Computational analysis of shrouded wind turbine configurations using a 3-dimensional RANS solver

RenewableEnergy75(2015)818e832

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Computationalanalysisofshroudedwindturbinecon gurationsusinga3-dimensionalRANSsolver

AniketC.Aranakea,*,kshminarayana,KarthikDuraisamyb

ab

DepartmentofAeronauticsandAstronautics,StanfordUniversity,Stanford,CA94035,USADepartmentofAerospaceEngineering,UniversityofMichigan,AnnArbor,MI,USA

articleinfo

Articlehistory:

Received7August2013Accepted18October2014

Availableonline15November2014Keywords:Windenergy

ShroudedwindturbinesDAWTCFD

Transitionmodel

abstract

Theuseofashroudisknowntoimproveperformance.Inthiswork,the owphysicsandperformanceofshroudedturbinesisassessedbysolvingtheReynoldsAveragedNaviereStokesequationssupplementedwithatransitionmodel.Shroudgeometriesareevaluatedfortheiraugmentationofmass owthroughtheturbine.Initialassessmentsareperformedusingaxisymmetriccalculationsofannularwingswithhigh-liftairfoilsascrosssections.Themass owampli cationfactorisde nedasaperformanceparameterandisfoundtoincreasenearlylinearlywithradialliftforce.Fromaselectionofconsideredairfoils,theSeligS1223high-liftairfoilisfoundtobestpromotemass owrate.Fullthree-dimensionalsimulationsofshroudedwindturbinesareperformedforselectedshroudgeometries.Theresultsarecomparedtoopenturbinesolutions.Augmentationratiosofupto1.9areachieved.Peakaugmentationoccursatthehighestwindspeedforwhichthe owoverthebladestaysattached.Flow eldsareexaminedindetailandthefollowingaspectsareinvestigated:regionswith owseparation,thedevelopmentofvelocitypro les,andtheinteractionbetweentheturbinewakeandshroudboundarylayer.Thesensitivityofthesolutionstorotationrateisexamined.

©2014ElsevierLtd.Allrightsreserved.

1.Introduction

Theprimaryobstaclepresentlyconfrontingthewidespreadadoptionofwindenergyiscost.Improvementsinwindturbineef ciencywouldlowerthecostofwindpowerandtherebyadvanceitssuitabilityforinvestmentsintheenergymarket.Atpresent,themostestablishedmachineforharvestingwindenergyisthehori-zontalaxiswindturbine(HAWT).Asstatedbythewell-knownBetzlimit,theef ciencyofconventionalHAWTsisboundedaboveby59.3%.Thislimitisbasedonanassumptionthatthefreestreamwindisnotdisturbedbyanyexternalforcepriortoitsinteractionwiththeturbine.Ithaslongbeenestablished,however,thatalteringthestreamtubeincidentontheturbinerotorviasomeexternalmechanismcanproduceef cienciesinexcessoftheBetzlimit.Ashroudedwindturbine,oftenreferredtoasadiffuseraugmentedwindturbine(DAWT)inpreviousliterature,makesuseofsuchamechanism.Inashroudedturbine,theturbineisencasedinashroudwhichacceleratestheincomingwind,signi cantlyincreasingthemassandpoweravailabletotheturbine.AccordingtoageneralizedversionoftheclassicBetzlimitproposedby

*Correspondingauthor.

E-mailaddress:aniketaranake@(A.C.

Aranake)./10.1016/j.renene.2014.10.049

0960-1481/©2014ElsevierLtd.Allrightsreserved.

Jamieson[1],thetheoreticallimitforpowerextractionforawindturbinewithaugmentationcanbeashighas88.89%oftheenergyavailableinthestreamtubeincidentontherotor.(Notethatthisisnotalimitofef ciencybythestandardde nition,astheupstreamareaofthestreamtubeinquestionwillingeneraldifferfromtheareaoftherotor.)

InadditiontothecapabilityofexceedingtheBetzlimit,shrou-dedturbinesalsoofferotherpotentialadvantages.Everywindturbineislimitedbyitsso-called‘cut-inspeed’,awindspeedbelowwhichthebladesdonotturn.Highcut-inspeedssigni cantlyhindertheabilityofawindoperationtocon dentlygeneratepo-werattimesofhighdemand,aproblemexacerbatedbytheinherentintermittencyofwind.Recentprogressinmaterialstechnologyhasenabledtheconstructionofmassivelylargeblades,whichalleviatesthechallengesduetocut-inspeedsfornewlargewindfarms.However,thissolutionisnotidealforallsituations.Forinstance,inurbanorremotesettings,asmallerandmoreportablesolutionisdesirable.Insuchcases,byacceleratingthewindbeforeitreachestheturbine,ashroudedturbineachievesalowercut-inspeedthanaconventionalopenrotorandcanoperateproduc-tivelyforalongerportionofitslifetime.

Fewresearchersinthepasthaveexaminedthebene tsandeconomicsofplacingadiffuseraroundawindturbine.Asurveyincludingacomprehensivehistoryofshroudedturbineshasbeen

A.C.Aranakeetal./RenewableEnergy75(2015)818e832819

recentlypublishedbyvanBussel[2].Theideaofashroudedturbinewas rstproposedbyLilleyandRainbird[3].Experimentalin-vestigationswereperformedinthe1980sbyGilbertandForeman[4]andbyIgra[5].AlthoughtheseexperimentsdiddemonstratethatpowerextractionbeyondtheclassicBetzlimitispossible,theshroudedturbinewasdeterminednottobepro tablecomparedtotraditionalwindturbinesandwasnotpursuedextensively.

Recentlyhowever,anincreaseinnumberofpublicationsonthetopicandattemptstocommercializetheideaindicatearenewedinterestinshroudedturbines.Researchershavecometoanagreementthatthereissigni cantpotentialforimprovementsinthisconceptandunderstandingthedetailsof owphysicsisoneofkeystoachievingthis.AninvestigationbyHansen[6]demonstratesusingbothlow- delitymomentumtheoryandCFDthatthepoweraugmentationofashroudedturbineisproportionaltotheincreaseinmass owratethroughtheturbineblades.Throughoutthepastdecade,theresearchgroupofOhya[7e10]haveperformedextensiveexperimentalandcomputationalworkonthistopicwhichhasledtothedevelopmentofahighperformanceso-called“ angeddiffuser”.

Duetothehighcostassociatedwithexperiments,computa-tionalstudiesareimportantforanin-depthunderstandingofthe owphysicsofshroudedturbines.However,totheauthorsknowledge,allthepreviousworkemploymodelingsimpli cationsthataffecttheaccuracyof ow eldpredictions.Earlywork[5,4]attemptedtomodelthebehaviorofashroudedwindturbineasaninternal owproblem,inwhichaturbinewasplacedinsideaductwithanimposedpressureconditionattheductoutlet.Morerecently,however,ithasbeendetermined[11,1]thatthisassump-tionisinaccurate,andamoreappropriateboundaryconditionisonewhichisimposedfardownstreamoftheentiresystem,i.e.,theshroudedturbinemustbetreatedasanexternal owproblem.

Insomeoftherecentcomputationalstudiesofshroudedtur-bines[6,7],anactuatordiskassumptionisusedtomodelthetur-bineblades.Insuchamodel,therotorisreplacedbyanin nitelythindiskacrosswhichapressuredropisexplicitlyimposed.Thissimpli cationoffersasubstantialcostsavingoverafullthree-dimensionalsolution,makingitausefultoolduringthepre-liminarydesignstage.Itdoesnot,however,capture nerandpotentionallyprofoundaspectsofthe owphysicsnearthebladeandinthewakeofashroudedturbine.Inaddition,theactuatordiskmodelcannotpredicttheeffectoftheturbinewakeon owsepa-rationalongthewallsoftheshroud.Thiseffect,the“swirlmixing

mechanism”mentionedinpreviousliterature[4],haspotentiallysigni cantconsequencesontheperformanceofashroudedturbine.

AfullyReynolds-Averaged-NaviereStokes(RANS)-basedCFDsimulationcanovercometheabovementionedmodelingissues.ThecurrentworkusesacompressibleRANSsolvertosimulateshroudedwindturbines.ThesolverhasbeenvalidatedpreviouslybyBaeder[12],Duraisamy[13,14]andLakshminarayan[15,16]foravarietyof owsincludingsingle,coaxial,andshroudedrotors,primarilyinthecontextofhelicopters.Whilethislegacyadmitsconsiderablecon denceintheaccuracyofthesolverforrotating ows,thesimulationofwindturbinespresentsanadditionalcomplication.Thisisduetofreetransitionfromlaminartoturbu-lent ow,whichoccursonwindturbinebladesduringtypicaloperatingconditionsandisfoundtosigni cantlyimpactbothpressureandviscousforces.Recently,authorsofthepresentworkhaveperformedadetailedvalidationof3Dsimulationsofanopenwindturbine[17]usingthegÀqtÀSAtransitionmodel[18],whichisdemonstratedtoaccuratelypredicttransitional owsintwoandthreedimensions.Apreliminaryinvestigationofshroudedturbineshasalsobeenperformedbypresentauthors[19].Theprimaryobjectiveofthepresentworkistoemploythispreviouslyvalidatedmethodologytoexaminetheperformanceand ow eldofshroudedwindturbines.

Theremainderofthispaperisorganizedasfollows.Section2outlinesthemethodologyemployedbythe owsolverOVER-TURNSwhichisusedthroughoutthiswork.InSection3,asum-maryofthevalidationispresented.InSection4,resultsarepresentedforabaselineshroudgeneratedfromaNACA0006airfoil.InSection5,axisymmetriccomputationsareperformedtoinves-tigateanumberofdifferentshroudgeometries.Basedonthisanalysis,thehigh-liftS1223airfoilshapeisselectedfordetailedinvestigation.InSection6,ashroudgeneratedfromaS1223airfoilsectionisexaminedindetail.Thebehavioroftheshroudedturbinewithvaryingtipspeedratioisinvestigated.Section7summarizesthekeyconclusionsandproposesfuturework.2.Methodology

Inthiswork,computationsareperformedusingtheoversetstructuredmeshsolverOVERTURNS[15].ThiscodesolvesthecompressibleRANSequationsusingapreconditioneddual-timeschemeinthediagonalizedapproximatefactorizationframework,describedbyBuelowetal.[20]andPandyaetal.[21].Inthisframework,timeisadvancedusingtheimplicitapproximatefactorizationmethoddevelopedbyPulliamandChaussee[22].LowMachpreconditioning,basedonthatdevelopedbyTurkel[23],isusedtoimprovebothconvergencepropertiesandtheaccuracyofthespatialdiscretization.InviscidtermsarecomputedusingathirdorderMUSCLschemeusingKoren'slimiterwithRoe's uxdiffer-encesplitting,andtheviscoustermsarecomputedusingsecondordercentraldifferencing.ForRANSclosure,theSpalarteAllmaras[24]turbulencemodelisemployed.Tocapturetheeffectsof owtransition,thegÀqtÀSA[18]modelisused.Thegoverningequationsaresolvedintherotatingframe,withno-slipconditionsaccountingforwallvelocitiesappliedonthesurfacesofboththeshroudandturbineblade.Animplicithole-cuttingtechniquedevelopedbyLee[25]andimprovedbyLakshminarayan[15]isusedtodeterminetheconnectivityinformationbetweenvariousoverlappingmeshes.

3.Assessmentoftransitionmodel

Inthissection,theperformanceofanewlyimplementedtran-sitionmodeliscarefullyevaluated.First,aperformance

prediction

820A.C.Aranakeetal./RenewableEnergy75(2015)818e832

validationisperformedonatwo-dimensional owovertheS809airfoil.Subsequently,fullthree-dimensionalcalculationsofthe owpasttheNRELPhaseVIturbineareperformed,andtheresultsarecomparedtotheexperimentalmeasurementsincludingper-formanceandsurfacepressuredata.3.1.S809airfoil

TheS809airfoil,employedbytheNRELPhaseVIwindturbine[26],isoneofafamilyofairfoilsdeliberatelydesignedtocontainaso-called“transition-ramp,”ashortregionofgentlepressurere-coveryalongtheuppersurfacewhichinducessmoothtransitionfromlaminartoturbulent ow[27].ComputationsareperformedusingstructuredC-mesheswithdimensions527Â101inthewrap-aroundandnormaldirectionsrespectively.TheairfoilandmeshareshowninFig.1.Tomatchexperimentalconditions,simulationsarerunataReynoldsnumberof2Â106,andalowMachnumberof0.1isselectedtoavoideffectsduetocompressibility.

Thepredictedliftanddragcoef cientsforasweepofanglesofattackareshowninFig.2.Resultsareplottedbothwiththetran-sitionmodelandwithout,inwhichcasethe owwasassumedtobefullyturbulent.ThegÀqtÀSAmodelsigni cantlyimprovesthepredictionofbothquantities,particularlyinthepost-stallre-gion.Theuseofthetransitionmodelalsoimprovesthepredictionofthepressurecoef cientsigni cantlyasshowninFig.3.Ata¼4 ,showninFig.3(a),theprominentdipinpressureontheupper

surfacewhichindicatestransitioniswell-capturedbythetransitionmodel.Furthermore,ata¼12 ,showninFig.3(b),whentheairfoilispartiallystalled,thepredictionoftheentirepressuredistributionissubstantiallyimprovedbytheinclusionofthetransitionmodel.ThelocationofthetransitionpointateachangleofattackisplottedagainstexperimentaldatainFig.4.Transitionistriggeredbylaminarseparationbubbles,whichspanseveralgridpointsintheCFDsolutions.Errorbarsareusedtorepresentthelocationandwidthofthesebubbles.Thetransitionlocationsarecapturedwellexceptintherangebetweena¼6 anda¼8 ontheuppersurfaceoftheairfoil.Thecalculationspredictthattheseparationbubbleontheuppersurfacejumpstotheleadingedgeoftheairfoilata¼9 .Inreality,thetransitionlocationishighlysensitivetosmalldis-turbancesinthemiddlerangeofangleofattack,andeventheexperimentaldataisnotsingle-valuedinthisregion.Nonetheless,thetransitionmodelsigni cantlyimprovesthepredictionofquantitiesfromtwo-dimensionalS809 ow eldsoverall.3.2.NRELphaseVIwindturbine

Forthree-dimensionalvalidation,theNRELPhaseVIservesasavaluabletestcase,ascomprehensiveperformanceandsurfacepressuredataisavailableforthiscon gurationfromtheUnsteadyAerodynamicsExperiment(UAE)[26].Thisexperimentwascon-ductedatalargescalewindtunnel(24.4mÂ36.6m)locatedatNASAAmes.Thetunnelmaintainsafreestream

turbulence

putationalmesh(527Â101)fortheS809windturbine

airfoil.

Fig.2.Liftanddragcoef cientsvsangleofattackforS809airfoil.

A.C.Aranakeetal./RenewableEnergy75(2015)818e832821

Fig.3.Pressurecoef cientdistributionforS809

airfoil.

intensityofbelow0.5%.Theturbineisa2-bladedcon gurationwitharadiusofR¼5.029m.ThetipMachnumberisMtip¼0.1135andtherotationalvelocityisu¼72rpm.Aspreviouslymentioned,theS809airfoilisusedforthebladecrosssections.MeasurementsofperformanceandsurfacepressurearetakenforwindspeedsrangingfromU∞¼3m/sto25m/s.

Inthiswork,theturbinebladeisrepresentedbyaCeOmeshwithdimensions257Â51Â51inthewrap-around,spanwise,andnormaldirectionsrespectively.AviewofthemeshnearthebladetipispresentedinFig.5(a).Thismeshisembeddedinabackgroundmeshwithdimensions201Â133Â164intheazimuthal,radial,andaxialdirectionsrespectively.Thebackgroundmeshissemi-cylindrical,andperiodicboundaryconditionsareappliedonlateralfacestosimulateafulltwo-bladedrotor.The uiddomainextends2Rupstreamand6.8Rdownstreamand3.4Rlaterally.Theazimuthalgridspacingstretchestogive neresolutionina15 patchinthevicinityoftheturbineblade.Characteristicboundaryconditionsareusedforthein ow,out ow,andfar eldfaces.Awallboundaryconditionimposesno-sliponthebladeintherotatingframe.Thesingularityattheaxisisavoidedbybeginningtheradialpointdistributionatasmallvalueofr¼0.03.Anextrapolationboundaryconditionisappliedatthislocation.ThebackgroundmeshandboundaryconditionsaredepictedinFig.5(b).Toevaluate

gridconvergence,a nermeshwithdimensions257Â201Â101forthebladeand180Â264Â228forthebackgroundmesh(withuniformazimuthalspacing)wererunforonewindspeed,andtheintegratedloadsandpressurecoef cientswereallfoundtodifferbylessthan3%.Sincethemeshsizeisalreadyveryhigh,wedidnotperformmoreextensivere nementstudies.However,allofthesemeshesweregeneratedusingpriorexperienceindealingwithanumberofhighlysimilarproblemswiththesamecodeoverthepast15years[12e16].

Fig.6(a)plotsthecomputedtorqueoftheopenturbineatvariouswindspeeds.Thesecomputationswereperformedby rstattemptingasteady-statesolution,whichingeneraldidnotconverge,andthenusingtheobtainedresulttorestartatime-accuratesolution.Errorbarsshowtheminandmaxvaluesattained.Forcomparison,computationalresultsfromPotsdametal.[28]arealsoincluded.Atlowspeedswherethe owremainsattached,computationalresultsagreequitewellwithmeasureddata.Atthesespeeds,transitionplaysasmallroleinthebehaviorofthe ow,andthetransitionmodelhaslittleeffectontheintegratedresults.AvisualizationofthewakefromonesuchresultisshowninFig.6(b),whichplotsisocontoursofq-criterion[29].Thehelicalwakevortexpersistsforseveralturnswithminimalnumericaldiffusion.Asthefreestreamvelocityincreases,transitionplaysamoresigni cantroleinthe ow.PressuredistributionsattwospanwisestationsareshowninFig.6(c)and(d)forawindspeedofU∞¼10m/s.Thetransitionmodelisseentocapturethepressuredistributionmoreaccuratelythanthefullyturbulentassumptionattheselocations.AtspeedsbeyondU∞¼10m/s,the owisdeeplystalledovermuchoftheturbineblade,leadingtoaninherentlyunsteady owforwhichRANSbasedcalculationsareinsuf cient.Indeed,undersuchconditions,interactionbetweenthebladesandtowerplaysasigni cantroleinthebladeloading,andthetowersurfacemustbeincludedincalculationstoproducereasonableresults[30].

4.Baseline(NACA0006)shroud

Aninitialassessmentoftheshroudedturbineconceptisper-formedbyenclosingtheNRELVIbladeinasimpleshroud.Thisshroud,referredtoas“baseline,”isgeneratedbyextrudingaNACA0006airfoilsectionatanangleofattackintoanannularwing.Thebladegridandbackgroundgridarethesameasthoseusedforvalidation.Thedimensionsfortheshroudgridare267Â201Â49inthewrap-around,spanwise,andnormaldirectionsrespectively.Fig.7(a)showssectionsofthebladeandshroudgrids,andFig.7

(b)

Fig.4.TransitionlocationsforS809airfoil.Thesquaresymbolsinthisplotareexperimentaldatapoints.ErrorbarsshowthesizeoftheseparationbubbleintheCFDsolution.

822A.C.Aranakeetal./RenewableEnergy75(2015)818e832

putationalmeshusedforopenturbine

simulations.

showsboththeshroudandbladeembeddedinthebackgroundgrid.Onceagain,periodicboundaryconditionsareusedonthelateralfacesofthedomain,andcharacteristicfar eldboundaryconditionsareusedontheupstream,downstream,andfar eldfacesofthecylinder.Ano-slipconditionisappliedonthebladeintherotatingframe,andaseparateno-slipconditionenforcingzerovelocityintheinertialframeisappliedontheshroudsurface.Theclearancebetweenthebladetipandshroudinterioris0.056Ror,expressedintheformcommoninturbomachinaryapplications,0.785ctwherectisthechordlengthatthebladetip.

ResultsforthebaselineshroudareshowninFig.8.Inthisplot,errorbarsareusedtorepresentthe uctuations(minandmax)duetounsteadyeffects.Fig.8(a)plotsthethrustcoef cientforboththeshroudedandopenturbines.Thecoef cientisnormalizedintwoways:accordingtotheturbineradius(solidline)andaccordingtothemaximumshroudradius(dashedline).Inthepresenceof

the

Fig.6.ValidationofcomputationalmethodologyincludingtransitionmodelonNRELPhaseVIturbine.

A.C.Aranakeetal./RenewableEnergy75(2015)818e832823

Fig.7.Oversetgridsusedfor3Dshroudedturbine

computation.

shroud,theloadsontheturbinearehigheratallwindspeeds.Thisistobeexpectedduetotheaccelerationinducedbytheshroudandshouldbeconsideredwhenselectingmaterialsforshroudedtur-bineblades.Inplaceofpowercoef cient,acommonlyused[4,5]measureofDAWTperformanceistheaugmentationratio,givenby

ra¼

Cp(1)

dimensionalbladeandshroud.Nonetheless,anaxisymmetricinvestigationprovidesconsiderableinsightintothepropertiesofashroudairfoil.

Onemeasureoftheeffectivenessofanairfoilasashroudpro leistheamountofmass owitinducesthroughtheinterioroftheshroud.Foranaxisymmetricshroud,anincreasedcirculationwouldcorrespondtoalargemass owrate.Themass owrateisgivenby

Rs2pZZ

whichissimplyaratioofthestandardpowercoef cienttotheBetz

limit.Thebaselineshroudisseentoaugmentpowersigni cantly,withamaximumaugmentationratioofra¼1.09atU∞¼5m/s.Inthiscase,powerextractionbeyondtheBetzlimitisachievedevenwithaverysimpleshroudgeometry.

_¼m

ruzrdrdq

(2)

whereuzistheaxialcomponentofvelocity.Anon-dimensionalmass owampli cationfactorisde nedas

5.2Danalysisofshroudairfoilsections

Inthepresentwork,onlyshroudswhicharesymmetricaboutanaxisofrotationareconsidered.Thispermitstheexpedientassess-mentofshroudsgeneratedbyseveraldifferentairfoilsectionsbysolvingtheaxisymmetricRANSequations.Followingtheworkofpreviousauthors[1,11],theeffectivenessofashroudisconsidereddecoupledfromthepropertiesofaturbine,andtheturbineisnotmodeledinthissection.Thisconclusionisbasedontheassump-tionsofinviscidmomentumtheory,whichwillreachtheirusefullimitswhenconsideringtheinteractionbetweenathree-

Mr¼

_m

:

∞s

(3)

wheretheshroudradiusRsismeasuredattheplaneoftheturbine.Inthepresentwork,forboth2Dand3Dsimulationstheturbinewasplacedatz/c¼10%,wherezistheaxialcoordinatefromtheleadingedgeandcistheshroudchordlength.

TheKuttaeJoukowksitheoremstatesthattheliftperspanisproportionaltothecirculationinducedinthesurrounding ow.Ahighcirculationcorrespondsdirectlytoahighmass

ow

Fig.8.TurbinethrustandaugmentationratiowithNACA0006shroud(u xed).Dashedlinesarenormalizedwithrespecttomaximumshroudradius.

824A.C.Aranakeetal./RenewableEnergy75(2015)818e832

Fig.9.High-liftsectionsshapesandperformance,blacklinesare

NACA0006.

ampli cation.Consideringthis,severalhigh-liftairfoilsareconsidered.Asampleofshapesconsideredareeachshownalong-sideaNACA0006pro leinFig.9(a).TheEpplerE423[31]isanairfoildesignedtomaximizeliftthroughtheuseofaconcavepressurerecoverywithnospeci cregardtotheeffectonmomentcoef cient.Themodi edNACA0006wasgeneratedbyrotatingthemeancamberlinedownwardby45 forthelast20%ofthechordlengthwhilemaintainingtheoriginalthicknessdistribution.Thisshapeisintendedtoresembleanairfoilwithade ectedcontrolsurface,anditalsoapproximatesthe angeddiffusershapeofAbeandOhya[7,10].TheSeligS1223isdesignedwiththesamehigh-liftdesignphilosophyastheEpplerE423[31],andisintendedtobelongtotheFXCL/MS-classofairfoils.TheFX74-CL4-140isahigh-liftairfoildesignedforaReynoldsnumberofRe¼106.

Two-dimensionaloversetgridsareonceagainusedforthesecomputations,with180Â218pointsinthebackgroundmeshand267Â61pointsinthenear-bodymesh.AsamplegridofanS1223sectionat10 angleofattackisshowninFig.10.TheReynolds

numberbasedonshroudlengthisRe¼3.2Â106,correspondingtoafreestreamvelocityof10m/sandshroudlengthof5.029m.TheaxisymmetricRANSequationsaresolvedalongwiththeSpa-larteAllmarasturbulencemodel.Thebackgroundmeshisextendedto25chordlengthsupstream,25chordlengthslaterally,and50chordlengthsdownstream.Asmallvalueofr¼0.001misusedattheboundarynearthesymmetryaxis,whereanextrapolationboundaryconditionisapplied.

Thecomputedperformanceoftheseairfoilsintermsofradialforceandmass owisshowninFig.9(b)and(c).AssuggestedbytheKuttaeJakouwskitheorem,themass owampli cationin-creasesnearlylinearlywithradialforcecoef cient.Theangleofattackforthisstudyisvariedfroma¼0 toa¼15 ,whichissuf cienttoobservethestallofeachairfoilconsidered.

TwolinesofthesameslopeareseenonthecurvesinFig.9(b).Thedeterminingfactorforwhichlineaparticulardatapointliesonisthelocationofthestagnationpoint.Forinstance,theliftcoef -cientfortheNACA0006sectionata ¼ 6 isnearlyequaltothat

of

Fig.10.Oversetgridsystemusedforviscousaxisymmetriccomputations.

A.C.Aranakeetal./RenewableEnergy75(2015)818e832825

Fig.11.Stagnationpointvisualizations,contoursof

pressure.

themodi edNACA0006ata¼0 ,yettheNACA0006inducesagreatermass ow.Thiscanbeseenbyexaminingthe ow elds.Fig.11(a)and(b)showcontoursofpressureandstreamlinesneartheleadingedgesofthesetwocases.InFig.11(a),thestagnationpointofthemodi edNACA0006islocatedsuchthatsomeofthe owapproachingtheleadingedgeoftheairfoilisdiverteddown-wardsandaroundtheshroud.Bycontrast,inFig.11(b),thelocationofthestagnationpointoftheNACA0006issuchthatapproaching owispushedupwardsandintotheshroud,increasingthetotalmass ow.Similarly,Fig.11(c)and(d)showtheleadingedgeoftheS1223airfoilata¼4 anda¼15 respectively.Onceagain,theradialliftforceisnearlyequalforthesetwocases.Howeverthereisgreatermass owfora¼15 duetothelocationofthestagnationpoint.

Oftheairfoilsconsidered,theSeligS1223achievesthegreatestmass owampli cation,andforthisreasonitisselectedfor3Danalysisinthenextsection.

6.High-liftshroudgeometry

Followingtheresultsoftheprevioussection,asecondthree-dimensionalshroudwasgeneratedwiththecrosssectionoftheSeligS1223airfoil.Thegriddimensionsandshroudradiuswereselectedtobeidenticaltothoseofthepreviouscases.

ComputationalresultsfortheperformanceareshowninFig.12alongsidetheresultsobtainedfromtheoriginalNACA0006ge-ometry.Thehigh-liftairfoilgreatlyimprovestheperformanceoftheshroudedturbine,yieldingamaximumaugmentationratioofr¼1.91atU∞¼5m/s.Thisisanincreaseinperformanceof75%overthebaselineNACA0006shroudatthewindspeedofpeakpowerextraction.TheturbinestillachievesdeepstallatU∞¼10m/s,andoncethishighspeedisreachedthepoweraugmentationoftheshroudedturbinescollapsetonearlythesamevalue.

TheeffectofbothshroudsonthespanwisedistributionofloadsontheturbinebladeisshowninFig.13.Twowindspeeds

are

Fig.12.Thrustcoef cientandaugmentationratioforNACA006shroudandSeligS1223shroud(u xed).Dashedlinesarenormalizedwithrespecttomaxshroudradius(asopposedtoturbineradius).

826A.C.Aranakeetal./RenewableEnergy75(2015)818e832

Fig.13.Spanwisethrustandtorquedistributiononopenandshrouded

blade.

shown,U∞¼5m/s,wherepeakaugmentationisobtained,andU∞¼7m/s,whichmarkstheonsetofseparationfortheshroudedcases.AtU∞¼5m/s,wheretheturbineismoreef cientinaug-mentingthepoweroutput,theincreaseinloadingissmooth,peakingnearr/R¼0.9.AtU∞¼7m/s,theforcecoef cientswhicharesmoothfortheopenrotorbegintodeterioratefromtheoutboardportionoftheblade.Thisisinpartduetothefactthattheresultisunsteady.

Tofurtherinvestigatetheirregularitiesseenintheoutboardportionoftheblade,Fig.14showscontoursofpressureandstreamlinesforabladecrosssectiontakenatr/R¼0.7forU∞¼7m/s.Whilethe owovertheopenturbineshowninFig.14(a)remainsfullyattached,the owovertheshroudedturbinebladeshowninFig.14(b)exhibits owseparation.ThisexplainstheirregularitiesseeninFig.13.Itcanalsobeseenintheseplotsthatthelocalangleofattackishigherinthepresenceoftheshrouded,indicatingthatthevelocitynormaltothebladesurface,i.e.thewindspeed,hasincreased.

6.1.Flow eld

Inordertobetterunderstandtheaerodynamicsofashroudedturbine,itisvaluabletoanalyzethe ow eldinsomedetail.First,

thecasewithU∞¼5m/sisconsidered,asitcorrespondstothewindspeedofgreatestpoweroutput.

Velocitypro les,from2rotorradiiupstreamto1radiidown-stream,areexaminedinFig.15fortheopenandshroudedrotor.Thesepro leshavebeenaveragedintheazimuthaldirection.Up-streamoftheturbine,atz/R¼2,thevelocitypro lesareconstantatthefreestreamvalueof5m/s.Atz/R¼0.5,ade citisseeninthewakeoftheopenturbinecorrespondingtoanexpansionofincomingstreamlines.Atthissameaxialstation,apeakisseeninthevicinityoftheshroud(nearr/R¼1),andthe owaccelerationduetotheshroudpreventsanupstreamwakede cit.Theincomingvelocityisgreaterthanthefreestreamvelocityacrosstheentirespanoftheturbine.Attheplaneoftheturbine,atz/R¼0.0,thewakeoftheopenrotorhasdeceleratedhalfwaytothedownstreamwake,agreeingwiththeresultfromactuatordisktheory.Theshroud,ontheotherhand,acceleratesthefreestreamwindconsiderably.Downstreamoftheturbinebutaheadoftheshroudtrailingedgeatz/R¼À0.5,thewakeoftheopenturbinehasreducedtoitsdownstreamvalue,whereastheshroudcontainsanddelayswakeexpansion.Fartherdownstream,thewakede citsaremuchlargerfortheshroudedturbineduetobothanincreaseinturbinethrustandmomentumextractionbytheshrouditself.Theshroudedturbinewakealsoexpandsfartherlaterallyinthisdownstreamregion,indicatingagreaterdiffusionof

momentum,

Fig.14.Contoursofp/p∞atradiallocationr/R¼0.7,U∞¼7m/s.

A.C.Aranakeetal./RenewableEnergy75(2015)818e832827

Fig.15.Azimuthallyaveragedvelocitydistribution,U∞¼5

m/s.

whichwouldreducethewakeeffectfeltbyadownstreamturbineinawindfarm.TheNRELPhaseVIbladeunderconsiderationinthepresentworkisdesignedtooperateinanearlyuniformvelocitydistributionlikethatseeninFig.15(a).Thelargedeviationfroma atpro leseeninFig.15(b)indicatesthataredesignofthebladeisneededforshroudedturbines.Sucharedesignisdeferredtofuturework.ContoursofthenormalizedpressureandskinfrictionalongtheturbinebladesareshowninFig.16forboththeopenandshroudedcases.Thedifferencesbetweenthecasesaremostpronouncedontheleewardsurface.Intheplotsontheleft,lowerpressuresareseentowardstheouterregionofthebladeintheshroudedcase,wheretheeffectoftheshroudsuctionpeakisfeltmoststrongly.Streamlinesindicatingthe owdirectionimmediatelyadjacent

to

Fig.16.Surfacepressureandskinfrictioncontours,U∞¼5m/s.

828A.C.Aranakeetal./RenewableEnergy75(2015)818e832

Fig.17.ContoursofvorticitymagnitudeforS1223

shroud.

minarseparationbubbles,predictedbythetransitionmodel,areseenasbluestripesacrosstheblades.Attheselocations,theskinfrictionisnegativeandlocallythe owdirectionisreversed.Substantialcross owandsomeleadingedgeseparationarepresentintheshroudedcase.

Interestingfeaturesareseenintheinteractionbetweentheturbinewakeandtheshroud.Fig.17(a)showscontoursofvorticitymagnitudetakenatanazimuthalcrosssectioninthevicinityofthebladeforU∞¼5m/s.Downstreamoftheturbine,thepassinghe-licalturbinewakeappearsincloseproximitytotheshroud,revealingtheexpansionofthehelixwithintheshroud.Thevorticityshedfromtheshroudboundarylayeroscillateswiththepassingwake.The uidintheboundarylayerdeceleratestowardpressurerecoverybutisintermittentlyreenergizedbythepassingturbinevortex.Thisphenomenonpromotesattached ow,andisquitebene cialinpreventingstallontheshroudinnersurface.Fig.17(b)showsasimilarplotforU∞¼7m/s.Inthiscase,thereissubstantial owseparationalongtheturbineblade,andalargeandirregularwakeisshedintotheshroud.

Thepressurecoef cientalongtheshroudisplottedinFig.18forwindspeedsofU∞¼5m/sandU∞¼7m/s.Pressurecoef cientdistributionsatvariousazimuthallocationsaredisplayedalongsidetheazimuthallyaverageddistribution.Thesolutionofa2Daxisymmetricsimulationisshownaswellforcomparison.AsdepictedinFig.18(c),thelocationwheretheazimuthq¼0 cor-respondstothelocationoftheturbineblade.Inbothcases,alargepeakinsuctionisseenatthislocation,asthesuctionpeakoftheshroudinteractswiththetipvortexshedbytheturbine.Theeffectofthepassinghelicalwakeisreadilyvisibleasoscillationsinthese

plots.

Fig.18.Pressurecoef cientalongshroud.Meanvalueshownalongsideseveralstations.Axisymmetricsolutionshownforcomparison.

A.C.Aranakeetal./RenewableEnergy75(2015)818e832829

6.2.Varyingtipspeedratioata xedwindspeed

Inpractice,itistypicallydesirabletoselectarotorrotationrate

utomatchtherateatwhichthegearboxandpowergenerator

performoptimally.Nonetheless,rotationrateplaysacrucialroleintheaerodynamicsofawindturbinesystem.InFig.8(b),thepowerproducedbytheshroudedturbinewhilemaintaininga xedudecreaseswithincreasingwindspeedbeginningatU∞¼7m/s.Atandbeyondthisspeed,thereis owseparationonthebladesurfaceowingtothehigherangleofattackexperiencedbyeachbladesection.Thegeometricangleofattackisgivenby

Performanceresultsfromcomputationsrunata xedspeedofU∞¼7m/satvarioustipspeedratiosareshowninFig.19.Thelowestvalueofl¼5.42correspondstotheoriginalconditionsoftheNRELPhaseVI,andsuccessiverunsincreasethisvalueby25%each.Fig.19(b)revealsthatthereisanoptimumvalueoflforpowerproduction.

Anexaminationofthe ow eldsrevealsthatthreephysicalphenomenonplayaroleinthepoweroutputofthissystem: Thepresenceandextentof owseparationontheturbineblade Pro ledragoftheblade

TheperformanceoftheshroudinthewakeoftheturbineThe rstoftheseitemscausesperformancetoimproveaslincreaseswhilethelattertwocauseperformancetodecreasewithincreasingl.

Considering rsttheextentof owseparation,Fig.20showsskinfrictionatthebladesurfacealongwithsurfacestreamlines.Atthebaselinetipspeedratio,showninFig.20(a),the owpasttheleewardsurfaceishighlyseparatedandmuchofitisreversed.Byincreasinglby25%,showninFig.20(b),theextentofseparation

is

aðrÞ¼fðrÞþtan

À1

1Rr

(4)

whereF(r)isthetwistoftheblade.Inspectionofthisrelationcon rmsthatthelocalangleofattackcanbereducedallalongthebladebyincreasingl.Inotherwords,itispossibletoreducetheextentofseparationonabladeandreduceitaltogetherbyincreasingtherotationrate.

Fig.19.Thrustcoef cientandpoweraugmentationforvaryingtipspeedratiosata xedfreestreamvelocityofU∞¼7

m/s.

Fig.20.Contoursofskinfrictionforvaryingtipspeedratios,S1223shroud.

830A.C.Aranakeetal./RenewableEnergy75(2015)818e832

Fig.21.Contoursofvorticitymagnitudeforvaryingtipspeedratios,S1223

shroud.

Fig.22.Pressurecoef cientalongshroud.Meanvalueshownalongsideseveralstations.Axisymmetricsolutionshownforcomparison.

A.C.Aranakeetal./RenewableEnergy75(2015)818e832831

greatlyreducedalthoughtherearestilllargeseparationregionsontheleewardsurfaceneartheleadingandtrailingedges.Byincreasinglfurther,inFig.20(c)and(d),separationiseliminatedexceptforthethinlaminarseparationbubblescuttingacrossthespanoftheblades.

Next,thepro ledragonashroudedturbineincreaseswithincreasingtipspeedratio.Neglectingthree-dimensionaleffects,thedragonasectionofaturbinebladeisgivenby

" 2

#d¼C1drðurÞ22þU∞¼C1Ulrdr2

∞þ1:

(5)

Thus,pro ledragisproportionaltol2andcanbeexpectedto

growsigni cantlyaslincreases.

Finally,themass owampli cationoftheshroudisalsoaffectedbyanincreaseinl.Asthetipspeedratioofawindturbinein-creases,thedistancesbetweensuccessivepassesofthetipvortexinthewakedecrease.Forashroudedturbine,thismeansthatmoreturnsofthewakearepresentintheshroudinterior.ThiscanbeseeninFig.21,whichplotsthemagnitudeofvorticityneartheshroudsurfaceinthevicinityoftheturbineblade.Increasingthetipspeedratiodrawswakevorticesclosertotheleadingedgeoftheshroudairfoilandincreasesthenumberoftimesthewakeinteractswiththeshroudboundarylayer.

TheimpactoftheincreasedinteractionbetweentheturbinewakeandtheshroudisexaminedinFig.22,whichshowsthepressuredistributionalongtheshroudforeachtipspeedratioconsidered.(ThelocationoftheazimuthalstationsintheseplotsisshowninFig.18(c)).TheareainsidethecurverepresentingthemeanCpdistributionmeasurestheliftforceontheshroudandisanindicatorofshroudperformance.Thisareadecreaseswithincreasingl,indicatingthattheimpactofthewakeistoreducetheperformanceoftheshroudasa owaccelerator.Indeed,exami-nationofvelocitypro les(notshown)forthesecasesrevealsthattheamountofaccelerationachievedbytheshroudattheturbineplanedecreaseswithincreasingl.

Atthehighesttipspeedratioconsidered,inFig.22(c),anespeciallyintensesuctionpeakappearsinthevicinityoftheblade(q¼0 ).However,themeansuctionislowerforthiscasethananyother.Thissigni esthepresenceofstrongpressuregradientsintheazimuthaldirectioninthiscase,andtheimpactofthishighsuctionpeakistosetthe owswirlingratherthandraw owaxiallythroughtheshroud.

Ofthethreephenomenadiscussedabove,theextentof owseparationoutweighstheothertwoinimportanceuptothepointwherethe owisfullyattachedontheblade.Beyondthispoint,increasinglfurtheronlyincreasesthepro ledragandreducestheshroudef ciency.

7.Conclusionsandfuturework

Theperformanceand ow eldofashroudedwindturbinehasbeencomputedandanalyzedindetail.High-liftairfoilshapesareconsideredforshroudgeometry.ValidationhasbeenperformedfortheRANSbasedmodelusedforcomputations,andtheuseofatransitionmodelisfoundtoimprovetheaccuracyofresults.

Aninvestigationofairfoilsectionsforshroudsofshroudedwindturbineshasbeenperformed.Ofanumberofairfoilshapesconsideredforshroudcrosssectionalpro les,theSeligS1223at-tainsthegreatestampli cationofmass ow.Three-dimensionalanalysisoffullshroudedturbinesystemsverifythebene tofincreasedmass owthroughtheplaneoftheturbine.Powerextractionupto90%beyondtheBetzlimitisachieved.Theimprovementinpowerextractionbeyondthebareturbineissub-stantial;theNACA0006shroudimprovespoweroverthebare

turbinebyafactorof1.93andtheS1223improvesitbyafactorof3.39atU∞¼5m/s.Theseresultsfurthersupporttheutilityoftheshroudedwindturbineasadevicethatcanbeusedeffectivelyatlowcut-inspeedsandofferpromisetosubstantiallyimprovetheenergycapturewhencomparedtoconventionalwindturbines.

Basedontheprecedinganalysis,thefollowingconclusionsaredrawnregardingthedesignofashroudedturbine:

Intheabsenceofaturbine,mass owampli cationthroughashroudincreasesapproximatelylinearlywithradialforce,andnonlinearbehavioroccursasthestagnationpointmovesfromtheinteriortotheexteriorsideoftheshroud.

Thewakeofashroudedturbineexpandsmorerapidlythananopenturbineandcanbetailoredtopromotemaximumpowerextraction.

Transitionhasanimportanteffectonthebehaviorofashroudedwindturbineandshouldbeaccountedforinanalysiswheneverfeasible.

Theshroudedturbinesystem'sperformanceismaximumataparticulartipspeedratio.Increasingthetipspeedratiobeyondthisoptimalvalueleadstolossesduetoseparation.

Theaboveconsiderationsprovideinsightforboththeanalysisanddesignofboththeshroudandtheturbineofashroudedtur-bine.Futureworkwillemployanoptimizationstrategytodesignashroudedturbinewithacontinueddetailedfocuson uidphysics.Thisprocedurewillincludearedesignofthetwist,chord,andthicknessdistributionoftheblade.Additionally,theresultspre-sentedhereshallbecomparedtolower-ordertheories,suchasthecommonly-usedactuatordiskmodel.Itwillalsobeinterestingtoconsidertheeffectofextremeloadsonsuchasystem,suchasahigh-speedgustatahighangleofattack.Acknowledgment

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