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Three-dimensional instability during vortex merging

Patrice Meunier, Thomas Leweke

To cite this version:

Patrice Meunier, Thomas Leweke. Three-dimensional instability during vortex merging. Physics of Fluids, American Institute of Physics, 2001, 13, n° 10, p. 2747-2750. �10.1063/1.1399033�. �hal- 00081686�

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Three-dimensional instability during vortex merging

Patrice Meunier y

and Thomas Leweke

Institut de Recherche sur les Phenomenes Hors

Equilibre,

UMR 6594 CNRS/Universites Aix-Marseille,

49 rue F. Joliot-Curie, F-13384 Marseille Cedex 13, France

Abstract

The interaction of two parallel vortices of equal circulation is observed experimentally. For low

Reynolds numbers (R e), the vortices remain two-dimensional and merge into a single one, when

their time-dependent core sizeexceeds approximately 30% of the vortex separation distance. At

higher R e, a three-dimensional instabilityis discovered, showing the characteristics of an elliptic

instability of the vortex cores. The instability rapidly generates small-scale turbulent motion,

which initiatesmerging forsmaller core sizes and produces a biggernal vortex thanfor laminar

2Dow.

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served in mixing layers [1], and in the wake of aircraft wings [2{4], and it may play an

activerole inturbulentows [5,6]. Inatheoretical study, Samanand Szeto[7]considered

two-dimensionalinviscidowandmodeledthe vortices astwosurfacesofconstant vorticity.

They found solutions,in which the two patches rotatearound eachother indenitely,when

their characteristic diameteris smaller than a certain fraction of their separation distance.

These solutionsare two-dimensionnallylinearly [8] and non-linearly[9]stable. Beyond this

limit,no such solutionexists, and Overman and Zabusky [10] showed numerically that the

twopatchesarerapidlydeformed,growingarmsofvorticityandmergingintoasinglevortex.

Experimental observations made by GriÆths and Hopnger [11] seem to be in agreement

with this size criterion for the evolution of corotating vortex pairs. However, the eect

of viscosity, which smoothes the vorticity distribution and makes the core size increase in

time,maysignicantlychangethis picture. Moreover, thepossibilityof athree-dimensional

instability of the vortex pair has so far not been taken into account. In recent experi-

ments [12], two distinct instabilities have been observed in counterrotating vortex pairs: a

long-wavelength instability, treated theoretically by Crow [13], and a short-wavelength in-

stability,identiedasthe so-calledellipticinstability[14{17]of thevortex cores. Whereas a

corotating vortex pairwas shown theoreticallyto bestable with respect to long-wavelength

perturbations [18], Le Dizes [19] recently predicted the persistence of the short-wavelength

instability. A dierent kindof three-dimensionalinstability,occurring in columnar corotat-

ing vortices, in a rapidly rotating stratied uid has recently been reported [20]. However,

in the absence of stratication and background rotation,the underlyingmechanism cannot

act inthe present ow. In this study, weinvestigate experimentallythe inuence of viscous

and three-dimensional eects onthe mergingof two corotating laminarvortices.

Theowisgeneratedinawatertankofdimensions5050130cm 3

,usingtwoatplates

with sharpened edges, impulsivelyrotated inasymmetric way by computer-controlledstep

motors. The vorticity created inthe boundarylayer of each plate rollsup into two starting

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or incross-sections. For quantitativevelocity measurements, ParticleImage Velocimetry is

used. Thewater isseededwithsmallreecting particlesand adigitalcameracapturespairs

of images insections perpendicular to the vortex axes. The images are then treated with a

cross-correlation algorithmto extract the velocity and vorticity eld [21]. The vortex pair

ischaracterizedby thecirculation ofeachvortex,the separationb between the twovortex

centers (locatedatthe maximaofvorticity) andthe vortex coresize a,dened asthe radius

forwhichthe averagedazimuthalvelocity ismaximum. Measurements haveshown that the

initialvorticity distribution of each vortex is approximately Gaussian(see Eq. (1)below).

Inthis paper, we consider pairsof equalvortices,for which the dynamicsdepend ontwo

non-dimensional parameters: 1)the Reynolds numberR e= = (varying between 700 and

4000 here), where is the kinematic viscosity of the uid, and 2) the initial ratio a

0

=b

0

of core size and separation distance (initial meaning at the end of the vortex formation),

which is in the range 0.1{0.2 in the present experiments. Two point vortices of the same

circulation and separated by a distance b

0

rotatearound each other by mutual induction

(the surrounding uid being irrotationnal) with a turnover period t

c

= 2

2

b 2

0

= . This

convective unit isusedtonon-dimensionalizetime t (t

=t=t

c

),startingatthe beginningof

the plate motion.

ForReynolds numbers lower than about 2000, the vortices remain two-dimensional and

laminar. They deform elliptically,get closer, and thenmerge intoa single vortex inarapid

transition,asshown by the vorticityeldsinFig. . The distanceb between the twomaxima

of vorticity is plotted in Fig. . The second curve (R e = 1506) corresponds to the elds of

Fig. . The evolution of the core size a of the two initialvortices beforemerging, and of the

nal vortex after merging is shown in Fig. . These measurements reveal that the merging

processcanbedecomposedintoseveraldierentstages,discussedinthefollowing(theinitial

phase of vortex roll-up and formationfor t

<0:2is not considered in further detail inthis

paper).

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in Fig. (a). The separation distance b remains approximately constant, and the period of

rotationis almost equalto the one for apointvortex pair (t

c

). The slight oscillationof the

separationdistance aroundb

0

isdue totheconnementof theowby thevortex-generating

plates,despite the optimizationof theirmotion proletoreduce this eect. The core size a

of anaxisymmetric vortex with Gaussianvorticity distribution

!(r)=

a 2

e r

2

=a 2

(1)

where = 1:2563, solution of the Navier-Stokes equations, grows by viscous diusion of

vorticity according to:

a 2

=4t+const: (2)

The solid lines of Fig. correspond to this linear law, which matches the experimentally

measured evolution in the rst stage of the ow. As the Reynolds number increases, the

vortices diuse slower and this stage lasts longer (in convective time units), as visible in

Fig. .

The secondstage begins when the vortices reach a criticalsize, scaledon the separation

distance b. At this point, two tips of vorticity are created atthe outer side of the vortices

(Fig. b), which are subsequently ejected radially to form two arms of vorticity wrapping

around the vortices (Fig. c). Meanwhile, the vortex centers get closer and rapidly merge

intoasinglecore. Theseparationdistancedecreasestozero(seeFig.)withinapproximately

a third of the initialrotation period, an interval which varies littlewith Reynolds number.

This indicates that this second stage, i.e. the actual merging, is mainlya convective pro-

cess, where diusion of vorticity plays a minor role. For the highest Reynolds number in

Fig. , however, the evolution is somewhat slower, because the onset of a three-dimensional

instability slightly modies the merging. During the merging, it is diÆcult to calculate a

core size since the velocityelds of the two vortices strongly interfere.

Thecritical ratio of coresize and separation distance, atwhichmergingbegins,is found

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a

c

b

0

=0:290:01: (3)

The uncertainty takesintoaccountthe error madein thedeterminationof the beginningof

merging,whichisdenedasthetimewhentheseparationdistancebstartstodecrease. This

criticalratiois closetothe value of0.32 abovewhich mergingof vortex patches isobserved,

as predicted by theoretical and numerical studies [7, 10]. It is surprising to see that this

criterion is little inuenced by the initial prole of vorticity, although the denitions of a

are necessarily dierent.

Inthe thirdstage, the vorticityarms rollup aroundthecentralpattern(Fig.c), forming

aspiralofvorticity,whichisspreadout andsmoothedbydiusion(Fig.d). The coresizeof

this nal vortex then keeps increasing similarlytothe Gaussianevolution(2), althoughthe

vortex is not exactlyGaussian aftermerging. The measurementsshow, by extrapolationof

the viscous core evolutions,that the area of the nalvortex isabout twice the area of each

initialvortex. This increase islarger (beyond measurement uncertainties) than the increase

by afactor of p

2predicted theoreticallyby Carnevaleet al. [22] onthe basis ofenergy and

vorticity conservation. In our case with Gaussian vortices, the vorticity seems to decrease

during the merging process. This eect, coupled to the conservation of the circulation,

leadstoanincrease ofthe experimentallymeasured coresize aftermerging,andmay bethe

explanation for the observed discrepancy.

WhenincreasingtheReynoldsnumberand/orreducingthe dimensionlessinitialcoresize

a

0

=b

0

,theviscousphasebeforemerginglastslonger,andthereistimeforathree-dimensional

instability to develop while the two vortices are still separated. This phenomenon is here

observed for the rst time inthe corotating vortex pair.

The sideview visualization in Fig. shows a clear observation of the three-dimensional

instability. At this particular instant, the vortices, which spin around each other, are in

a plane perpendicular to the view direction. The vortex centers are deformed sinusoidally

withawavelengthclosetoone vortex separationb,andthe perturbationsonbothvortices

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counterrotating vortex pairs [12], where it was found to be a consequence of a cooperative

ellipticinstabilityofthevortexcores. Itisinterestingtoseethatthisinstabilitystilldevelops

in a corotating conguration, whereas the long-wavelength instability is suppressed by the

rotation of the vortex pair [18]. Elliptic instability occurs in ows with locally elliptic

streamlines [14, 16], resultinghere from the interaction between the vorticity of one vortex

and the external strain induced by the opposite vortex. Theory states that the wavelength

scales on the vortex core size a. In our experiment, the most amplied wavelength was

measured to be =a = 3:10:3. It is in agreement with the value of 3:5 found for the

elliptic instability of a Gaussian vortex in a rotating external strain, which can be derived

from theory [19, 17]. This is the rst experimental observation of an elliptic instability in

such aow.

Furthervisualizations and measurements were carried out to determine the growth rate

of the instability. By simultaneously illuminatingtwo cross-cut planes B-B and C-C (Fig.

),separated by halfa wavelengthand locatedat the maximum and minimumof the vortex

centerline displacement, one obtains an image as in the inset of Fig. (shown here as a

negative). First, these visualizations conrm that the instability mode is stationary (not

rotating orpropagating) in the rotatingframe of reference of the vortex pair, and that the

planes of the wavy centerline deformations are aligned with the stretching directionof the

mutually induced strain (at 45 Æ

with respect to the linejoining the vortices), as predicted

theoretically [16]. Second, from a series of such images, the amplitude A of the centerline

displacement of each vortex can be obtained as function of time. This displacement is

proportional to the amplitude of the unstable mode for small values. The result is shown

in Fig. . The growth is indeed exponential over a certain period, and anappropriate least-

squares t allows the determinationof the growth rate . The result corresponding to the

straight lineof Fig. is:

="=1:60:2; (4)

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where"= =2b isthemutuallyinducedexternalstrainatthevortexcenters. Atheoretical

prediction can be obtained, using the resultsin [19] which extend Baily's theory [15] tothe

case of anelliptic instability with a rotating strain s, and those in[23] which calculatethe

value of the strain s at a Gaussian vortex center as a function of the strain far from the

center (equal to " in our experiment). For the present case, with the experimental values

of R e and a=b, a growth rate of =" = 1:757 is predicted, which is reasonably close to the

measured value in (4).

All this evidence indicates that the three-dimensional instability observed in Fig. is

indeed anelliptic instability ofthe strainedvortical owin the cores.

When the perturbation amplitude gets suÆciently large, the organized spatial structure

breaks down. At the locations where the vortex centers are most deformed, layers of uid

initially orbiting one vortex are drawn around the respectively other vortex in a periodic

interlocking fashion. The corresponding tongues of dye are faintly visible in Fig. . These

tongues contain vorticity, which is reoriented and stretched into perpendicular secondary

vortices wrapping around the primary pair. This exchange of uid (and vorticity) between

the vortices has two consequences: 1) the vortices are drawn closer together, initiating a

premature merging, and 2) the interaction between primary and secondary vortices leads

to the almost explosive breakdown of the ow intosmall-scalemotion during this merging.

Eventually the ow relaminarizes, and one nds again a single viscous vortex at the end.

Despite the 3D perturbation and intermittentturbulence, one can stilldene eective core

sizes. The resulting measurements in Fig. show that merging sets inmuch earlier,i.e. for

lowera=b, than intwo-dimensionalow(althoughthe actualmergingphasemaybeslightly

longer,accordingtoFig.),andthat thenalvortexappearstobebiggerthanitwouldhave

been without the three-dimensional instability.

Insummary, we have presented experimental results concerning the interaction between

twoidentical,parallel,initiallylaminarvortices. Unlikeforvorticesininviscidows,merging

alwayshappens, sincevorticitydiusionmakesthe vortex coresize increaseintime,eventu-

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the separationdistance, 2)the actualmerging,i.e. the kinematicreorganisationof vorticity

intoa single core andspiral arms, with littleinuenceof viscosity, and3) the axisymmetri-

sation and viscous diusion of the nal vortex, which again depends on Reynolds number.

In addition, the measurements show a larger increase in vortex core size during merging

thantheoreticallypredictedforinviscidpatches. AthighR e, athree-dimensionalinstability

is discovered, showing the characteristic features of a cooperative elliptic instability of the

vortex cores. The existence of this instability in a situation with rotating external strain

is here demonstrated experimentallyfor the rst time. The subsequent stage of small-scale

turbulent owmakes the vortices mergefor smaller core sizes and into alarger nal vortex

than inthe absence of instability. Experimentsare presently underway to quantify inmore

detail the eect of three-dimensional instability onvortex merging.

REFERENCES

y

meunier@marius.univ-mrs.fr

[1] C.D.WinantandF.K.Browand,\VortexPairing: theMechanismofTurbulentMixing

Layer Growth atModerateReynolds Number", J.Fluid Mech. 63(2),237 (1974).

[2] S. A. Brandt and J. D. Iversen, \Merging of Aircraft Trailing Vortices", J. Aircraft

14(12), 1212 (1977).

[3] A. L.Chen, J. D. Jacob,and



O.Savas, \Dynamics ofa CorotatingVortex Pairsinthe

Wakes of Flapped Airfoils", J. FluidMech. 378, 155 (1999).

[4] W. J.Devenport, C. M.Vogel, and J.S.Zsoldos, J.FluidMech. 394, \FlowStructure

Produced by the Interaction and Merger of a Pairof Co-rotatingWing-Tip Vortices",

357 (1999).

[5] A. Vincent and M. Meneguzzi, \The Spatial Structures and Statistical Properties of

Homogeneous Turbulence" , J. FluidMech. 225, 1(1991).

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in Three-DimensionalTurbulent Shear Flow", Phys. Fluids7(3), 630 (1995).

[7] P.G. Samanand R. Szeto, \Equilibriumofa Pair of EqualUniformVortices",Phys.

Fluids 23(12), 2339 (1980).

[8] D. G. Dritschel, \The Stability and Energetics of Corotating Uniform Vortices", J.

Fluid Mech. 157, 95(1985).

[9] D. G. Dritschel, \A Moment Model for Vortex Interactions of the Two-Dimensional

Euler Equations.Part 1.ComputationalValidationof a HamiltonianEllipticalRepre-

sentation", J.Fluid Mech. 172, 157 (1986).

[10] E. A. Overman and N. J. Zabusky, \Evolution and Merger of Isolated Vortex Struc-

tures", Phys. Fluids 25(8),1297 (1982).

[11] R. W. GriÆths and E. J. Hopnger, \Coalescing of Geostrophic Vortices", J. Fluid

Mech. 178, 73(1987).

[12] T. LewekeandC. H.K.Williamson,\CooperativeEllipticInstabilityofaVortexPair",

J. Fluid Mech.360, 85 (1998).

[13] S. C. Crow, \Stability Theory for a Pair of Trailing Vortices", AIAA J. 8(12), 2172

(1970).

[14] C.-Y. Tsai and S. E. Widnall, \The Stability of Short Waves on a Straight Vortex

FilamentinaWeakExternallyImposedStrainField",J.FluidMech.73(4),721(1976).

[15] B.J.Baily,\Three-DimensionalInstabilityofEllipticalFlow",Phys.Rev.Lett.57(17),

2160 (1986).

[16] F. Walee, \On the Three-Dimensional Instability of Strained Vortices", Phys Fluids

A 2, 76(1990).

[17] C. Eloy and S. Le Dizes, \Three-Dimensional Instability of Burgers and Lamb-Oseen

Vortices ina Strain Field",J.Fluid Mech. 378, 145 (1999).

[18] J. Jimenez, \Stability of a Pair of Co-Rotating Vortices", Phys. Fluids 18(11), 1580

(1975).

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[20] D. G. Dritschel and M. de la Torre Juarez, \The Instability and Breakdown of Tall

Columnar Vortices ina Quasi-GeostrophicFluid" , J.Fluid Mech.328, 129 (1996).

[21] M. Rael, C. Willert, and J. Kompenhans, Particle Image Velocimetry: a Practical

Guide (Springer-Verlag,1998).

[22] G. F. Carnevale, J. C. McWilliams, Y. Pomeau, J.B. Weiss, and W. R. Young, \Evo-

lution of Vortex Statisticsin Two-Dimensional Turbulence", Phys. Rev. Lett. 66(21),

2735 (1991).

[23] S. Le Dizes, \Non-Axisymmetric Vortices in Two-DimensionalFlows", J. Fluid Mech.

406, 175 (2000).

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Figure 1: Vorticity elds during the merging of two corotating vortices for R e =1506

and a

0

=b

0

=0:18. (a) t

=0:7, (b) t

=1:2, (c) t

=1:6, (d) t

=2. Dierencebetween two

vorticity contours: 0:3s 1

.

Figure 2:Evolution of the separation distance b for dierentReynolds numbers before

merging.

Figure 3: Evolution of the square of the core size a before and after merging. Solid

lines: evolution for Gaussianvortices. Broken line: a=b

o

=0:29.

Figure 4: Side view of the elliptic instability on a corotating vortex pair. R e = 4140,

a

0

=b

0

0:15.

Figure 5: Amplitudeof the centerline oscillation. Inset: simultaneous view of the two

cross-cut sections B-B and C-C of Fig. (negativeimage). R e=2780, =a=3:1.

Figure 6:Evolutionofthesquareofthecoresizeawithandwithoutinstability. Broken

line: a=b

o

=0:2:

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-2 0 2 4 x [cm]

-4 -2 0 2 4

y[cm]

(a) -2 x [cm]0 2 4

-4 -2 0 2 4

y[cm]

(b) -2 x [cm]0 2 4

-4 -2 0 2 4

y[cm]

(c) -2 x [cm]0 2 4

-4 -2 0 2 4

y[cm]

(d)

Figure 1, Meunier, Phys. Fluids

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0.0 0.5 1.0 2.0

0.0 0.5 1.0 1.5 2.0

Re = 742 Re = 1506 Re = 2258

b / b

o

t*

Figure 2, Meunier, Phys. Fluids

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0.0 0.1 0.2 0.3

0.00 0.05 0.15

Re=742 Re=1506 Re=2258

a /b

2 o2

(8 π

2

/ Re) · t*

2 vortices merging 1 vortex

~×2

Figure 3, Meunier, Phys. Fluids

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B

B

C

C

Figure 4, Meunier, Phys. Fluids

(17)

0.1 1

10 12 14 16

A [cm]

t [s]

2A ϕ

b

Figure 5, Meunier, Phys. Fluids

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0.0 0.1 0.2 0.3

0.00 0.05 0.10

Re=1506 Re=3460

a /b

2 2o

(8 π

2

/ Re) t*

merging without instability

×3.5

×2 merging

with instability

Figure 6, Meunier, Phys. Fluids

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