Dipolar vortices in a strain flow · Dipolar vortices in a strain ﬂow R. R. Trieling,a) J. M. A....

Dipolar vortices in a strain flow

Citation for published version (APA):Trieling, R. R., Wesenbeeck, van, J. M. A., & Heijst, van, G. J. F. (1998). Dipolar vortices in a strain flow.Physics of Fluids, 10(1), 144-159. https://doi.org/10.1063/1.869556

DOI:10.1063/1.869556

Document status and date:Published: 01/01/1998

Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne

Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.

Download date: 22. May. 2021

https://doi.org/10.1063/1.869556

https://doi.org/10.1063/1.869556

https://research.tue.nl/en/publications/dipolar-vortices-in-a-strain-flow(09415451-dad5-4a37-a4b6-61625eda0ce5).html

Down

Dipolar vortices in a strain flowR. R. Trieling,a) J. M. A. van Wesenbeeck, and G. J. F. van HeijstFluid Dynamics Laboratory, Department of Technical Physics, Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands

~Received 12 June 1997; accepted 20 August 1997!

The evolution characteristics of dipolar vortices in a strain flow were investigated bothexperimentally and theoretically. The laboratory experiments were performed in a stratified fluid,the strain flow being generated by four rotating horizontal discs, whereas the dipolar vortex wascreated by a pulsed injection of a small amount of fluid. Dye-visualization studies andparticle-tracking techniques were used to obtain qualitative and quantitative information about thehorizontal flow field. Depending on the initial orientation of the dipole, either a head–tail structureor a pair of elliptic-like monopolar vortices was formed. In the former case, the distance between thevortex centers was observed to remain nearly constant due to the opposing effects of strain andlateral diffusion, while in the latter case, the vortex centers were passively advected by the ambientflow. The head–tail formation could be explained kinematically by a simple point-vortex model.Full-numerical simulations based on the quasi-two-dimensional vorticity equation revealed a verygood agreement with the laboratory observations. ©1998 American Institute of Physics.@S1070-6631~97!03112-7#

drbean

icho

ow

lyineeoD

rero

ot-ncao

acve

bytryr-

oleas

on-theinred

m-itu-

s-seetthertexore,

eren-of

ed

sivehereo-

nu-is

op-ivecu-ue

jstnd

B

I. INTRODUCTION

Two-dimensional vortices, such as those presentlarge-scale geophysical flow systems, may be deformedto their mutual interaction or by the presence of some atrary background flow. Based on their topological shapthese vortices can be classified into monopolar, dipolarother multipolar vortex structures.

The most common type is the monopolar vortex, whusually has a circular or elliptic shape. The deformationmonopolar vortices in straining or shearing ambient flfields has been investigated in analytical,1,2 numerical3–6 andexperimental7–11 studies.

The dipolar vortex is characterized by two closepacked patches of oppositely signed vorticity and contanet linear momentum. Although less frequently observthan monopolar vortices, the dipolar vortex is important bcause of its self-propelling motion which implies transportscalar properties like heat, salt and other constituents.poles have been observed both in nature~see, e.g., Refs. 12and 13! and in the laboratory~see, e.g., Ref. 14!. An impor-tant question concerns the stability of these vortex structuin a deforming ambient flow. For example, in the atmosphdipolar vortices may lead to a phenomenon called ‘‘atmspheric blocking’’~see Refs. 15 and 16!, which means thatthe global west–east circulation is locally hindered‘‘blocked’’ by two pressure cells of opposite circulation. Amospheric blocking systems may persist for a relatively lotime and consequently may have a large effect on loweather conditions. Therefore, it is essential to investigthe effect of a deforming background flow on the stabilitythese systems.

In contrast to monopolar vortices, the evolution charteristics of dipolar vortex structures in ambient flows ha

a!Present address: Royal Netherlands Meteorological Institute, P.O.201, 3730 AE De Bilt, The Netherlands.

144 Phys. Fluids 10 (1), January 1998 1070-6631/98/1

loaded 05 Sep 2011 to 131.155.151.8. Redistribution subject to AIP licens

inuei-s,d

f

sd-fi-

ese-

r

galtef

-

hardly been addressed in literature. In a numerical studyKida et al.,17 it was observed that the front–back symmeof a two-dimensional dipolar vortex is broken when the votex is submitted to a strain flow that pushes both diphalves together. As a result, a head–tail structure wformed. However, this behavior has as yet not been cfirmed by experimental observations. For that reason,strain-induced evolution of dipolar vortices is investigatedthe present study. In fact, two different cases are considein which the vortex centers of the dipole are either copressed or separated by the ambient flow. In the former sation, see Fig. 1~a!, the strain flow will be referred to as‘‘cooperative’’ in view of its progressive effect on the tranlational motion of the dipole, whereas in the latter case,Fig. 1~b!, the strain flow will be termed ‘‘adverse’’ since iopposes the dipole’s self-induced motion. As far aspresent authors are aware, the evolution of a dipolar voin an adverse strain flow has not been considered befneither in numerical studies nor in experimental work.

The laboratory experiments described in this paper wcarried out in a stratified fluid. The dipolar vortex was geerated by a pulsed horizontal injection of a small amountfluid, whereas the strain flow was continuously forcthrough four rotating discs~see Ref. 9!. The horizontal flowcharacteristics were measured by tracking small pastracer particles which were floating at a specific level in tstratified fluid. Furthermore, dye-visualization studies weused to obtain qualitative information about the dipole evlution.

The laboratory observations are compared with twomerical models. As a first approach, the dipolar vortexrepresented by two point vortices of equal strength, butposite in sign, surrounded by a single contour of passtracers. The time evolution of the passive contour is callated by the method of contour kinematics. This techniqhas been applied successfully by Meleshko and van Hei18

to study the stirring properties of interacting monopolar aox

0(1)/144/16/$10.00 © 1998 American Institute of Physics

e or copyright; see http://pof.aip.org/about/rights_and_permissions

deor

ipbailaa

difthec

ieraofthd

aycecye

,t

al

anthtao

oth

f

adehe

n

ed

-ud19

gionr-

r-ne

sivea

x-ere

lysis

ityntal

de-

the

-

Down

dipolar vortices. As a second approach, both the spatialtribution of vorticity and the effect of viscous diffusion artaken into account by solving the quasi-two-dimensional vticity equation with a finite-difference method.

This paper is organized as follows. In Sec. II, a descrtion of the experimental set-up is given. After that, the oserved evolution of the dipolar vortex in a cooperative strflow is discussed in Sec. III. Furthermore, a kinematic expnation of the head–tail formation is given, followed bycomparison with the numerical results obtained by theferent calculation techniques. Likewise, the behavior ofdipolar vortex in an adverse strain flow is examined in SIV. Finally, the main conclusions are given in Sec. V.

II. EXPERIMENTAL ARRANGEMENT

The experiments were carried out in a Perspex tank, whorizontal dimensions 1003100 cm and depth 30 cm. Thtank was filled with a two-layer fluid in which fresh watewas lying over salty water, and the total fluid depth wapproximately 26 cm. Owing to vertical mass diffusionsalt, a layer of typically 5 cm thickness was formed nearmid-plane of the tank in which the density rapidly changeSince this sharp density gradient persisted for several dmass diffusion of salt is believed to be of minor importanduring the evolution of the experiment. The buoyanfrequencyN near the mid-plane of the tank was varied btween 3.5 and 4.5 rad s21. Here, N is defined byN5(2g/ r )(dr/dz), with g the gravitational accelerationr the mean density anddr/dz the vertical density gradienaveraged over the thickness of the diffused interface.

The strain flow was realized with four rotating horizontdiscs~diameter 10 cm, thickness 0.5 cm!, which were posi-tioned at the corners of a square in the mid-plane of the t~see Fig. 2!. The diagonal distance between the centers ofdiscs was 60 cm, and the discs were rotated with a consrotation speed of typically 3.0 r.p.m. This generation methis identical to that described by Trielinget al.9 Typically onehour after the forcing was started, a quasi-steady strain flwas formed that was close to uniform near the center oftank.

The dipolar vortex was created by a pulsed injection osmall amount of fluid through a thin nozzle~see, e.g., Ref.19!. The injection nozzle was positioned in the same planethe rotating discs, along one of the strain axes, and thesity of the injected fluid was chosen equal to that of t

FIG. 1. Schematic drawing of a dipolar vortex in~a! a cooperative and~b!an adverse strain flow.

Phys. Fluids, Vol. 10, No. 1, January 1998


is-

-

--n-

-e.

th

s

e.s,

-

kentd

we

a

sn-

ambient fluid at the plane of injection. A computer-driveinjection mechanism was used to control the injection rateQ(55.6 ml s21) and the forcing perioddt (51.2 s!. The Rey-nolds number Re5Uin jd/n was equal to 2400 and was bason the injection speedUin j , the nozzle diameterd (52 mm!and the kinematic viscosityn of the injected fluid~whichwas equal to 1.08531022 cm2 s21 with the temperature being 19 °C!. As a result, a three-dimensional turbulent clowas created by the injected fluid. It was shown in Refs.and 14 that in the subsequent stage, the turbulent requickly collapsed under gravity, leading to the gradual fomation of a flat dipolar vortex structure.

Quantitative information about the horizontal flow chaacteristics was obtained by monitoring small polystyreparticles of density 1.04 g cm23 with a video camera, whichwas mounted at some distance above the tank. The pasparticles were distributed around the injection level withinlayer of approximately 1 cm thickness. After the entire eperiment was recorded on video tape, the video images wdigitized and subsequently processed by the image anasystem DigImage.20 A particle-tracking technique, which ispart of this system, was used to obtain the local velocvectors at successive times. As a next step, the horizovelocity field was calculated on a rectangular 65365 grid bya spline-interpolation method,21,22 from which the values ofthe vertical vorticityv and the stream functionc could becalculated in each grid point. The stream function wasfined byuh52k3¹c, with uh the horizontal velocity andkthe unit vector in the upward direction. In other cases,

FIG. 2. ~a! Top view and~b! side view of the experimental set-up to generate the strain flow.

145Trieling, van Wesenbeeck, and van Heijst


in

a-e

idrve

ratthhiewug

otho-

arncth

fo

thagd

easepao

exinhen-he

i

of

nts

ere

aine

int-

ow

t is

a. It

Down

flow was visualized by adding fluorescent dye of matchdensity to the injection fluid.

III. DIPOLAR VORTEX IN COOPERATIVE STRAINFLOW

A. Qualitative observations

The typical evolution of the dipolar vortex in a coopertive strain flow is shown by the sequence of video imagpresented in Fig. 3. Initially, the motion of the injected fluis essentially three-dimensional turbulent, as can be obsein Fig. 3~a!. Owing to the gravitational collapse and thmerging of like-signed eddies~see Ref. 14!, this turbulentregion eventually transforms into a dipolar vortex structu@see Figs. 3~b!–3~d!#. Although special care was taken ththe injection nozzle was accurately aligned with one ofstrain axes, a slightly asymmetric dipole was formed in tcase. Symmetric dipoles were obtained on only very foccasions, which is most likely due to the irregular distribtion of vorticity during the dipole formation. The deformineffect of the straining flow can be observed in Figs. 3~e!–3~h!: the dipole is compressed and elongated as a whresulting in the formation of two tails at its rear. Becausedipole is slightly asymmetric, the lower tail is more prnounced than the other.

Note that the long dye filaments behind the dipolejust remnants of the formation process of the vortex ashould not be confused with the tails of the dipole. In fathese filamentary structures have also been observed bedipolar vortices in a quiescent ambient fluid~see Ref. 14!.Furthermore, it can be seen that shortly after the dipolemation, considerable entrainment of~irrotational! ambientfluid occurs at the rear side of the dipole, which leads tospiral shape in the dye distributions. In the subsequent sthowever, the entrainment process is counteracted by thetrainment of dyed fluid.

B. Kinematic explanation of the head–tail formation

The formation of a head–tail structure can be explainby a simple kinematic approach. For this, the flow issumed to be two-dimensional and the dipolar vortex is rresented by a configuration of two point vortices of equbut oppositely signed strengths. Supposing that these pvortices with strengths 2g and g are located at(x,y)5(0,2b) and (x,y)5(0, b), respectively, the point-vortex dipole will move steadily along thex-axis with atranslation speedU5g/4pb. Relative to a co-moving framewith velocity U ~in which the flow is steady!, the streamfunction associated with the vortex pair is given by

c~x,y!5g

4plnFx21~y2b!2

x21~y1b!2G2Uy. ~1!

Figure 4~a! shows the steady flow pattern of the point-vortdipole, in which two types of streamlines may be distguished: ‘‘closed’’ streamlines, which surround one of tpoint vortices, and ‘‘open’’ streamlines, which extend to ifinity. It is obvious that fluid particles are trapped inside tregion of closed streamlines. This region of trapped fluid

146 Phys. Fluids, Vol. 10, No. 1, January 1998


g

s

ed

e

es

-

le,e

ed,ind

r-

ee,e-

d--

l,int

-

s

usually called the ‘‘atmosphere’’ of the dipole. The shapethe atmosphere boundary is given by the expression~see Ref.23!

x252yb cothS y

2bD2y22b2, ~2!

and can be approximated by an ellipse with axes 2.09b and1.73b. Furthermore, the flow contains two stagnation poilocated at (x,y)5(2bA3,0) and (x,y)5(bA3,0), respec-tively, which are the points of intersection of the atmosphboundary and the streamliney50.

When the point-vortex dipole is exposed to a pure strflow, the analytical velocity field in a co-moving referencframe is given by

u~x,y!52g

2p

y2b

x21~y2b!21

g

2p

y1b

x21~y1b!21ex2Uy,

~3!

v~x,y!5g

2p

x

x21~y2b!22

g

2p

x

x21~y1b!22ey, ~4!

with u andv the velocity components in the Cartesianx- andy-directions, respectively, ande the rate of strain. It shouldbe noted that the relations~3! and ~4! contain an implicittime dependence: owing to the interaction between the povortex dipole and the strain flow, the value ofb varies ac-cording tob52eb, or

b~ t !5b0exp~2et!, ~5!

with b0[b(0). As aconsequence, the distance 2b betweenthe point vortices will decrease in a cooperative strain fl(e.0). The stream function associated with~3! and~4! canbe written as

c~x,y!5g

4plnFx21~y2b!2

x21~y1b!2G1exy2gy

4pb, ~6!

whereU5g/4pb has been used.In order to find the corresponding stagnation points, i

convenient to introduce the complex velocityV[u1 iv, sothat relations~3! and ~4! can be combined into

V* 521

2p i

g

z2 ib1

1

2p i

g

z1 ib2

g

4pb1ez, ~7!

where V* represents the complex conjugate ofV, andz[x1 iy ~with i 5A21). On puttingV50, and defining thedimensionless valuesz[z/b andk[g/4peb2, a cubic equa-tion is obtained of the form

z32kz22z12k50, ~8!

of which the three roots can be found by standard algebris worth mentioning that the dimensionless parameterk rep-resents the ratio between the dipole’s translation speedU andthe characteristic velocityeb induced by the strain flow. Bydefining

k05pS 592

31

32

3~343!1/2D 1/2

'62.4, ~9!

Trieling, van Wesenbeeck, and van Heijst


Down

FIG. 3. Sequence of video images showing the evolution of a dye-visualized dipolar vortex in a cooperative strain flow, withe50.4531022 s21. The imageswere taken at~a! t510 s, ~b! 20 s, ~c! 30 s, ~d! 40 s, ~e! 50 s, ~f! 60 s, ~g! 70 s and~h! 80 s. Experimental parameters:Q55.6 ml s21, dt51.2 s andN53.5 rad s21.

ta

e

all

lie

three different cases can be considered. Ifk.k0, then theroots are real and unequal, which means that the three snation points are located at the symmetry axis@see Fig. 4~b!#.If k5k0, there will be three real roots of which two ar



g-equal. In other words, the symmetry axis will containstagnation points, of which two are degenerate@see Fig.4~c!#. Finally, if k,k0 there will be one real root and twoconjugate imaginary roots, i.e., one stagnation point will



ut the

Down

FIG. 4. Streamline patterns associated with a point-vortex dipole in a co-moving reference frame for four different values ofk ~see the text!: ~a! k5` ~nostrain flow!, ~b! k580, ~c! k5k0562.4, and~d! k540. The dashed lines correspond to the atmosphere boundary of the point-vortex dipole withopresence of a strain flow. The stagnation points are marked by a diamond.

re

e

ot

boline

oenurieex

igrd

lyur

la

la-thehea-alf.nu-

i-etryialat

cu-e.bu-b-

sonsandbyhe

tialhater,-

by

on the symmetry axis, whereas the two others are mirrowith respect to the symmetry axis@see Fig. 4~d!#. Note thateach conjugate imaginary root is associated with the sintersection of a streamline.

Also shown in Figs. 4~b!–4~d! is the boundary of theatmosphere associated with the point-vortex dipole in anerwise quiescent fluid~dashed lines!. If the strain flow wereabsent, all fluid particles inside this atmosphere wouldtrapped and would consequently move along with the dipHowever, if the point-vortex pair is submitted to a straflow, part of the tracers enclosed by the original atmosphboundary~i.e., with the strain flow being absent! will belocated in the exterior region. In other words, a new ‘‘atmsphere’’ is formed which only partly catches the tracersclosed by the dashed line. Consequently, part of the flenclosed by the original atmosphere boundary is caraway by the ambient flow at the rear of the point-vortdipole. It is important to note that the velocity field~7! isunsteady, so that the streamline patterns as shown in F4~b!–4~d! only indicate the instantaneous direction of paticle movement. Nevertheless, the streamline patternspicted in Figs. 4~b!–4~d! clearly explain the formation of ahead–tail structure.

C. Quantitative observations

The formation of the head–tail structure is also niceillustrated by the experimentally obtained vorticity contoplots presented in Figs. 5~a!–5~d!, in which the vorticity con-tours around the rotating discs have been eliminated for c



d

lf-

h-

ee.

re

--

idd

s.-e-

r-

ity. Note that in this case, and those that follow, the transtional motion of the dipole has been chosen alongvertical axis. Like in the dye-visualization experiment, tdipole is elongated in the axial direction, while simultneously a tail is formed at the rear side of each dipole hThese results are in good qualitative agreement with themerical simulations obtained by Kidaet al.17 Likewise, asimilar head–tail asymmetry was observed by Flo´r and vanHeijst14 in their study on laminar-injection dipoles in a quescent ambient fluid. However, in their case, the asymmand the deformation of the vortex were due to the initforcing with a jet. Moreover, their experiments revealed ththe dipole gradually relaxed towards an approximately cirlar shape, whereas the present results show the opposit

Figure 6~a! shows a sequence of cross-sectional distritions of vorticity which correspond to the experimental oservations depicted in Figs. 5~b!–5~d!. The vorticity profileassociated with Fig. 5~a! was excluded since the dipole wanot completely developed by that time. The cross-sectiwere taken along a line intersecting both vortex centers,the vorticity values between the grid points were obtainedbilinear interpolation. The vorticity has been scaled with tmaximum vorticity valuev0 at t540 s @Fig. 5~b!#, whereasthe spatial coordinate has been normalized with the inidistanced0 between the vortex centers. It can be seen tthe profiles are approximately self-similar in time. Moreovthe instantaneous distanced between the vortex centers remains nearly constant until aboutt570 s, as can be inferredfrom Fig. 7. If the vortex centers were passively advected



of,

boundary

Down

FIG. 5. Experimental and numerical results for a dipolar vortex in a cooperative strain flow of strengthe50.4531022 s21. Panels~a!–~d! show the observedevolution of the spatial vorticity distribution, with the contour labels presented in units of s21. Experimental parameters as in Fig. 3. The time evolutionthe Lamb–Chaplygin vortex as calculated by the finite-difference method is depicted in panels~e!–~h!; for each plot, the contour levels correspond to 1%10%, 30%, 50%, 70% and 90% of the instantaneous extremal vorticity, respectively. Numerical parameters:a53.7 cm, ULC50.98 cm s21 andn51.08531022 cm2 s21. Along the horizontal lines depicted in frames~d! and~h!, the cross-sectional distributions of vorticity were determined~see Fig. 8!.Panels~i!–~l! show the time evolution of passive tracers as obtained by the contour kinematics technique. The tracers were initially located at theof the vortex-pair atmosphere; see~i!. The numerical parameters area52b53.7 cm,C5173/1.09 cm2 s21/2 andt585 s.

or hetynter,

the strain flow,d would approximately decrease by a fact1.6 during the course of the experiment~i.e., d/d0'0.64,based on the strain rate constante50.4531022 s21 and thetypical time span of the experiment, which is 100 s!. It is



therefore most likely that the compressive effect of tstraining flow is counteracted by lateral diffusion of vorticiand also by the initial entrainment of irrotational ambiefluid at the rear side of the dipole. In the final stage, howev



gty

Down

FIG. 6. Experimental and numerical results for a dipolar vortex in a cooperative strain flow. Panel~a! shows the cross-sectional distributions of vorticity alona line through the vortex centers of the laboratory dipoles which are shown in Figs. 5~b!–5~d!. The vorticity has been scaled with the initial maximum vorticiv051.04 s21 ~at t540 s!, whereas the spatial coordinate has been normalized by the corresponding distance between the vortex centersd056.58 cm@see Fig.5~b!#. Panel~b! shows similar vorticity profiles which are based on the finite-difference results depicted in Figs. 5~f!–5~h!, with v051.03 s21 andd053.70cm at t540 s @see Fig. 5~f!#.

gnit

am

reitys

ndr-r-

ac-

tin

the

ore,ith

the

isat

n ofarengtheide

rle

ua-

.the

en

the distance between the vortex centers decreases sicantly as a result of the non-uniformity of the strain flow:was shown by Trielinget al.9 that the local rate of strainincreases at larger radii from the center of the tank. Appently, the strain rate is eventually large enough to overcothe effect of lateral diffusion.

Additional cross-sectional distributions of vorticity wetaken across the tails of the dipolar vortex. A typical vorticprofile is shown in Fig. 8~a!, where the cross-section wataken along the horizontal line depicted in Fig. 5~d!. Obvi-

FIG. 7. The measured distanced/d0 plotted as a function of time. Thedistanced between the points of maximum and minimum vorticity has bescaled with the initial distanced056.58 cm~at t520 s!. The experimentaldata are based on the experiment depicted in Figs. 5~a!–5~d!.



ifi-

r-e

ously, the left and right tail are associated with positive anegative vorticity, respectively. The asymmetry in the voticity distributions is due to the asymmetric shedding of voticity at the rear side of the vortex.

Some more information about the dipolar vortex charteristics can be obtained by plotting the vorticityv againstthe stream functionc in a so-called ‘‘scatter plot’’ for a largenumber of grid points. Figure 9~a! shows the scatter plocorresponding to the head of the dipolar vortex depictedFig. 5~d!. The stream function has been corrected fortranslation speed (Ux ,Uy) of the dipolar vortex according toc85c2Uxy1Uyx, whereUx and Uy have been obtainedfrom the displacements of the vortex centers. Furthermthe vorticity and the stream function have been scaled wtheir extremal valuesvm and cm8 , respectively. Obviously,the positive and negative branches are related to one ofdipole halves, whereas the horizontal band aroundv50 isassociated with the exterior potential flow. The scattermore pronounced for weaker vorticity values, indicating ththe scatter is mainly caused by the continuous deformatiothe dipole by the strain flow, i.e., most parts of the scatterrelated to the edge of the dipole where vorticity is beiremoved and being shed in the form of two tails. Despitescatter, an approximately linear relationship is present insthe dipolar vortex.

A theoretical model of a dipolar vortex with a linea(v,c8)-relationship is the so-called Lamb–Chaplygin dipomodel.24–26 This model is based on the steady Euler eqtions, and assumes a linear relationshipv5k2c8 ~with kbeing a constant! inside a circular region with radiusa, whilein the exterior region (r .a) an irrotational flow is assumedThe stream function and the vorticity associated withLamb–Chaplygin dipole are given by



sen the

Down

FIG. 8. ~a! Measured cross-sectional distribution of vorticity across the tails of the dipolar vortex depicted in Fig. 5~d!. The vorticity has been scaled with itextremal valuevm50.38 s21 ~corresponding to one of the vortex centers!, whereas the spatial coordinate has been normalized by the distance betwevortex centersd55.62 cm.~b! Corresponding calculated profiles, see Fig. 5~h!, with vm50.40 s21 andd53.90 cm.

ly,yed

-

venear

c8~r ,u!55 22ULC

kJ0~ka!J1~kr !sinu, if r<a,

ULCS r 2a2

rD sinu, if r .a,

~10!

v~r ,u!5H 22ULCk

J0~ka!J1~kr !sinu, if r<a,

0, if r .a,

~11!

respectively, where polar coordinates (r ,u) have been usedfor convenience. The functionsJ0 and J1 are the first- and



second-order Bessel function of the first kind, respectiveand ULC represents the uniform fluid velocity at infinit~which is equal to the dipole’s translation speed in a fixframe of reference!. Continuity of velocity atr 5a requiresthat J1(ka)50, i.e., ka53.83. Furthermore, it can be derived from ~11! that the circulationgLC in each dipole half isequal to 6.83ULCa, and that the distanced between thepoints of extremal vorticity is equal to 0.96a ~so thatkd53.68).

In Fig. 9~a!, the data points associated with the positiand negative branches were least-square fitted with the lirelationship v5k2c8, which yielded the slope

o
FIG. 9. ~a! Characteristic (v,c8)-scatter plot of the laboratory vortex shown in Fig. 5~d!, where the stream functionc has been corrected according tc85c2Uxy1Uyx, with (Ux ,Uy) the translation speed of the dipole. The vorticity and the stream function have been scaled by their extremal valuesvm andcm8 , respectively. Experimental values:vm50.39 s21, cm8 51.09 cm2 s21 and (Ux ,Uy)5(20.03,0.32) cm s21. ~b! The numerically obtained (v,c8)-scatterplot corresponding to Fig. 5~h!. Numerical values:vm50.40 s21, cm8 51.61 cm2 s21 and (Ux ,Uy)5(0,0.235) cm s21.


s-le

cal.

thain

ikn-th

tebi

rs

ep-iv

ybynd

-

-

itetio

nasa

io

ari-l

ua-

i-d,

y

ion

len-

x

li-ig.the-s

teral

toined

ity

ory

nged,.

bo-

lsingat-oryon-

sle

.yer

sh. In

eal

Down

k250.4060.10 cm2. By measuring the corresponding ditanced between the vortex centers from the vorticity profiin Fig. 6~a! ~indicated by the crosses!, the valuekd53.660.6 was obtained, which is close to the theoretivalue kd53.68 related with the Lamb–Chaplygin modeThus, despite the deformation of the dipolar vortex,Lamb–Chaplygin model apparently still applies to the mpart of the dipolar vortex.

In contrast, it was shown by Flo´r and van Heijst14 thatturbulent-injection dipoles are characterized by a sinh-l(v,c8)-relationship. In their experimental study, this nonliear relationship was ascribed to the weak linking betweenpositive and negative vorticity patches; that is, the vorcenters were further apart than expected from the LamChaplygin model. Since the cross-sectional distributionsFig. 6~a! show a strong linking between the vortex centeand the nonlinearity of the (v,c8)-relationship in Fig. 9~a! isless pronounced than in the experimental observationsFlor and van Heijst,14 it may be concluded that th(v,c8)-relationship of turbulent-injection dipoles in a cooerative strain flow is more linear due to the compressaction of the ambient flow.

D. Numerical simulations

1. Finite-difference method

Stratified fluid experiments by Flo´r and van Heijst14

have shown that dipolar vortices are characterized bpancake-like shape. The same authors demonstratedscaling analysis that in a thin region around the mid-pla(z50) vortex tilting by the vertical shear is negligible, anthat the vertical vorticityv is to leading order governed by

]v

]t1J~v,c!5n¹h

2v1n]2v

]z2, ~12!

wherec satisfies the Poisson equationv52¹h2c with ap-

propriate boundary conditions,¹h2 is the horizontal Laplace

operator andJ represents the Jacobian.In order to solve~12! in only two dimensions, i.e., in the

mid-planez50, the latter term in~12! is modelled by theanalytical expression

]2v

]z2Uz50

52v

2tUz50

, ~13!

which applies fort.0 andz50 ~see Ref. 9 for more details!. This expression is based on the assumption that att50the vorticity is distributed according tov5v2D(x,y)d(z),with v2D(x,y) the horizontal distribution of the vertical vorticity and d the Dirac delta function~see Ref. 27!. As aresult, the vertical diffusion term in~12! is independent ofzand can be easily treated by a two-dimensional findifference code. In the present paper, the vorticity equa~12! with the vertical diffusion being modelled by~13! isreferred to as the quasi-two-dimensional vorticity equatio

The quasi-two-dimensional vorticity equation wsolved by a finite-difference method that is second-ordercurate both in space and in time~see Ref. 28!. The samenumerical method has been used successfully in prev



l

e

e

ex–n,

by

e

aa

e

-n

.

c-

us

studies9,10 to model the evolution of monopolar vortices instrain flow. The time integration was performed with a vaable time step, such thatCFL50.5, and the computationadomain was represented by a 2563256 grid. The strain flowwas included as a boundary condition for the Poisson eqtion.

As an initial condition, the vorticity distribution assocated with the Lamb–Chaplygin dipole model was usewhere the relevant initial parameters~i.e., the radiusa andthe translation speedULC! were based on typical laboratorvalues. Considering the singularity att50 and the typicaltime scale on which the dipole was formed, the simulatwas initiated att510 s.

The calculated evolution of the Lamb–Chaplygin dipoin a cooperative strain flow is shown by the vorticity cotours plotted in Figs. 5~e!–5~h!, and the formation of thehead–tail structure is obvious. Figure 6~b! shows the distri-butions of vorticity along the line intersecting both vortecenters, which correspond to Figs. 5~f!–5~h!. The numeri-cally obtained vorticity distributions are in very good quatative agreement with the observed profiles given in F6~a!. In both cases, the profiles have similar shapes, anddistanced between the positions of extremal vorticity remains virtually constant. Additional numerical simulationhave shown that the balance between advection and ladiffusion of vorticity does not hold for all values ofe: forlarger strain ratese, the vortex centers were observedapproach each other. Similar numerical results were obtaby Kida et al.17

The calculated cross-sectional distribution of vorticacross the tails of the dipolar vortex are shown in Fig. 8~b!,and are in good qualitative agreement with the laboratobservations@Fig. 8~a!#. The magnitude of vorticity is ex-perimentally larger than in the numerical simulation: owito the non-uniformity of the experimental strain flow thshedding of vorticity at the dipole boundary is intensifieleading to higher vorticity values in the tails of the dipole

In Fig. 9~b!, the numerically obtained (v,c8)-scatterplot is provided for the dipolar vortex shown in Fig. 5~h!,and a good qualitative agreement is obtained with the laratory observations depicted in Fig. 9~a!. Close to the vortexcenters the (v,c8)-relation is linear, whereas at lower leveof vorticity considerable scatter occurs due to the sheddof low-level vorticity. Besides experimental errors, the scter in Fig. 9~a! is more pronounced because the laboratvortex experienced a larger strain rate due to the nuniformity of the experimental strain flow.

The scatter in Fig. 9~b! may also be caused by viscoueffects. Since the initially imposed Lamb–Chaplygin dipois discontinuous in]v/]r on the circler 5a, the kink in thevorticity profile will be smoothed out by viscous diffusionAs a result, the Jacobian becomes non-zero in a thin lanear the circular boundaryr 5a. However, it was shown byKida et al.17 that this nonlinear effect is small as long at!Td , with Td5a2/n the characteristic time scale on whicthe viscous layer spreads over the whole vortex structurethe present study, the dipole radiusa was typically 5 cm,implying that Td'2500 s, which is much larger than thduration of the numerical simulation. Moreover, addition



Down

FIG. 10. Sequence of video images illustrating the evolution of a dye-visualized dipolar vortex in an adverse strain flow withe50.7031022 s21. The imageswere taken at~a! t510 s,~b! 30 s,~c! 50 s,~d! 90 s,~e! 130 s and~f! 170 s. Experimental parameters:Q55.6 ml s21, dt51.2 s andN53.5 rad s21.

ttbitythx

tace

o-foa

itedotearble

di

in

ees

n,lcayIn

cu-ipi-r-isd-alanton.so-ent

om

ailent

finite-difference simulations have revealed much less scaat the weaker vorticity levels when the strain flow was asent. Yet, the continuous removal of the weaker vorticlevels may increase the spatial vorticity gradient atboundary, leading to an accelerated outward diffusion flu6

2. Contour kinematics method

It was shown above that the formation of a head–structure can be explained by considering the strain-induevolution of a point-vortex dipole and its original atmsphere. For that reason it is interesting to calculate the demation of the atmosphere boundary and to make a compson with both the laboratory observations and the findifference results. For this, the contour kinematics methoadopted here, which implies that a material contour is flowed in an analytically prescribed velocity field. The marial line is composed of a large number of markers, whichdisplaced in time by a Runge–Kutta method with variatime step and order~see Ref. 29!. For a detailed descriptionof the contour kinematics technique, the reader is referreRef. 18. In the present study, the analytical velocity fieldrepresented by that of a point-vortex dipole in a strainflow @see~3! and ~4!#.

In order to compare the point-vortex pair with thLamb–Chaplygin dipole, two important physical quantiti



er-

e.

ild

r-ri--isl--e

tosg

were taken to be the same: the total impulse~being 2bgrand 2pa2rULC , respectively! and the translation speed~be-ing g/4pb and ULC , respectively!. As a result,a52b andgLC51.09g. In order to account for the decay of circulatiowhich is mainly due to diffusion of vorticity in the verticadirection,g was made time dependent according to the deof circulation as obtained by the finite-difference method.fact, the calculated decay of circulation could be very acrately described by the functional relationsh(C/At)exp(2t/t), with t a characteristic time scale assocated with horizontal diffusion due to the cancellation of voticity along the symmetry axis. This analytical relationshipbased on the so-called ‘‘vertical diffusion model’’ introduceby Flor et al.27 with the horizontal flow based on the viscously decaying Lamb–Chaplygin dipole. Both the initiposition of the point-vortex pair and the strain rate constwere taken the same as in the finite-difference simulatiThe tracers were initially distributed on the separatrix asciated with the point-vortex dipole in an otherwise quiescfluid.

Figures 5~i!–5~l! show that the initially elliptic materialline is deformed into a head–tail structure, as expected frthe streamline patterns depicted in Figs. 4~b!–4~d!. Since theinitial contour encloses both point vortices, only one tshows up in the contour kinematics simulation. It is appar



of%,

initially

Down

FIG. 11. Experimental and numerical results for a dipolar vortex in an adverse strain flow of strengthe50.7031022 s21. Panels~a!–~d! show the observedevolution of the spatial vorticity distribution, with the contour labels displayed in units of s21. Experimental parameters as in Fig. 10. The time evolutionthe Lamb–Chaplygin vortex as calculated by the finite-difference method is depicted in panels~e!–~h!; for each plot, the contour levels correspond to 1030%, 50%, 70% and 90% of the instantaneous extremal vorticity, respectively. The numerical parameters area55.3 cm, ULC50.88 cm s21 andn51.08531022 cm2 s21. Panels~i!–~l! show the time evolution of passive tracers as obtained by the contour kinematics technique. The tracers weredistributed on the separatrix associated with the point-vortex dipole in an otherwise quiescent fluid. The numerical parameters area52b55.3 cm andC5102/1.09 cm2 s21/2.

mlt

arse,

ctedof

that the translation speed of the vortex pair is larger copared to both the experimental and finite-difference resuThis can be explained by noting that the point vorticespassively advected by the strain flow according to~5!, so that



-s.e

the self-induced translation speedU5g/4pb of the point-vortex dipole increases more rapidly than in the viscous cawhere the convergence of the dipole centers is counteraby horizontal diffusion of vorticity. Likewise, the absence



e

Down

FIG. 12. Experimental and numerical results for a dipolar vortex in an adverse strain flow. In panels~a!–~d!, the cross-sectional distributions of vorticity arplotted along a line through the vortex centers of the laboratory dipoles which are shown in Figs. 11~a!–11~d!, respectively. In each plot, the vorticityv hasbeen scaled with its extremal valuevm , whereas the spatial coordinate has been normalized by the initial distance between the vortex centersd058.85 cm~at t520 s!. The extremal vorticity ranged fromvm51.14 s21 at t520 s tovm50.14 s21 at t5200 s. In frames~e!–~h!, similar normalized vorticity profilesare shown related to the finite-difference results depicted in Figs. 11~e!–11~h!. The extremal vorticityvm varied from 1.21 s21 at t520 s to 0.17 s21 at t5200s. Furthermore,d055.20 cm~at t520 s!.

155Phys. Fluids, Vol. 10, No. 1, January 1998 Trieling, van Wesenbeeck, and van Heijst

loaded 05 Sep 2011 to 131.155.151.8. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions

thheon

t

d-i

ol-in

-i.eerleisnwsm

dh

os

lin

a

ib

i-tan

ane

ed

tel

e

n

e-

-tal

lar

texen

ma-di-

f these

fre

pre-en

at

the

uml

Down

viscous effects may explain why the tail associated withcontour kinematics simulation is much thinner than in tlaboratory experiments and the finite-difference simulatiNevertheless, this simple approach clearly demonstratesformation of a head–tail structure.

IV. DIPOLAR VORTEX IN ADVERSE STRAIN FLOW

A. Laboratory observations

The qualitative behavior of a dipolar vortex in an averse strain flow is displayed by the video images shownFig. 10. Shortly after the turbulent-injected fluid has clapsed under gravity, a dipolar vortex appears which entraa considerable amount of non-colored ambient fluid@see~a!–~c!#. In the next stage, see~d!–~f!, the dipole halves gradually separate by the action of the background strain flow,the dipole breaks up into two monopolar vortices. It is intesting to note that both monopoles are oriented at an angapproximately 45° with respect to the horizontal strain axSimilar observations were made in analytical, numerical aexperimental studies on monopolar vortices in strain flo~see, e.g., Refs. 2, 4, and 9!, where it was shown that thiorientation corresponds to the quasi-stationary state of anopolar vortex in a strain flow.

The measured vorticity fields are shown in Figs. 11~a!–11~d! at four different times, and a similar behavior is founto that observed in the dye-visualization experiments. Tseparation of the vortex centers is also clear from the crsectional distributions of vorticity depicted in Figs. 12~a!–12~d!, where the cross-sections were taken along theintersecting both vortex centers.

In order to examine whether the dipole halves were psively advected by the strain flow, the distanced between thevortex centers was determined from cross-sectional distrtions as shown in Figs. 12~a!–12~d!. In Fig. 13, the distanced/d0 is plotted logarithmically as a function of time, withd0

the distanced at t520 s. The exponential increase is obvous, at least up tot5150 s. Hence, the experimental dacorresponding tot<150 s were least-square fitted with aexponential function of the form exp(bt) @see~5!#, which isdisplayed by a straight line in Fig. 13. The parameterb wasfound to be equal to (0.7360.05)31022 s21, which is inexcellent agreement with the strain rate conste5(0.7060.01)31022 s21. The final accelerated increasof d is due to the non-uniformity of the strain flow~see Ref.9 for more details!. In view of the above results, it may bconcluded that the vortex centers are passively advectethe strain flow.

The monopolar vortex characteristics were investigaby plotting the distribution of vorticity along a horizontaline through one vortex center. Figure 14~a! shows such avorticity profile corresponding to the monopolar vortex dpicted on the left-hand side of Fig. 11~c!, which obviouslyhas only single-signed vorticity. A possible model for a noisolated vortex is the so-called Lamb vortex~see Ref. 24!, forwhich the radial distributions of vorticity and azimuthal vlocity are given by



e

.he

n

s

.,-of.ds

o-

es-

e

s-

u-

t

by

d

-

-

v~r !5gL

pR2exp~2r 2/R2!, ~14!

vu~r !5gL

2pr@12exp~2r 2/R2!#, ~15!

respectively, withgL the total circulation of the Lamb vortex, andR a characteristic length scale. The experimendata in Fig. 14~a! were least-square fitted with~14! and showa very good agreement with the Lamb-vortex model.

In addition, the scatter plot of the observed monopovortex is displayed in Fig. 15~a!, where the stream functionhas been corrected for the translation velocity of the vorcenter. The vorticity and the stream function have bescaled with their extremal valuesvm and cm8 , respectively.The scatter is probably caused by the continuous defortion of the monopole and the measurement errors. Also incated is the analytical (v,c8)-relationship according to theLamb-vortex model@the solid line in Fig. 15~a!#. Since thetheoretical and measured (v,c8)-relationships are in goodagreement, it can be concluded that the characteristics oelliptic-shaped monopolar vortices are very similar to thoof the Lamb vortex.

B. Numerical simulations

Figures 11~e!–11~h! provide the calculated evolution oa Lamb–Chaplygin vortex in an adverse strain flow, whethe quasi-two-dimensional vorticity equation~12! was solvedby the same finite-difference method as described in theceding section. Likewise, the initial conditions were takfrom the experiments, and the simulation was startedt510 s. Comparison with Figs. 11~a!–11~d! reveals that thenumerical results are in good qualitative agreement with

FIG. 13. The measured distanced/d0 plotted logarithmically as a functionof time, whered represents the distance between the points of maximand minimum vorticity, andd5d058.85 cm att520 s. The experimentadata have been obtained from the experiment observations@Figs. 11~a!–11~d!# and were least-square fitted fort<150 s by an exponential functionof the form exp(bt), with b a free parameter.



g. 11rtex

ding

Down

FIG. 14. ~a! Measured cross-sectional distribution of vorticity along a horizontal line through the vortex center displayed on the left-hand side of Fi~c!.The vorticityv has been scaled with its maximum valuevm50.20 s21, while the spatial coordinatex8 has been scaled with the distance between the vocentersd519.7 cm. The solid line corresponds to~14! and has been least-square fitted to the experimental data.~b! The corresponding vorticity profile asobtained by the finite-difference calculation@see Fig. 11~g!#, with vm50.259 s21 andd513.0 cm, and a least-square fit through the numerical data accorto ~14!.

tio

tr

b

inn

ase

enre-ele.

onherfea-el-nts,

laboratory observations, which is evident from the separaas well as the orientation of both dipole halves.

Also shown are the corresponding cross-sectional disbutions of vorticity through the vortex centers@see Figs.12~e!–12~h!#, which have similar characteristics to those otained experimentally@see Figs. 12~a!–12~d!#.

The characteristics of the monopolar vortices werevestigated numerically by taking cross-sectional distributioof vorticity along a line through a single vortex center,shown in Fig. 14~b!, and by plotting the vorticity versus th



n

i-

-

-s

stream function in a scatter plot, as displayed in Fig. 15~b!.The vorticity profiles as well as the scatter plots have becompared with the Lamb-vortex model. The numericalsults definitely confirm the formation of two Lamb-likmonopoles after the breakup of the Lamb–Chaplygin dipo

Since the point-vortex model can explain the formatiof a head–tail structure, it is interesting to examine whetthis simple model can also describe the characteristictures of a dipolar vortex in an adverse strain flow. The revant initial parameters were taken from the experime

rlar

FIG. 15. ~a! Characteristic (v,c8)-scatter plot corresponding to the monopolar vortex depicted on the left-hand side of Fig. 11~c!, as well as the(v,c8)-relationship of a Lamb vortex~solid line!. The stream function c has been corrected according toc85c2Uxy1Uyx, with(Ux ,Uy)5(20.072,20.021) cm s21 the drift speed of the monopole. The vorticityv and the corrected stream functionc8 have been normalized by theiextremal valuesvm50.22 s21 andcm8 51.51 cm2 s21, respectively.~b! Numerically obtained (v,c8)-scatter plot corresponding to the left-hand monopovortex displayed in Fig. 11~g! after correction for the monopole’s translation speed (Ux ,Uy)5(20.047,0.009) cm s21. The extremal values ofv andc8 arevm50.256 s21 andcm8 51.29 cm2 s21, respectively. The solid line denotes the (v,c8)-relationship according to the Lamb-vortex model.



cti-u

thoe

ex

lren-v

yelato

got

ioa

ora

erg

byn

floraa

nuer

b–teeth-

d–x

fople

veno

ita

rac-a

talby

thee

e-toorse

c.

.

of

e

in

s-

o-

lar

: An

e,’’ J.

entinrt

r

ric

tic

ure

aler-

-

Down

and the same correction was made as in the previous seto link the point-vortex model to the Lamb–Chaplygin dpole. The circulation of each point vortex was again modlated by the functional relationship (C/At)exp(2t/t), whichproved to describe the decay of circulation as obtained byfinite-difference simulations very well. Due to the breakupthe dipolar vortex, the cancellation of vorticity along thsymmetry axis appeared to be negligibly small~i.e., t wasmuch larger than the typical time span of the laboratoryperiment! so that the exponential term exp(2t/t) could beexcluded.

Figures 11~i!–11~l! display the evolution of the materiacontour, which initially corresponded to the atmospheboundary of a vortex pair in an otherwise still fluid. Thagreement with the full-numerical simulation is striking cosidering the translation speed, the distance between thetex centers, and the orientation of the~passive! contoursaround the vortex centers. Moreover, both the dvisualization experiment and the contour kinematics simution show the entrainment of ambient fluid, which leadsthe spiral shape of the passive tracer distributions. Owinhorizontal diffusion of vorticity, such a spiral structure is npresent in the vorticity distributions@see Figs. 11~a!–11~h!#.Since the dipole halves are weakly linked, the translatspeed of the point-vortex pair is approximately the samein the finite-difference simulation.

V. CONCLUSIONS

In this paper, the evolution characteristics of dipolar vtices have been studied both in a cooperative and in anverse strain flow. Experimentally, the strain flow was genated in the mid-plane of a stratified fluid by four rotatinhorizontal discs, whereas the dipolar vortex was createdpulsed injection of a small amount of fluid. Dye-visualizatiostudies and quantitative measurements of the planarfield revealed that the evolution of the dipole in a coopetive strain flow was characterized by the formation ofhead–tail structure. Similar results were obtained in americal study by Kidaet al.17 Despite the presence of thstrain flow, the vorticity distribution in the core of the dipolavortex could be accurately described by the LamChaplygin dipole model. The distance between the vorcenters was observed to remain nearly constant becauscompressive effect of the strain flow was cancelled bylateral diffusion of vorticity as well as by the initial entrainment of ambient fluid. A kinematic explanation of the heatail formation was given by modelling the dipolar vortewith a point-vortex pair and its atmosphere, where the demation of the atmosphere boundary was calculated by aping the contour kinematics technique. However, this kinmatic model was not suitable for making a qualitaticomparison with the experimental results. On the other hathe full-numerical simulations based on the quasi-twdimensional vorticity equation revealed a very good qualtive agreement with the laboratory observations.

When the dipole was embedded in an adverse stflow, the vortex was observed to split up into two elliptilike quasi-stationary Lamb vortices, which were oriented



ion

-

ef

-

e

or-

--

to

ns

-d--

a

w-

-

xthee

r-y--

d,--

in

t

an angle of approximately 45° with respect to the horizonstrain axis. The vortex centers were passively advectedthe strain flow. Both the contour kinematics results andfull-numerical simulations were in excellent qualitativagreement with the experimental observations.

ACKNOWLEDGMENTS

We would like to thank Professor Vyacheslav V. Mleshko and Dr. Alexei Galaktionov for providing softwareperform contour kinematics simulations. One of the auth~R.R.T.! gratefully acknowledges financial support by thFoundation for Fundamental Research on Matter~FOM! ofthe Netherlands Organisation for Pure Research~NWO!.

1S. A. Chaplygin ‘‘On a pulsating cylindrical vortex,’’ Trans. Phys. SeImperial Moscow Soc. Friends Nat. Sci.10, 13 ~1899! ~in Russian!.

2S. Kida, ‘‘Motion of an elliptic vortex in a uniform shear flow,’’ J. PhysSoc. Jpn.50, 3517~1981!.

3D. G. Dritschel, ‘‘Strain-induced vortex stripping,’’ inMathematical As-pects of Vortex Dynamics,edited by R. E. Caflisch~Society of Industrialand Applied Mathematics, Philadelphia, 1989!, pp. 107–119.

4B. Legras and D. G. Dritschel, ‘‘Vortex stripping and the generationhigh vorticity gradients in two-dimensional flows,’’ Appl. Sci. Res.51,445 ~1993!.

5B. Legras and D. G. Dritschel, ‘‘Vortex stripping,’’ inModelling of Oce-anic Vortices,edited by G. J. F. van Heijst~North Holland, Amsterdam,1994!, pp. 51–59.

6A. Mariotti, B. Legras, and D. G. Dritschel, ‘‘Vortex stripping and therosion of coherent structures in two-dimensional flows,’’ Phys. Fluids6,3954 ~1994!.

7D. Brickman and B. R. Ruddick, ‘‘The behavior and stability of a lensa strain field,’’ J. Geophys. Res.95, 9657~1990!.

8G. J. F. van Heijst and R. R. Trieling, ‘‘Laboratory modelling of geophyical vortices,’’ in Theoretical and Applied Mechanics 1996,edited by T.Tatsumi, E. Watanabe, and T. Kambe~Elsevier, Amsterdam, 1997!, pp.105–119.

9R. R. Trieling, M. Beckers, and G. J. F. van Heijst, ‘‘Dynamics of mnopolar vortices in a strain flow,’’ J. Fluid Mech.345, 165 ~1997!.

10R. R. Trieling and G. J. F. van Heijst, ‘‘Kinematic properties of monopovortices in a strain flow,’’ submitted to Fluid Dyn. Res.

11O. Paireau, P. Tabeling, and B. Legras, ‘‘A vortex subjected to a shearexperimental study,’’ to appear in J. Fluid Mech.

12K. Ahlnas, T. C. Royer, and T. H. George, ‘‘Multiple dipole eddies in thAlaska Coastal Current detected with Landsat Thematic Mapper dataGeophys. Res.92, 13041~1987!.

13K. N. Fedorov and A. I. Ginsburg, ‘‘Mushroom-like currents~vortex di-poles!: One of the most wide-spread forms of non-stationary cohermotions in the ocean,’’ inMesoscale/Synoptic Coherent StructuresGeophysical Turbulence,edited by J. C. J. Nihoul and B. M. Jama~Elsevier, Amsterdam, 1989!, pp. 15–24.

14J. B. Flor and G. J. F. van Heijst, ‘‘An experimental study of dipolavortex structures in a stratified fluid,’’ J. Fluid Mech.279, 101 ~1994!.

15J. C. McWilliams, ‘‘An application of equivalent modons to atmospheblocking,’’ Dyn. Atmos. Oceans5, 43 ~1980!.

16N. Butchart, K. Haines, and J. C. Marshall, ‘‘A theory and diagnosstudy of solitary waves and atmospheric blocking,’’ J. Atmos. Sci.46,2063 ~1989!.

17S. Kida, M. Takaoka, and F. Hussain, ‘‘Formation of head–tail structin a two-dimensional uniform straining flow,’’ Phys. Fluids A3, 2688~1991!.

18V. V. Meleshko and G. J. F. van Heijst, ‘‘Interacting two-dimensionvortex structures: Point vortices, contour kinematics and stirring propties,’’ Chaos Solitons Fract.4, 977 ~1994!.

19G. J. F. van Heijst and J. B. Flo´r, ‘‘Dipole formation and collisions in astratified fluid,’’ Nature340, 212 ~1989!.

20S. Dalziel,DigImage. Image Processing for Fluid Dynamics~CambridgeEnvironmental Research Consultants Ltd., Cambridge, 1992!.

21L. Paihua Montes, ‘‘Methodes nume´riques pour le calcul de fonctions



o

-

ofh.

x

Down

spline aune ou plusieurs variables,’’ The`se de 3e cycle, Universite´ deGrenoble, France.

22T. Nguyen Duc and J. Sommeria, ‘‘Experimental characterizationsteady two-dimensional vortex couples,’’ J. Fluid Mech.192, 175 ~1988!.

23W. Thomson, ‘‘On vortex atoms,’’ Philos. Mag.34, 15 ~1867!.24H. Lamb, Hydrodynamics,6th ed. ~Cambridge University Press, Cam

bridge, 1932!.25G. K. Batchelor,An Introduction to Fluid Dynamics~Cambridge Univer-

sity Press, Cambridge, 1967!.



f

26V. V. Meleshko and G. J. F. van Heijst, ‘‘On Chaplygin’s investigationstwo-dimensional vortex structures in an inviscid fluid,’’ J. Fluid Mec272, 157 ~1994!.

27J. B. Flor, G. J. F. van Heijst, and R. Delfos, ‘‘Decay of dipolar vortestructures in a stratified fluid,’’ Phys. Fluids7, 374 ~1995!.

28P. Orlandi, ‘‘Vortex dipole rebound from a wall,’’ Phys. Fluids A2, 1429~1990!.

29E. Hairer, S. P. No”rsett, and G. Wanner,Solving Ordinary DifferentialEquations—I: Nonstiff Problems~Springer, Berlin, 1987!.



Date post:	21-Jan-2021
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Dipolar vortices in a strain flow · Dipolar vortices in a strain ﬂow R. R. Trieling,a) J. M. A....

Documents