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UNSTEADYFLOWANALYSISINHYDRAULICTURBOMACHINERYALBERTRUPRECHTINSTITUTEOFFLUIDMECHANICSANDHYDRAULICMACHINERYUNIVERSITYOFSTUTTGART,GERMANYABSTRACTINTHEFIELDOFHYDRAULICMACHINERYCOMPUTATIONALFLUIDDYNAMICSCFDISROUTINELYUSEDTODAYINRESEARCHANDDEVELOPMENTASWELLASINDESIGNATTHATNEARLYALWAYSSTEADYSTATESIMULATIONSAREAPPLIEDINTHISPAPER,HOWEVER,UNSTEADYSIMULATIONSARESHOWNFORDIFFERENTEXAMPLESTHEPRESENTEDEXAMPLESCONTAINAPPLICATIONSWITHSELFEXCITEDUNSTEADINESS,EGVORTEXSHEDDINGORVORTEXROPEINTHEDRAFTTUBE,ASWELLASAPPLICATIONSWITHEXTERNALLYFORCEDUNSTEADINESSBYCHANGINGORMOVINGGEOMETRIES,EGROTORSTATORINTERACTIONSFORTHESEEXAMPLESTHEREQUIREMENTS,POTENTIALANDLIMITATIONSOFUNSTEADYFLOWANALYSISASSESSEDPARTICULARLYTHEDEMANDSONTHETURBULENCEMODELSANDTHENECESSARYCOMPUTATIONALEFFORTSAREDISCUSSEDINTRODUCTIONFORMORETHANADECADECOMPUTATIONALFLUIDDYNAMICSCFDISUSEDINTHEFIELDOFHYDRAULICMACHINERYINRESEARCHANDDEVELOPMENTASWELLASINTHEDAILYDESIGNBUSINESSEARLYSUCCESSFULDEMONSTRATIONSAREGIVENEGINTHEGAMMWORKSHOP1THEAPPLICATIONSARESTEADILYINCREASINGTHISISEXPRESSEDINFIG1,WHERETHEPERCENTAGEOFPAPERSDEALINGWITHCFDISSHOWN,WHICHWEREPRESENTEDATTHEIAHRSYMPOSIUMONHYDRAULICMACHINERYANDCAVITATIONSTARTINGWITHQ3DEULERAND3DEULERTODAYUSUALLYTHEREYNOLDSAVERAGEDNAVIERSTOKESEQUATIONSTOGETHERWITHAROBUSTMODELOFTURBULENCEUSUALLYTHEKMODELISUSEDITISCOMMONPRACTICETOAPPLYSTEADYSTATESIMULATIONS,THEUNSTEADINESSINCONSEQUENCEOFTHEROTORSTATORINTERACTIONSISADDRESSEDBYAVERAGINGPROCEDURESBYTHISMETHODACCURATERESULTSAREOBTAINEDFORMANYQUESTIONSINTHEDESIGNOFCOMPONENTSHOWEVER,DIFFERENTPROBLEMSINTURBOMACHINERYARISEFROMUNSTEADYFLOWPHENOMENAINORDERTOGETINFORMATIONONTHISPHENOMENAORSOLUTIONSTOTHEPROBLEMSANUNSTEADYFLOWANALYSISISNECESSARYTHISREQUIRESAMUCHHIGHERCOMPUTATIONALEFFORT,ROUGHLYAFACTOR510COMPAREDTOSTEADYSTATE,DEPENDINGOFTHEPROBLEMANDOFTHEDEGREEOFMODELINGASSUMPTIONSWITHTODAYSCOMPUTERSANDSOFTWARE,HOWEVER,UNSTEADYPROBLEMSCANBESOLVEDFIG1PERCENTAGEOFPAPERSATTHEIAHRSYMPOSIUMDEALINGWITHCFDTWOMAJORGROUPSOFUNSTEADYPROBLEMSCANBEDISTINGUISHEDTHEFIRSTGROUPAREFLOWSWITHANEXTERNALLYFORCEDUNSTEADINESSTHISCANBECAUSEDBYUNSTEADYBOUNDARYCONDITIONSORBYCHANGINGOFTHEGEOMETRYWITHTIMEEXAMPLESARETHECLOSUREOFAVALVE,THECHANGEOFTHEFLOWDOMAININAPISTONPUMP,ORTHEROTORSTATORINTERACTIONSTHESECONDGROUPAREFLOWSWITHSELFEXCITEDUNSTEADINESS,WHICHAREEGTURBULENTMOTION,VORTEXSHEDDINGKARMANVORTEXSTREETORUNSTEADYVORTEXBEHAVIOREGVORTEXROPEINADRAFTTUBEHERETHEUNSTEADINESSISOBTAINEDWITHOUTANYCHANGEOFTHEBOUNDARYCONDITIONSOROFTHEGEOMETRYTHERECANALSOOCCURACOMBINATIONOFBOTHGROUPSEGFLOWINDUCEDVIBRATIONS,CHANGEOFGEOMETRYCAUSEDBYVORTEXSHEDDINGALLTHESEPHENOMENACANTAKEPLACEINATURBINEORPUMPANDREQUIREDIFFERENTSOLUTIONPROCEDURESBASICEQUATIONSANDNUMERICALPROCEDURESINHYDRAULICTURBOMACHINERYTODAYUSUALLYTHEREYNOLDSAVERAGEDNAVIERSTOKESEQUATIONSFORANINCOMPRESSIBLEFLOWAREAPPLIEDCOMPAREDTOTHESTEADYSTATETHEMOMENTUMEQUATIONSCONTAINANADDITIONALTERMPRESCRIBINGTHEUNSTEADYCHANGE0XUXUXXP1XUUTUIJIJJIJIJIJIGF7GF7GF8GF6GE7GE7GE8GE6GF7GF7GF8GF6GE7GE7GE8GE61IJARETHEREYNOLDSSTRESSES,WHICHARECALCULATEDFROMTHETURBULENCEMODELTHECONTINUITYEQUATIONFORINCOMPRESSIBLEFLOWREADS0XUII2ANDDOESNOTCONTAINATIMEDEPENDINGTERMITHASTOBEEMPHASIZEDTHATTHEEQUATIONS1AND2BEHAVESDIFFERENTINTIMEANDINSPACEINSPACETHEYSHOWELLIPTICBEHAVIOR,THEREFORETHEYREQUIREBOUNDARYCONDITIONSONALLSURFACESINTIME,HOWEVER,THEYAREOFPARABOLICNATURE,WHICHMEANTHATTHEREISNOFEEDBACKFROMTHEFUTURETOTHEPRESENTORPASTBECAUSEOFTHATNOBOUNDARYCONDITIONSAREREQUIREDINTHEFUTURETHISISSCHEMATICALLYSHOWNINFIG2THISISTHEREASON,WHYTHETIMEDISCRETIZATIONISGENERALLYCARRIEDOUTINADIFFERENTWAYTHANTHESPATIALDISCRETIZATIONFORSPATIALDISCRETIZATIONUSUALLYAFINITEVOLUMEORAFINITEELEMENTAPPROXIMATIONISAPPLIEDFORTIMEDISCRETIZATION,HOWEVER,MOSTLYTHEFINITEDIFFERENCEMETHODISUSEDAFEWOFTHEMOSTPOPULARFINITEDIFFERENCEAPPROXIMATIONSARESHOWNINFIG3INADDITIONEXPLICITMULTIPOINTSCHEMESOFRUNGEKUTTATYPEORPREDICTORCORRECTORSCHEMESAREOFTENAPPLIEDFIG2BOUNDARYANDINITIALCONDITIONSFIG3TIMEDISCRETIZATIONSCHEMESITHASTOBEMENTIONEDTHATTHEEXPLICITMETHODSREQUIREARESTRICTIONOFTHETIMESTEPACCORDINGTOSTABILITYCRITERIACFLCRITERIA,WHICHDEPENDONTHELOCALVELOCITIESANDTHELOCALGRIDSIZETHEIMPLICITMETHODS,INCONTRARY,AREALWAYSSTABLE,THEREISNORESTRICTIONOFTHETIMESTEPITCANBECHOSENONLYACCORDINGTOTHEPHYSICALREQUIREMENTSINORDERTOOBTAINACCURATESOLUTIONSTHETIMEDISCRETIZATIONSHOULDBEATLEASTOF2NDORDER,SIMILARTOTHESPATIALDISCRETIZATIONOTHERWISEEXTREMELYSMALLTIMESTEPSWOULDBEREQUIREDTHEABOVEDESCRIPTIONOFTHEFLOWINTHEEULERIANCOORDINATESCANBEAPPLIEDFORUNSTEADYBOUNDARYCONDITIONPROBLEMSASWELLASFORSELFEXCITEDUNSTEADINESSHOWEVER,TOEXPRESSPROBLEMSWITHMOVINGGEOMETRIESINEULERIANCOORDINATESISMOREDIFFICULTATTHEMOVINGBOUNDARYALAGRANGIANDESCRIPTIONCANBEAPPLIEDVERYEASILYSINCETHEFLUIDPARTICLESCANBETRACEDBYTHISMETHODCOMBININGTHESETWOMETHODSANARBITRARYLAGRANGIANEULERIANALEMETHODCANBEUTILIZEDTHISMETHODISSUITABLEFORTHESOLUTIONOFPROBLEMSWITHMOVINGBOUNDARIESINTHEALEMETHODTHEREFERENCECOORDINATESCANBECHOSENARBITRARYINTHISREFERENTIALCOORDINATESYSTEMTHEMATERIALDERIVATIVECANBEDESCRIBEDASJEIJJRILIXT,XFWUTT,XFTT,XF3WITHTHECOORDINATESSCOODDINATEEULERIANXSCOODDINATELREFERENTIAXSCOODDINATELAGRANGIANXEIRILIANDWIREFERENCEVELOCITYTHEMOMENTUMEQUATIONSINTHEALEFORMULATIONCANBEWRITTENASFOLLOWS0XUXUXXP1XUWUTUIJIJJIJIJIJJIGF7GF7GF8GF6GE7GE7GE8GE6GF7GF7GF8GF6GE7GE7GE8GE64THEMOVINGOFTHEREFERENCESYSTEMWICANBECHOSENARBITRARYIFWIISEQUALTOZEROONEGETSTHEEULERIANDESCRIPTION,ONTHEOTHERHAND,IFWIISEQUALTOTHEVELOCITYOFTHEFLUIDPARTICLETHELAGRANGIANFORMULATIONISOBTAINEDTHECONVECTIVETERMINTHETRANSPORTEQUATIONSFORSCALARQUANTITIESCHANGESINTHESAMEWAYTHANINTHEMOMENTUMEQUATIONSTHISAPPLIESALSOTOTHEKANDEQUATIONSTHENUMERICALREALIZATIONOFMOVINGORCHANGINGGRIDSCANEITHERBEOBTAINEDBYDEFORMATIONOFANEXISTINGMESHINEACHTIMESTEPFORLARGEDEFORMATIONSTHISREQUIRESANAUTOMATICGRIDSMOOTHINGALGORITHMOREVENANAUTOMATICREMESHINGAFTERAFEWTIMESTEPSANOTHERMETHODISTHEUSEOFDIFFERENTEMBEDDEDGRIDS,WHICHCANMOVEAGAINSTEACHOTHERINTHISCASEASLIDINGINTERFACEBETWEENTHENONMATCHINGGRIDSISREQUIREDTHISPROCEDUREISSCHEMATICALLYSHOWNINFIG4FORTWODIFFERENTPROBLEMS,NAMELYROTORSTATORINTERACTIONANDVIBRATIONOFACYLINDERINAFLUIDINFENFLOSS,THECOMPUTERCODEDEVELOPEDATOURINSTITUTEATUNIVERSITYOFSTUTTGART,THESECONDAPPROACHISAPPLIEDTHEINTERFACEBETWEENTHEGRIDSISREALIZEDBYMEANSOFDYNAMICBOUNDARYCONDITIONS,WHEREDOWNSTREAMTHENODEVALUESVELOCITIESANDTURBULENCEQUANTITIESAREPRESCRIBEDANDUPSTREAMPRESSUREANDFLUXESAREINTRODUCEDASSURFACECONDITIONSABRIEFOVERVIEWONTHENUMERICALPROCEDURESISGIVENIN2,FORMOREDETAILSTHEREADERISREFERREDTO3,4ONEPOINTHASTOBEEMPHASIZEDSINCETHEUNSTEADYSIMULATIONSREQUIREASEVEREINCREASEOFCOMPUTATIONALEFFORTCOMPAREDTOSTEADYSTATESOLUTIONS,PARALLELPROCEDURESARENECESSARYINTHISCASETHEALEFORMULATIONWITHMOVINGGRIDSLEADSTOADYNAMICCHANGEOFCOMMUNICATIONBECAUSETHELOCATIONOFEXCHANGEBOUNDARIESVARIESWITHTIMEANDCANTHEREFORECHANGETHECOMPUTATIONALDOMAINOFTHEPROCESSORS,SEE2INFENFLOSSANIMPLICITSOLUTIONALGORITHMISAPPLIEDASALREADYMENTIONEDTHISHASTHEADVANTAGETHATTHEREISNOSTABILITYLIMITATIONFORTHETIMESTEPTHEOVERALLSOLUTIONPROCEDUREINCLUDINGTHEFLUIDSTRUCTUREINTERACTIONISSHOWNINFIG5IFTHEMOVEMENTOFTHEGRIDDOESNOTDEPENDONTHEFLOWSITUATIONTHEFLUIDSTRUCTURELOOPVANISHESFIG5FLOWCHARTOFFENFLOSSINCLUDINGFLUIDSTRUCTUREINTERACTIONFIG4MOVINGGRIDEXAMPLESAPPLICATIONSINTHEFOLLOWINGSELECTEDAPPLICATIONSARESHOWNANDTHESPECIFICPROBLEMSFORTHISEXAMPLESAREDISCUSSEDFIRSTLYSOMECASESWITHSELFEXCITEDUNSTEADINESSAREPRESENTEDVORTEXSHEDDINGATTHEINLETOFAPOWERPLANTPROBLEMDESCRIPTIONTHEFIRSTEXAMPLESHOWSTHEFLOWBEHAVIORATTHEINLETOFALOWHEADPOWERPLANTITISANEXISTINGPLANTWITHTWOIDENTICALBULBTURBINESDURINGOPERATIONTHEINNERTURBINESHOWEDSEVEREBEARINGPROBLEMSWHEREASTHEOUTERTURBINEOPERATESSMOOTHLYTHEREASONWASEXPECTEDTOBEVORTEXSHEDDINGATTHEINLETBYNUMERICALANALYSISTHEPROBLEMWASINVESTIGATEDANDITWASTRIEDTOFINDASOLUTIONTOTHEPROBLEMINFIG6THEGEOMETRYISSHOWNTHECALCULATIONHASBEENCARRIEDOUTIN2DASWELLASIN3DFIRSTLYITWASTRIEDTOCARRYOUTASTEADYSTATESIMULATION,HOWEVER,NOCONVERGEDSOLUTIONCOULDBEOBTAINEDTHEREFOREANUNSTEADYSIMULATIONWASUNDERTAKENTHERESULTSINDICATEASTRONGUNSTEADYMOTIONINFIG7THEVELOCITYDISTRIBUTIONATACERTAINTIMESTEPISPRESENTEDCLEARLYVISIBLEARETHEVORTICES,SHEDDINGFROMTHEINLETANDMOVINGDOWNSTREAMINTOTHEINNERTURBINETHISISTHEREASONOFTHEDESTRUCTIONOFTHEBEARINGSINORDERTOIMPROVETHEFLOWBEHAVIORAMODIFIEDGEOMETRYWASSUGGESTEDTHISGEOMETRY,SHOWNINFIG8,HASBEENBUILTINTHEMEANTIMETHEREARENOLONGERPROBLEMSWITHVORTEXSHEDDINGFURTHERDETAILSABOUTTHISAPPLICATIONCANBEFOUNDIN5,6DISCUSSIONTHEPHYSICALUNSTEADINESSOFTHEFLOWHASBEENINDICATEDBYTHEINABILITYTOACHIEVEACONVERGEDSTEADYSTATESOLUTIONTHISISVERYOFTENTHECASEWITHFLOWSSHOWINGVORTEXSHEDDINGINREALITYFIG6GEOMETRYOFPOWERPLANTINLETFIG7INSTANTANEOUSVELOCITYVECTORS,VORTEXSHEDDINGATTHEINLETPIERFIG8MODIFIEDGEOMETRYANECESSARYCONDITIONFORTHATIS,THATTHENUMERICALSCHEMEDOESNOTCONTAINSERIOUSARTIFICIALDIFFUSION,WHICHWOULDSUPPRESSTHEUNSTEADYMOTIONTHESAMEAPPLIESTOTHEUSEDTURBULENCEMODELTHESTANDARDKMODELUSUALLYPRODUCESATOOHIGHEDDYVISCOSITY,ESPECIALLYINSWIRLINGFLOWS,ANDTHEREFOREITVERYOFTENSUPPRESSESTHEUNSTEADYMOTIONTHISWILLBEDISCUSSEDAGAININOTHERAPPLICATIONSFORMANYCASESATLEASTASTREAMLINECURVATURECORRECTIONOREVENANONLINEAREDDYVISCOSITYFORMULATIONISNECESSARYINORDERTOAVOIDATOOHIGHTURBULENCEPRODUCTIONANOTHERPOINTINTURBULENCEMODELINGISTHETREATMENTOFTHENEARWALLFLOWITISWELLKNOWNTHATTHEUSEOFWALLFUNCTIONSUSUALLYTENDSTOPREDICTAFLOWSEPARATIONTOOLATEINCASEOFVORTEXSHEDDINGTHISCANCAUSEASEVEREREDUCTIONOFTHEVORTEXSIZESOREVENACOMPLETESUPPRESSIONOFTHEVORTICESMOREACCURATERESULTSCANBEOBTAINEDBYSOLVINGTHEFLOWUPTOTHEWALLIFPOSSIBLEBYALOWREYNOLDSORATWOLAYERMODELTHERESULTSSHOWNABOVEAREACHIEVEDBYANALGEBRAICTURBULENCEMODELBALDWINLOMAXTYPEWHERETHEFLOWISRESOLVEDUPTOTHEWALLVORTEXROPEINADRAFTTUBEPROBLEMDESCRIPTIONASANOTHERSELFEXCITEDUNSTEADYFLOWEXAMPLETHESIMULATIONOFAVORTEXROPEINADRAFTTUBEISSHOWNHEREASTRAIGHTAXISYMMETRICALDIFFUSERISCONSIDEREDTHEINFLOWCONDITIONSTOTHEDIFFUSERARECHOSENACCORDINGTOTHEPARTLOADOPERATIONOFAFRANCISTURBINETHISMEANSTHATTHEFLOWSHOWSASTRONGSWIRLCOMPONENTTHEINLETVELOCITYDISTRIBUTIONANDTHEGEOMETRYAREPRESENTEDINFIG9THEINSTANTANEOUSFLOWFORACERTAINTIMESTEPISGIVENINFIG10,WHEREANISOPRESSURESURFACEASWELLASTHESECONDARYVELOCITYVECTORSINTHREECROSSSECTIONSAREPLOTTEDCLEARLYTHECORKSCREWTYPEFLOWWITHANUNSYMMETRICALFORMISVISIBLE,ALTHOUGHTHEGEOMETRYANDTHEBOUNDARYCONDITIONSARECOMPLETELYAXISYMMETRICALFIG9GEOMETRYANDINLETCONDITIONSFIG10ISOPRESSUREANDSECONDARYFLOWOFAVORTEXROPEINFIG11THESECONDARYVELOCITYANDTHELOWPRESSUREREGION,WHICHREPRESENTSTHEVORTEXCENTER,ISSHOWNINTHECROSSSECTIONS,INDICATEDINFIG9,FORCERTAINTIMESTEPSCLEARLYTHEREVOLUTIONOFTHEVORTEXCENTERCANBEOBSERVEDTHIS,OFCOURSE,CAUSESPRESSUREFLUCTUATIONSANDTHEREFOREDYNAMICALFORCESONTHEDRAFTTUBESURFACEFIG11SECONDARYMOTIONANDLOWPRESSUREREGIONFORDIFFERENTTIMESTEPSDISCUSSIONCONCERNINGTHENUMERICALSCHEMEANDTHETURBULENCEMODELSTHEDISCUSSIONABOVEALSOAPPLIESHERE,EGAPPLICATIONOFTHESTANDARDKMODELLEADSTOASTEADYSTATE,SYMMETRICALSOLUTIONTHISISALSOREPORTEDIN7THERESULTSSHOWNABOVEAREACHIEVEDBYAPPLYINGTHEMULTISCALEKMODELOFKIM8TOGETHERWITHASTREAMLINECURVATURECORRECTIONTHISMODELSHOWSAMUCHLOWEREDDYVISCOSITYTHANTHESTANDARDMODEL,ESPECIALLYINSWIRLINGFLOWSTHEAPPLICATIONOFWALLFUNCTIONSDOESNOTGIVEANYPROBLEMSHERE,SINCETHEFLOWINSTABILITYHASITSORIGININTHECENTERANDISNOTAFFECTEDBYTHEPREDICTIONOFTHENEARWALLREGIONVORTEXINSTABILITYINAPIPETRIFURCATIONPROBLEMDESCRIPTIONINTHEFOLLOWINGANOTHERPROBLEMCAUSEDBYAVORTEXINSTABILITYISSHOWNITISAPIPETRIFURCATION,WHICHISESTABLISHEDINAPOWERPLANTINNEPALTHETRIFURCATIONDISTRIBUTESTHEWATERFROMTHEPENSTOCKTOTHETHREETURBINEUNITSTHEPROBLEMINTHISPLANTARISESFROMSEVEREFLUCTUATIONSOFTHEPOWEROUTPUTOFTHEBOTHOUTERTURBINESBYFIELDMEASUREMENTSTHETRIFURCATIONWASDISCOVEREDASTHEREASONFORTHEFLUCTUATIONSBYMEANSOFCFDANDBYMODELTESTS,CARRIEDOUTATASTROEINGRAZ,THEFLOWBEHAVIORSHOULDBEANALYZEDANDACUREOFTHEPROBLEMSHOULDBEFOUNDTHEGEOMETRYOFTHETRIFURCATIONISSHOWNINFIG12ITHASASPHERICALSHAPETHEFLUCTUATIONINTHETRIFURCATIONISCAUSEDBYASTRONGVORTEX,WHICHTENDSTOBEUNSTABLEITSKIPSBETWEENTHETWOSITUATIONS,SKETCHEDINFIG13INTHEMODELTESTSTHESECONDARYVELOCITYOFTHEVORTEXCOULDBEFOUNDTOBE30TIMESHIGHERTHANTHETRANSPORTVELOCITYTHEREASONISTHATATTHETOPOFTHESPHERETHEREISENOUGHSPACEFORAHUGEVORTEXTOFORMTHISVORTEXCONCENTRATESINTHESIDEBRANCHESANDTHEREFOREINCREASESTHESWIRLINTENSITYBECAUSEOFTHISSTRONGSECONDARYMOTIONTHEREARESTRONGLOSSESATTHEINLETOFTHEBRANCH,WHICHREDUCESTHEHEADOFTHETURBINEANDTHEREFORECAUSESTHEREDUCTIONOFPOWEROUTPUTDURINGTHEPROJECTITWASTRIEDTOOBTAINTHEUNSTEADYBEHAVIORBYAKSIMULATIONONRELATIVELYCOARSEGRIDS200300000NODESHOWEVER,THESECALCULATIONSDIDNOTSHOWTHEVORTEXINSTABILITYMERELYAVORTEXFORMSWHICHEXTENDSFROMONESIDEBRANCHTOTHEOTHERTHESWIRLINTENSITYWASUNDERPREDICTEDBYMORETHANAFACTORFIVEBECAUSEOFTHELOWSWIRLRATETHEVORTEXISCOMPLETELYSTABLEANDHASNOTENDENCYOFSKIPPINGBETWEENDIFFERENTSTATIONSEVENBYADYNAMICALEXCITATIONCAUSEDBYCHANGESOFTHEOUTLETBOUNDARYCONDITIONOFONEBRANCHTHEPREDICTEDVORTEXDIDNOTCHANGEITSPOSITIONONLYWHENAPPLYINGFINERGRIDSANDANOTHERTURBULENCEMODELTHEPREDICTEDSWIRLINTENSITYCOULDBEINCREASEDHEREANALGEBRAICTURBULENCEMODELWITHALIMITATIONOFTHEEDDYVISCOSITYISAPPLIEDTHEUSEDGRIDSCONSISTSOFABOUT500000NODESASACONSEQUENCETHISLEADSTOANINSTABILITYOFTHEVORTEXINTHEPREDICTIONTHEVORTEXSKIPSBETWEENTHETWOSTRUCTURESSHOWNINFIG14ONEOFTHESESTRUCTURESCORRESPONDSQUITEWELLWITHTHESTRUCTUREOBSERVEDINTHEMODELTESTSINTHESECONDSITUATIONTHEVORTEXEXPENDSFROMONESIDEBRANCHTOTHEOTHERTHISCOMPLIESWITHTHEABOVEMENTIONEDSTABLERESULTSTHECALCULATEDSWIRLINTENSITYISSTILLMORETHANTWOTIMESLOWERCOMPAREDTOTHERESULTSOFTHEMODELTESTSTHEREFOREFURTHERINVESTIGAFIG12GEOMETRYOFTHETRIFURCATIONFIG13VORTEXSTRUCTURETIONSWITHOTHERTURBULENCEMODELSANDWITHFINERGRIDSARENECESSARYANDWILLBECARRIEDOUTINFUTUREFIG14PREDICTEDVORTEXSTRUCTURESFORCOMPLETENESSTHESOLUTIONTOTHEPROBLEMISSHOWNITCONSISTSOFTHEINSTALLATIONOFTWOPLATESINTHEUPPERANDLOWERPARTOFTHESPHERETHISISSHOWNINFIG15HENCENOFREESPACEISAVAILABLE,WHERETHEVORTEXCANFORMCONSEQUENTLYTHEINTENSITYOFTHEVORTEXISDRAMATICALLYREDUCEDANDTHEVORTEXISCOMPLETELYSTABLEINTHEMEANTIMETHERECONSTRUCTIONWASCARRIEDOUTANDTHEFLUCTUATIONOFTHEPOWEROUTPUTVANISHEDASABYPRODUCTTHELOSSESINTHETRIFURCATIONARESEVERELYREDUCED,WHICHRESULTSINANINCREASEOFPOWEROUTPUTOFAPPROXIMATELY5FURTHERDETAILSOFTHISPROBLEMCANBEFOUNDIN9,10DISCUSSIONASALREADYMENTIONEDTHECALCULATIONSUSINGTHEKMODELWERENOTSUCCESSFULITISWELLKNOWNTHATTHISMODELISNOTABLETOPREDICTHIGHLYSWIRLINGFLOWSACCURATELYTHEUNSTEADYMOTIONOFTHEVORTICESESPECIALLYOFVERYSLIMVORTICES,HOWEVER,VERYMUCHDEPENDSONTHESWIRLINTENSITYINORDERTOPRESCRIBESUCHTYPESOFFLOWWITHSUFFICIENTACCURACYITISNECESSARYTOHAVEHIGHLYSOPHISTICATEDTURBULENCEMODELSANDVERYFINEGRIDS,MAYBETHEONLYWAYTOACHIEVEITISTHEAPPLICATIONOFLARGEEDDYSIMULATIONROTORSTATORINTERACTIONINANAXIALTUBINETHEFOLLOWINGEXAMPLEBELONGSTOTHESECONDGROUP,THEUNSTEADINESSISFORCEDBYMOVINGGEOMETRIESTHEPROBLEMINQUESTIONISTHEFIG15MODIFIEDGEOMETRYFIG16GEOMETRYOFTHEINVESTIGATEDAXIALTURBINEFLOWINANAXIALTURBINETHESPECIALITYOFTHISTURBINEISITSRELATIVELYLOWSPECIFICSPEEDITHASBEENDESIGNEDFORPRESSURERECUPERATIONINPIPINGSYSTEMSTHEADVANTAGEISTHATTHEDISCHARGEISNEARLYINDEPENDENTOFTHESPEED,BECAUSEOFTHATTHETURBINECANNOTINTRODUCEWATERHAMMERSINTHESYSTEMTHEGEOMETRYOFTHETURBINEISSHOWNINFIG16ITCONSISTSOFTHEINLETCONFUSER,12FIXEDGUIDEVANES,15RUNNERBLADESANDTHEDRAFTTUBETHESTATORANDROTORPARTISSHOWNINMOREDETAILINFIG17FORTHESIMULATIONTHECOMPLETETURBINEISCONSIDEREDINCLUDINGALLFLOWCHANNELSINTHEGUIDEVANESANDINTHERUNNER,ALTHOUGHASYMMETRYCONDITIONOF120COULDBEUSEDTHEREASONIS,THATALSOAVARIANTWITHUNSYMMETRICALOUTLETHASBEENINVESTIGATEDTHECOMPUTATIONALMESHCONSISTSOFMORETHAN2MILLIONGRIDNODES,PARTOFTHEGRIDISSHOWNINFIG18THESEAREROUGHLY60000NODESPERFLOWCHANNELITISARATHERCOARSEGRID,CONSIDERINGTHATTHECLEARANCEBETWEENRUNNERBLADESANDCASINGHASTOBEINCLUDEDINTHEMODEL,WHICHISNECESSARYSINCETHECLEARANCEFLOWVERYMUCHAFFECTSTHECHANNELFLOWBECAUSEOFTHESHORTRUNNERBLADESTHECALCULATIONSARECARRIEDOUTUSINGTHESTANDARDKMODELINTHEFOLLOWINGSOMERESULTSOFTHECALCULATIONWILLBESHOWNINFIG19THEINSTANTANEOUSFLOWINTHERUNNERISPRESENTEDTHEFIGURESHOWSTHEPRESSUREDISTRIBUTIONOFTHERUNNERSURFACEASWELLASSTREAMLINESSTARTEDATDIFFERENTLOCATIONSLOOKINGATTHEPRESSUREONECLEARLYSEESTHESTAGNATIONPOINTATTHELEADINGEDGETHELOCATIONOFTHEDRAFTTUBEGUIDEVANESRUNNERFIG18PARTOFTHECOMPUTATIONALMESHFIG17GEOMETRYOFROTORANDSTATORFIG19INSTANTANEOUSFLOWINTHERUNNERSTAGNATIONPOINTVARIESSLIGHTLYWITHTHERUNNERPOSITIONGENERALLYTHEINLETFLOWANGLESEEMSTOBESLIGHTLYTOOFLATTHEREFORETHESTAGNATIONPOINTISSHIFTEDTOWARDSTHESUCTIONSIDECONSIDERINGTHEFLOWINTHETIPCLEARANCEONECANOBSERVETHATATTHEINLETTHESHEARFORCESDOMINATETHEFLOWTENDSTOGOFROMTHESUCTIONTOTHEPRESSURESIDEINTHESECONDHALFOFTHEBLADETHEPRESSUREFORCESDOMINATETHEFLOWINTHECLEARANCEGOESFROMTHEPRESSURETOTHESUCTIONSIDEITCANALREADYBESEENBYTHISRESULTSTHATTHEDESIGNOFTHERUNNERISNOTOPTIMALTHISISAFIRSTVERSION,INTHEMEANTIMEAMUCHBETTERRUNNERHASBEENDESIGNEDHOWEVERTHISGEOMETRYISNUMERICALLYINVESTIGATEDSINCEEXTENSIVEMEASUREMENTSHAVEBEENCARRIEDOUTFORTHISCONFIGURATIONANDTHENUMERICALRESULTSCANBEVALIDATEDINFIG20AGAINTHEINSTANTANEOUSPRESSUREFORACERTAINTIMESTEPISSHOWNONECANOBSERVETHELOWPRESSUREREGIONONTHESUCTIONSIDEATTHETOPOFTHERUNNERBLADESCLEARLYVISIBLEISTHEVARIATIONOFTHEPRESSUREWITHTHEPOSITIONTHELOWPRESSUREREGIONCORRESPONDSQUITEWELLWITHTHECAVITATIONOBSERVATIONATTHETESTRIG,SEEFIG21THEREONEALSOCANOBSERVETHEVARIATIONOFTHECAVITATIONBUBBLESACCORDINGTOTHERUNNERPOSITIONASAQUANTITATIVECOMPARISONTHEPRESSUREATTWOLOCATIONSISSHOWNINFIG22POSITION1ISLOCATEDINFRONTOFTHEGUIDEVANESANDTHESECONDPOSITIONLIESBETWEENTHEGUIDEVANESANDTHERUNNERATBOTHLOCATIONSTHEMEASUREDANDTHECALCULATEDPRESSURECORRESPONDSQUITEWELLONECANSEETHATEVENINFRONTOFTHEGUIDEVANESPRESSUREFLUCTUATIONSCANBEOBSERVEDBETWEENTHESTATORFIG20CALCULATEDPRESSUREDISTRIBUTIONFORACERTAINRUNNERPOSITIONFIG21CAVITATIONOBSERVATIONINTHERUNNERFIG22PRESSUREDISTRIBUTIONATTWOSPOTPOINTSANDTHEROTORFLUCTUATIONSOFNEARLY25OFTHEHEADOFTHETURBINECANBESEENTHIS,OFCOURSE,LEADSTODYNAMICALFORCESONTHEBLADESINFIG23THETORQUEONONERUNNERBLADEASWELLASTHETORQUEOFTHECOMPLETERUNNERISSHOWNTHECALCULATEDTORQUEFLUCTUATIONONASINGLEBLADEARENEARLY30OFTHEAVERAGEDTORQUETHISISADYNAMICALFORCEONTHEBLADINGTHETOTALTORQUE,HOWEVER,ISNEARLYCONSTANTDUETOTHEGREATNUMBEROFBLADESANDDUETODIFFERENTPHASESOFTHEFLUCTUATIO