A generalized relationship for swirl decay in laminar

SˉadhanˉaVol.35,Part2,April2010,pp.129–137.?IndianAcademyofSciences

Ageneralizedrelationshipforswirldecayinlaminarpipe?ow

TFAYINDE

MechanicalEngineeringDepartment,KingFahdUniversityofPetroleumand

Minerals,Dhahran31261,SaudiArabia

e-mail:ayinde@kfupm.edu.sa

MSreceived24July2008;revised28January2010;accepted1February2010

Abstract.Swirling?owisofgreatimportanceinheatandmasstransferenhance-mentsandin?owmeasurements.Inthisstudy,laminarswirling?owinastraightpipewasconsidered.Steadythree-dimensionalaxisymmetricNavier–Stokesequa-tionsweresolvednumericallyusingacontrolvolumeapproach.Theswirlnumberdistributionalongthepipelengthwascomputed.Itwasfoundthattheswirlnumberatanylocationalongthepipelengthdependsontheswirlnumberatinlet,the?owReynoldsnumber,thedistancefromthepipeinlet,thepipediameterandthenatureoftheinletswirl.Ageneralizedrelationshipforswirldecayasafunctionoftheseparameterswasthenobtainedbycurve-?ttingtechnique.

Keywords.

freevortex.

1.Introduction

Theconceptofswirling?owinapipeisimportantbecausetherearenumerousapplicationsinwhichswirliseitherdesiredoritisanuisance.Theuseofswirl?owhasbeenrecognizedasoneofthemostpromisingtechniquesforheattransferaugmentation(Chang&Dhir1995,Bali1998,Li&Tomita1994)aswellasmasstransferenhancement(Yapicietal1994).Ontheotherhand,swirliscreditedwithcausingsigni?canterrorsin?owmeasurementsemployingori?ceplates,nozzles,andventuritubes(Reader-Harris1994,Parchen&Steenbergen1998).Chang&Dhir(1995)experimentallyinvestigatedheattransferenhancementresultingfromintroductionofswirl.Theirresultsshowedthattheheattransfercoef?cientincreasedwithswirlintensity.Thiswasattributedtothehighmagnitudeofmaximumaxialvelocitynearthewall(whichproducedhighrateofheat?uxfromthewall)andhighturbulencelevelinthemiddleregion(whichimproved?owmixing).Similarly,Bali(1998)reportedheattransferenhancement,aswellashigherpressuredrop,inthe?owwhenswirlwasintroduced.Li&Tomita(1994)experimentallyobtainedcorrelations,intermsofswirllevel,forstatic,dynamicandwallpressures.

Inspiteofthesigni?canceofswirlingpipe?owinsomeindustrialprocesses,therearenocleargeneralizedmethodsintheliteraturetocomputethedecayofswirl.Whiletheauthors

129Laminarpipe?ow;axisymmetric;swirlnumber;forcedvortex;

130TF

Ayinde

Figure1.Aschematicviewofthepipeshowingthecoor-dinatesystemandthecomputa-tionaldomain(hatchedplane).

Li&Tomita(1994);Parchen&Steenbergen(1998)correlatedtheobtainedswirldecayasafunctionofaxialpositionnormalizedbypipediameteronly,Reader-Harris(1994)arguedthatswirlwasalsoafunctionofthepipefrictionfactor.Inordertoobtainageneralizedrelationshipforswirldecayinlaminarpipe?ow,numericalcomputationsareperformedforfourdifferenttangentialvelocitydistributionsatpipeinlet,fourdifferentlevelsofinletswirlnumbersandawiderangeofReynoldsnumber(inthelaminarregime).Suchageneralizedformulacanbeusedasinputintheexistingcorrelationsforheattransferenhancementinpipe?ow(Chang&Dhir1995,Bali1998,Li&Tomita1994).Itcanalsoserveasapredictivetoolfordeterminingthedownstreamlocationwheretheswirlwouldhavereducedtosuchanacceptablelevelthat?owmeasurementcanbeperformedwithstandard?owmeteringdevices.

2.Mathematicalmodelling

2.1Flowdomain

Thesteadylaminarincompressibleandaxisymmetric?owinastraightconstant-diameterpipeisconsidered.Theschematicviewofthecomputationaldomainisshownin?gure1.Thepipelengthanddiameterareselectedas6·4mand80mmrespectively,givingalength-to-diameterratioofL/D=80.

2.2Governingequations

Thegoverningequations,incylindricalcoordinatessystem,areasfollows:

Continuityequation:?VV?U++=0.?x?rr

Themomentumequationinthexdirection:

????21?U?U1?p?U?2U?U.++V=?+ν+U?x?rρ?x?x2r?r?r2

Themomentumequationintherdirection:

?VW21?p?V+V?=?+νU?x?rrρ?r??1?V?2V?2VV+?+?xr?rr?r??.(3)(1)(2)

Ageneralizedrelationshipforswirldecayinlaminarpipe?ow

Themomentumequationintheφdirection:

??2???WVW?WW1?W?2W?W+V+=ν?2+.+U?x?rr?x2r?rr?r2

2.3Boundaryconditions131(4)

Atinlet,aswirlcomponentissuperimposedonPoiseuille?ow.Theswirlisformedthroughacombinationofforcedvortexinthecoreandfreevortexintheannulus.Thisissimilartothedistributionsthatwereexperimentallyrealizedinthepreviousstudies(Bali1998,Parchen&Steenbergen1998).Thus,theboundaryconditionsatpipeinlet(x=0)are:

????r??2??.(5)U(0,r,φ)=Umax1?R

V(0,r,φ)=0.

??(6)

(7)Wmaxtrans,r<rtrans????W(0,r,φ)=,r≥rtrans.Wmaxtrans

Inequation(7),rtransistheradiallocationatwhichtransitionfromforcedtofreevortextakesplace.Thebracketedterminthefreevortexvelocitydistributionisincludedtoensurethatthevelocitygoestozeroonthepipewall.

Atexit(x=L),afully-developed?owconditionwasassumed.Thenoslipcondition(U=V=W=0)wasimposedatthepipewall.

2.4Swirlanalysis

Asuitablemeasureoftheswirlinpipe?owistheswirlnumber,de?nedastheratioofthetotal?uxofangularmomentumtotheaxialmomentum?ux(Bali1998,Parchen&Steenbergen1998).Itisexpressedasfollows:

??R2π0U(rW)rdr.(8)S=2πR3Uav

3.Methodofsolution

Thedynamicsofcon?nedswirl?owisacomplexonebecauseoftheco-existenceoftheaxialandtangentialcomponentsofvelocitiesatanypointofthe?ow?eldandtheboundarylayersatwallarethree-dimensional.The?owisthereforenoteasilyamenabletoanalyticalsolution,exceptinthecoreregionwherethe?owcanbeconsideredtobeinvscid.Here,applicationofEuler’sequation(Croweetal2005)showsfavourablepressuregradienttowardsthevortexcenter.Thisresultsintheaccelerationoftheradial?owtowardsthecenter,which,inturn,leadstodecelerationoftheaxial?owinordertosatisfycontinuityequation.Asthestrengthoftheswirlweakensdownstreamduetoviscousdissipation,theoriginalPoiseuillepro?leoftheaxial?owisgraduallyrecovered.

Themodelequations1–7weresolvednumerically.Duetotheaxisymmetric?owsituation,thecomputationaldomainbecomestwo-dimensionalandthesymmetryaxispassesthrough

132TFAyinde

thepipecenterasshownin?gure1.Arectangulargridsystemwasemployedandthegrid-independencetestwasconducted,whichyieldedagridindependentsolutionfor40×600gridpoints.

Computationwasmadeusingthe?nitevolumemethod.Astaggeredgridarrangementwasusedinordertopreventa‘checkerboard’pressure?eld(Patankar1980,Versteeg&Malalasekera1995).TheSIMPLEalgorithm(Patankar1980,VersteegandMalalasekera1995)wasusedtoobtainthenumericalsolution.

DifferentinletswirlnumberswererealizedbyvaryingthevaluesofWmaxinequation(7).Theeffectofthenatureoftheinletswirldistributionwasinvestigatedbyvaryingthevalueofrtrans.

4.Resultsanddiscussions

Inthepresentstudy,thedecayofswirlinlaminarpipe?owwasinvestigatednumerically.Thepipedimensionswereselectedas80mmdiameter,6·4mlength.Computationswereper-formedforsixdifferentReynoldsnumbers(Re=of800,1000,1200,1400,1600,1800)andfourinletswirlnumbers(So=0·5,1·0,1·5,2·5),withfourdifferentinlettangentialvelocitydistributions(rtrans/ro=0·5,0·6,0·75,0·9).Intheresultstobepresentedin?gures2–6,thesymbolsareincludedforvisualaidonly;theydonotrepresentthenumberofnodesusedinthecomputations.

TheaxialvelocitydistributionsinthepipeforswirlnumberSo=1·0,Re=1000andrtrans/ro=0·75arepresentedin?gure2,whereitisrevealedthatthefully-developed(Poiseuille)velocitydistributionatinletisaltereddownstreamduetotheintroductionofswirl.Thisisconsistentwiththeobservationmadeatthebeginningofsection3.The?owgraduallyrecoversfromthedestabilizingeffectofswirl(occasionallybyadversepressuregradientintheaxialdirection)andtheinitialpro?leisalmostfullyrecoveredatthepipe

exit.

Figure2.Axialvelocitydistri-butionsinthepipeforSo=1·0,Re=1000andrtrans/ro=0·75.

Ageneralizedrelationshipforswirldecayinlaminarpipe?ow

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Figure3.Tangentialvelocitydis-tributionsinthepipeforSo=1·0,Re=1000andrtrans/ro=0·75.

SimilartrendsareobtainedforothervaluesofSo,Reandrtrans/robutarenotshownhere.Theactualfully-developedpro?lewillberecoveredonlyiftheswirlcompletelydisappearsandthisrequiresanin?nitelylongpipe.Apreliminaryinvestigationrevealedthatthecom-putedswirldecay,whichisthesubjectofthisstudy,isnotaffectedbythe?nitelengthofthecomputationaldomainifL/D>60.ThevalueofL/D=80usedinthisstudyisthereforeconsideredtobe