Measurement of Lagrangian velocity in fully developed turbul

We have developed a new experimental technique to measure the Lagrangian velocity of tracer particles in a turbulent flow, based on ultrasonic Doppler tracking. This method yields a direct access to the velocity of a single particule at a turbulent Reynold

MeasurementofLagrangianvelocityinfullydevelopedturbulence

N.Mordant(1),P.Metz(1),O.Michel(2),J.-F.Pinton(1)

CNRS&LaboratoiredePhysique,´EcoleNormaleSup´erieure,

46all´eed’Italie,F-69007Lyon,FranceLaboratoired’Astrophysique,Universit´edeNice

ParcValrose,F-06108,Nice,France

(1)

(2)

arXiv:physics/0103084v2 [physics.flu-dyn] 30 Jul 2001

WehavedevelopedanewexperimentaltechniquetomeasuretheLagrangianvelocityoftracerparticlesinaturbulent ow,basedonultrasonicDopplertracking.ThismethodyieldsadirectaccesstothevelocityofasingleparticleataturbulentReynoldsnumberRλ=740.Itsdynamicsisanalyzedwithtwodecadesoftimeresolution,belowtheLagrangiancorrelationtime.Weobserve

2

thattheLagrangianvelocityspectrumhasaLorentzianformEL(ω)=u2rmsTL/(1+(TLω)),inagreementwithaKolmogorov-likescalingintheinertialrange.Theprobabilitydensityfunction(PDF)ofthevelocitytimeincrementsdisplaysachangeofshapefromquasi-Gaussianaintegraltimescaletostretchedexponentialtailsatthesmallesttimeincrements.Thisintermittency,whenmeasuredfromrelativescalingexponentsofstructurefunctions,ismorepronouncedthanintheEulerianframework.

PACSnumbers:47.27.Gs,43.58.+z,02.50.Fz

Lagrangiancharacteristicsof uidmotionareoffun-damentalimportanceintheunderstandingoftransportandmixing.Itisanaturalapproachforreacting owsorpollutantcontaminationproblemstoanalyzethemotionofindividual uidparticles[1].Anothercharacteristicofmixing owsistheirhighdegreeofturbulence.Forprac-ticalreasons,mostoftheexperimentalworkconcerninghighReynoldsnumber grangianmeasurementsarechal-lengingbecausetheyinvolvethetrackingofparticletra-jectories:enoughtimeresolution,bothatsmallandlargescales,isrequiredtodescribetheturbulent uctuations.EarlyLagrangianinformationhavebeenextractedfromthedispersionofparticles,followingTaylor’sap-proach.Recentlynumericalandexperimentalstudieshavefocusedonresolvingthemotionofindividual uidortracerparticles.Theemergingpictureisasfollows.Theone-componentvelocityauto-correlationfunctionisquasi-exponentialwithacharacteristictimeoftheorderoftheenergyinjectionscale[2,3,4].ThevelocitypowerspectrumisexpectedtohaveascalingEL(ω)∝ω 2,asrecentlyreported[5,6]andexpectedfromaKolmogorovsimilarityarguments.Inthesamespirit,thesecondor-L

derstructurefunctionshouldscaleasD2(τ)=C0 τ,where isthethepowerdissipation.Measurementsofatmosphericballoons[7]havegivenC0=4±2andalimitC0→7hasbeensuggestedinstochasticmodels[8].Recentexperiments[9,usinghighspeedopticaltech-niqueshaveshownthatthestatisticsoftheLagrangianaccelerationarestronglynon-Gaussian.

Wehavedevelopedanewexperimentalmethod,basedonsonartechniques[11],inordertostudyinalaboratoryexperimenttheLagrangianvelocityacrosstheinertialrangeoftimescales.Weobtainthe rstmeasurementofsingleparticlevelocityfortimesuptothe owlargescaleturnovertime,athighReynoldsnumber.InthisLetter,wereporttheresultsofthismeasurementsandcomparewithpreviousobservationsandnumericalpredictions.Ourtechniqueisbasedontheprincipleofacontinu-ousDopplersonar.Asmall(2mm×2mm)emittercon-tinuouslyinsoni esthe owwithapuresinewave,atfrequencyf0=2.5MHz(inwater).Themovingparticlebackscatterstheultrasoundtowardsanarrayofreceiv-ingtransducers,withaDopplerfrequencyshiftrelatedtothevelocityoftheparticle:2π f=q.v.Thescatter-ingwavevectorqisequaltothedi erencebetweentheincidentandscattereddirections.Anumericaldemod-ulationofthetimeevolutionoftheDopplershiftgivesthecomponentoftheparticlevelocityalongthescat-teringwavevectorq.Itisperformedusingahighresolu-tionparametricmethodwhichreliesonanApproximatedMaximumLikelihoodschemecoupledwithageneralizedKalman lterThestudyreportedhereismadewithasinglearrayoftransducerssothatonlyoneLagrangianvelocitycomponentismeasured.

Theturbulent owisproducedinthegapbetweentwocounter-rotatingdiscs[12].Thissetuphastheadvan-tagetogenerateastrongturbulenceinacompactregionofspace,withnomeanadvection.Inthisway,parti-clescanbetrackedduringtimescomparabletothelargeeddyturnovertime.DiscsofradiusR=9.5cmareusedtosetwaterintomotioninsideacylindricalvesselofheightH=18cm.Toensureinertialentrainment,thediscsare ttedwith8bladeswithheighthb=5mm.Inthemeasurementreportedhere,thepowerinputis =25W/kg.Itismeasuredontheexperimentcool-ingsystem,fromtheinjection-dissipationbalance.TheintegralReynoldsnumberisRe=R2 /ν=6.5104,where istherotationfrequencyofthediscs(7.2Hz),andν=10 6m2/sisthekinematicviscosityofwa-ter.AconventionalturbulentReynoldsnumbercanbecomputedfromthemeasuredrmsamplitudeofveloc-

We have developed a new experimental technique to measure the Lagrangian velocity of tracer particles in a turbulent flow, based on ultrasonic Doppler tracking. This method yields a direct access to the velocity of a single particule at a turbulent Reynold

ity uctuations(urms=0.98(λ=

m/s)andanestimateoftheTaylormicroscaleν/ =0.2ms)

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