A generalized relationship for swirl decay in laminar(2)

adequate.

Figure4.DecayoftheswirlnumberalongthepipeforSo=1·0andrtrans/ro=0·75.

134TF

Ayinde

Figure5.DecayoftheswirlnumberalongthepipeforRe=1000andrtrans/ro=0·75.

Figure3showsthedistributionsofthetangentialvelocityinthepipe.TheFigurerevealsthatasthe?owprogressesdownstream,theswirldecays,thecoreregion(forforcedvortex)shrinkswhiletheannularregion(forfreevortex)expandsThistrendwasalsoreportedbyChang&Dhir(1995)andBali(1998)forturbulentswirling?ow.

Thevariationoftheswirlnumberalongthepipelengthisshownin?gure4forReynoldsnumbersfrom800to1800andSo=1·0.Itcouldbeseenfromthe?gurethatafteran

initial

Figure6.DecayofswirlalongthepipeatRe=1200andSo=1·0fordifferentdistributionsofinlettangentialvelocity.

Ageneralizedrelationshipforswirldecayinlaminarpipe?ow135rapiddecayoftheswirl,itthencontinuestodecayexponentiallytowardsthedownstream.The?gurealsoshowsthattherateofdecaydecreasesastheReynoldsnumberincreases.TheswirldecayatRe=1000forfourlevelsofinletswirlnumbersareshownin?gure5.The?gureshowsthattheswirlnumberatpipeinletdoesnotaffecttheswirldecayrate,butitsmemorypersistsinde?nitely.Figure6showsthedecayofswirlalongthepipefordifferentinlettangentialvelocitydistributions.Here,itisrevealedthattheswirlnumberatanydownstreamlocationdependsonthenatureoftheinlettangentialvelocitydistribution.From?gures4to6itisapparentthataftertheinitialnon-lineardecay,theswirldecayslinearlyonasemi-logplot.Thislinearportionisfoundtobestartingatx/D=16inalltheinletconditionsaccommodatedinthesimulations.Theswirldistributioncan,therefore,bemodelledintheformgivenbelow:

ln(S/So)=lnC?mx/D.(9)Byusinglinearregressionanalysis(Montgomery1997),theconstantsCandmweredetermined.ItwasfoundedthatmisafunctionofReonlyanditsvariation?tsonapowerfunction.CvarieswithSo,Reandrtrans/ro.ForeachpairofRe,rtrans/ro,thedependenceofConSowasinvestigated,andapowerfunctionwaschosenasthebest?t.TheconstantsinthepowerfunctionweremodelledascombinationsoflinearfunctionsofReandrtrans/ro.Theseweredeterminedbylinearregression.Theequationforswirldecayis,therefore,presentedasfollows:

S/So=Ce?mx/D

wheremandCarede?nedas

m=25Re?0·92

BC=ASo.(10)(11)(12)

(13)

(14)A=7×10?5Re?0·78(rtrans/ro)+1·2.B=2×10?5Re?0·17.

Startingfromx/D=16,theswirldistributionde?nedbyequations(10)–(14)hasbeenfoundtomatchtheresultsofournumericalcomputation(forallSo,Reandrtrans/ro)toamaximumerrorof1%.

5.Conclusions

Theswirldecayinlaminarpipe?owwithinletswirlhasbeenexaminedthroughanumericalcomputationofthe?ow?eldforfourdifferentinletswirlnumbers,sixvaluesofReynoldsnumberandfourdifferenttangentialvelocitydistributionsatpipeinlet.Theswirlnumberdistributionalongthepipewascomputed.Ageneralizedrelationshipforswirldecaywasthenobtainedbycurve-?ttingtechnique.Thespeci?cconclusionsderivedfromthepresentstudymaybelistedasfollows:

(i)Theintroductionofswirlintoafully-developedlaminarpipe?owdistortstheusualparabolicvelocitypro?leinthepipe.Thepro?leisgraduallyrecoveredasswirldecaystowardsdownstream.

136TFAyinde

(ii)Fromthetangentialvelocitypro?le,theswirl?owcanbedividedintoacoreregionand

anannularregion,characterizedbyforced-vortexandfree-vortextypes,respectively.Asthe?owprogressestowardsthedownstream,thecoreregionshrinks(reducesinsize)whiletheannularregionoffreevortexexpands.

(iii)Theswirlnumberatanylocationinthedownstreamdependsontheinletswirlnumber,

the?owReynoldsnumber,thedistancefromthepipeinlet,thepipediameterandthenatureoftheinlettangentialvelocitydistribution.Theswirldecaysexponentiallyalongthepipelengthstartingfromx/D=16.Thedecayiscorrelatedwithageneralizedrelationshipasde?nedbyequations(10)–(14).

ThesupportprovidedbytheKingFahdUniversityofPetroleumandMineralsincompletingthisworkisacknowledged.TheauthorisindebtedtoProf.BSYilbasforhisencouragementandguidancethroughoutthework.

Listofsymbols

DLPRrRe

SUVWxPipediameter[m]Pipelength[m]Pressure[Pa]Piperadius[m]radialcoordinate[m]Reynoldsnumber(=UD/ν)Swirlnumberaxialvelocitycomponent[m/s]radialvelocitycomponent[m/s]tangentialvelocitycomponent[m/s]axialcoordinate[m]

Greeksymbols

φ

ρνcircumferentialcoordinate?uiddensity[Kg/m3]

kinematicviscosity[m2/s]

Subscripts

av

max

o

transaveragemaximuminlettransitionpoint(fromforcedtofreevortex)

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