弯管二次流

Anoteonsecondaryflowinbendsandbendcombinations

H.E.Fiedler

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AbstractInthispaperanexplanationfortheformationofswirlpastthree-dimensionalbendcombinationsinpipeflowsisgiven.Supportiveandcomparativeswirlmeasurementspastadoublebendcombinationandpastacombinationofanasymmetricdiaphragmandafollowinganglebendarepresentedanddiscussed.

1

Introduction

Aflowthroughathree-dimensionalcombinationoftwo

90°-bendswheretheexitplaneofthesecondoneisnotparalleltotheintakeplaneofthefirstoneexperiencesaswirl.Thisphenomenonhasoftenbeendescribed(e.g.Miller1978);asketchoftheconfigurationisgiveninFig.2).

Whilethecharacteristicdoublevortexdevelopingintheflowthroughasinglebendiswellknownandequallywellunder-stood(seee.g.Schlichting(1958)andFig.1a),nosimpleexplanationappearstobeathandforthecaseofadoublebendandindeedtheoftensuggestedsuperpositionofthesuccessivepairsofsecondaryvorticesisnotonlyquestionablebutdoesnotprovideanexplanationeither.Thereis,however,asimpleexplanationforwhichthesecondaryvorticesproducedbythefirstbendareofnoeffect:

Ofbasicimportancefortheflowpastabendistheveloc-itydistributionoftheflowinitsentranceplane.Thus,thesecondarytwin-vorticesareaconsequenceoftheprimaryvelocitydistributionbeinginhomogeneousbutsymmetricalaboutthex—y-plane.Asymmetriesofthevelocityprofileintheentranceaboutthex—y-planewillbynecessityleadtoasym-metriesintheensuingsecondaryflow,i.e.tounequalstrengthsofthecounter-rotatingvortices.Avelocityprofilepastafirstbendisasymmetricinthex󰁀—y󰁀-plane.Forasecondbend,turnedbyanangle0°:󰀁:180°againstthefirstone,thisprofileisasymmetricinitsy—z-planewhichcausesasymmetryintheensuingsecondaryflow.Thisasymmetryisinmostcasesstrongenoughtosuppresstheformationofthesecond,smaller,vortex,leadingalmostimmediatelytoasinglevortex,

Received:21August1996/Accepted:12February1997H.E.FiedlerHermann-Fottinger-InstitutfurStromungsmechanik,TechnischeUniversitatBerlin,D-10623Berlin,GermanyThemeasurementsweredonebyD.Obermann.AdditionalhelpwasprovidedbyDipl.-Ing.R.Nagel.Theircontributionisgratefullyacknowledged.

thesenseofrotationofwhichisobviousfromFig.1b:Theswirlvectoroftheflowpastthesecondbendpointsinthedirectionofthevectorproductoftheprofileexcentricityzandthe

maximumvelocitycofthevelocityprofileinthemy—z-planein

theentranceofthembend(z;c).Thus,asymmetryofthe

primaryvelocityprofiletowardsmmpositivez-value(asinFig.1b)causesformationofaclockwiseswirlintheexitarea(y—z-plane)andviceversa—aconsequenceofthefactthatavorticityvectorretainsitsdirectionwhengoingthroughabend.

2

Experiments

Experimentswerecarriedoutinsupportofthisexplanation.ForthispurposeacommercialplasticpipeofinnerdiameterD:70mmandlengthL:700mmwasfittedwitharoundedinletmountedinsideapressurechamber.Commercialbendswereconnectedtothepipeexit.TheradiusofcurvatureofthecenterlineofthebendsusedinthisexperimentwasR+0.8D:56mm(seeFig.2).Alldataweretakenmean!*

withvelocities(averagedoverthepipecross-section)ofcm+10m/s,correspondingtoRe:cNodetailedvelocitymeasurementsm

D/󰀃+50000.weredoneaswewereonlyinterestedinbulkeffects.ForthispurposeabulkswirlmeterwithopticalRPMcounterwasinsertedinthepipeimmediatelydownstreamofsecondbend.Itconsistedofapaddle-wheeloftworectangularplatesalignedparalleltothepipeaxis,whicharemountedonalightlyrotableshaftinthepipeaxis.TheRPM-valuesS*measuredwiththisinstrumentprovidesonlyacomparativemeasureforthestrengthoftheswirlinthepipe,whichinthiscontextshouldsuffice.InparallelflowS*:0.ThisbulkquantityS*isrelatedtotherealswirlnumberSoftheflow.Anexactrelationshipwas,forlackofdetailedvelocitymeasurements,notevaluated.

FirstweinvestigatedtheflowpasttwosuccessivebendsfordifferentdistancesHbetweenbends.Figure2givesasketchwithdefinitions.ThemeasuredswirlnumberS*versus

angle󰀁isplottedinFig.3.Thisdistributionhasasine-likeappearancewithmaximumvaluesbetween󰀁:<70°and󰀁:<90°dependingonH.

Thefirstbendwasthenreplacedbyanasymmetricallyinsertedpartialdiaphragm,whichcreatesanasymmetricvelocityprofilesimilartothatpastasinglebendwithoutproducingsecondaryvortices.ThearrangementissketchedanddefinedinFig.4.Figures5and6givedistributionsofS*versusangle󰀁󰁀fordifferentrelativedistancesH󰁀/Dandrelativeinsertiondepthsb/Dofthediaphragm.Someofthose

Fig.1a,b.Secondaryflowinabendasaconsequenceofvelocityprofileintheentranceplane.Situationsforasingleandbdoublebend.Coordinatesforsecondbendare(x,y,z);forfirstbend(x󰁀,y󰁀,z󰁀)

Fig.2.Doublebendcombinationanddefinitions

Fig.3.BulkswirlS*pastdoublebendversusangle󰀁fordifferentdistanceH/D

measurementswerethenrepeatedwithaperforateddia-phragmof90%solidity,wheretheswirldistributionwasfoundtobemorelikethatinFig.3(Fig.7).Whenusedwitharotablesleevethisdiaphragmprovidesasimpleandflexiblemeanstoproduceswirlinapipeofvariablestrengthandsignwithoutnecessitationofgeometricalchangesaswouldbeneededinadoublebendarrangement.

Fig.4.Replacementoffirstbendbypartial(asymmetric)diaphragm;definitionsofdiaphragm-bendcombination

Fig.5.BulkswirlS*asfunctionofangle󰀁󰁀andrelativeinsertionb/D;H󰁀/D:1

Fig.6.BulkswirlS*asfunctionofangle󰀁󰁀andrelativeinsertionb/D;H󰁀/D:2

3

Summary

Theinvestigationhasshownthatasimpleexplanationfortheswirlpastadoublebendcanbegiven.Furtherobservationsare:

—Swirlofcomparablestrengthaswithadoublebendisalsoachievedwhenthefirstbendissubstitutedbyasidewayspartiallyinserteddiaphragm

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Fig.7.BulkswirlS*asfunctionofangle󰀁󰁀andvariableb/DandH󰁀/D;perforateddiaphragm

—Perforateddiaphragmscausesmootherswirldistributions,moreakintothedoublebendcase

—Forthepurposeofcreatingswirlofvariablestrengthandsignthediaphragm-bendcombinationoffersmoreflexibilitythanthedoublebendcombination

—Partialsuppressionofswirlasfoundbehindacombinationoftwobendsispossiblebypositioningapartiallyinserteddiaphragmbetweenthetwobends.

References

SchlichtingH(1958)Grenzschichttheorie.Karlsruhe:G.Braun

MillerDS(1978)InternalFlowSystems.BedfordUK:BritishHydro-mechanicsResearchAssociation,pp50—51

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