Streamline topology and dilute particle dynamics i

arXiv:physics/0411191v2[physics.flu-dyn]23Nov2004StreamlinetopologyanddiluteparticledynamicsinaK´arm´anvortexstreetflowZuo-BingWuStateKeyLaboratoryofNonlinearMechanics(LNM),InstituteofMechanics,AcademiaSinica,Beijing100080,China(February2,2008)AbstractThreetypesofstreamlinetopologyinaK´arm´anvortexstreetflowareshownunderthevariationofspatialparameters.ForthemotionofdiluteparticlesintheK´arm´anvortexstreetflow,thereexistarouteofbifurcationtoachaoticorbitandmoreattractorsinabifurcationdiagramfortheproportionofpar-ticledensitytofluiddensity.Alongwiththeincreaseofspatialparametersintheflowfiled,thebifurcationprocessissuspended,aswellasmoreandmoreattractorsemerge.Inthemotionofdiluteparticles,adragtermandgravitytermdominateandresultinthebifurcationphenomenon.PACSnumber(s):47.52.+j,47.32.Cc,05.45.-aTypesetusingREVTEX1I.INTRODUCTIONThemotionsofparticlesinanonuniformflowhavewidetechnologicalapplications,suchastoforecastchemicalreactionsandenvironmentalpollution.DuetoparticlemotioninthelowReynoldsnumbercategory,theequationofmotionforasmallrigidsphereinanonuniformflowfieldisdeduced[1].Whenthebackgroundflowismainlydominatedbylargescalestructures,thefluidviscosityisnotincludedinthegoverningflowequation[2].Relatedstudiesshowthatevenwhenthebackgroundflowfieldsareverysimple,themotionscanhaveabundantphenomena.InaperiodicStuartvortexflow,dependingonthevaluesofparameters,theparticlesasymptoticallyconcentratealongperiodic,quasiperiodicorchaoticopentrajectory[3,4].Inacellularflowfield,aerosolparticlesalsomergeintoisolatedasymptotictrajectories,whicharedescribedbyslowmanifolds[5,6].Moreover,themethodduetotheLagrangeviewofparticlescanalsobeappliedtoinvestigateeffectsofparticledispersiononstreamwisebraidvorticesinaplanemixinglayer[7,8].Inaplanewakeflowbehindacircularcylinder,aregularvortexstreetstructurewasinvestigatedatRe=60−5000.Alongwiththeadvanceofexperimentaltechnique,evenatRe=O(104),theregularvortexstreetisobtainedbyusingthephase-averagemethod[9].Inparticular,thephenomenarelatingtoorganizedvortexstructure,suchas,reconnectionofvortexstreet[10]andemergenceofthreedimensionalvortexstructure[11],arousewideinterestforthetransitionofaplanewakeflow.Recently,particlefocusinginnarrowbandsneartheperipheriesofthevortexstructuresfortheparticledispersioninaplanewakeflowisobservedexperimentally[12].ByconsideringStokesdrag,thephenomenonofparticlefocusingisstudiedontwodimensionalcentremanifolds[13].TheregularK´arm´anvortexstreetflowasamodeltoapproachtheplanewakeflowandinvestigatetheabovephenomenaplaysanimportantrole.Forthemotionofparticlesinregularvortexstreetflow,K´arm´anvortexspacinginfluencesontopologicalstructureofbackgroundflowfield.Atthesametime,adensityratioasabasicparametermayhaveawiderange.Inthispaper,wewillconsiderstreamlinetopology2anddiluteparticledynamicsintheK´arm´anvortexstreetflowinarangeofdensityratio.InSect.II,itisshownthatfortheK´arm´anvortexstreetflow,thereexistthreetypesofglobaltopologicalstructuredependingonthespatialparametersinflowfield.DiluteparticledynamicsintheK´arm´anvortexstreetflowrelatedtothedensityratioisinvestigatedinSect.III.EffectsofspatialparametersinflowfieldondiluteparticledispersionisdeterminedinSect.IV.Finally,abriefsummaryisgiveninSect.V.II.STREAMLINETOPOLOGYThestreamfunctionofK´arm´anvortexstreetflow[14]isΨ(x,y)=Γ4πlnch2πl(y−h/2)−cos2πlxch2πl(y+h/2)+cos2πlx+Γy2lthπhl,(1)whereΓisthestrengthofvortices,landharethestreamwiseandtransversespacingofvortices,respectively.Thedimensionlessquantitiesdenotedbyasterisksareintroducedasx∗=x/l,y∗=y/l,h∗=h/l,u∗=u/U∞,Γ∗=Γ/(U∞l)andΨ∗=Ψ/(U∞l).Thestreamfunction(1)canberepresentedasΨ∗(x∗,y∗)=Γ∗4πlnch2π(y∗−h∗/2)−cos2πx∗ch2π(y∗+h∗/2)+cos2πx∗+Γ∗y∗2thπh∗.(2)Fromnowon,theasterisks”*”forthedimensionlessquantitiesinthissectionareomittedforconvenience.Thestreamfunction(2)hassymmetries:Ψ(x+1/2,−y)=−Ψ(x,y),Ψ(x+1,y)=Ψ(x,y).Theassociatedvelocityfiledisgivenbyux=∂Ψ∂y=Γ2[sh2π(y−h/2)ch2π(y−h/2)−cos2πx−sh2π(y+h/2)ch2π(y+h/2)+cos2πx]+Γ2thπh,uy=−∂Ψ∂x=−Γ2[1ch2π(y−h/2)−cos2πx+1ch2π(y+h/2)+cos2πx]sin2πx,(3)whichhassingularityatthevortexcenters.InthecaseofremovingthesingularitybytheRankinevortex[15],freestagnationpointsinvelocityfield(3)consistofcentersandsaddlepoints,whichcanberespectivelydescribedby(0,h2),(12,−h2)and(0,−12πlnc,(12,12πlnc),wherec=2/shπh+shπh+q[2/shπh+shπh]2+1.3InthesymmetryΨ(x+1/2,−y)=−Ψ(x,y),itcanbeshownthatthereexistsazerostreamlinebetweentwoneighboringcenters(0,h2)and(12,−h2).Underthevariationofspatialparameterh,wetakezerostreamlinestoinvestigatetheevolutionofglobaltopologicalstructure.Thepointsy0,correspondingtozerostreamlinespassingthroughtheycoordinate,satisfyΨ(0,y0)=Γ4πlnch2π(y0−h/2)−1ch2π(y0+h/2)+1+Γy02thπh=0.(4)UsingtheNewton-Raphsonandbisectionmethods[16],wesolveEq.(4)andplottherelationy0∼hinFig.1.Whenh=0,theK´arm´anvortexstreetreturnstoaperiodicrowofvortexpair(ux=Γcos2πxsh2πych22πy−cos22πxanduy=−Γsin2πxch2πych22πy−cos22πx).Inthiscase,azerostreamlineexistsiny0=∞,aswellasfreestagnationpointsareonlycenters.Allstreamlinesincludinganembeddedzerostreamlineforh=0.001,whichapproachtothoseforh=0,aredrawninFig.2