −1
zX zXb −1
i(mz0 +q) 2π ·s ri i(mz +q) 2π ·s
e zb F̂i s,m · = e 0 zb F̂ s,m · cos (θS + γ − θ)
s=0
r s=0
∞
X
= zb · F̂ m · δ (mz0 + q − m0 zb ) (E.9)
m0 =−∞
assuming that the amplitude of the unsteady loading on each blade is the same F s,m =
F m .
Moreover, in the following lines, the problem is restricted to the case where r > R1
leading to the simplification in the denominator rS ≈ r.
∞ ∞
F (t − rS /co ) ri X X 0 F̂ m · zb −im0 zb 2πnt im0 zb θ0
· ≈ (−i)m zb −mzo e e
rS r m=−∞ m0 =−∞
r
0
· ei(mzo −m zb )θ eiko r Jm0 zb −mzo (ko R1 ) (E.10)
On the one hand, according to eq. E.4, the pressure due to the RSI is written:
∞ ∞
1 X X 0 F̂ m · zb m0 zb 2πn −im0 zb 2πnt im0 zb θ0
0
p (x, t) ≈ (−i)m zb −mzo +1 e e
4π m=−∞ m0 =−∞ r co
0
· ei(mz0 −m zb )θ eiko r Jm0 zb −mzo (ko R1 ) (E.11)
On the other hand, from eq. E.1 and using eq. E.10, one may write the pressure due
to the RSI as:
zXb −1
0 1 ∂ Fi (t − rS /co )S
p (x, t) = −
s=0
4π ∂xi rS
zXb −1
1 ∂ Fi (t − rS /co )S ∂r
= − ·
s=0
4π ∂r rS ∂xi
zXb −1
1 1 ∂Fi (t − rS /co )S 1 ∂ri ∂r
= − − 2
s=0
4π rS ∂r rS ∂r ∂xi
1 1 ∂Fi (t − rS /co ) ri
≈ −
4π rS ∂r r
∞ ∞
1 X X 0 F̂ m · zb −im0 zb 2πnt im0 zb θ0
≈ (−i)m zb −mzo +1 ko e e
4π m=−∞ m0 =−∞ r
0
· ei(mz0 −m zb )θ eiko r Jm0 zb −mzo (ko R1 ) (E.12)
m0 zb 2πn
ko = (E.13)
co
Therefore, the pressure fluctuations due to the RSI may be written as follows:
∞ X
∞
X 0 F̂ m · zb m0 zb 2πn −im0 zb 2πnt ±im0 zb θ0
0
p (x, t) ≈ (−i)m zb −mzo +1 e e
m=0 m0 =1
r co
0
i(mz0 ±m0 zb )θ iko r m zb 2πn
· e e Jm0 zb ±mzo R1 (E.15)
co
The pressure fluctuations may then be expressed as the summation of the m and m0
harmonics:
∞ X
X ∞
0
p (x, t) ≈ (p0 (x, t))m,m0 (E.16)
m=0 m0 =1
where
!
F̂ m · zb 2m0 zb 2πn
(p0 (x, t))m,m0 = ·
r co
0
m zb 2πn
R1 · cos k+ · θ − m0 zb 2πn · t + φk+
(Jk+
c
0 o
m zb 2πn
R1 · cos k− · θ − m0 zb 2πn · t + φk− )
+ Jk− (E.17)
co
with k+ = m0 zb + mzo and k− = m0 zb − mzo being the number of diametrical nodes of the
spinning modes.
The pressure fluctuations monitored at the angular positions corresponding to two
adjacent guide vanes are phase shifted by an angle ∆φk expressed as follows:
k
∆φk = 2π , (E.18)
zo
where k refers either to k− or k+ .
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Curriculum Vitae
Steven Roth
Avenue de Montoie 37
1007 Lausanne
Suisse
+41 76 421 88 74 Suisse et Anglais
steven.roth13@gmail.com Né le 13 janvier 1984
FORMATION
20082012 Doctorat ès sciences techniques
Ecole Polytechnique Fédérale de Lausanne (EPFL), Suisse
EXPERIENCES PROFESSIONNELLES
20052009 Ecole Polytechnique Fédérale de Lausanne (EPFL), Suisse
Laboratoire de Machines Hydrauliques (LMH)
Activités de recherche:
Thèse de Doctorat: Etude expérimentale du couplage fluidestructure dans la grille
d’aubes directrices d’un modèle réduit de pompeturbine
• Mesures de pressions pariétales
• Mesures des vibrations
Rédaction de publications scientifiques et participation à diverses conférences
Activités d’enseignement:
• Encadrement de projets de Masters
• Préparation de séances d’exercices et d’examens
LANGUES
Français Langue maternelle
Anglais Avancé
Allemand Avancé
List of Publications
Journal papers
1. Roth S., Hasmatuchi V., Botero F., Dreyer M., Farhat M. and Avellan F. FluidStructure Coupling
Effects on the Dynamic Response of PumpTurbine Guide Vanes. Journal of Fluids and Structures,
Submitted for publication
2. Hasmatuchi V., Farhat M., Roth S., Botero F. and Avellan F. Experimental Evidence of Rotating
Stall in a PumpTurbine at OffDesign Conditions in Generating Mode. Transactions American
Society of Mechanical Engineers Journal of Fluids Engineering, Vol. 133, num. 5, 2011.
Conference papers
1. Hasmatuchi V., Roth S., Botero F., Farhat M. and Avellan F. Hydrodynamics of a PumpTurbine
Operation at OffDesign Conditions in Generating Mode. Simhydro, Nice, France, 2012.
2. Müller A., Bullani A., Dreyer M., Roth S., Favrel A., Landry C. and Avellan F. Interaction of
a pulsating vortex rope with the local velocity field in a Francis turbine draft tube. 26th IAHR
Symposium on Hydraulic Machinery and Systems, Beijing, China, 2012.
3. Hasmatuchi V., Farhat M., Roth S., Botero F. and Avellan F. Hydrodynamics of a PumpTurbine
Operating at OffDesign Conditions in Generating Mode: Experimental Investigation. SHF Con
ference on Cavitation and Hydraulic Machines, Lausanne, Switzerland, 2011.
4. Landry C., Alligné S., Hasmatuchi V., Roth S., Müller A. and Avellan F. NonLinear Stability
Analysis of a Reduced Scale Model PumpTurbine at OffDesign Operation. 4th IAHR Inter
national Meeting on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems,
Belgrade, Serbia, 2011.
5. Roth S., Hasmatuchi V., Botero F., Farhat M. and Avellan F. Influence of the PumpTurbine
Guide Vanes Vibrations on the Pressure Fluctuations in the RotorStator Vaneless Gap. 4th
IAHR International Meeting on Cavitation and Dynamic Problems in Hydraulic Machinery and
Systems, Belgrade, Serbia, 2011.
6. Hasmatuchi V., Roth S., Botero F., Farhat M. and Avellan F. Hydrodynamics of a PumpTurbine
at OffDesign Operating Conditions: Numerical Simulation. ASMEJSMEKSME Joint Fluids
Engineering Conference 2011  7th International Symposium on Pumping Machinery, Hamamatsu,
Japan, 2011.
7. Hübner B., Seidel U. and Roth S. Application of FluidStructure Coupling to Predict the Dynamic
Behavior of Turbine Components. 25th IAHR Symposium on Hydraulic Machinery and Systems,
Timisoara, Romania, 2010.
8. Roth S., Hasmatuchi V., Botero F., Farhat M. and Avellan F. FluidStructure Coupling in the
Guide Vanes Cascade of a PumpTurbine Scale Model. 25th IAHR Symposium on Hydraulic
Machinery and Systems, Timisoara, Roumania, 2010.
9. Hasmatuchi V., Roth S., Botero F., Avellan F. and Farhat M. Highspeed flow visualization in
a pumpturbine under offdesign operating conditions. 25th IAHR Symposium on Hydraulic Ma
chinery and Systems, Timisoara, Romania, 2010.
10. Roth S., Hasmatuchi V., Botero F., Farhat M. and Avellan F. Advanced Instrumentation for
Measuring FluidStructure Coupling Phenomena in the Guide Vanes Cascade of a PumpTurbine
Scale Model. ASME 2010 7th International Symposium on FluidStructure Interactions, Flow
Sound Interactions, and FlowInduced Vibration & Noise, Montreal, Québec, Canada, 2010.
11. Roth S., Calmon M., Farhat M., Münch C. and Hübner B. and Avellan F. Hydrodynamic Damping
Identification from an Impulse Response of a Vibrating Blade. 3rd IAHR International Meeting of
the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Brno,
2009.