To link to this article : DOI: 10.1016/j.cherd.2013.05.002
http://doi.wiley.com/10.1016/j.cherd.2013.05.002
To cite this version: Liné, Alain and Gabelle, Jean-Christophe and
Morchain, Jérôme and Anne-Archard, Dominique and Augier, Frédéric On
POD analysis of PIV measurements applied to mixing in a stirred vessel with
a shear thinning fluid. (2013) Chemical Engineering Research and Design,
vol. 91 (n° 11). pp. 2073-2083. ISSN 0263-8762
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On
POD
analysis
of
PIV
measurements
applied
to
mixing
in
a
stirred
vessel
with
a
shear
thinning
fluid
A.
Liné
a,b,c,∗,
J.-C.
Gabelle
a,b,c,d,e,
J.
Morchain
a,b,c,
D.
Anne-Archard
f,
F.
Augier
eaUniversitédeToulouse,INSA,LISBP,135AvenuedeRangueil,F-31077Toulouse,France bINRAUMR792IngénieriedesSystèmesBiologiquesetdesProcédés,Toulouse,France cCNRS,UMR5504,F-31400Toulouse,France
dADEMEFrenchEnvironmentandEnergyManagementAgency,20avenueduGrésillé,BP90406,49004AngersCedex01,France eIFPEnergiesnouvelles,Rond-pointdel’échangeurdeSolaize,BP3,69360Solaize,France
fUniversitédeToulouse,INPT,UPS,IMFT,AlléeC.Soula,F-31400Toulouse,France
a
b
s
t
r
a
c
t
P.O.D.techniqueisappliedto2DP.I.V.datainthefieldofhydrodynamicsinamixingtankwithaRushtonturbine andashearthinningfluid.Classicaleigen-valuespectrumispresentedandphaseportraitofP.O.D.coefficientsare plottedandanalyzedintermsoftrailingvortices.Aspectrumofdissipationrateofkineticenergyisintroducedand discussed.LengthscalesassociatedtoeachP.O.D.modesareproposed.
Keywords: Mixing;Particleimagevelocimetry;Properorthogonaldecomposition;Eigenvaluespectrum;Dissipation rate;Stirredtank
1.
Introduction
Advanced experimentaltechniques such as particle image velocimetry(P.I.V.)areavailablenowadays,andtheygeneratea hugeamountofdata,intermsofinstantaneousvelocityfields (inaplaneasfaras2DP.I.V.isconcerned).Indeed,advanced dataprocessingtoolsarerequiredtoextractmoreandmore informationfrom this sourceof experimentaldata. Proper orthogonaldecomposition(P.O.D.)orKarhunen–Loevemethod thatisknowntobeefficienttoisolatecoherentstructuresfrom aseriesofinstantaneousvelocityfields(Berkoozetal.,1993) willbeusedinthispaper.P.O.D.isamodaldecompositionof instantaneousvelocityfields,themodesbeingorthogonalto eachothers.Itcanbeexpressedas:
−→ Vk(x,y,z,t)= N
X
I=1 −→ V(I)k(x,y,z,t)= NX
I=1 a(I)k(t)−→˚(I)(x,y,z) (1)where −V→k is the kth instantaneous event of velocity field
measurement and −→
V(I)k is the Ith component of the P.O.D.
∗ Correspondingauthorat:UniversitédeToulouse,INSA,LISBP,135AvenuedeRangueil,F-31077Toulouse,France.Tel.:+33561559786; fax:+33561559760.
E-mailaddress:alain.line@insa-toulouse.fr(A.Liné).
decomposition.ForeachmodeI,theP.O.D.methodgenerates temporal coefficientsa(I)k andspatialmodes−→˚(I).a(I)
k (t) isan
instantaneous(k)ortemporal(t)scalar(calledP.O.D.coefficient ofP.O.D.modeI);itisindependentofspace.−→˚(I)(x,y,z)isa spa-tialeigenfunctionofmodeIanditisindependentoftime(or instantaneouseventk).Modescanbederivedfromthe Fred-holmeigenvalueintegralequation,adaptedbySirovich(1987)
as:
Z
Z
˝
R(x,y,z,x′,y′,z′)−→˚(I)(x′,y′,z′)dx′dy′dz′=(I)−→˚(I)(x,y,z) (2)
whereRisthecross-correlationtensorand(I)isthe
eigenval-ues.Followingthisapproach,theeigenvaluesareexpressed inm5/s2inafull3Danalysis.Indeed,onecanpointoutthat
thedimensioncorrespondstoakineticenergy(m2/s2)times
a volume(dxdydz). In2DPIV problem,the eigenvaluesare expressedinm4/s2,correspondingtoakineticenergy(m2/s2)
times a surface (dxdz). Usually eigenvalue sequences are rankedindecreasingorder.Clearly,thisorderingof eigenval-uesensuresthatthefirstP.O.D.modesarethemostenergetic.
Given its eigenvalue, (1) and its eigenfunction −→˚(I)(x,y,z),
it ispossibleto reconstructthe velocity field associatedto themode1(
−−→
V(1)k (x,y,z,t))whichwillbeconfirmedto repre-sentthemeanflow.Consideringmodes2and3,Oudheusden et al. (2005) have shown that, when their P.O.D. eigenval-ues are close to each other, the POD modes are coupled: usually,theycorrespondtoorthogonalcomponentsofa peri-odicprocess (coherentstructures induced bythe bladesof the impeller). Higher modes correspond to lower eigenval-ues that are associated tosmaller scale structures. Knight and Sirovich (1990) have shown that the eigenvalue spec-trum(plotofeigenvaluesversusmodenumberI)exhibitI−11/9
trendintheinertialsubrangeofturbulence.Thistrendwas confirmed by De Angelis et al. (2003) and Hoisiadas et al. (2005)(among others).Theanalysis ofthe differentmodes willbeaddressedinthis paper, inthe field ofmixingin a stirredtankwithaRushtonturbineandashearthinningfluid. Inaddition,aspectrumofdissipationrateofkineticenergy associatedtoeachmodeI,similartothespectrumof eigen-values,willbepresentedanddiscussed.Characteristiclength scales (wave length) associatedto each mode will be also proposed.
As highlighted by many authors, P.O.D. eigenfunctions presentflow-field-likestructuresandtheiranalysismustbe drivenwithcare.However,thesignificanceofP.O.D.modescan becomeclearerbythedetailedanalysisofP.O.D.coefficients a(I)k.SomerelationbetweentwomodesIandJcanbetracked downbyplottingthetwocoefficientsseries[ak(I),a(J)k],k=1,N inthe(a(I),a(J))plane.Suchaplotiscalleda2Dprojectionof
thephaseportrait.Hydrodynamiccoherentstructurescanbe revealedwhen characteristicpatternsappearinthis plane, suchascircle,ellipseorLissajousfigures thatcanbe asso-ciated to periodic or intermittent structures. In particular, circularshapescansuggestcyclicvariationsofmodesIand
Jrelatedtovorticesorcoherentstructures.Inthiscase,(a(I),
a(J))seriescanbemodeledassine–cosinefunctionswithsame
period.Suchphaseportraitswillbeplottedinthispaperin order to emphasizethe couplingbetween modes 2 and 3, associatedtotrailingvorticesgeneratedbythebladesofthe impeller.
P.O.D. was appliedto hydrodynamicsin mixing tankby
MoreauandLiné(2006),Doulgerakis(2010),andGabelleetal. (2013). SnapshotP.O.D. studies werereviewedbyTabiband Joshi (2008) in the field ofChemical Engineering. Thegoal of this paper is to contribute to extract information from 2D P.I.V. data by P.O.D. technique in the field of mixing. Classical P.O.D. data processing is performed in terms of eigenvalueproblem,spectrumofeigenvaluesandphase por-trait analysis. New data processing is proposed in terms of dissipation rates and wavelengths associated to each modeI.
2.
Materials
and
methods
2.1. Mixingtank
2D-P.I.V.measurementswereperformedina70Lstirredtank equippedwithfourequallyspacedbaffles.ARushtonturbine (D/T=0.33) was used. Thevessel was standard (T=0.45m); theimpeller clearancewasC=T/3.Moreinformationabout the mixing tank dimensions is provided in Gabelle et al. (2013).
Fig.1–ApparentviscosityofZetag7587solutionsversus shearrate.
2.2. P.I.V.system
TheP.I.V.systemusedinthisstudyconsistedofalaserand animageacquisitionsystemprovidedbyLaVision(LaVision GmbH, Goettingen, Germany).Thesystem includeda laser Nd-Yag(Quantel,10Hz,200×2mJ),asynchronizationsystem andacharge-coupled-device(CCD)camera(ImagerIntense,12 bits,1376×1040pixels).FastFouriertransform(FFT)cross cor-relationwasusedtointerrogatethetwoimages,whichwere dividedintointerrogationarea(32×32or16×16pixels).For allexperiments,500–700imagepairswererecordedand sta-tisticalconvergenceofthevelocitywaschecked.Inorderto studytheinfluenceofspatialresolutiononthevicinityofthe impeller,twospatialresolutions1x(0.7–1mm)were tested. Thedelaybetweeneachframeinanimagepairwaschosen inrelationtotheimpellerrotationalspeed.Fourzonesinthe vesselwereinvestigated:oneintheimpellersteam,twoabove andonebelow.PODwasonlycarriedoutfortheimageatthe impellerlevel.
2.3. Workingfluid
The shear thinning fluid Zetag7587 (BASF, Ludvigshafen, Germany)wasused.Itoffersgoodtransparencyevenathigh concentration (0.4%). Rheological measurements were car-ried out using aHaake Mars III rheometer (ThermoHaake, Germany).FlowcurvesofZetag7587solutionsarepresented inFig.1forshearratesrangingfrom0.001to1000s−1.
Rhe-ologicaldatacanbefittedtothemodelofOstwalddeWaele (powerlaw)accordingto:
a=K
˙
n−1
(3)
withKthefluidconsistencyfactorandntheflowindex.The rheologicalparametersarereportedinTable1.Zetag7587is knowntobevisco-elastic(EscudierandSmith,2001).However, thechoiceofZetag7587atahighconcentration0.4%wasmade becauseathighconcentrationtheviscosityishigh;itisthus easiertomeasureturbulentflowcharacteristicsclosetothe impeller,theturbulentmicroscalesbeinglargeratrelatively
Table1–Rheologicalparametersfittedtothepowerlaw equation.
Fluid n K(Pasn) Shearrate
range(s−1)
Zetag75870.1% 0.422 0.30 1–350 Zetag75870.2% 0.376 0.74 1–600 Zetag75870.4% 0.349 1.65 1–1000
lowReynoldsnumberandthespatialresolutionofPIV mea-surementneededtocatchthephysicsbeinglarger.
2.4. Experimentalconditions
TheimpellerspeedNis205rpm.Theglobalshearrate˙ MOis
estimatedfromMetzner–Ottocorrelation:
˙ MO=kSN (4)
where the constant kS depends on the impeller type (kS=11.5foraRushtonturbine).Theshearrateisthusequal
to 40s−1. Given the rheological law of the shear thinning
fluid,theapparentviscosityis0.15Pas.TheReynolds num-beristhusequalto530,correspondingtothetransitionregion betweenlaminarandturbulentregime.Thevolumeaveraged valueofthedissipationrateofkineticenergyhεicanbe esti-mated;itisequaltohεi=0.2m2/s3 orW/kg.Theassociated
valueofglobalshearrateis h i =37s−1.Indeed,theflowis
turbulentclosetotheimpellerandlessandlessturbulentfar fromit.TheP.I.V.datawereacquiredinadomainclosetothe impellertip(6cm×6cm),ensuringturbulentflowconditions asitwillbeshownintheresultspresentationanddiscussion.
3.
Data
processing
Properorthogonaldecomposition(P.O.D.)isalinearprocedure, whichdecomposesasetofinstantaneousvelocityfieldsinto amodalbase(seeSection 3.1). Weproposeinthis work to investigateP.O.D.intermsofreconstructionofinstantaneous velocity fields in a mixing tank. This methodology allows separatingthe organized and the turbulent motions with-outthenecessitytocollectangularphaseddata(Moreauand Liné,2006).PODiscarriedoutusingsnapshotmethodwith all the sets of instantaneous velocityfields directly issued from the measurements. The result is then an orthonor-malbasisofeigenfunctionsandassociatedeigenvalues.The instantaneous velocityfield can be projectedon each POD eigenfunction. From these projections it is possible to get P.O.D.coefficientsandtoreconstructthevelocityfield(see Sec-tion3.2).Dissipationrateofkineticenergycanbeestimated fromthe2Dvelocityfield(Section3.3).Finally,characteristic lengthscales mayalsobedeterminedfrom thePIV experi-ments.
3.1. Snapshotmethod
P.I.V.measurementsareperformedinthe(x,z)plane,ona regu-larmeshwithLrowsandCcolumns,leadingtoNP=LCpoints per plane. Each instantaneous velocity field measurement constitutesasnapshotoftheflow.Thestatisticalanalysisis performedonNsnapshotstakeninthesameplane(P.I.V.data).
ThenumberofgridpointsbeingLC,oneobtainsthematrixof instantaneousvelocityvectordataas:
−−→ V(1)k x=
− → Vk(x1,z1) −V→k(x1,z2) −→Vk(x1,zC) − → Vk(x2,z1) −V→k(x2,z2) −→Vk(x2,zC) −→ Vk(xL,z1) −V→k(xL,z2) −V→k(xL,zC)
(5)wherekistheindexoftheinstantaneousevent(k=1,N).This matrixofvectorscanbereshapedtobuildavector−→Vkwith2LC
rowsasfollows: −→ Vk=
uk(x1,z1) uk(x2,z1) uk(xL,zC) wk(x1,z1) wk(x2,z1) wk(xL,zC)
(6)Thesnapshotmethodadoptedinthiseigenvalueproblem was proposed by Sirovich (1987). This methodis based on thesnapshotmatrixMcorrespondingtotheNinstantaneous velocityfields.ThematrixMcanbeexpressedas
M=
u1(x1,z1) u2(x1,z1) ... uN(x1,z1) u1(x2,z1) u2(x2,z1) ... uN(x2,z1) ... u1(xL,zC) u2(xL,zC) ... uN(xL,zC) w1(x1,z1) w2(x1,z1) ... wN(x1,z1) w1(x2,z1) w2(x2,z1) ... wN(x2,z1) ... w1(xL,zC) w2(xL,zC) ... wN(xL,zC)
(7)ThematrixMhas2LCrowsandNcolumns,eachcolumnof thematrixMrepresentingthektheventoftheinstantaneous velocityfield.Theauto-covariancetensorRcanbederivedas:
R= 1 NM.M T=
u2(x1,z1) ... w2(xL,zC)
(8)TheFredholmintegraleigenvalueproblemisrepresented inthe2Ddomain˝ofmeasurementby:
Z
Z
˝
R(x,z,x′,z′)−→˚(I)(x′,z′) dx′dz′=(I)−→˚(I)(x,z) (9)
where(I)istheItheigenvalueand−→˚(I)x(x,z)istheIth
asso-ciatedeigenfunction.Thecorrelationtensorcanbeexpressed in the orthonormalbasisas adiagonal matrix.The eigen-valuesofR,(I)R, areexpressedinm2/s2 andare definedas:
(I)R =(I)/(dxdz).
The dimension of M.MT matrix is (2LC)2 whereas the
are−→˚(I)(x,z);theycanbesimplyrelatedtotheeigenvaluesof
MT.M,noted−−→˚′(I)(x,z),as:
−−→
˚′(I)(x,z)=MT−→˚(I)(x,z) or −→˚(I)(x,z)=M−−→˚′(I)(x,z) (10)
Thismustbekeptinmindinordertosavecomputingtime sinceM.MTandMT.Mhaveidenticaleigenvalues,butmayhave
significantlydifferentsizesifN2≪(2LC)2.Inaddition,the
spa-tialmodesconstituteanorthonormalbasis,thus: −→
˚(I).−→˚(J)=ıIJ (11)
3.2. Modaldecompositionofthevelocityfield
Eachinstantaneousvelocityfieldorsnapshotoftheflowcan beexpandedinaseriesofP.O.D.modes.TheP.O.D.coefficients a(I)k areobtainedbyprojectingeachinstantaneousvelocityfield ontheItheigenfunction−→˚(I):
a(I)k =−V→k.
−→
˚(I) (12)
TheIthvelocityvector −→
V(I)k correspondingtotheIth decompo-sitionisgivenby:
−→
Vk(I)=a(I)k−→˚(I) (13)
Onecanthusreconstructeachinstantaneousvelocityfield givenbyEq.(1).Onecaneasilyderive:
1 N N
X
k=1 a(I)ka(J)k =(I)RıIJ (14)Thisrelationshowsthateachmodemakesanindependent contributiontothetotalkineticenergy.
3.3. Modaldecompositionofthedissipationrateof kineticenergy
Therateofviscousdissipationofkineticenergyisdefinedas:
ε=2aS:S (15)
whereaistheapparentviscosityofthefluidandSisthestrain
rate(orstretching)tensor,definedas:
(I)∝−11/3 (16)
Consideringthedecompositionofthevelocityfieldinthe sumofIthcomponents,onecanassociatetoeachIth com-ponentofthevelocityV(I)i,kacomponentS(I)ij,kofthestrainrate tensor: S(I)ij,k= a (I) k 2 · ∂˚(I)i ∂xj + ∂˚(I)j ∂xi
!
(17)AccountingforEqs.(11)and(14),onecanthusderivetheIth componentoftherateofviscousdissipationofkineticenergy:
ε(I)=a(˙ ) (I)R 2 3
X
i=1 3X
j=1 ∂˚(I)i ∂xj + ∂˚(I)j ∂xi!
2 (18)Itisinterestingtorealizethatthetotaldissipationrateof kineticenergyisthesumoftheIthcomponentoftherateof viscousdissipationofkineticenergy:ε(x,z,t)=
P
NI=1ε(I)(x,z,t).RecallthattheeigenvalueIisindependentofspace.It repre-sentsthecontributionofeachmodetothetotalkineticenergy intheplaneofmeasurement.Onecanalsoestimatethe dis-sipationrateofkineticenergy
ε(I)
averagedintheplane:
ε(I)= 1 LC L
X
l=1 CX
k=1 ε(I)(xl,zk) (19)Itisthuspossibletodrawaplotsimilartothespectrumof eigenvalues; itisthespectrumofdissipationrateofkinetic energy (plot ofdissipation rate
ε(I)
associatedto mode I
versusmodenumberI).Thisspectrumwillbepresentedand discussed.
Inaddition,thestrainrateofsmallscalesofturbulenceis largeandcanbeexpressedass′
ij≈u/.RecallingEq.(17)and
consideringthattheP.O.D.coefficienta(I)k havethedimension ofavelocity(m/s),theinverseofalengthscalecouldbe asso-ciatedtothegradientsoftheeigen-function−→˚(I).Thiswillbe
analyzedindepthinthediscussion.
3.4. Reshapedeigenfunctions
Theeigenfunctionsarevectorswith2LCrows.Thesevectors canbereshapedtoforma(L,C)matrixoflocalvectors:
−→ ˚(I)k =
˚x,k(x1,z1) ˚x,k(x2,z1) ˚x,k(xL,zC) ˚z,k(x1,z1) ˚z,k(x2,z1) ˚z,k(xL,zC)
−→ ˚(I)k =
−→ ˚k(x1,z1) −˚→k(x1,z2) −˚→k(x1,zC) −→ ˚k(x2,z1) −˚→k(x2,z2) −˚→k(x2,zC) −→ ˚k(xL,z1) −˚→k(xL,z2) −˚→k(xL,zC)
(20)Thereshaped eigenfunctioncan beplottedintheplane (x,z).Itcorrespondstoa2Dvectorfield,eachvector˚−→k(xi,zj)
havingahorizontal˚x,k(xi,zj)andavertical˚z,k(xi,zj)
compo-nent.
3.5. Commentson2DP.I.V.measurements
Theflowfieldis3Dinthemixingtank.However,theanalysis performedwith2D-P.I.V.islimitedtoa2Dplaneof measure-ment.Intermsofmeanflow,itisclearthatthetangentialflow isnotmeasurednortakenintoaccountinthepresent analy-sis.Indeed,thegoalofthestudyis2-folds:onetheonehand, itistoextractfromthemeasurementstheorganizedmotion induced bytheimpellerrotation. Itwas showninprevious studies(MoreauandLiné,2006,amongothers)thatPODcan enabletodecomposetheorganizedmotionfromthe instanta-neousmeasurement,withoutcollectingangularphaseddata. Itwasalsoshownbymanyauthorsthat2Dmeasurementin averticalplaneenabletoextracttrailingvorticesinducedby aRushtonturbine.Onetheotherhand,thegoalofthisstudy istoanalyzeturbulentscales.Intermsofturbulence,itwas alsoshownbymanyauthorsthatthemeasurementof instan-taneousvelocityfieldinasingleplaneofmeasurementwas
sufficienttoestimatethelocaldissipationrateofturbulent kineticenergyasfarasthespatialresolutionofthePIVsystem ishighenough.Thisisrelatedtotheisotropyofsmallestscales ofturbulencecoupledtotheefficiencyofthemixingtank pro-cess.Keepingthisinmind,wemustbeawareofthefactthat thetangentialmeanflowisnotmeasuredandthatassociated largescalesarenotmeasuredtoo.IntermsofPODanalysis, the2DlimitationofPIVmeasurementsmayaffectthelargest scalesofmotion(intermediatemodesofPODdecomposition).
4.
Results
Theeigenvaluescorrespondtothecontributionofeachmode tothetotalkineticenergyinthe2Ddomainofmeasurement (Section4.1).Mode1isstructurallyclosetothe mean (sta-tisticalaverage)velocityfield(Section4.2).Whentwoofthe followingmodesare closetoeach other intermsof eigen-values,theenergybroughtbythesemodesisthesame.Such modesmayreveallargescalecoherentstructures(Sections4.4 and4.5).Highermodesmayberelatedtoturbulence(Sections
4.5and4.6).
4.1. Eigen-valuespectrum
Theeigen-valuespectrumisplottedinFig.2.Theeigen-values arenormalizedbythesumoftheeigen-values;these normal-izedeigen-valuescorrespondtothecontributionofeachmode
Fig.2–Eigenvaluespectrum.
tothetotalkineticenergyintheplaneofmeasurement.One cansee thatthe firstmodeexplainsmorethan 80%ofthe totalkineticenergy;itischaracteristicofmeanflow contribu-tion.Onecanobservethatmodes2and3andmodes4and5 areclosetoeachother.Theyrespectivelycontributeto3–5% and0.6–0.8%ofthetotalkineticenergy.Theuppermodes con-tributetolowerlevelsofkineticenergy.Onecannoticethat
Fig.3–FirstPODmode(a)eigenfunction,(b)pdfofnormalizedPODcoefficient,and(c)comparisonoffirstmodetomean flow.
uppermodesfollowalineartrendinlog–logplot(−11/9);this trendwillbeexplainedlaterinthediscussion.
4.2. Firstmodeanalysis
Thereshapedeigen-function−˚−→(1)(x,z)correspondingtomode
1isplottedinFig. 3a. Itlookslike avelocityfield but it is not, sincethe eigen-functions have no dimension. Indeed, theinstantaneousvelocityfieldassociatedtothefirstmode isgivenby
−−→
Vk(1)(x,z,t)=ak(1)(t)−˚−→(1)(x,z) (21)
foreachkrealizationofthevelocity.OnecanplotinFig.3b theprobability densityfunctionofthecoefficientsa(1)k . The coefficientsa(1)k werenormalizedbythesquarerootofthefirst eigen-value(1)R expressedinm2/s2.Thefirstremarkisthat
thepdfiscenteredontheunityvalue;thesecondremarkis thedispersionoftheNrealizationsa(1)k isweak,meaningthat a(1)k isalmostconstant.Onecanthusreconstructthevelocity fieldassociatedtothefirstmodeas:
−−→ Vk(1)(x,z,t)=
q
(1)R −˚−→(1)(x,z) ∼=cst ∀k (22)
Onecan compare this velocityfield (reconstructed with thefirstmode)tothestatisticalaverageoftheinstantaneous velocityfield,classicallydefinedas:
− → ¯V (x,z,t)= 1 N N
X
k=1 −→ Vkx(x,z,t) (23)Thevertical profileof the horizontal component ofthe velocityfieldreconstructedwiththefirstmode
−−→
Vk(1)(x,z,t)is comparedtotheverticalprofileofthehorizontalcomponent themeanflow−→¯V (x,z,t)inFig.3c,ataradialpositioncloseto theimpellertip.Thetwoprofilesareidentical.Onecanthus confirmthatthevelocityfieldassociatedtothefirstmodeisa steady-statefieldcorrespondingtothemeanflow.
4.3. Organizedmotionassociatedtomodes2and3 Asaforementioned,theeigen-values(2)R and(3)R have simi-larordersofmagnitude,suggestingthattheymaycorrespond toorganizedmotions.Thisresultisexpectedinmixingwith Rushton turbine and may correspond to trailing vortices induced bytheimpeller blades. Theeigen-functions corre-sponding to mode 2, −˚−→(2)(x,z), and 3, −˚−→(3)(x,z), are plotted
inFig.4aand b. Thesetwomodes exhibitvortices, almost symmetricalwithrespecttotheplaneoftheimpeller(z=0), suggestingsomeorganizationinthestructureofthesemodes. Onceagain,theseplotsare notvelocityfields.Thevelocity fieldassociatedtothesumofthesemodescanbewritten:
−−−→ Vk(2,3)(x,z,t)= 3
X
I=2 −→ V(I)k(x,z,t)=a(2)k (t)−˚−→(2)(x,z)+a(3)k (t)−˚−→(3)(x,z) (24)Inordertohighlightthelinkbetweenmodes2and3,thek
realizationsofP.O.D.coefficientsa(2)k wereplottedversusthek
realizationsofP.O.D.coefficientsa(3)k inFig.5a.A2Dprojection
Fig.4–PODeigenfunctionsof(a)mode2and(b)mode3.
ofthephaseportraitinthe(a(2)k ,a(3)k )planeisthusobtained. Thesetofdatashowanorganizedshape;forsakeofsimplicity, itwillbeconsideredasanellipse.Thecoefficientscanthusbe relatedbytheexpression:
a(2)2k 2(2)R + a (3)2 k 2(3)R =1 (25)
wheretheeigen-valuesareexpressedinm2/s2.Theperiodic
natureofthese coherentstructuresassociatedwithmodes 2and 3could beexpressedassinusoidal variationsa(2)(ϕ
k)
and a(3)(ϕ
k)asshown inpreviouspapers(Duccietal.,2007; Doulgerakisetal.,2011;Gabelleetal.,2013):
a(2)(ϕk)=
q
2(2)R cos(ϕk) and a(3)(ϕk)=
q
2(3)R sin(ϕk) (26)
whereϕkisthephaseangle.Theparametersa(2)(ϕk)anda(3)(ϕk)
havethedimensionofvelocities(m/s).Itisthuspossibleto reconstructthetemporalevolutionoftheorganizedmotion associatedtomodes2and3:
−−−→ V(2,3)k (x,z,t)=
q
2(2)R cos(ϕk) −−→ ˚(2)(x,z)+q
2(3)R sin(ϕk) −−→ ˚(3)(x,z) (27)Fig.5–Modes2and3:(a)portraitofphase,(b)pdfofnormalizedPODcoefficientofmode2,and(c)pdfofnormalizedPOD coefficientofmode3.
OnecanplotinFig.5canddtheprobabilitydensity func-tionsofthenormalizedcoefficientsa(2)k /
q
(2)R anda(3)k /
q
(3)R . Thedistributionsarecenteredontheoriginandtheirshape canberelatedtosinusoidal distributions,asexpectedfrom thepreviousanalysis.
4.4. Organizedmotionassociatedtomodes4and5 Asaforementioned,theeigen-values (4)R and (5)R havealso similarordersofmagnitudeandmaythuscontributeto orga-nizedmotions.Theeigen-functionscorrespondingtomode4, −−→
˚(4)(x,z),andmode5,−˚−→(5)(x,z),are plottedinFig.6aandb. Comparedtomodes2and3,modes4and5showagain vor-ticalstructuresbutthelength-scalesseemtoberoughlyhalf thelength-scalesofmodes2and3.Similarlytotheprevious analysisofmodes2and3,thekrealizationsofa(4)k wereplotted versusthekrealizationsofa(5)k inFig.7aandtheprobability densityfunctionsofthecoefficientsa(4)k /
q
(4)R anda(5)k /
q
(5)R wereplottedinFig.7bandc,theeigenvaluesbeingexpressed inm2/s2.Fig.7aexhibitsastrongrelationbetweenmodes4
and5whereasthepdfdistributionsreveal noisysinusoidal distributions.Inordertogoindeeperdepth,themodes4and 5havebeenplottedversus modes2and 3inFig.8a–d.2D projectionsofthephaseportraitarethusplottedfor differ-entbases.Theorganizedstructuresrevealedbytheseplots exhibittwolobes(contrarytothepreviousportraitofphases
plottedinFig.5,withonlyonelobe)andcanbeanalyzedas follows:
• the relationbetweenmode 2andmode 4(Fig. 8a) corre-spondsroughlytoaplotofsin(2ϕk−(/2))versuscos(ϕk),
thatis−cos(2ϕk)versuscos(ϕk);
• therelationbetweenmode 2andmode 5(Fig.8b) corre-spondsroughlytoaplotofsin(2ϕk)versuscos(ϕk);
• the relationbetweenmode3and mode4(Fig.8c) corre-spondsroughlytoaplotofsin(2ϕk+(/2))versuscos(ϕk),that
iscos(2ϕk)versuscos(ϕk);
• therelationbetweenmode 2andmode 5(Fig.8b) corre-spondsagaintoaplotofsin(2ϕk)versuscos(ϕk).
Modes4and5followthusasinusoidaltrend,withaperiod whichishalftheperiodofmodes2and3.Itisthuspossible toreconstructthetemporalevolutionoftheorganizedmotion associatedtomodes4and5:
−−−→ V(4,5)k (x,z,t)=−
q
2(4)R cos(2ϕk) −−→ ˚(4)(x,z) +q
2(5)R sin(2ϕk) −−→ ˚(5)(x,z) (28)Thestructuresrevealedbymodes2to5canbecompared tovortexsheddingbehindbluffbodiesthathavebeen investi-gatedbymanyworkerssincethepioneeringpaperofRoshko (1955).SimilarLissajousfigureswereobtainedintheirworks.
Fig.6–PODeigenfunctionsof(a)mode4and(b)mode5.
4.5. Turbulentmotion
Followingthedecompositionmethod,areconstructionofthe turbulenceisbasedonallthemodeshigherthan6:
−−−−→ Vk(turb)(x,z,t)= N
X
I=6 −→ Vk(I)(x,z,t)= NX
I=6 a(I)k(t)−→˚(I)(x,z) (29)Theprobabilitydensityfunctionofhorizontal(u′
k)component
oftheturbulentvelocityvectorgivenbyEq.(29)isplottedin
Fig.9a,inapointlocatedinthejetoftheimpeller.Thepdf aresignificantlydifferentfromthepreviousones;aGaussian functionbasedonthemeanandrmsvaluesofthisdistribution isplottedandfitsperfectlythedata.
4.6. Kineticenergydissipationratespectrum
Asafore-mentioned,onecanderivetheIthcomponentofthe rateofviscousdissipationofkineticenergyaveragedinthe plane,
ε(I)
,expressedbyEq.(19).Itisthuspossibletoplotthe distributionofrateofviscousdissipationassociatedtoeach
Imodeversusthemodenumber,similarlytotheeigen-value spectrum(Fig.2).Thiskineticenergydissipationratespectrum
isplottedinFig.9b.Onecannoticethatuppermodesfollowa lineartrendinlog–logplot(−5/9);thistrendisrelatedtothe −11/9slopeoftheeigen-valuespectrumplottedinFig.2,asit willbeexplainedlaterinthediscussion.
5.
Discussion
5.1. Eigenvaluespectrum
Theeigen-value spectrumexhibitsa−11/9slopeinlog–log plotof(I) versustheeigen-valuenumberI.Thisresultwas
explainedbyKnightand Sirovich(1990)arguing thatitisa characteristicofinertialrangeofturbulence.Asexposedby theseauthors,inturbulentflows,theturbulentkineticenergy (tke)canberelatedtotheenergydensityspectrumofvelocity
EV(K),pervectorwavenumberK,as:
tke= 1 ϑ
Z
Z
Z
u2dv=Z
Z
Z
EV(K)dK (30)Clearly,thedimensionoftheenergydensityspectrumof velocityEV(K)isdim[EV]=l5/t2orl2/t2timesl3.Itcorresponds
toakineticenergyinavolume.Intheinertialrangeof turbu-lence,assumingturbulenceisotropy,onecanwritetheenergy density spectrumofvelocityEV per vectorwavenumberin terms of the energydensity spectrum ofvelocity ES() per scalarwavenumber:EV=ES()/42.Thetrendoftheenergy
densityspectrumofvelocityES()perscalarwavenumberis
well knownES()∝ε2/3−5/3. Onecanthusderivetheenergy densityspectrumofvelocityEV()pervectorwavenumberas
EV∝ε2/3−11/3.
KnightandSirovich(1990)consideredthenthePOD anal-ysis.AsaforementionedinSection3.1,theeigenvalue(I)is
expressedinm5/s2inafull3Danalysis,similarlytotheenergy
densityspectrumofvelocityEVpervectorwavenumber.Thus,
KnightandSirovich(1990)consideredthateacheigenvalueis ageneralizationoftheenergydensityspectrumofvelocityEV, carryingthesamephysicaldimensions.Itfollows(I)∝−11/3.
In addition, Knight and Sirovich (1990) stipulated that the wave number maybe relatedtothe eigenvaluenumber as ∝I1/3.Thus,theyobtainedthe−11/9trend(I)∝I−11/9.This
trendisobservedinourdataprocessing(Fig.2).Thisresult issurprisingatfirstappearance,sincetheReynoldsnumber oftheflowinthetankislow(itisequalto530), correspond-ingtothe“transition”regionofturbulence.Itisknownthat localisotropyonlyexistifthelocalReynoldsnumberislarge enough.ThefirstargumentisthattheglobalReynolds num-ber islow but the hydrodynamicsis heterogeneousin the tank.Thelevelofturbulenceishigherclosetotheimpeller (wherethedatawhereacquired)thanfarfromit.Secondly, Knight and Sirovichhaveshownthat aninertial rangecan exist at very modestReynolds numbers.Such −11/9 trend was also observed inthe same mixing tank with another shearthinningfluid(withyieldstress,Carbopol,Gabelleetal., 2013) but this trend was limited to the higher Reynolds number.
Onemustalsohighlightthatthe−11/9trendcorresponds to modeslarger than20. Thisrelativelylarge valuecanbe explainedbythelimitationof2Dmeasurementofa3D veloc-ityfield.Asaforementioned,thevelocitycomponentnormal tothe 2Dplaneofmeasurement isnotmeasured.In addi-tion,thechoiceofZetagfluidwasdictatedinordertolimit theturbulencelevelclosetotheimpeller;itwasthus possi-bletomeasurealmostalltheturbulentscales.Therefore,the
Fig.7–Modes4and5:(a)portraitofphase,(b)pdfofnormalizedPODcoefficientofmode4,and(c)pdfofnormalizedPOD coefficientofmode5.
mainissueofthisworkistoexhibitthe−11/9trendofthe eigen-valuespectrumforlargevaluesofmodes, correspond-ingtoturbulentmotion.
5.2. Dissipationratespectrum
Onecan extend the development proposed by Knight and Sirovich(1990)totheanalysisofthespectrumofkineticenergy dissipationrate,plottedinFig.9b.Thisspectrumexhibitsa lin-eartrendinlog–logplotwithaslope−5/9.Thisresultcanbe explainedasfollows.Thespectrumofthedissipationinscalar wavenumbercanbeexpressedas:
ε=
Z
D()d (31)
Thedissipation density spectrum can be relatedto the energydensityspectrumbyD()≈2E()∝−5/3.Considering
oncemorethatthescalarwavenumberisrelatedtothe eigen-value number by ∝I1/3, we obtain the trend−5/9that is observedintheplotof
ε(I),normalizedbythe total dissi-pationrate(summationforallmodes).Thistotaldissipation rateisequalto0.42W/kg,almosttwicethevolumeaveraged valueofthedissipationrateofkineticenergyhεiwhichisequal to0.2W/kg.
Hereagain,the −5/9trendcorresponds tomodeslarger than20.However,themainissueofthis workistoexplain
the−5/9trendofthedissipationspectrumforlargevaluesof modes,correspondingtoturbulentmotion.
5.3. WavenumberassociatedtothemodeI
Thelast questionisrelated totheassumptionstatingthat ∝I1/3. Recall first thatthe reconstruction process isgiven
byEq.(1).Thedimensionofthecoefficienta(I)k isl/twhereas theeigen-function−→˚(I)(x,z)hasnodimension.Inaddition,the coefficient a(I)k isconstantinthe domainand onlydepends onthetimewhereastheeigen-function−→˚(I)(x,z)isconstant withtimebutitvariesinthespatialcoordinates(x,z). Conse-quently,anylength-scaleassociatedtotheIthmodecanonly berelatedtotheeigen-function−→˚(I)(x,z)andcannotberelated totheP.O.D.coefficientwhichisconstantforeachmodeinthe plane.Itisthusforeseeablethatthewavenumberassociated toeachImodedependsontheeigen-function−→˚(I)(x,z)and/or onitsgradients.If(Vx,Vz)arethehorizontalandvertical
com-ponentsofthevelocityvector −→V,and[˚(I)x(x,z),˚(I)z (x,z)]are
thehorizontalandverticalcomponentsoftheeigen-function −→
˚(I)(x,z),onecanexpressthecross-correlationfunctionAxxof thehorizontalvelocityVxbetween(x,z)and(x+L,z)as:
Axx(L)=Vx(x,z)Vx(x+L,z)= 1 N N
X
k=1 Vx,k(x,z)Vx,k(x+L,z) (32)Fig.8–Portraitofphase:(a)base2-4,(b)base2-5,(c)base3-4,and(d)base3-5.
Afterderivation,basedonthereconstructionprocessand onEq.(14),oneobtains:
Axx(L)=Vx(x,z)Vx(x+L,z)= N
X
I=1
(I)R˚(I)x(x,z)˚(I)x(x+L,z) (33)
Inotherwords,foreachmodeI,onecandefinea cross-correlationfunctionA(I)xxintermsofA(I)xx(L)=(I)˚(I)x(x,z)˚(I)x(x+
L,z)orinnon-dimensionalform:
R(I)xx(L)=
A(I)xx(L)
(I)R
=˚(I)x(x,z)˚(I)x(x+L,z) (34)
ATaylordevelopmentattheorigingives
R(I)xx(L)=1− x2 2
∂˚(I)x ∂x 2 =1−(I)2x x2 (35)Thus,awavenumbercaneasilyberelatedtothespatial gradientoftheeigen-function−→˚(I)(x,z).
Indeed,thekinetic energydissipation rateassociatedto each modeIwasexpressed byEq.(18). Dimensionally,one
canconsiderawavenumberassociatedtoeachImodeand definedby: (I)(x l,zk)=
v
u
u
u
t
3X
i=1 3X
j=1 ∂˚(I)i (xl,zk) ∂xj + ∂˚(I)j (xl,zk) ∂xi!
2 (36)This wave numbers can easily be derived from the eigen-function gradients. In factthis wavenumber variesin the planeofmeasurement.Sincetheeigenvalueisanaverageof thekineticenergyintheplaneofmeasurementassociatedto modeIand
ε(I)isalsoaveragedintheplaneofmeasurement, onecanestimateanaverageofthewavenumberintheplane
(I)
similarlytoEq.(19).
This averaged wave number
(I)
isnormalized by the wavenumberofthefirstmodeandplottedversusthe eigen-valuenumberIinFig.10.ThecurveI∝(I)
3
isalsoplotted. The normalized averaged wave numbers issued from the eigen-functionsgradientsbehavesasthewavenumberpower 1/3,asstipulatedbyKnightandSirovich.Thedefinitionofthe wavenumberassociatedtotheIthmode,(I),isa
generaliza-tionofthissimplifieddevelopmentgivenbyEq.(35).Further analysisshouldbedoneonthepdfofgradientsofeigen func-tionsintheplaneofmeasurementforeachImodetobetter understandthephysicalmeaningofthisscale.
Fig.9–Turbulence(a)pdfofturbulentvelocityfluctuations and(b)spectrumofdissipationrateofkineticenergy.
Fig.10–Wavenumberversuseigenmodenumber.
6.
Conclusion
P.O.D.technique wasappliedto2DP.I.V. data.Each instan-taneousvelocityfieldcanbeexpandedinaseriesofP.O.D. modes. The first mode is associated to mean flow. The
followingpairsofmodesare associatedtotrailingvortices. Thehighermodesarerelatedtoturbulence.The−11/9trend oftheeigen-valuespectrumisverified.Thespectrumof dis-sipationrateofkineticenergyisplottedandthe−5/9trendis shownandexplained.Lengthscalesassociatedtoeachmode aredeterminedandplottedversusmodenumber.Thepower 3trendisshown.Thegoalofthispaperwastoshowthe dif-ferenttrends(−11/9trendoftheeigen-valuespectrum,−5/9 forthespectrumofdissipationrateofkineticenergy,and3for thelengthscalesassociatedtoeachmode)correspondingto turbulentmotions.Clearlythesetrendsarelimitedtomodes largerthan20.Theanalysisofthefull3Dflowshouldbe com-pletedtoimprovetheunderstandingandthequalityofboth measurementandprocessingoftheP.O.D.modeslowerthan 20.
Acknowledgement
This work was supported by the French Environment and EnergyManagementAgency(ADEME)andIFPEnergies Nou-velles.
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