Measurement of Lagrangian velocity in fully developed turbul(3)

～40inagreementwithaccelerationmea-surementsin[10].

10

L

q

D10

10

DL

10

FIG.5:ESSplotsofthestructurefunctionvariation(indou-blelogcoordinates).Thesolidcurvesarebestlinear tswith

slopesequaltoξL

q=0.56±0.01,1.34±0.02,1.56±0.06,1.8±0.2forp=1,3,4,5fromtoptobottom.Coordinatesinarbitraryunits.

Moregenerally,onecanchoosetodescribetheevolu-tionofthePDFsbythebehavioroftheirmoments(or

‘structurefunctions’)DL

changeofq(τ)= |δτv|q .Indeed,acon-sequenceoftheshapeofthePDFswithscaleisthattheirmoments,asthe atnessfactorabove,varywithscale.ClassicallyintheEulerianpicture,oneex-pectsscalingintheinertialrange,DE

q(r)∝rζq,atleastinthelimitofverylargeReynoldsnumbers.Atthe -niteReynoldsnumberwheremostexperimentsaremade,thelackofatrueinertialrangeisusuallycompensatedbystudyingtherelativescalingofthestructurefunc-tions–theESSansatz[16].Weusethesecondorderstructurefunctionasareference.Indeedthedimensional

estimationofDL2(asthatofDE

3)dependslinearlyontheincrementandonthedissipation.Fig.5showsthat,asintheEulerianframe,arelativescalingisobservedfortheLagrangianstructurefunctionsoforders1to5,DLq(τ)∝DL2(τ)ξq.Weobservethattherelativeexpo-nentsfollowasequencecloseto,butmoreintermittentthanthecorrespondingEulerianquantity.Indeed,we

obtain:ξLL

L/ξLξ1/ξ3=0.42,ξ3=0.75,ξL/ξLL3=1.17,5

/ξL

2

43=1.28tobecomparedtothecommonlyac-4

ceptedEulerianvalues[17]ξEξ1/ξE3=0.36,ξE2/ξE

3=0.70,E4/ξE3=1.28,ξE5/ξE

3=1.53.

Inconclusion,usinganewexperimentaltechnique,wehaveobtainedaLagrangianvelocitymeasurementthatcoverstheinertialrangeofscales.OurresultsareconsistentwithKolmogorov-likedimensionalpredictionsforsecondorderstatisticalquantities.Athigherorders,theobservedintermittencyisverystrong.HowtheLagrangianintermittencyisrelatedtothestatisticalpropertiesoftheenergytransfersisanopenquestion.Fromadynamicalpointofview,theNavier-StokesequationinLagrangiancoordinatescouldbemodeledusingstochasticequations.WorkiscurrentlyunderwaytocomparethedynamicsoftheLagrangianvelocitytopredictionsofLangevin-likemodels.

acknowledgements:WethankBernardCastaingforinterestingdiscussionsandVermonCorporationforthedesignoftheultrasonictransducers.Thisworkissup-portedbygrantACINo.2226fromtheFrenchMinist`eredelaRecherche.

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