B. ZoppLaboratoire des Ecoulements GophysiquesIndustriels,B.P.
53,38041 Grenoble Cdex 9, FranceC. PelloneCentre National de la
Recherche Scientique,Grenoble, FranceT. MaitreInstitut National
PolytechniqueGrenoble, FranceP. LeroyALSTOM Power Hydro,B.P.
75,38041 Grenoble, FranceFlow Analysis Inside a PeltonTurbine
Bucket1The aim of this work is to provide a detailed experimental
and numerical analysis of theowinaxedbucketofaPeltonturbine.
Thehead, jetincidence, andowratehavebeenvariedtocoverawiderangeof
theturbinefunctioningpoints. Theexperimentalanalysis provides
measurements of pressure and torque as well as ow visualization.
Thenumerical analysis is performed with the FLUENT code using the
two-phase ow
volumeofuidmethod.Theresultspresentagoodconsistencywithexperimentaldata.Inpar-ticular,
the pressure distribution is very well predicted for the whole
range of the studiedparameters. A detailed analysis of torque and
thrust allows evaluating the losses due tothe edge and the cutout
of the bucket. These results give insight into the benet we
canexpect of steady owcalculations throughthe optimizationprocess
of the designofPelton turbines. DOI: 10.1115/1.21843501
IntroductionUptonow,
Peltonturbineshavebeendesignedusingexperi-mental techniques and
semi-empirical methods. The reason is
thattheowinthebucketisunsteady, separatedfromairbyanun-known free
surface two-phase ow, and developed within mov-ingboundaries.
Thesefeatures concernmainlytheideal
rst-orderinviscidoweldinvolvedinlossmechanisms, suchasbucket
backsplashingorjet interference.
Thepredictionofthisowrepresentsagreatchallengefornumericalmodeling.
A sec-ondgroupofdifcultiesdealswiththeactualow.
Wemention,forinstance,theenlargementoratomizationofthejetandwaterlayers,
thesecondaryoweldat theinjector outlet, thewakeeffect behind the
injector nozzle, the gravity deviation of the wa-ter, etc. These
phenomena, depending on Froude, Weber, andReynoldsnumbers,
areintimatelylinkedtolossmechanismsinthe turbine. They also depend
on the turbine design. Though theyareof secondorder
comparedtotherst-order aforementionedow, their
understandingisnecessarytoimprovetheefciencypredictions,
particularlyinthecaseofmodeltoprototypetrans-position
laws.Nowadays, the performance of computers allows numerical
in-vestigationsof theowinbothxedandrotatingpartsof
thePeltonturbine. Concerningtheinternal
viscousowintheup-streamguidingpipes,
ReynoldsaverageNavier-StokesRANSapproachhasproveditsrelevance.
Asanexample, wenotethatthe calculations precisely predicted the
secondary velocity eld atthe outlet of the injector 1. Concerning
the external ow in thebucket, much work has been performed during
the last few yearsusing two kinds of sheet description. The rst one
uses discrete distribution of particles,spherical pellets,
orstripstodiscretizethewatersheet.The corresponding methods have
been applied for a two-dimensional 2D xed and rotating at plate 2
and forthree-dimensional3Drotatingbuckets35. Inthesemethods, no
grid is needed and the air ow is not calcu-lated. The second
corresponds to more classical grid-basedcomputational uid dynamics
CFD approaches using afree-surface tracking method generally, the
well-knownvolume of uid VOF method. Steady buckets were cal-culated
by several authors 610. Rotating buckets
werealsomodeledwithdifferent levels of approximations.Some
calculations were performed on a xed grid with amovingjet at the
inlet 6,11. Inthis case, onlyonebucket is considered and the
cutting of the jet inlet by
thefollowingbucketisnotmodeled.Othercalculationsareperformed with a
stationary grid zone stator containingthejet inlet
andarotatingzonerotor containingthebuckets.
Theslidingmeshtechniqueallowsconnectingthe two regions. Mack and
Moser 12 Mach et al. 13,andZopp
14consideredthreeadjacentbuckets.Thisapproachprovidestheconditionsinthemiddlebucketthat
are common to all revolving buckets. The only limi-tation seems to
appear at a high number of injectors 46, wherejet
interferencephenomenaoccurs 15. Byassuming a periodic ow between
two successive injec-tors, Perrigetal. 16avoidsthislimitation.
Itisnotedthat onlyMackcomparedcalculatedpressurewithex-perimental
time-dependent pressure
signals.Itisnotedthatmuchworkhasbeendoneintheeldofbucketow
modeling, but very few of them are compared to experimen-tal
measurements. For example, onlyKvicinsky11 comparedthe calculated
pressure distribution to experimental data. For thisreason, the
object of this work is to perform RANS modeling of asteadybucket
comparedtoglobal andlocal measurementsforasignicant rangeof
functioningparameters. Vizualizationswerealso planned to support a
better understanding of the ow. A
sec-ondobjectoftheworkistousenumericalmodelingtoquantifythedifferent
causesof thrust lossinthebucket separately. Theresults, presentedat
theendof this paper,
areusedtoprovideinsightintothebenetwecanexpectofsteady-stateowcalcu-lations
through the global process of Pelton design optimization.2
Experimental StudyTheexperimental
studywasconductedinthelaboratoriesofAlstomPowerHydro Grenoble,
France.Thebucket, character-istic of a Pelton turbine, is placed in
the uniform ow created bya cylindrical jet. This study mainly
provides a cartography of pres-sures inside the bucket and total
forces values on the bucket. Thebucket is placed at various
incidence angles; several jet diametersand head heights are
used.2.1 TestImplementation. ThetestingrigisschematizedinFig. 1. A
centrifugal pump, drivenbyavariablespeed250 KW1The bucket geometry
is partially provided due to
condentiality.ContributedbytheTurbomachineryDivisionofASMEfor
publicationintheJOURNALOF TURBOMACHINERY. Manuscript received
November 16, 2004; nal manu-script received January 9, 2006. Review
conducted by R. L. Davis.500 / Vol. 128, JULY 2006 Copyright 2006
by ASME Transactions of the ASMEpowered motor, supplies the test
bench by means of a 200 mm diapipe. For a given series of tests,
the speed of rotation of the pumpis maintained constant. The ow
rate inside the pipe is measuredby means of two owmeters: an
electromagnetic owmeterKrohneandamagnehelicgage Brooks.
Themeasurement ofthe rst allows the verication of the second. In
order to adjust thejet diameter, an orice is placed at the pipe
outlet.Thestatichead, correspondingtothepressuredifferencebe-tween
the interior of the pipe and the atmosphere is measured
bymeansoftwodifferentialpressuresensorsRosemountDP27andE22locatedjust
upstreamfromthetestingbench. Thepressureand velocity upstream of
the orice are designated byp1and U1crosssection1onFig.1.
Theatmosphericpressureandthejetoutlet
velocityaredesignatedbypatmandU. Thenet headHn=U2/ 2g g=9.81 m/
s2is the gravitational acceleration, the staticheadHs=p1patm / g
=998 kg/ m3isthewaterdensityandthe dynamical head Hd=U12/ 2g are
simply connected by theBernoulli relation Hn=Hs+Hd.The measurement
of the ow rateQgives the velocityU1 soHd, the measurement of p1
gives the static head Hs, which leadsto the Hn value and thus
provides an experimental measurement ofthe jet velocity U. The
adjustment of Hn is ensured using the twovalves.A photograph of the
test bench is presented in Fig. 2. It consistsof the steel frame,
the water jet intake, the water sheet extractors,the water jet
extractor for safety, and the instrumented bucket. Inorder to limit
the ow disturbances related to the singularities ofthe testing rig
elbows, valves, etc, the bench is placed at the endof a rectilinear
pipe, 3.5 mlong.Thewater sheet extractors arecurvedpipes of
arectangularcross section. Theymakeit possibletodirect thewater
sheetsowingout the bucket towardthe collectingcontainer
locatedunderthetestarea. Theextractorspositionisadjustedaccordingto
the bucket incidence. The quantity of water that these elementsmust
direct is signicant; in fact, protection was added in order
toreduce the splashes near the measurement zone andthe back-ows on
the jet.Theoriceddiameter, narrowsthewaterjet toaminimumvalue of D.
One designates by S=D2/ 4 the cross section of
thejetupstreamofthebucket.Thejetmustbeminimallydisturbedand spoiled
by the contact of its free surface with the air. With
thisintention, a convergent nozzle is placed at the pipe outlet,
whichallows reducing the jet length between the orice and the
bucket.The bucket, made out of bronze, La=150 mmwide, is fur-nished
with a handle, allowing it to be attached via two axes tothe test
bench frame Fig. 3. The rst axis point on Fig. 4a,locatedat
thearmend, servesastherotationaxisfortheentirewheel. The bucket
rotation in reference to this axis denes
theincidenceanglebetweenthebucketandthejet.
Thesecondaxismeasurement axis of the moment M maintains the
incidenceangle. The bucket edge hatched surface drawn on the Fig.
4b issituated in thexoy plane namedz0plane. Theyoz plane,
perpen-dicular to thez0plane, is the symmetry plane of the bucket
con-tainingthebucketsplitter.
Theprojectionofthesplitterontheoy-axis gives the AB segment, 103 mm
long. A direct orthonormalreference frame oxyz is dened, having for
origin the middle pointOof theABsegment. The xoz plane is calledthe
referenceplane. In order to have a progressive jet inlet ow, the
bucket istruncatedinthe vicinityof point A. This zone constitutes
thecutout of the bucket. The zone close to point B constitutes
theFig. 1 Diagram of the testing rigFig. 2 Testing benchJournal of
Turbomachinery JULY 2006, Vol. 128 / 501back part of the
bucket.Three nondimensional numbers are classically used in the
studyof the ows within a Pelton turbine: TheReynoldsnumber Re=UD/ ,
withwater mo-lecular viscosity equal to 1.002103kg/ m s. Thisnumber
represents the ratio of the inertia forces with re-spect to the
viscosity forces.The Froude number Fr=U/gLa. This number
representsthe ratio of the inertia forces with respect to the
gravita-tional forces. The Weber number We=U2La/ , withwater
surfacetension equal to 0.074 N/ m. This number represents theratio
of the inertia forces with respect to the surface ten-sion
forces.2.2 PressureMeasurement.
Thepressurepiismeasuredin21pointsofthewettedsurface
innersurfaceofthebucketar-ranged as indicated in Fig. 5. The ve
numbered pressure
intakes15,locatedontherighthalfofthebucket,arethesymmetricalones of
the ve corresponding measurement points of the left part.These ve
measurement points are used to ensure the jet-centeringcontrol and
the ow symmetry with respect to the bucket
splitter.Theintakesareplacedatthepointsofaregularorthogonalnet-work;
thex-axisspacingisof15 mmandthey-axisspacingof20
mm.Toworkthesepressureintakes, thebucket isboredorthogo-nallyonits
surface. Fine pipes are weldedontothe externalsurface in front of
each orice Fig. 3; each tube is connected tothe pressure
transducer. Measurements of pressure are carried
outusingadoublemultiplexer
ScannivalveDSS,24channelscon-nectedtoadifferential
pressuretransducer Rosemount DP27.Theinstrument Scannivalve makesit
possibletomeasurethepressureat the21points
usingasinglepressuretransducer. Itoperates as abarrel that connects
thesensor withthepressureintakes, one after another. This device,
requiring only one calibra-tion, provides homogeneous measurement
uncertainties. Hi=pipatm / 1/ 2U2designates the measured relative
pressure atpoint of index i, the unit of measurement being the
water columnmeter mCE.2.3 Thrust and Torque Measurement.
Measurements
relatetothedrivingforcePeltonFzforcecreatingtheenginetorqueand the
bending moment Mw with respect to the wheel axis Fig.4.
TheFzforceisthecomponent, perpendicular
tothebuckethandle,ofthejetforceexertedonthebucket.
Theforceandthemoment aremeasuredusingeight
straingagesmountedonthebucket handle Figs. 4b and 4c. In order to
increase the handledeformations, the gage region is intentionally
weakened. Bridge 1Fig. 4b, madeupof four unidirectional gages,
measuresthebending momentM. Bridge 2 Fig. 4c, made up of four
semi-conductor gages assembled in a differential manner between
twocrosssections, measuresthedrivingforceFz.
Theassemblyperpairofgages, locatedoneachsideofthehandle,
allowsonetoeliminate by cancellation the deformations interference
due to, forinstance, dilation, radial force, or torsion. The
measurement error,thus, comes primarily from the gage calibration
and amplicationFig. 3 Bucket experimental devicesFig. 4 Schematic
views of the bucketFig. 5 Location of the pressure intakes502 /
Vol. 128, JULY 2006 Transactions of the ASMEquality. The use of
semiconductor gages provides, for bridge 2,
arelativeerrorcomparabletothat ofbridge1andequal to2%Table
1.FromFzandMmeasurements, onededucesthemoment Mwand the shift
distance between the origin of force and the ref-erence plane by
the relations:Mw = M + FzL1 = Fz + LS1 + L1 1L1=122
mmisthedistancebetweentheaxisofthemoment Mand the rotation
axis.LS1=121.7 mm is the distance between theaxis of the moment
Mand the reference plane. The Pelton diam-eter is then dened by
DP=2LS1+L1 =487.5 mm.2.4 Experimental Tests. Table 1 indicates the
relative uncer-tainties recordedfor the measurements of the owrate,
statichead,pressure,drivingforce,andthemoment. ThenetheadHn,the
orice diameterd, and the incidence are the three
varyingparameters.Four diameters of the orice were used: d=38.1
mm,50.1 mm, 56.0 mm, and61.5 mm. Theincidencevariesfrom60 deg to
120 deg in 10 deg steps. The three chosen net heads areHn=30 m, 40
m, and 50 m. The corresponding velocities have therespective
values: U=24.26 m/ s, 28.01 m/ s, and 31.32 m/ s.
Foreachcoupleofvalues oricediameter-incidence, Table2indi-cates the
tested net heads, the total number of tests being 56.
Table3indicatesthemeanvolumeowratemeasured literspersec-ond for the
three net heads and the four diameters of the orice.Because of the
jet contraction, the orice d is higher than the jetdiameter D. The
jet diameter, ow rate, and net head are bound bytherelationD=4Q/
2gHn. Forthewholeofthetests Table3, theReynolds number Reis
includedinvalues 3.6106to4.7106. In consideration of high values of
the Reynolds number,the contraction coefcient of the jet is
constant; consequently, foraxedvalueof theoricediameteror jet
diameter theratioQ/Hnremainsthesameone.
ThevaluesofthediameterDareobtained with a relative error of 2.5%.
The usual nondimensionalmagnitude D*indicated in Table 3 is dened
by D*=La/ D.2.5 FlowVisualization(Fig. 6).
Thephotographsaretakenwith a numerical camera provided with a ash.
A droplets fog isalways present in the enclosure. It is accentuated
when the head orthe jet diameter increases. No particular effect of
the head on theowinthebucket wasnoted. For thisreason,
thephotographswere made in the case of a slight head, so one
reaches the maxi-mum quality.Table 1 Measured magnitudesMagnitudes
InstrumentationRelativeuncertaintyFlow-rate QElectromagnetic
ow-meter or turbine ow-meter0,4%Static head HsDifferential pressure
transducer0,2%Pressure HiDifferential pressure transducer
Scanivalve0,2%Driving force Fz andBending momentMStrain
gages2,0%Table 2 Hn head values. Total of 56 tests deg d=38.1 mm
d=50.1, 1 mm d=56 mm d=61.5 mm60 deg 3050 m 3050 m 3050 m70 deg
3050 m 3050 m 3050 m80 deg 30 - 40 - 50 m 304050 m 304050 m 304050
m90 deg 30 - 40 - 50 m 304050 m 304050 m 304050 m100 deg 30 - 50 m
3050 m 3050 m 3050 m110 deg 3050 m 3050 m 3050 m120 deg 3050 m 3050
m 3050 mTable3 MeanvolumeowrateQ, inletvelocityU, Reynoldsnumber
ReD*dmmHn=30 mQ l/sHn=40 mQ l/sHn=50 mQ l/s5.07 38.1 16.7 19.3
21.53.84 50.1 29.1 33.6 37.53.44 56.0 36.2 41.8 46.83.13 61.5 43.8
50.5 56.5Inlet velocity U 24.3 m/ s 28.0 m/ s 31.3 m/ sReynolds
number Re3.61064.21064.7106Fig. 6 Different side and top views of
the sheets of waterJournal of Turbomachinery JULY 2006, Vol. 128 /
503Therst panel of Fig. 6presents theowobtainedwithanorice of 38.1
mm and an incidence of 90 deg. After jet impact
inthebucketoccurs,theowattheexitofthebucketturnsintoasheet. These
sheets of water appear of a white and opaque color,typical of an
air-water mixture. Furthermore, downstreamthebucket, thesheetsof
water breakupindroplets. At thebucketexit,
thestreamlinesdeviationanglesaremoresignicant at theends backpart
andcutoutthaninthevicinityofthereferenceplane. This
phenomenoninvolves thecontractionof thesheetsdownstreamthebucket.
Thefthpanel ofFig. 6 d=50.1 mmand=60 deg presents a top view of the
ow, the white arrowsindicatingthevariousdirectionsofthesheet
ofwaterat exit ofbucket. Thevarious photographs of Fig. 6highlight
thewaterquantities leaving by the cutout. This phenomenon was
observedinthevariouscongurationsobtainedwhilevaryingtheoricediameter
andtheincidenceangle. Imagesonetofour inFig. 6correspondtodifferent
jet diameters for anincidence xedat90 deg. The uid enters entirely
in the bucket. It is observed thatthe cutout leakage ow rate
increases with the diameter.Images ve to eight in Fig. 6 correspond
to different angles ofincidence for an orice diameter d xed at 50.1
mm. Let us notethat, for incidence =60 deg, all of the jet does not
enter into thebucket.
ThepartofthejetthatdoesnotcomeintothebucketisexpelledoutsidethePeltonwheel.Beyond80
deg,thejetentersentirely into the bucket. A part of the jet,
strongly increasing
withtheincidence,leavesthebucketbythecutoutwhilehavingcov-eredonlyasmall
distanceinsideit. Inall cases, aleakageowrate is noted at the
cutout.2.6 Net Head Effect on Pressure. The pressure coefcient
isgiven by Cp=ppatm/ 1/ 2U2 =Hi/ Hn. For the 16 pressure in-takes
points621, themeasurementofthepressurecoefcientwas realized for
each of the three net heads. No signicant
varia-tionofthepressurecoefcientisnotedwiththenethead. Asanexample,
forajet diameterD*=3.44andanincidenceangleof90 deg,
themaximumrelativegapis0.1%. Asaresult, inthefollowing, the results
are presented only for Hn=30 m.2.7 Symmetry Checking. The symmetry
of the ow relativetotheyoz planewas checkedfor
thepairednumberedpoints:1,19, 2,21, 3,16, 4,11, and 5,13. The
measurements showthat,fortheincidenceangleofvalue90
deg,thepressuressym-metry is realized. For these ve points, the
maximum relative gapis0.8%.3 Numerical StudyThe numerical study was
conducted in the laboratory of LEGIGrenoble, France.
Thesoftwareusedis FLUENT. TheNavier-Stokes solver solves the mean
equations of turbulence RANS.Torepresent thetwo-phaseows,
thereareEulerianmethodsand Lagrangian methods 17. The rst consists
of assuming eachphaseas continuous. Informationbetweenphases is
carriedbyinterfaceconditions.Thesecondassumesthatoneofthephaseswater
disperses itself in the other air. Within the framework ofthis
study, the Eulerian methods are well adapted. The free surfaceis
then modeled using a multiuid model or a homogeneous
two-phasemodeloravolumeofuid VOFmethod.
Themultiuidmodelisacompleteandprecisemodelbutrequiresmuchcom-putingtime.
Thehomogeneous model is rather usedfor owswhere one of the phases
is uniformly distributed in the other, suchas theows withbubbles.
Under thepresent congurationtheVOF method is completely appropriate
18. It consists of repre-sentingtheuidvolumebythewater
volumefraction. Thevalue ofis 1 when the cell is lled with water
and 0 when thecell is empty. The determination of requires an
additional equa-tion, thus, the advection equation of the uid. The
free surface isthe set of points for which the volume fraction is
equal to0,0being included in values 0 to
1.Thestudiedcasescorrespondtosevenjet diameterswithanincidencexedat
=90 degandtosevenincidenceswithajetdiameter xedat D*=3.44.
Therangeof diametersusedat
thetimeoftheexperimentalstudywassupplementedbythreeaddi-tionalones:D*=5.91,4.39,2.90.
Theincidencesareidenticaltothe experimental ones.3.1 Numerical
Modeling.
Apreliminarystudywasinitiallyperformedconcerningthe2Dand3Djetsimpactonaatplate.ThemajorobjectivewastoevaluatetheabilitiesoftheFLUENTVOFmodel.
Thenumerical resultswerecomparedtoanalyticalresults for the 2D and
experimental results for the 3D. The com-parisons with regard to
the sheet of water thickness and the pres-sure are excellent
14.3.1.1 Discretization. The simulation of the ow in the
bucketrequiresthesettingof thecontrol volume,
theboundarycondi-tions, and the 3D mesh. Because of the ow symmetry
relative toyoz plane, only the space of a half bucket geometry is
considered.Figure 7 illustrates the composition of the control
volume. It con-sists of four parts: the jet inlet domain, the
bucket, the edge, andthe cutout. This partition makes it possible
to modify only the jetinlet domain when incidence is changed. The
inlet domain is builtin order to include all of the jet whatever
the diameter value. Foreach case, the jet inlet face is taken
parallel to the reference plane.Consequently, thejet inlet
borderisahalfcirclefor =90 degand a half ellipse for the other
incidences. This face is located at50 mm above the reference plane.
The outlet region of the bucketedge is 20 mm high above z0 plane
red zone in Fig.
7.Theboundaryconditionsareazerovelocityconditiononthebottomandtheedgeofthebucket,asymmetryconditiononthefaces
belonging to the symmetry plane, a uniform velocity condi-tion on
the jet inlet face, and a constant pressure condition for allthe
faces in contact with the air.The mesh construction required a
preliminary study in order todetermine the type and the density of
cells to be used. Amaximum3
mmsizeofthecellsstabilizestheresultsincomparisontotherenement of
the mesh. Table 4 shows that, for a number of cellshigher
than180,000, the thrust andtorque become insensitivewith the cell
numbers. For all the treated cases, a number of cellsapproximately
equal to 300,000 corresponds to the criterion of themaximum size as
well as the numerical stability.The cells constituting the hybrid
mesh are hexahedral, tetrahe-Table 4 Numerical stability testCells
numberThrust FzNTorqueMwNm37,500 1178 28994,500 1188 291183,000
1189 293342,000 1189 293Fig. 7 Diagram of the calculation blocks504
/ Vol. 128, JULY 2006 Transactions of the ASMEdral, or pyramidal in
shape. The pyramidal cells allow the
connec-tionbetweendomainsmeshedwiththetwoothertypesofcells.The jet
inlet face is paved in a nonstructured way with quadrilat-erals.
The inlet domain mesh blue block is built using theCooper method.
The domain relative to the cutout outlet ismeshed with
tetrahedrons. Figure 8 illustrates the meshes obtainedfor a xed
diameter for three cases of incidence.3.1.2 CalculationParameters.
The Navier-Stokes equationsare discretizedbya nite volumes method.
The discretizationscheme used to model the uid advection is of the
second order,upstream centered. The free surfaces are characterized
by the vol-umefractionvalue0=0.5.
ThePLICmethodpiecewiselinearinterface calculation 19 is used for
the geometrical reconstruc-tionoftheinterface.
Turbulenceistakenintoaccount usingthek- standard model with wall
functions. On the jet inlet face the
kandvaluesareexpressedaccordingtothemeancharacteristicsof the ow,
namely, turbulence intensity and characteristic length.The
turbulence intensity is taken equal to 5% and the characteris-tic
length to the jet diameter value.A 3D boundary layer calculation 20
carried out on the bucketwith a 90 deg jet incidence highlights
that the viscosity forces arevery weak compared to the inertia
forces. Compared to the 2QUvalue of the ideal force, the three
components of the viscous forcehavethefollowingvalues:
0.26%onthex-axis, 0.02%onthey-axis, and0.3%onthez-axis.
Consideringthesevalues, inarst stage, a laminar calculation has
been performed. In this case,numerical instabilitiesoccur.
Theseinstabilitiesdonot
originatefromthenear-wallregionbutareduetothestrongvelocitygra-dients
close tothe interfaceair entrainment due tothe watermotion. The use
of a turbulence model considerably reducesthese gradients and,
thus, stabilizes the calculation. Consequently,the modeling of the
boundary layer and the renement of the
gridnearthewallarenotreallynecessary.Thevaluesofthedimen-sionless
near wall distanceY+are between 250 and 600, and theminimumvalueof
thevelocityat therst nodeof thewall is13 m/ scomparedtothe25 m/
svalueofthejetvelocity.Itissignicant tonotethat, for themeshused,
FLUENTcalculationgives viscous forces about those given by the 3D
boundary
layercalculation,namely:1%onthex-axis,0.04%onthey-axis,and0.4%onthez-axis.Finally,theresolutionoftheviscousowisnot
optimal, but it is of no importance because the viscous effectsare
very weak. This is conrmed by the good comparison betweenthe
numerical and experimental pressures see Fig. 14.Carried-out
calculations are unsteadily converging toward asteady state. Time
integration is performed using the implicit Eu-ler methodof
thesecondorder. Thetwonumerical criteriaofconvergence are the
stabilization of the force exerted by the jet onthebucket
andtheequalityoftheinowandoutowvalues. Ittakes 35 hof time
consumption bi-processor PC AMD Athlon2000
toperformthe4500timesteps requiredtoinsuretheconvergence.Thepresent
experimental congurationsallowneglectingtheforce of gravity
compared to the force of pressure. Indeed, accord-ingtoTable3,
minimumvelocityofthejet is 24 m/ s, whichgives aminimumFroudenumber
of 24. Inthesameway, thesurface tension effects are neglected the
minimum Weber numberis equal to 1.2106.3.2 Numerical Results and
Comparison3.2.1 FixedAngleof Incidence=90 deg.
Figure9showsthefreesurfaceinthecaseofthreeexperimental jet
diameters.Inside the bucket, the wetted surface increases with the
jet diam-eter.Thethicknessoftheoutgoingsheetofwater
atthebucketedgelevel increases backandforwards except for
thelargestdiameter D*=3.13, for which the sheet of water is thinner
in thereferenceplaneareathanat thebucket ends. Nowater
leakageowthroughthecut-out isnotedexcept forthecasewhereD*=3.13.
For the three studied diameters, the experimental tests re-veal a
low leakage at the cutout outlet Fig. 6.In the reference plane,
Fig. 10 shows the water thickness. Insidethe bucket, the water
thickness e, measured according to thebucket normal, is denedas the
distance froma point of thebucket surface to the free face. This
distance nondimensionalizedwith respect to the bucket width is
given by:e*=e/ La. The non-dimensional curvilinear abscissa s*is
worth 0 for the splitter pointand1fortheedgepoint.
Forthelowestjetdiametersthewaterthickness decreases regularly
inside the bucket. A water
accumu-lationaroundthecommonvalues*=0.60appearsonlywhenD*Fig. 8
View of the meshesFig. 9 Free surface of the jetJournal of
Turbomachinery JULY 2006, Vol. 128 / 5053.84. It is characterized
by a stage that becomes more and moreclear as the jet diameter
increases. This phenomenon is related totheoverpressureinthebucket
bottom, whichcausesavelocityreduction in the sheet of water core
and consequently increases itsthickness.Figure 11 illustrates a
cartography of the pressure coefcient Cpon the left and of the
water volume fraction on the right. Thethree presented cases
correspond to the three cases of Fig. 9. Thezonehavingthestrongest
pressuresCp0.9 islocatedinthebucket bottomandshiftedtowardtheedge.
This zoneextendswhenthejet diameter increases. That isinagreement
withthepressureeffectsonthesheet of water thickness
aspreviouslycommented.3.2.2 Fixed Diameter D*=3.44. Figure 12
illustrates the wa-ter distribution inside the bucket for three
incidences. For =60 deg, water spreadsout over most of theinner
surfaceandconcentratesonthebackedgeat thebucket exit. Part
ofthejetdoes not penetrate inside the bucket: the corresponding
waterquantitydoesnotactonthebucket. For =90 deg,
thesheetofwaterextendstowardthecutout edgeandbecomesthinner.
Theentirejet entersthebucket andtheentiresheet ofwateriscon-tained
in the bucket. For =120 deg, the sheet of water is particu-larly
wide along the bucket edge. A signicant part of the jet goesout of
the cutout, which implies, for this case, a signicant loss
offorce.TheexperimentalviewsinFig.6conrm,qualitatively,totheprecedingnumerical
results. One, however, notes inthis gurethat, whatever the diameter
and incidence, there is always a quan-tityof more or less signicant
water that vacates throughtheFig. 10 Thickness of the sheet of
water in the reference planeFig. 11 Pressure coefcient and water
volume fractionFig. 12 Free surface of the jetFig. 13 Pressure
coefcient and water volume fraction506 / Vol. 128, JULY 2006
Transactions of the ASMEFig. 14 Pressure coefcient: numerical and
experimental resultsJournal of Turbomachinery JULY 2006, Vol. 128 /
507cutout.Figure 13represents the pressure
eldcorrespondingtothethreepreviouscases. Onenotes,
inconformitywiththeexperi-mental results, the displacement of the
overpressure zone towardthecutout whentheincidenceincreases. Under
thecaseof the120 deg incidence, a signicant nonwetted zone is
localized nearthe rear of the bucket.3.2.3 Comparison of the
Measured and CalculatedMagnitudes. Figure14presentstheexperimental
andnumericalpressure coefcients in the planes y1, y0, y1, and y2
indicated onFig. 5. The four gures in the left-hand column are
related to thecaseof incidence=90 deg.
Eachgurepresentsthethreejetdiameter values: D*=3.13, 3.84, and
5.07. The four gures of theright-hand column are related to the
case of jet diameter D*=3.44. Each gure presents four incidence
values: =60 deg,80 deg, 100 deg, and120 deg. Theexperimental
pointsandthecorresponding numerical curves have the same color.The
agreement of the results with numerical calculations is verygood.
However, small deviations appear for the points of planes
y1andy2underthecaseofthetwoextremeincidences60 degand120 deg. These
are the two incidences for which a
nonnegligiblequantityofuidgoesout throughthecutout
andforwhichtheowundergoesthemostsignicantchangeofdirectionclosetothe
cutout.With the incidence xed at 90 deg curves on the left side, it
isclearthatthepressureincreaseswiththediameter. Intheplanesy1, y0,
andy1, themaximumpressureisaroundthenondimen-sional abscissa x*=0.3
and in the planey2around x*=0.2. Thesepoints are situatedinthe
deepest zone of the bucket.
Ineachplane,noshapevariationofthecurvesisnotedbychangingthediameter.
Intheplaney2, thenotedpressuredecit comesfromthe proximity of the
cutout.Concerning the total force, the momentum theorem is applied
tothe closed uid domain illustrated in Fig. 15. This domain
consistsof the following boundary surfaces: the crosssection Sof
the jet inlet. the free surface Sjof the jet. the wetted surface Sb
located inside the bucket. the surface Se, obtained by the
intersection of the sheet ofwater and a plane parallel with the z0
plane. This sectionis located at an external vicinity of the bucket
edge. the surface Sco, obtained by the intersection of the
sheetofwaterandaplaneperpendiculartotheunitvector n.Thevector nis
directedoutsidethedomainandcon-tained in the symmetry plane yoz
plane. The direction nisselectedsothat all of thewater
exitingthroughthecutout crosses the plane. In almost all
congurations, thechoiceofthedirection nparallelwiththedirection
yissufcienttoensurethepreviouscondition.Thissectionis located at an
external vicinity of the
cutout.Whenprojectingonthez-axisexaminingtheforceexertedbythe
bucket interior on the uidone obtains the expression for thePelton
driving forceFz = QU sin + SeVz2dS + ScoVn2dS
2Q=USistheowratethroughajetsection. VzandVn, respec-tively,
indicatethevelocitycomponentsaccordingtothez-axisand direction n.
The maximum force is obtained for the ideal
case.Thiscasecorrespondstoabucket that wouldforcethestream-lines,
at exit, tobe perpendicular tothe z0plane andwithoutvelocity loss.
In fact, the exit velocity on surface Se is equal to U,which gives
the following maximum force:Fzmax= QU1 + sin = 2gD2Hn1 + sin 3The
nondimensional driving force is dened by Fz*=Fz/ Fzmax. Inorder
togiveapractical evaluationof theloss of forceonanactual bucket
comparedtoanideal bucket, oneintroduces therelative loss of
thrustFz*=FzmaxFz / Fzmax=1Fz*. This equa-tion represents the
relative difference between the force generatedby the ow and that
which this ow, in the ideal case, would haveproduced.
Byusingtherelation 2therelativelossofthrust iswritten as the sum of
the loss due to the edge and the loss due tothe cutoutFz*= Fz*Edge+
Fz*Cutout4Fz*Edge= 1S1 + sin SeVzU1 + VzUdS 5Fz*Cutout=1S1 + sin
ScoVnU1 + VzUdS 6Inasimilar way, themoment Mwrelativetothewheel
axisisnondimensionalized with respect to the valueMwmax
correspond-ing to the ideal case. In this case, the shift is taken
equal to zero,which provides Mwmax=DP/ 2Fzmax. The moment relative
to themeasurement axis is dened byM*=M/ Mwmax.Figures 16a and 17a
present respectively the torque
M*andthethrustFz*versustheincidenceangleofthebucketforame-dium jet
size D*=3.44. A good agreement between the numericaland
experimental results is noted. It is observed that the
maximumthrust is obtained at 90 deg and maximum torque at 110 deg.
Thisisduetothemaximumpressuredisplacementtowardthecutoutwhen
incidence exceeds 90 deg cf. Sec. 2.2, Fig. 5.Below80 deg, the
experimental torque and thrust decreaseregularly, although
numerical ones present a plateau and then
de-creasestrongly.Thesediscrepanciescanbeexplainedasfollow-ing.
Between 70 deg and 80 deg, the numerical calculation
under-estimatestheleakageowexitingthecutout. Thisishighlightedby
Fig. 18. Consequently, the thrust and torque
experimentallossesduetothecutout
leakageowarehigherthanthecorre-spondingnumerical losses.
Inthiscase, thestrongcurvatureofstreamlines and the water sheet
thinness are difcult to model.Letusnotethat, above80 deg,
whentheincidenceincreases,thesedifferencestendtowardzeroinspiteofthegrowthofthecutout
leakage ow. In this case, the sheets of water that leave thecutout
become increasingly thick and have increasingly small cur-vatures.
Thus, numerical calculationprovides a verygoodap-proximation of the
actual ow.Below 70 deg, the previous experimental analysis shows
that apart of the jet does not enter in the bucket Fig. 6 and ows
closetoitsrear. Thisow,
probablycreatesalow-pressurezonethatcontributestoatorqueandthrust
increase. Thiseffect
hasbeendemonstratedonrotatingbucketcalculationsbyZopp 14andMack 13.
Because the rear zone is not considered in the presentFig. 15
Schematic view of the boundary surfaces508 / Vol. 128, JULY 2006
Transactions of the ASMEmodeling,thismechanismisnotpredicted.
Thus,thetorqueandthrust values provided by calculation are smaller
than the experi-mental ones.Figures 16b and 17b show torque and
thrust variations ver-susthejetsizeD*forajetincidence=90
deg.Numericalandexperimental values t very well. Both the torque
and thrust val-ues aremaximumnear D*=3.44. This valuematches
approxi-mately the optimum jet size of the corresponding Pelton
turbine.Figures 19a and 19b show the decit of thrust compared
toanidealbucketforthejetsizeandincidencevariations,respec-tively.
On each gure, the contributions of the edge and the cutoutrelation
4 are presented. In Fig. 19a, the edge loss presents aminimumequal
to0.06at the100 degvalueof incidenceandnever exceeds0.105.
Thecutout lossbeginsbelow80 degandabove 90 deg, and increases
strongly. This behavior conrms theprevious analysis on the ow-rate
loss close to the cutout.InFig. 19b, at the90 degxedvalue,
thecutout lossesoccur only for the large jet diameters though they
always exist intheexperiments seeFig. 6.
TheedgelossesdecreasewithD*.Forthesmallerjets
higherD*,thisdecreaseisattributedtoanincrease of the outlet mean
velocity at the edge. This is a typicalviscous effect
14.ForthelargerjetsD*3. 44, theedgelossbecomesnearlyconstant. This
last tendency is explained by the kinematic devia-tion of the uid
compared to the edge bucket. In order to evaluatethis deviation,
one considers, in projection in the reference plane,the angle
between the velocity vector and the plane tangential
tothebucketsurfaceattheedge. Inthecaseofthelargestjet D*=2, 9,
numerical calculation provides a deviation angle value ofFig. 16
Total torqueFig. 17 Total thrustFig. 18 Cut-out leakage ow
rateJournal of Turbomachinery JULY 2006, Vol. 128 / 5092 deg
everywhere on the edge except for the bucket ends. Usingthe
relation 5 in the 2D conguration case, this deviation anglegives an
increase of 0.01 on the edge loss Fz*Edge.Figure 20 represents the
graph of four nondimensional magni-tudes Fznum*, Fzexp*, Fzv*, P,
versus D*. The rst two correspond
tothepreviousforcesobtained,respectively,bythecalculationandtheexperiment.
ThePmagnitudeindicates thePeltonturbineefciency.Thenondimensional
forceFzv*=1Fzv/
FzmaxisobtainedbykeepingonlythecontributionFzvoftheviscouseffectstothetotal
loss. Assuming that the boundary layer is little disturbed bythe
D*variationandthat thewettedsurfacevariationremainsweak,
FzvisconstantandnotvaryingwithD*. KeepingFzmaxproportional
toD2relation3,
oneobtainsarelativeviscouslossproportionaltoD*2andthusFzv*=1KD*2.Ontheassump-tionthat,
at point P0D*=5.07, thelossesareonlyofviscousorigin, wehaveFzv*P0
=Fzexp*P0. Thisrelationdeterminesthecoefcient K. It is noteworthy
that the second experimental pointcorresponding to D*=3.84
isexactly on the curve Fzv*. This pointseparatestheinertialzonefrom
theviscouszone. Withregardtothe inertial zone, the experimental
force Fzexp*decreases morequicklythanFzv*, indicatingthat
theinertial lossesbecomenon-negligible. They consist of losses due
to the cutout leakage ow aswell as losses due to the streamlines
deviation at the bucket edgeexit.The efciency curve of the Pelton
turbine was translated so thatthe experimental point P0 belongs to
it. It is similar to the curve ofthe experimental force Fzexp*, in
particular, with a maximum in thesame area. This similarity occurs
owing to the fact that, in the caseof a moving bucket, the maximum
of thrust is obtained when thejet is approximatelyperpendicular
tothe bucket. Finally, it isnoted that the curve of the calculated
forceFznum*presents also amaximumat the same point but with weaker
gradients.
Thatcomesowingtothefactthatthelossesestimationisnotpreciseenough.
In practice, it is difcult to reduce the viscous losses. Infact,
the maximum of thrust is limited by the curve Fzv*.Consequently, if
one wants to reduce the losses by a change ofdesign of the xed
bucket, the zone of action to be considered canbe onlyinthe
inertial zone. Inthe case of the nominal pointD*=3.44, the
maximumgain of thrust is 1%. This gainquickly increases in the case
of larger jets: for example, it is 3%when D*=3.1.In consideration
of the previous analysis, the results relative tothe xed bucket
provide a rst approximation of what one couldgain on the efciency
of a Pelton turbine.4 ConclusionTheexperimental andnumerical
studiesof theowinsideaPelton turbine bucket under xed conguration
were carried out.Three heads, four jet diameters, and seven bucket
incidences werestudied in order to cover the range of the operating
parameters ofa rotating bucket.The main results of the experimental
study are as follows:
Thevarioustestedheadsleadtothesamepressuredis-tribution on the
bucket. Moreover, not any particular in-uence of the head on the
jet trajectory inside the bucketwas noted. In all the cases of
varying incidence and diameter, a
leak-ageowthroughthecutoutisfound.Thisowrapidlyincreases with the
jet diameter and the bucket incidence.
Thepressureforceoriginislocatednear
thereferenceplaneexceptforhighincidencesforwhichitmovesto-ward the
cutout.The numerical modeling quality is demonstrated by the low
rela-tive difference between the calculated pressures and the
measuredpressures.Itisconrmedbytheresultsregardingthetotalofallforces.
The only difference relates to the ow rate loss through thecutout.
Thenumerical
processunderestimatesthisleakageowrate.Theanalysisofthelossesofforceduetotheedgeandthecutout
reveals the following points:
Thelossesduetoedgeslightlyvarywiththeincidenceand decrease with the
jet diameter. The losses due to cutout are lower than those due to
edgeexcept at extreme incidences, at which they
becomedominating.The variation in losses according to the jet
diameter highlights aninertial zonelargediameters
andaviscouszonesmall diam-Fig. 19 Edge and cut-out lossesFig. 20
Non-dimensional forces and Pelton turbine efciency510 / Vol. 128,
JULY 2006 Transactions of the ASMEeters. Intheinertialzone,
theanalysisshowsthatonecangainfrom 1% to 3% on the bucket
thrust.The analogy between the Pelton turbine efciency and the
forceon the xed bucket shows that one can carry out part of the
opti-mizationoftherotatingbucket
usinganalysisperformedonthexedbucket edgeandcutout.
Therotatingbucketoptimizationrequiresfurther
studyrelatingtothefollowingphenomena: un-steadyfeeding, centrifugal
andCoriolis forces, backsplashing,and interference of the sheets of
water.NomenclatureCppressure coefcientDjet diameter mDpPelton
diameter mFzPelton driving force NFzmaxPelton driving force for an
ideal bucket Ng Gravitational acceleration m/ s2Hddynamical head
mHnnet head mHsstatic head mLabucket width mLS1distance between the
measurement axis ofMand the reference plane mL1distance between the
measurement axis ofMand the Pelton wheel axis mMwmoment of force Fz
relative to the Peltonwheel axis NmMmoment of force Fz relative to
the measure-ment axis NmMwmaxmomentMw for an ideal bucket Nmp
pressure Papatmatmospheric pressure PaQjet cross section ow rate
m3/ sS cross section of the narrowed jet m2Ujet velocity m/sOx
x-axis perpendicular to the plane of symme-try mOy y-axis
perpendicular to the reference planemOz z-axis perpendicular to the
edge plane ms curvilinear abscissa in the reference
planemypnear-wall distance mincidence angle degdorice diameter
mwater molecular viscosity Kg/ m s water density Kg/ m3ushear
velocity m/swater volume fractionD*=La/ Dnondimensional magnitude
of the jetdiameterFz*=Fz/ Fzmaxnondimensional driving forceM*=M/
Mwmaxnondimensional moment of forcex*=x/ Lanondimensional
abscissay*=y/ Lanondimensional ordinatez*=z/ Lanondimensional
zvalues*nondimensional curvilinear abscissaY+=ypu/ dimensionless
near-wall distanceRe=UD/ Reynolds numberFr=U/gLaFroude
numberReferences1 Parkison, E., Garcin, H., Bissel, C., Muggli, F.,
andBraune, A., 2002, De-scriptionof PeltonFlowPatterns
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