Study-on-the-Vortex-Ring-and-Its-Phenomena

FluidDynamicsTermPaperRuobingBai1StudyontheVortexRingandItsPhenomenaRuobingBaiAbstract:Inthispaper,webeginwiththebasicideaofthevortexfilament.ByusingtheBiot-SavartLaw,wetherebydeducetheinducedvelocityfieldofthevortexfilament.Thisresultleadstoadiscussiononthemotionofvortexrings,andwefinallysuccessfullyexplainthephenomenaofvortexringmotioninatheoreticalway.Keywords:vortexring;Biot-Savartlaw1.IntroductionAvortexring,alsocalledatoroidalvortex,isadoughnutshapedvortexinafluid;thatis,aregionwherethefluidmostlyspinsaroundanimaginaryaxislinethatformsaclosedloop[1].Vortexringscanbeidentifiedinmanysituations,fromstartingjetstovolcaniceruptionsorthepropulsiveactionofsomeaquaticcreatures,aswellasthedischargeofbloodfromtheleftatriumtotheleftventricularcavityinthehumanheart[2].Besidesthese,perhapsthemostcommonvortexringsweseeindailylifearetheringsthatoccurwhenadropofcoloredliquidfallsintoacupofwater,orthesmokeringsexpelledfromasmoker’smouth.Figure1.Left:smokeringscreatedbyacigarettesmoker;Right:anirregularbubbleringcreatedbyadolphin.Inanengineer’sinterest,onemightcitethefactthatcavitatedringsareusedforunderwaterdrilling,havepotentialuseinfightingoilwellfiresandareusedinmodelingthedownburst,ahazardtoaircraft[3].Therefore,thevortexringisaverycommonphenomenoninnatureandhasmanypotentialusagesorharms(suchasthedownburstandthecauseofturbulence),whichhasbeenstudiedbyscientistsandengineersfornearlyacentury.Inthispaper,we’lldiscussaveryinterestingphenomenonthatinvolvesthemotionoftwocoaxialrings.ByintroducingtheideaofvortexfilamentandusingtheBiot–SavartLawinfluidmechanics,wecanfinallydeducethevelocityfieldandexplainittheoretically.2.ModelBuildingandCalculationFluidDynamicsTermPaperRuobingBai22.1VortexFilamentandtheBiot–SavartLawConsideranidealflowwithnoexternalforce.TaketheRankinevortexmodelasanexample.Theoutsideflowfieldisirrotationalbutthecirculationonthecurvesurroundingthecentralpointisnonzero.Inthiscase,we’veknownwecanapproximatethattheoutsideflowiscausedbyavortextubeat0rwhichhasaconstantvorticityvalue.Similarly,inalotofrealworkingsituations,vorticityisoftendistributedwithinacertainvolume.Ifsuchavolumeisaverythinvortextube,withacross-sectionallengthmuchsmallerthanthecharacteristicscale,wecanconsideritasavortexfilament.Assumethevorticityperunitvolumeis.Consideravolumedwithareaandlengthdl.Thenddldlwheredlistheunitlengthvector,whichhasamagnitudeofdlandadirectionsameasthevorticity.Thestrengthofvortexfilamentisdefinedas0limFigure2.AnexampleofthevortexfilamentByusingtheRankinevortexapproximationideainamoregeneralway,ifgivenacertaindistributionofvortexfilament,wecansaytheflowfieldisgeneratedbythevortexfilament.Asimilarmethodofdoingsuchaproblemisstudiedinelectromagnetism,whichgivestheBiot-SavartLawbyanalogy[4]:34ddrlrvTheBiot-SavartLawtellsusthevelocitydvcausedbythecurvedvortexfilamentdl.Inthisway,wecancalculateanyvelocityfieldifweknowthepropertyofthevortexfilamentoftheflow.2.2VelocityGeneratedbyaVortexFilamentCurveFluidDynamicsTermPaperRuobingBai3Figure3.ThemodelcoordinatesConsiderageneralvortexfilament,whichhasaconstantstrength.NowwecalculatethevelocitygeneratedbythefilamentatapointPnearby.Thecoordinatesarechosenas123Oxxx,asshowninFigure3.,,tnbaretheunitvectorsofthetangent,normalandbinormal,respectively.AssumePisonthenormalplanethatcontainsO.Then23pxxrnbAmovingpointMonthevortexfilamenthasapositionvector21cossin2MllllrtntnwhereisthecurvatureofthefilamentatpointO.Therefore,()dldlltn2231()2pMlxlxrrrtnbPlugthisintotheBiot-SavartLawwecanget2332222243/22321()214[(1)]4xlxxlddlxxlxltnbvSincewearedealingwiththepointPnearthevortexfilament,we’llnotworryaboutthecontributionmadebythevortexfilamentelementsfarawayfromO,whichissmallenoughcomparedwiththecontributionfromthenearbyfilamentelements.Withthisidea,wechooseacharacteristiclengthL.Lisonlyusedtoanalyzetheresultmoreconveniently,anditsvalueisundetermined.Therefore,theintegralregionwetakehereisLlL.Let22223xx,lm,andconsider2cosxand3sinx,wehaveFluidDynamicsTermPaperRuobingBai411222243/21sinsincos214[1(1cos)]4LLmmdmmmtnbbvNow,ifwewanttoknowthevelocityofthefluidclosetothevortexfilament,we’lllet0anduseadominantbalanceweknowthebottompartoftheintegralwillapproximate23/2(1)m(Infact,wedonotneedtoworryaboutthisapproximationsincewecanalwaysselecttheproperLfordifferenttomakethedominantbalanceconsistent.).Integrateit,wehave21/2121/221/221/21[(1)sin(1)(cossin){(1)ln[(1)]}]42LLmmmmmmmvtbnbNotethat/22/ln(1)ln(2)2lnConst.LLLLmmWecanfinallygettheapproximateinducedvelocity(cossin)lnConst.24Lvbnb3.AnalysisoftheResult3.1SingleVortexFilament/VortexRingInthefirstsightoftheresult,wemaypointoutthatwhen0,thevelocityoffluidonthevortexfilament,orsaying,thevelocityofthefilamentwillgotoinfinity.Thisismainlybecausewechosethethicknessofthefilamenttobezero,whileintherealsituationtherewillalwaysbeavortextubewithnonzerothicknessinsteadoftheidealfilament.Thus,wedonotneedtowor