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Patrice Meunier, Emmanuel Villermaux
To cite this version:
Patrice Meunier, Emmanuel Villermaux. How vortices mix. Journal of Fluid Mechanics, Cambridge
University Press (CUP), 2003, 476, pp.213 - 222. �hal-00014833�
How vortices mix
By P. Meunier and E. Villermaux
IRPHE,UniversitédeProvence,AixMarseille1
TechnopôledeChâteau-Gombert
49,rueFrédéricJoliot-Curie13384 MarseilleCedex13, France
(Received29October2002)
The advection of a passive scalar blob in the deformation eld of an axisymmetric
vortexisasimplemixingprotocolforwhichtheadvection-diusionproblemisamenable
toanear-exactdescription.Theblobrolls-upin aspiralwhich ultimatelyfadesawayin
thedilutingmedium.Thecompletetransientconcentrationeldinthespiralisaccessible
from theFourierequationsin aproperlychosenframe. Theconcentrationhistogram of
thescalarwrappedin thespiralpresentsunexpectedsingulartransientfeaturesand its
longtimepropertiesarediscussedin connectionwithmixturesfromtherealworld.
1. Introduction
A central question in scalar mixing consists in oering a satisfactory description of
thehistogram, orProbability Density Function (PDF)P(c) ofthe concentrationlevels
cofthesubstancebeingmixed.Thequestionisparticularlyinteresting,andrelevantto
manyapplicationswhenthesubstrateisstirredsincein thatcasemoleculardiusion is
altered,andinmostcasesenhanced,bytheunderlyingsubstratemotions.
The interplay between molecular diusion and simple deformation elds is a classi-
cal problem. It is solvedin a closed form in a variety of situations such asthe saddle
point ow, the simple shear in two dimensions (Ranz(1979), Moatt(1983)), in three
dimensions(Villermaux&Rehab(2000)),andin theaxisymmetricpointvortex(Rhines
&Young(1983),Flohr&Vassilicos(1997))orspreadingvortexow(Marble(1988),Bajer
etal.(2001)).
Mostoftheattentionhasfocussedonthekineticsofthediusionprocessinthepres-
enceofstirringmotion,particularlyitsdependenceonthesubstraterateofdeformation
,anddiusionpropertiesofthescalar(diusivityD).Regardingthecharacteristictime
t
s
after which uctuations startto decay from an initial scalar spatial distribution, of
crucialimportance istherateatwhichmateriallinesgrowintimedueto thesubstrate
motions (Villermaux(2002)).If material lines grow like t,as it is the case in a point
vortex ow,the mixing timeof, say,a scalarblobof initial size s
0 is t
s
1
Pe 1=3
; if
materialsurfacesinthreedimensionsgrowlike(t) 2
,thent
s
1
Pe 1=5
andifmaterial
linesare exponentiallystretched likee t
,then t
s (2)
1
logPe wherePe=s 2
0
=D is
aPécletnumber.
Thetimes t
s
given aboveare therelevantmixing times assoonastheinverseof the
elongationrate 1
issmallerthanthediusivetimeoftheblobconstructedonitsinitial
size s 2
0
=D, that is for Pe> 1.In thelimit Pe 1,t
s
is essentiallygivenby the time
needed to deform the blob 1
and molecular diusion, although acrucial step in the
ultimateuniformization,playsonlyaweakcorrectionroleinthekineticsoftheprocess.
Figure1.Roll-upofablobofuorescentdyeinapointvortexatt=0(upperleft),t=2sec
(upperright),t =5sec (lowerleft) and t = 10sec (lower right). Each picture covers a eld
4:84:8cm 2
wideandthecirculationofthevortexis14:2cm 2
=s.Thedatacomefromexperi-
mentsdescribedinsection2.
andaremostlylimitedtoshorttimes(i.e.t.t
s
),thereforereectingmorethekinemat-
icsoftheowthanitsmixing properties(see,howeverCetegen&Mohamad(1993)and
Verzicco&Orlandi(1995)).
Basedonaspatiallyandtemporallyresolvedexperiment,westudythemixingchronol-
ogyofablobofdyeembeddedinthedisplacementeldofadiusing,LambOseentype
vortex.Theprocessisdescribed,fromtheinitialsegregationoftheblobtoastatewhere
it is almost completely diluted in the surrounding medium, through the evolution of
the spatial scalar eld, and associated transient evolution of the overall concentration
distributionP(c).
2. A diusive spiral
2.1. Chronology
Thephenomenon weanalyzeisillustratedonFigure1.Auniformblob ofdye(thedark
patch shown on Fig. 1(a))is deposed in a still transparent medium. Then a vortex is
(a)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2
x [cm]
y [cm]
(b)
0.00 0.05 0.10
0 3 6 9
vθa0/Γ
r/a0 (c)
0.0 0.4 0.8 1.2
0.0 0.4 0.8 1.2
a2 (cm2)
4νt (cm2) a02
Figure2.(a)Velocityeldintheplaneofthe vortexatt=10sec .(b)Radialprolesof the
azimuthalvelocitymeasuredatt=5sec(Æ),t=10sec()andt=20sec(M).Solidlinescorre-
spondtotheprolesexpectedfromaLamb-Oseenvortexdenedby(2.1)with =14:2cm 2
/s
anda
0
=0:3cm.Thedashedlinecorrespondstoapointvortexdenedby(3.1).(c)Coresizeof
theLamb-Oseenvortexmeasuredbyaleast-squaretofthetwo-dimensionalmeasuredvelocity
eldandcomparedtoEq.(2.2)(solidline).
thevortexasseenonFigure1(b).Althoughithasbeenbroughttoathintransversesize,
mostoftheuidparticlesconstitutiveoftheblobstillbeartheinitialconcentration.The
blobdeforms inaspiralshapeand afterfour turns(Fig.1(c)),thedyeconcentrationis
nomore uniform alongthe spiral: it isweakernear thecenter ofthe vortex where the
spiralisverythin,andstillclosetotheinjectionconcentrationintheouterregionofthe
spiralwhichisthickerthere.OnFig.1(d),thespiralhasmademorethanseventurnsand
isabouttovanishinthedilutingmedium.Thethicknessofthespiralisfairlyconstant.
Moleculardiusion hasclearly beenenhancedby thevortexmotion. The timelapse
betweengures1(a)and1(d) is10seconds, whenthetimescale ofpurediusion based
ontheinitialsize s
0
of theblobs 2
0
=Disabout10 3
seconds.
2.2. Floweld
The vortex is formed by the impulsive ap motion of along atplate in a largetank
ofwaterinitially at rest. Thevorticitylayerformed at thesurfaceofthe plate rolls-up
and detaches at the plate end, producing an axisymmetric vortex which remains two-
dimensionallongafterthedyehasbeenmixed.A thinuniform Argon-Ionlasersheet is
shed through the tank perpendicular to the plate, and the two-dimensional motion of
thevortexisanalyzedbyParticleImageVelocimetry(PIV)usingaKodak10081018
pixelsdigitalcameraaimedperpendiculartothelasersheet.Furtherinformationonthe
set-up and PIV techniques canbefound in Meunier & Leweke(2002a)and Meunier&
Leweke(2002b)respectively.
Thedyeisintroduced,priortotheformationofthevortex,byasmalltubepositioned
below the lasersheet, and forming aslowly ascending column ofdye, alignedwith the
vortexaxis.Thedyeconcentrationeld(disodiumFluoresceinewithinitialconcentration
c
0 10
3
mol=l) is recordedwith the samecamera and stored on a disk. The overall
framingrateallowsacompleteroll-upsequencetobetemporallyresolved.Theimagesare
digitizedon8bitsandtheresultingbackgroundsubtractedgreylevelsareproportional
tothedyeconcentration.
Figure2(a)showsanexampleoftheaxisymmetricvelocityeldobtainedbyPIVafter
thevortexcreation.Theradialprolesofazimuthalvelocityv
shownonFig.2(b)agree
v
=
2r
1 e r
2
=a 2
(2.1)
Here, =14:2cm 2
=s isthecirculationof thevortex,andaitscoresize.This vortex
isanexactsolutionoftheNavier-Stokesequationsprovidedthat
a 2
=a 2
0
+4t (2.2)
where is the kinematic viscosity of the uid, a law in close agreement with the
observedgrowth(Fig.2(c)),a
0
beingtheinitialvortexradiusequalto0.3cm.
The dashed line in Fig. 2(b) is the velocity prole of a point vortex with the same
circulation,denedby(3.1).Itistangenttothemeasuredvelocityprolesforlargeradii
(r=a
0
>3).
Willingtodecoupletheproblemofmixingfrom the(trivial)problemofthetemporal
evolutionof thevelocityeld itself, wehavesystematicallydeposed theblobof dyefar
enough from the vortex core so that the velocity eld remains that of a steady, point
vortex,throughoutthewholemixing process.
3. Concentration eld along the spiral
Weconsider theevolution of ablob of dyeof initial size s
0
, in the two-dimensional,
incompressible ow of a point vortex of circulation (see Fig. 3a), whose azimuthal
velocityis
v
=
2r
(3.1)
Werstdescribethe kinematicsof theblob deformation.Auidparticle ofthe blob
locatedatadistance rfromthecenterofthevortexturnsduring timetbyanangle
(r;t)= Z
t
0 v
r dt=
t
2r 2
(3.2)
A scalar strip of initial length dr, located at a distance r from the vortex center
(Fig.3(a))is stretchedsothat itslengthequals attimet
dX= p
dr 2
+(rd ) 2
=dr s
1+r 2
d
dr
2
=dr r
1+ 2
t 2
2
r 4
(3.3)
Meanwhile, the transverse, or striation thickness s(t) of the strip, in the absence of
diusion,decreasessothatthesurfaces(t)dXremainsconstantin thistwodimensional
ow
s(t)= s
0 dr
dX
= s
0
q
1+ 2
t 2
2
r 4
(3.4)
Wenowdescribethescalardissipationoftheblob.Thedisplacementeldresultslocally
in acompression perpendicular to the strip, and in an extension along the strip. It is
Γ
s
0dr
s(t) dX
s
0X Y O
(a) (b)
Figure3.Schematicofthescalarblobelongation.(a)initialstateand(b)attimet
withthespiralasshownonFig. 3(b). Inthatframe, thevelocityeld isprescribedby
thetemporalevolutionofthestriationthicknesss(t)as
U = X
s ds
dt
and V =
Y
s ds
dt
(3.5)
Theevolutionequationforthedyeconcentrationcistheconvectiondiusionequation
inthe(X;Y)coordinates
@c
@t +U
@c
@X +V
@c
@Y
=D
@ 2
c
@X 2
+
@ 2
c
@Y 2
(3.6)
TheratioofthetwoconvectivetermsV@c=@Y andU@c=@X isin magnitudepropor-
tional to thestrip aspect ratio1+( 2
t 2
)=(
2
r 4
):the concentration varies moreslowly
alongthespiralthanin itstransversedirectionfor t=r 2
>1sothat Eq.(3.6)becomes
@c
@t +
Y
s ds
d t
@c
@Y
=D
@ 2
c
@Y 2
(3.7)
Achangeofvariables(seee.g.Ranz(1979),Marble(1988),Villermaux&Rehab(2000))
consisting in counting transverse distances in units of the striation thickness s(t) and
time in units of the current diusion time s(t) 2
=D transforms Eq. (3.7) into a simple
diusionequation
with = Y
s(t)
and (r)= Z
t
0 Ddt
0
s(t 0
) 2
= Dt
s 2
0 +
D 2
t 3
3 2
r 4
s 2
0
giving
@c
@
=
@ 2
c
@ 2
(3.8)
Ifc
0
istheinitialconcentrationofthedye,theinitialconditionsat =0are
c=c
0
for jj<1=2
c=0 for jj>1=2
(3.9)
Theconcentrationproleatanytimeandradialpositionalongthespiralis
c(;)= c
0
erf
+1=2
p
erf
1=2
p
(3.10)
(a)
0.0 0.5 1.0
0 2 4 6 8
c/c0
r / a0
(b)
0.1 1.0
0.5 1.0
c/c0
t / ts(r/a0=4.4)
Figure 4.Comparisonof themaximaldyeconcentrationsobtainedexperimentally(symbols)
and theoretically(solidlines) by Eq.(3.11). (a)Radialdependenceat t=5sec(Æ), t=10sec
()andt=20sec(M).(b)Temporaldependenceforr=a0=4:4
Themaximalconcentrationisobtainedat theprolecenter =0
c
M
(r;t)=c
0 erf
1
4 p
=c
0 erf
0
@
1
4 q
Dt
s 2
0 +
D 2
t 3
3 2
r 4
s 2
0 1
A
(3.11)
This relationcanbeexamined from theexperiment ( =14:2cm 2
/s,D =510 6
cm 2
/sands
0
0:22cm).Figure4(a)showsthemaximaldyeconcentrationsasafunction
oftheradiusr ataxedtime, forthree dierenttimes.Theconcentrationfallsto zero
morerapidlycloser tothe spiralcenter sincethe rateofelongationis higherthere (see
Eq.(3.3)).
Conversely,thetemporalevolutionof theconcentrationataxed r locationiscon-
stant(Fig.4(b))uptothemixingtimet
s
(r).Thistimemakestheargumentoftheerror
functionin Eq.(3.11)oforderunityi.e. =O(1)
t
s (r)=
r 2
3 2
16
1=3
s
0
r
2=3
D
1=3
(3.12)
anddisplaystheexpectedPécletnumberdependencePe 1=3
,with Pe= =D charac-
teristicofowswheremateriallinesgrowasymptoticallylinearlyintime(seeEq.(3.3)).
Afterthemixingtime,themaximalconcentrationc
M
decreasesliket 3=2
,incloseagree-
mentwiththetrendshownonFig.4(b).
4. Probability Density Function
IfA isthetotalsurfaceareaofthespiral bearing anon-zeroconcentrationlevel,the
Probability DensityFunction(PDF) ofthe scalarP(c)is thefraction of thetotalarea
whoseconcentrationliesintheinterval[c; c+dc].ItisconvenienttocomputeP(c)inthe
(r;)coordinateswhere is dened in (3.8)so that with dX = p
1+( 2
t 2
)=(
2
r 4
) dr
anddY=sd=s d=
p
1+( 2
t 2
)=(
2
r 4
),onehas
0.5 1
1.5 2
−0.02
−0.01 0 0.01 0.02
0 0.2 0.4 0.6 0.8 1
r [cm]
Y=s.ξ [cm]
c/c0
(a)
(b) r [cm]
Y=s.ξ [cm]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−0.03
−0.02
−0.01 0 0.01 0.02 0.03
(c)
Figure 5. (a) Perspective view and (b) contour plot of the concentration prole given in
Eq. (4.2). The white band corresponds to an iso-concentration c=c
0
= 0:6. (c) Zoom of the
endofthespiralonFig. 1andsameconstruction.
P(c)dc= ZZ
c(X ;Y)2[c;c+dc]
dXdY
A
= ZZ
c(r;)2[c;c+dc]
s
0 drd
A
(4.1)
The scalar spatial distribution is given in Eq. (3.10) as the dierence of two error
functions.However,afterthemixingtime,thatiswhenthespiralthicknessisverythin,
this dierence approximates the derivativeof the error function, providing aGaussian
concentrationprole
c(;r)=c
0 erf
1
4 p
(r)
!
e
2
=2 2
(4.2)
where (r) is given by Eq. (3.8) and
(r) is the standarddeviation of the original
prolec(Y)giveninEq. (3.10)
2
= R
Y 2
c(Y)dY
R
c(Y)dY
=s 2
(t) R
2
c()d
R
c()d
=s 2
(t)
1+24(r)
12
; or
2
=
1+24(r)
12
(4.3)
Notethat the`spiralthickness' rst decreasesast 1
,reachesaminimumatt =t
s
and re-increasesas t 1=2
after the mixing time,when thespiralis locally nearly parallel
tothevortexstreamlines.
The shape of the iso-concentration lines c(r;) = c in the (r;) plane is shown in
Fig.5
(r;c)=
(r)
r
2log h
erf
1=4 p
(r) i
2log(c=c
0
) (4.4)
Thiscurveisdened forr>r
1
(c)only,that isabovethesmallestradiusbearingthe
(a)
1.0 10.0
0.4 0.6 0.8 1.0
c/c0
(b)
1.0 10.0
0.4 0.6 0.8 1.0
c/c0
(c)
1.0 10.0
0.4 0.6 0.8 1.0
P(c/c0)
c/c0
(d)
1.0 10.0
0.4 0.6 0.8 1.0
P(c/c0)
c/c0
Figure6. ProbabilityDensityFunctionsat(a)t=5sec, (b)t=8sec,(c)t=10secand(d)
t=13sec . Solidlines correspond to the theoreticalpredictiongivenbyEq.(4.6) and dashed
linescorrespondtothePDFofthespatialmaximaofconcentration,denedbyEq.(4.7).
r
1 (c)=
"
16
3 2
D 2
t 3
s 2
0
erf 1
(c=c
0 )
2
16Dt
#
1=4
(4.5)
Ifthescalarblobwasinitiallydelimitedbetweentheradiir
1 andr
2
,theconcentration
PDFis
P(c)= 2s
0
A Z
r
2
max[r1;r
1 (c)]
@c
@
1
dr (4.6)
Theconcentrationproleacrossthespiral,and theevolutionof themaximalconcen-
trationalongthespiralsettheglobalPDF.
TheaboverelationiscomparedonFig.6withtheexperimental histogramsrecorded
with a blob initially located between r
1
= 1:65cm and r
2
= 2:1cm. At early stages,
(Fig.6a),aslongasmostoftheuidparticlesconstitutiveofthespiralhavenotreached
themixingtime yet,thePDF isthat ofaGaussianspatial prole1=c p
log(c=c
M )with
c =c displayingacharacteristic[shape.
Assoonasdiusionbecomeseective,thePDFnucleatesacusplocatedatthemaximal
concentrationc
M (r
1
)obtainedattheinnerend ofthespiral.Theshapeof thePDF for
c
M (r
1
)<c<c
M (r
2
)resultsfromthesuperpositionoftherightbranchesofthe[shaped
distributionsparameterizedbyc
M
(r)withr
1
<r<r
2
(Fig.6b,c,d)andweightedbythe
probabilityofndingthemaximalconcentrationc
M
,namelyQ(c
M
).Thisdistributionis
thefractionofthespirallengthdXwhoseconcentrationisintheinterval[c
M
; c
M +dc
M ]
Q(c
M )=
1
L
dc
M
dX
1
(4.7)
where L is thespiral lengthL = R
r
2
r1
dX.It is dened in the range[c
M (r
1 ); c
M (r
2 )]
andshownasthedottedlineonFig.6.Atshorttimes,P(c)andQ(c
M
)areverydierent
because the low concentration levelsat a small radii r and = 0 are as numerous as
the same levels at the edges of the Gaussian transverse prole ( 6= 0) at ahigher r.
The spatial distribution c() contaminates the whole distribution P(c), inducing the
characteristic[shape.Atlaterstages(Fig.6d),thelowlevelsofconcentrationfromthe
edgesoftheGaussianproleatlargeradiiarerareincomparisontothoseat thecenter
ofthe spiraland =0. Therefore,Q(c
M
)becomesadecreasingfunction of c and gets
closertoP(c).Inthenal stages,when t=r 2
1and fort
s
(r)>1forallr,thesetwo
distributionsarebothgivenby
P(c)Q(c
M
=c)
~ r 4
s 2
0
D 2
t 3
1=4
1
c 3=2
(4.8)
wherer~standsfor(1=r
1 +1=r
2 )
1
.
5. Conclusions and implications
Inthesimpledisplacementeldofatwo-dimensionalvortex,adirectconnectionexists
between the microscopic equations of diusion, and the resulting global statistics of
themixture throughthe scalarconcentration PDF P(c) which, therefore,appearsasa
reformulationofthemicroscopicconvectiondiusionproblem.
Thisone-to-oneconnectionispossiblebecausetheowsolelyresultsinaspatialmap-
ping of the uid particles with no interaction between the particles themselves. The
concentrationofagivenuidelementevolvesduetomoleculardiusionandnotbecause
itinteracts withanearbyelement;indeed, thearmsofthespiral neverreconnect.This
situationwouldleadtoacompletelydierentroutefortheevolutionofP(c).Itis,tothis
respect,usefulto learnthat thedistribution Q(c
M
)tendsasymptoticallytowardsP(c),
ahiddenassumption madewhen consideringmixtures evolutionbyparticleinteraction
(Curl(1963),Pope(1985),Pumiretal.(1991),Villermaux(2002)).
Thesimplestirringprotocol consideredherealsoprovidesanexactestimation ofthe
scalardissipationrate = d
dt hc
2
i=2Dh(rc) 2
i,a quantity sometimesmodeled in an
ad-hocway.Herehidenotesaspatialintegration,therefore
=2D Z
r
2
r1 dX
s(t) Z
+1
1
@c
@
2
d (5.1)
