UNIVERSIDAD CARLOS III DE MADRID EscuelaPolitécnicaSuperior ABSOLUTE INSTABILITY IN THE NEAR FIELD OF LOW-DENSITY JETS Tesis Doctoral Autor WilfriedCoenen Director AlejandroSevillaSantiago DEPARTAMENTODEINGENIERÍATÉRMICAYDEFLUIDOS Leganés,abril 2010 DEPARTAMENTODEINGENIERÍATÉRMICAYDEFLUIDOS EscuelaPolitécnicaSuperior ABSOLUTE INSTABILITY IN THE NEAR FIELD OF LOW-DENSITY JETS Autor WilfriedCoenen DirectordeTesis AlejandroSevillaSantiago Leganés,abril 2010 Tomyparents,mybrother,andtoAna Tesisdoctoral ABSOLUTEINSTABILITYINTHENEARFIELDOFLOW-DENSITYJETS Autor:WilfriedCoenen DirectordeTesis:AlejandroSevillaSantiago FirmadelTribunalCalificador: Firma Presidente: D.NormanRiley Vocal: D.RamónFernándezFeria Vocal: D.JoséManuelGordilloAriasdeSaavedra Vocal: D.CarlosMartínezBazán Secretario: D.AntonioLuisSánchezPérez Suplente: D.EnriqueSanmiguelRojas Suplente: D.CarlosdelPinoPeñas Calificación: Leganés,5 de abril de 2010 Abstract Variabledensityjetsareknowntosupportself-sustainedoscillationswhenthejet-to- ambientdensityratioissufficientlysmall. Thischangeindynamicalresponsetosmall perturbationsisassociatedwithatransitionfromconvectivetoabsoluteinstabilityof theunderlyingunperturbedbaseflow. Thefocusofthisdissertationliesintheuseof linear stability theory to describe the convective to absolute instability transition of buoyancy-free low-density jets emerging from a circular injector tube at moderately highReynoldsnumbersandlowMachnumbers. Particularinterestisgiventothein- fluenceofthelengthoftheinjectortubeonthestabilitycharacteristicsoftheresulting jetflow,whosebasevelocityprofileatthejetexitiscomputedintermsofthenondimen- sionaltubelengthLt byintegratingtheboundarylayerequationsalongtheinjector. Webeginwiththeinvestigationofinviscidaxisymmetricandhelicalmodesofin- stabilityinaheatedjetfordifferentvaluesofthejet-to-ambientdensityratio.Forshort tubesLt ≪ 1thebasevelocityprofileatthetubeexitisuniformexceptinathinsur- roundingboundarylayer. Correspondingly,thestabilityanalysisreproducesprevious resultsofuniformvelocityjets,accordingtowhichthejetbecomesabsolutelyunstable toaxisymmetricmodesforacriticaldensityratioSc ≃ 0.66,andtohelicalmodesfor Sc ≃ 0.35. Fortubesofincreasinglengththeanalysisrevealsthatbothmodesexhibit absolutelyunstableregionsforallvaluesofLt andsmallenoughvaluesofthedensity ratio. Inthecaseofthehelicalmode,wefindthatSc increasesmonotonicallywithLt, reachingitsmaximumvalueSc ≃0.5astheexitvelocityapproachesthePoiseuillepro- fileforLt ≫1.Concerningtheaxisymmetricmode,itsassociatedvalueofSc achieves amaximumvalueSc ≃ 0.9forLt ≃ 0.04andthendecreasestoapproachSc ≃ 0.7for Lt ≫1.Theabsolutegrowthratesinthislimitingcaseofnear-Poiseuillejetprofilesare howeverextremelysmallform =0,inagreementwiththefactthataxisymmetricdis- turbancesofajetwithparabolicprofileareneutrallystable.Asaresult,forS <0.5the absolutegrowthrateofthehelicalmodebecomeslargerthanthatoftheaxisymmetric modeforsufficientlylargevaluesofLt,suggestingthatthehelicalmodemayprevailin theinstabilitydevelopmentofverylightjetsissuingfromlonginjectors. Asecondpartofthisdissertationisdevotedtotheviscouslinearinstabilityofpar- allelgasflowswithpiecewiseconstantbaseprofilesinthelimitoflowMachnumbers, bothforplanarandaxisymmetricgeometriessuchasmixinglayers,jetsandwakes.Our resultsgeneralizethoseofDrazin(J.FluidMech.vol.10,1961,p.571),bycontemplating i ii Abstract thepossibilityofarbitraryjumpsindensityandtransportpropertiesbetweentwouni- formstreamsseparatedbyavortexsheet. Theeigenfunctions,obtainedanalyticallyin theregionsofuniformflow,arematchedthroughanappropriatesetofjumpconditions atthediscontinuityofthebasicflow,whicharederivedbyrepeatedintegrationofthe linearized conservation equations in their primitive variable form. The development leadstoanalgebraicdispersionrelationthatisvalidatedthroughcomparisonswithsta- bilitycalculationsperformedwithcontinuousprofilesandisapplied,inparticular,to studytheeffectsofmoleculartransportonthespatiotemporalstabilityofparallelnon- isothermalgaseousjetsandwakeswithverythinshearlayers. Finallywegobacktothestabilityanalysisoflow-densityjetsemergingfromcircu- larnozzlesortubes,thistimeconsideringviscousperturbationssothattheReynolds numberentersthestabilityproblem.Weconsiderseparatelythetwoparticularcasesof ahotgasjetdischargingintoacolderambientofthesamegas,aswellastheisothermal dischargeofajetofgaswithmolecularweightsmallerthanthatoftheambientgas.In bothcases,weconsiderthedetaileddownstreamevolutionofthelocalstabilityprop- ertiesinthenearfieldofthejetwiththeaimatestablishingtheconvectiveorabsolute natureoftheinstability.Wediscusstherelationshipofourresultswiththoseobtainedin previousworkswithusemadeofparametricvelocityanddensityprofiles,andcompare bothapproacheswiththeactualglobaltransitionobservedinexperimentsperformed withhotandlightjets.