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 WithComputational FlowSimulations,Symposium Hydropower, Turkey, Nov 47, 2002.2 Nakanishi, Y., Kubota, T., and Shin, T., 2002, Numerical Simulation of FlowsonPeltonBuckets byParticleMethod: FlowonaStationary/RotatingFlatPlate, Proceedings of 21th IAHR Symposium, Lausanne, Sept. 912.3 Agarwal,A.K.,Harwani,L.K.,andRamanthan,S.M.,2004,CustomDe-sign of Pelton Turbine Runners by Numerical Analysis, Proceedings of 22thIAHR Symposium, Stockholm, June 29July 2.4 Kubota, T., Jinjong, X., Masuda, J., andNakanishi, Y, 1998, NumericalAnalysisofFreeWaterSheetFlowofPeltonBuckets,Proceedingsof19thIAHR Symposium, Singapore, pp. 316329.5 Liu, J., Han, F., Kubota, T., and Masuda, J., 2002, Effect of Free Jet Enlarge-ment on the Bucket Flow in Pelton Turbine, Proceedings of 21th IAHR Sym-posium, Lausanne, Sept. 912.6 Hana, M., 1999, Numerical Analysis of Non-Stationary Free Surface Flow ina Pelton Bucket, Ph.D. thesis, Norwegian University of Science and Technol-ogy.7 Janetzky,B.,Gde,E.,Ruprecht,A.,Keck,H.,andSchrer,C.,1998,Nu-merical Simulation of the Flow in a Pelton Bucket, Proceedings of 19th IAHRSymposium, Singapore, pp. 276283.8 Avellan, F., Dupont, Ph., Kvicinsky, S, Chapuis, L., Parkinson, E., andVul-lioud,G.,1998,FlowCalculationsinPeltonturbinesPart2:FreeSurfaceFlows, Proceedings of 19th IAHR Symposium, Singapore, pp. 294305.9 Kvicinsky, S., Kueny, J. L., andAvellan, F., 2002, Numerical andExperi-mentalAnalysisofFreeSurfaceFlowina3DNonrotatingPeltonBucket,Proceedings of 9thInternational SymposiumonTransport PhenomenaandDynamics of Rotating Machinery, Honolulu, Feb. 1014.10 Traversaz, M., Leroy, P., Zopp, B., and Maitre, T., 2002, Numerical Study ofPelton Bucket Flow: Comparison ofFLUENT andCFX Results, Proceedings of21th IAHR Symposium, Lausanne, Sept. 912.11 Kvicinsky,S.,2002,MthodedAnalysedescoulements3DSurfaceLi-bre: Applicationaux TurbinesPelton,ThseNo.2526,EcolePolytechniqueFdrale de Lausanne.12 Mack, R., andMoser, W., 2002, Numerical InvestigationoftheFlowinaPelton Turbine, Proceedings of 21th IAHR Symposium, Lausanne, Sept 912.13 Mack, R., Aschenbrenner, T., Rohne, W., and Farhat, M., 2004, Validation ofBucket FlowSimulationUsingDynamicPressureMeasurements,Proceed-ings of 22th IAHR Symposium, Stockholm, June 29July 2.14 Zopp, B., 2004, Simulation Numrique et Analyse de lcoulement dans lesAugets des Turbines Pelton, Thse, Institut National Polytechnique deGrenoble, France.15 Kubota, T., 1989, Observationof Jet Interferencein6-NozzlePeltonTur-bine, J. Hydraul. Res., 276, pp. 753767.16 Perrig, A., Farhat, M., Avellan, F., Parkison, E., Garcin, H., Bissel, C., Valle,M., and Favre, J., 2004, Numerical FlowAnalysis in a Pelton TurbineBucket, Proceedings of 22th IAHR Symposium, Stockholm, June 29July 2.17 Ishii, M., 1975, Thermo Fluid DynamicTheory of Two-Phase Flow, Collec-tion Direction Etudes Recherches Electrcit de France, Eyrolles.18 Hirt, C. W., andNichols, B. D., 1981, Volume of FluidMethodfor theDynamics of Free Boundaries, J. Comput. Phys., 391, pp. 201225.19 Youngs, D. L., 1982, Time Dependent Multi-Material Flow With Large FluidDistorsion,Numerical MethodsforFluidDynamics,AcademicPress, NewYork.20 Cousteix, J., 1986, Three-Dimensional and Unsteady Boundary Layer Com-putation, Annu. Rev. Fluid Mech., 18, pp. 173196.Journal of Turbomachinery JULY 2006, Vol. 128 / 511