javier jimenez, h.k. moffatt and carlos vasco- the structure of the vortices in in freely decaying...

8/3/2019 Javier Jimenez, H.K. Moffatt and Carlos Vasco- The structure of the vortices in in freely decaying two-dimensional tu

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210 J. Jimdnez,H. K . Moffatt and C. Vasco

(x*, *, * ) ,and of a triaxial strain(ax*, y', yz*). The three principal strains satisfya + 3 + = 0, and y is assumed to be positive. The vorticityo*(x*, *, * ) s parallelto the z-axis.

Lundgren's transformation states that there is a related two-dimensional flow in

which the axial strainy is absent, defined by(1.1)= (y t ) -1 /2 (u* , ' ) + s(yt)-'(x, -y), 0 = ( y t ) - ' o * , s = ( a - P)/2yt,

where the coordinates and time are transformed by

In essence, the two flows behave similarly, but the velocities and the vorticities of thestrained flow are amplified, its distances contract, a nd its time, which is proportional

to the eddy turnover period, runs faster.In the case of a Burgers' vortex, in whicha = p = - y / 2 and the three-dimensional

solution is

a * = CO; exp(-yre2/4v), (1.3)

CO = ( o ; / y t )exp(-r2/4vt). (1.4)

the transformed two-dimensional flow is the viscous spreading vortex

For triaxially strained vortices, the correspondence with two-dimensional vortices ina plane strain is not complete, since it follows from(1.1) that the two-dimensionalstrain should decrease in inverse proportion to time. The reason becomes clearonce the governing dimensionless parameters are considered. It is found in MK094that the relevant perturbation parameter for the strained vortex is, to lowest order,

= (v /T) p - ) / ? , which is proportional to( p - ) / o ; . The triaxial strain(a , p , y)can be understood as the superposition of an axisymmetric stretching(-y/2, -y/2, ) ,which can be eliminated using Lundgren's formula, an d a plane equatorial component(S I - p, p - , 0)/2. The perturbation parameter is the ratio of this equatorial strainto the characteristic vorticity of the vortex core.

This is the same parameter identified by Ting& Tung (1965) for the two-dimensionaldiffusing vortex but, in that case, as the vortex diffuses and its core vorticity decays,the driving strain has to decrease continuously if similarity is to be maintained.Alternatively, if the vortex diffuses in a constan t s train ing flow, it suffers an increasingdeformation as it spreads and becomes weaker.

Because the internal timescale of the vortex core is much faster than the viscoustime, the spreading vortex can be treated as quasi-steady, and corresponds to thetriaxial case to lowest order, but the difference appears at higher orders. It is foundin MK094 that, for orders beyondO(E! ) ,he two factorsy / o { and ( p - a ) / ? enterindependently in the solution, instead of as the single productE ' .

The asymptotic analysis of the structure of a two-dimensional vortex ina weakstrain is developed in the next two sections. The expansion within the vortex corefollows closely that for the triaxial case in MK094, and will be discussed in thecontext of that paper. Itis recast, however, in a slightly different form, to eliminate anapparent inconsistency in the ordering of the different perturbation termsin the farfield of the vortex, and the reason for that inconsistency is discussed. The predictionsof the model are then compared to the resultsof the turbulence simulation.


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Vortices in two-dimensional turbulence 21 1

2. The asymptotic formulationConsider a two-dimensional vortex of circulationr , ubject to a straining velocity

s( x, -y). The velocity of the vortex and of the strain are com parable at a distance oforder R, = (T / s ) ' l 2 , while the vortex radius spreads, over characteristic timesO(s-'),

to be of orderR, = ( v / s ) ~ / ~ .ts characteristic vorticity is thenw, = r/q:. We a 2interested in the weakly perturbed limit in which

If we normalize length withR, and time withs-l, the vorticity equation becomes

V l p = - O ,

where the velocities are related to the stream function byU =

d y / a y,U =

-dy/ilx,and y - y at infinity. Th e vortex has unit circulation and spread s only slowly, overt = O(e- ' /2) , under the action of viscosity. For shorter times, its radius is small, andit looks like a point at the scale ofR,. The streamlines of such a point vortex floware plotted in figure1, where the vortex rotates counterclockwise, and the strainingvelocity is outgoing along the x-axis. The 'cat's eye' is aligned at 45" to the strainaxes, and the streamlines are circular near the vortex and become more elliptical asthey move away from it. Since the vortex radius is onlyO(e ' / ' ) (see equation (2.1)),its vorticity is almost totally confined well inside the dividing streamline, and is onlybled slowly along the outgoing direction of the forcing strain. The situation is exactlyequivalent t o that in M K 094 , where very long-lived vortices were found to exist in thecase of a triaxial strain , even when one of the equatorial eigenvalues was extensional.

Two-dimensional strained vortices have been studied often. The stability of ellipticaluniform-vorticity patches was analysed by Moore& Saffman (1971), who concludedthat the strain would tear the core if, in our notation,e > 0.15. The stability analysiswas extended to unsteady strains by Dritschel (1990), resulting in a broader class ofbehaviour. The full initial value problem for elliptical patches was studiedby Kida(1981), and generalized to the case of vortices constructed from an arbitrary numberof nested elliptical patches by Legras& Dritschel (1991) and Dritschel& Legras

(1991). The same authors studied numerically the interaction of an external shearwith a non-un iform vortex, an d concluded that the weak vorticity a t the edge of thevortex is stripped away, to the level at w hichw /s = O . l l ) ,while the core itself remainscoherent (Legras& Dritschel 1993). All these results support the idea that the vortexbreaks down only when its vorticity becomes substantial in the neighbourhood of thedividing streamline in figure1, and are consistent with the vortex being stable forE. < 1.

In the units of (2.2) the core vorticity is0(1/e). If inner variables were defined inwhich both the vorticity and the vortex radius are0(1), quation ( 6 ) would remainunchanged. The corresponding timescale is the fast eddy-turnover time of the vortexcore. We will be interested in solutions in which the structure of the vortex does notchange on that timescale, although it may doso on the slower one of the externalflow. We therefore choose inner variables,

(A,9 )= e-ll2(x, y), t ^ = t , 3 = y , ( 2 . 3 )

(2.4)

in which the equations of motion become" 2 A

[I$,;] = L&, v y = -6,


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212 J . Jimdnez,H. K . Moffatt and C. Vasco

FIGURE . Streamlinesof a point vortex in a pure strain whose principal axesare aligned to the coordinate axes.

where [I$,&] = a($,&)/a(?,9 ) is the Jacobian, and L= d ; - 9'. The circulationof the vortex is still unity, but3 - 2 9 at large distances. This equa tion isequivalent to equation (2.7) of M K 094, with the time derivative of the right-hand

side substituting for the stretching operator of the three-dimensional case, as inLundgren's transformation. We have chosen to keep the straining motion inside thedefinition of3 , whereas in M K 0 9 4 it was m ade explicit in the right-hand side. Thiswill make the structure of the perturbation somewhat clearer.

We recall the analysis in MK094. It is immediately obvious from (2.4) that, to thelowest approximation in6 ,

W O , 6 0 1 = 0 , 6 0 = Go(do), (2.5)where we are assuming an expansionG = do + c G1+ . . ., and a similar one for3 .We will only consider cases in which this lowest-order motion is axisymmetric,

do(?),in polar coordinates(?, 8) centred on the vortex.The first-order perturbation is

Averaging over8, we arrive at the compatibility condition

whose solution corresponding to a point vortex att = 0 is

0 o 1 ,-P2/4;

4ntSubstituting (2.7) into (2.6) we also get

a linear homogeneous equation which is however forced by the boundary condition


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Vortices in two-dimensional turbulence

6 -

-P 4 -S2 -

213

I I

for at infinity. It can be solved to give

where the functionf , as given in MK094, satisfies

(2.10)

(2.11)

and the boundary conditionsf = O(c2) fo r 5 + 0, and f = O(c-2) for [ + CO. Apossible axisymmetric component vanishes because of the compatibility condition at

The resulting streamlines are elliptical, with their major axes aligned at 45" to thestrain, in line with the axisof the cat's-eye pattern in figure1, and w ith an ellipticitywhich does not vanish at the origin. If we denote the major and minor semi-axes asa and b, the ellipticity is

O(E.2).

(2.12)

where 0 is the (dimensional) maximum vorticity a t the centre of the vortex. Thequantity p Q / s is plotted in figure 2. It tends to 2.5259. . . at the origin, and increasesfor large radii, to merge into the inner streamlines of figure1.

3. The far fieldThe procedure outlined in the previous section is the one followed in M K 09 4 , and

results in an expression for the perturbed vorticity which behaves at large radii as6 - exp(-t2/4t)( l + O(EP ) + 0 ( e 2 ^) + . .), and which becomes inconsistent wheni = O(e-lI4). Since this is a narrow er range than the characteristic size of the cat's-eye,t = O (E -' /~ ), t raises some concern abo ut the uniform validity of the perturbationscheme (MK094). In particular, the asymptotic series inside the parentheses hasthe same ordering of terms as exp(&), which could dominate the exponent at longdistances, and compromise the exponential decay of the vorticity. What is needed


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214 J. Jimdnez,H. K . MofSatt and C. Vasco

is an intermediate asymptotic procedure for the construction of an exponentiallydecaying vorticity in the full range of radii1


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Vortices in two-dimensional turbulence 215

where the partial derivatives assume that everything is expressed in terms of( R , 0),and where the operator L also has to be expressed in terms of the new variables.Because the Laplacian in L is a second-order operator, and because the changeto new variables is given only implicitly by(3.6), this step involves some algebraiccomplication, but it is easily programmed in a symbolic manipulator. The resultsgiven below were obtained using Maple. We define the expansions

n=l n=O

and separateH,, into axisymmetric and non-axisymmetric parts:

H,, = H,, + H i , H,, = g [ ,, do. (3.9)As in the previous section, theE" term of (3.7) separates into

-H n - l = 0 (3.10)

and

J,, + Hi-, d0 = funct.(R, ), (3.11)where the right-hand side of(3.11) is a free integration 'constant', andJ , comes fromthe series expansion of$ ( r , 0) in terms of the new variables. It takes the form

J n = $OR&

+n) &OR + * . , (3.12)

where the extra terms involve onlyRi fo r i < n. That is also true ofH n P l ,and (3.11)can therefore be used to determineR,, as

Rn = &(R, 0, t) + Q n ( k t), (3.13)with the arbitrary integration functionQn.The axisymmetric part(3.10)has the form

R-'H,, = Lo&, = r.h.s.(Qn, . ), (3.14)

where LO= d, - Vi ,and V iis the Laplacian operator in the variables( R , 0) although,as it is always applied to the functions&,,(R, ), only its radial part is of interest. Atthis stage the functionQn can be chosen to control the growth ofd n with n at largeR.

We will now indicate the structure of the first few orders of the expansion, andthe results for the particular case of the streamfunction(3.4).The 0(1) equations areidentical to those in the previous section. Our choiceof & = R makes J o identicallyzero, and the first axisymmetric equation is

R-lH - L ~ G ~0, m0 =- -R2/4t,4nt

(3.15)

There is no angular component at this order, and the non-axisymmetric equation atO(c) is

(3.16)J l e = % ( $ O R & + 1 ) = 0, Ri = - $ 1 / $ 0 ~ +Qi(R , ) .For the streamfunction(3.4),

RI = n(R3 +2A2/R) in 20 + Q1(R, ) . (3.17)


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216

The rest of the equation at this order is

J . J imknez,H . K .Moffatt and C. Vasco

(3.18)

where the right-hand side is a linear and homogeneous functional of RI. Since theangular average of$1 vanishes, it follows from (3.16) that the average of R1 isthe arbitrary additive functionQ1, and that the right-hand side of the axisymmetriccomponent of (3.16) is homogeneous inQ1. e can therefore choose Q1= 6 1 = 0.

The first order for which this is not possible isO ( e 2 ) .The non-axisymmetric partof the equa tion needs no special treatm ent. Fo r the particular case (3.4) the leadingterms are

R2 = - ( @ ~ R R I $ ~ R R R ; / ~ ) / $ O R. . = Q2 f z 2 R 5 (3/t)COS28 - COS48)+ 0 ( R 3 ) .(3.19)

In the axisymmetric equation, however, the right-hand side contains q uad ratic termsin R I which have non-zero angular average, and the equation for6 2 is no longerhomogeneous. For our special case we obtain 0

e-R2/4t

Lob2 = - 77t2R6- R 2 Q 2 ~ O ( R 4 ) ) .32nt3 (3.20)The solution of (3.18) would result in6 2 - R6exp(-R2/4t), and would limit theconvergence of the expansion tor^ = O ( E - ' / ~ ) . e can now useQ 2 to cancel as man yof the high powers of R in the right-hand side of the equation as needed,so thatthe leading term of the solution is at most proportional to R4exp(-R2/4t), and theexpansion remains valid to the desired distance. In the present case it is enough totake

Q2 = 7z2R5/20. (3.21)Note that Rn/R- R2" a t large distances, guaranteeing a uniform ordering of theterms of the coordinate deformation out toi = O ( E - ' / ~ ) .The convergence of theseries for 6 as already been discussed.

It is also clear how our result can be matched to the inner expansion of MK094.We can invert our expression fori to give

(3.22)= ^ - i3n: in 28 + . . ,and expand the leading term of our series

(3.23)

which is equivalent to the expansion in MK094 forr^ >> 1.The success of the coordinate deformation in controlling the non-uniformities of

the expansion is not surprising. The perturbation scheme used here and in MK094is a variant of the averaging method which has been known in different forms atleast since the work of Lagrange in the eighteenth century. It was first developed

for Hamiltonian perturbations, and this application is reviewed by Arnold (1978).It was soon understood that the use of appropriate coordinates was important forthe success of the scheme, which depends on the approximate correspondence ofdifferent types of averaging. I t is of interest to no te tha t the left-hand side of(2.4) has a Hamiltonian form, with either the streamfunction or the vorticity acting asHamiltonians. Its right-hand side does not preserve that structure and the use of strictHamiltonian techniques, although indicative, is not necessary. The use of coordinatedeformations in more general applications is discussed by Van Dyke (1975). Thea


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posteriori scheme that led to (3.3) is identified there as due to Pritulo (1962), whilethe m ore general scheme used later is associated with Poincark. There is, in principle,no reason why the latter scheme could not be used to obtain a uniformly validexpansion at all distances within the cats-eye, but the nonlinearity associated withthe coupling of the streamfunction to the vorticity makes the algebra unwieldy atthe higher orders inside the vortex core. The appro ach used here is sufficient todisplay the physical character of the solution, while keeping algebraic complicationto a minimum. A som ewhat similar approach, applied to uniform vortex patchesand using conformal deformations to regularize the expansion, was used by Jimenez

(1988).

4. Numerical experiments

The predictions in the previous sections are checked next against the structureof vortices in a numerical simulation of decaying two-dimensional turbulence in aperiodic square domain at high Reynolds number. The code is a standard spectralFourier approximation to the vorticity equation, fully de-aliased, with a third-orderRunga-Kutta method for time advancem ent. It uses New tonian viscosity, and theReynolds number, based on the side of the domainL and on the initial root-mean-square fluctuation velocityU, s 5.5 x i04 .

The num erical resolution is 10242 Fourier modes before de-aliasing, correspondingto a maximum numerical wavenumberk,,, = 341. The initial conditions are chosenwith a random phase, and with an energy spectrum decaying slowly (likekt3I2)

fo r k < 200, and much faster(k t7) for higher wavenumbers. Afterut /L e 0.2,the vorticity concentrates into compact vortices (McWilliams 1984) scattered over amuch weaker background. The results used here are compiled atut /L e 1.65-1.85.By that time the total enstrophy has decayed by a factor of 400 from the initialcondition, while the energy has only decayed by 40%, and there are approximately100 identifiable vortices in the com putational box.

The mean radius of the vortices is? = 0.015L, which agrees in order of magnitudewith the radius of a viscous vortex spreading from an initial point,r , = 2(vt) l I2 w0.01L. It shou ld be com pared to the larger average sepa ration between vortices,D w 0.1~5. The mean Reynolds number of individual vortices is

r/vNN 3500.

The timescale for the evolution of the vortex distribution isD/u , shorter than theturnover time, ,!,/U, but still longer than the rotation time of individual vortices,w; = F2/T w 5 x 10M3L/u.Therefore the scale separation assumed in the previoussections applies, and the results of the analysis should hold.

A catalogue of vortices is compiled at regular intervals from the computed flowsfields. Candidates are identified as connected regions in which the topological dis-criminant (Weiss 1991)

(4.1)

is more negative than a given threshold, which is chosen as a small multiple of thestandard deviation ofQ over the whole field(Qt /Q = 2.5). The influence of thisthreshold was checked, and found to be small, although a good choice is important inminimizing the work involved in the subsequent filtering of the results. Each vortexcandidate is characterized by its circulation, its area, its centre of gravity, and thetensor of inertia of its vorticity distributionJ J w(x, - X,)(x, - a,) dS. This tensoris used to compute the two principal moments of inertia and the orientation of theprincipal axes.

Q = s2 - w2/4


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218 J . Jimknez,H. . MofSattand C . Vasco

0

FIGURE . Averaged distribution of vorticity for all the vortices of the numerical catalogue, withrespect to their radial elliptical coordinate. Solid line is equation (4.2).

Each vortex is assumed to have a Gaussian vorticity distribution in ellipticalcoordinates aligned with those axes,

COM = s2 exp(-xI2/a2 - 2 / b 2 ) , a 2 b. (4.2)The three parameters,a, b, s2, and the dimensionless radiusp of the original patch,

are obtained by equating the measured values for the area, the circulation, and theprincipal m oments to those of a n elliptical vortex of the form (4.2), contained insidexI2/a2+ yI2/b2< p2 . The result is a set of positions, areasS = nab, circulationsr = S O , and orientations of the major axes4,. Candidates with no solution for themodel parameters, and those originating from patches formed by less than four gridpoints, are eliminated at this stage.

The assum ption (4.2) for the vorticity distribution is tested next. Assum ing anelliptical Gaussian vortex, it is easy to show that the contour for whichQ = 0 shouldcorrespond to p = 1 and, in consequence, only vortices with0.5 < p < 1.5 arekept in the database. The fit between the model (4.2) an d the ac tual flow vorticityis also tested directly. The mean-square deviatione% = (CO - C O ~ ) ~s computedover all the points of the original patch, and candidates for whiche,/Q > 0.2 arediscarded. This procedure was found to work reasonably well in identifying vortices,while missing only a few percent of those which could be identified visually, usuallyvery deformed ones involved in close interactions. The to tal enstrophy containedin our vortex catalogue is about70% of the total, concentrated in5 % of the totalarea. A comparison of the Gaussian model with the mean and standard deviationsof the measured vorticities is given in figure3. Note that only points inside theoriginal patch, for whichQ < Qt x. 0, are used in the computation ofe,, and thatthe Gaussian fit outsidexI2 /a2+ yI2/b2= 1 is not guaranteed by the mean-squaredeviation test. The total sam ple includes about 104 vortices over160 different flowfields, corresponding to about100 different vortex histories. Only70% of them haveellipticities that are high enough( a / b > 1.1) for the o rientation of the m ajor axesto be reliably defined, and those are the only ones used in our statistics. A roughlysimilar procedure for generating a vortex catalogue was presented by McWilliams(1990)but no d at a were given in tha t case for the enstrophy and area fractions whichcan be compared to the present ones.


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0 ' I I I

Aylsign (a) IQ1PIS- 0 -4 5 0 45 90 0 2 4 6 8

FIGURE . Probability density functions for the offset angle between the major axes of the vortexwith respect to the extensional axes of the ambient strain, and for the reduced ellipticity of thecores, defined as in figure 2.

The next step is to determine an 'ambient' strains at the position of each vortex.Several methods were tried, an d the one selected was to use the s train generated at thecentre of gravity of each vortex by all the other vortices in the database. The resultchanges only slightly if the other vortices are treated as points or as full Gaussianmodels, with the correlation coefficient between the results of both methods beingover 0.95. In contrast, the simple method of subtracting the Gaussian model forthe vortex in question and computing the strain generated by the rest of the flowproved to be inadequate. A typical value for the ratios/Q is 0.05, and the smallvorticity residuals left by the difference between the actual and the modelled vorticitydistributions are enough to mask the ambient strain generated by the rest of the flow.For each vortexa strain magnitude and an orientation$s of the extensional axis aregenerated in this manner.

These data allow us to check the results obtained in $2 for the orientation andellipticity of the vorticity distribution with respect to the driving strain . In particular,the axes of the vortices should be offset with respect to thoseof the strain by45" nthe sense of rota tion of the vortex, an d their ellipticity, defined as in (2.12), shou ld

be close to 2.52s/a2.Both conclusions are tested in figure4. Although there is0 considerable statistical scatter, it is clear that the maxima of both histograms agreewell with the theoretical analysis.

The assumptions on the timescales of the flow can also be now checked directly.A measure of the rate of change of the imposed strain isc = ldsll/dtl/lsl, where s11is one of the components of the imposed strain tensor. From the vortex cataloguea = 21s on the average,so that the evolution time of the strain is of the same orderas the straining time itself. They are both, however, longer than the rotation time,and a/s = 0.1.

Du ring the review process of this paper one of the referees remarked tha t the steadystrain used in this paper is only a pa rticular case of the general one in which the strainrotates and varies with time, and that a more realistic model for the deformations intwo dimensions would include an additional solid-body rotation component ofO(s).The results in the previous paragraph lend strength to that remark, and pose thequestion of the reason for the agreement in figure4.

It is however easy to see that the addition of an axisymmetric component to theperturbation streamfunction does not change the results of the analysis in $$2-3.


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220 J . J i m h e z ,H . K. Moflatt and C . Vasco

(4

0.46

0

(b)

1.27

-1.27

FIGURE . Typical total strain ( a ) and vorticity ( b )fields for the two-dimensional turbulent flowused in the text.


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Vortices in two-dimensional turbulence 22 1

Because (2.6) an d (2.9) are insensitive to com ponents of$1 which are only functionsof 5, he equation satisfied byf ( 5 ) is not changed, and the shape of the vorticitydistribution is not modified toO ( E ) .To this order the vortex follows the imposedstrain even if the latter rotates on a timescale comparable to its own deformationtime. In essence, while the effect of the stra in is visible to first order because thelowest-order axisymmetric vortex generates no radial velocities which could competewith it, the effect of a comparable rotation appears only atO ( E )because it competesdirectly with the much stronger azimuthal velocity of the vortex.

We have dealt up to now with the ambient strain which would be present at thelocation of the vortex if the vortex were not present. One of the striking results inMK094 was the structure of thetotal strain (or equivalently dissipation) producedby the driving flow and by the vortex itself. In a circular core the total strain iszero at infinity and at the centre of the core, and is maximum in an annulus at theperiphery of the vortex. When the axisymmetry is broken by the driving strain, the

crown of maxima is deformed into two crescents aligned with the major axis of thevortex ellipse, while the central minimum splits into two minima aligned to the minoraxis (see figure7 of MK094). The same structure can be seen many times in themap of total strain for a typical turbulent field in figure5(a) .A comparison with thevorticity map in figure5(b )confirms that the strain structures do indeed correspondto the elliptical vortices.

I

5 . Discussion

In the first part of this paper we have given an asymptotic expansion for thestructure of a two-dimensional viscous vortex in a weak plane strain. The results aresimilar to those of Ting& Tung (1965) an d in M K 09 4, but the expansion is extendedup to distancesi = O ( E - / ~ ) , hich is the scale of the cats-eye streamline separatingthe elliptic points dominated by the effect of the vortex from the hyperbolic regiondominated by the driving strain. In the limit studied here,Rer >> 1, the radius ofthe core itself is much smaller, and it behaves like a point vortex with respect to theexternal flow. The non-uniformity mentioned above refers to distances much greaterthan the core radius but smaller than the cats-eye, and was present in the expansionsin both of the papers cited, although it was only explicitly recognized in the secondone. It is removed here by using a coordinate deformation in conjunction with thebasic averaging technique of the previous studies.

The results confirm that the vorticity is approximately Gaussian in distorted coor-dinates, which coincide roughly with the flow streamlines. It is therefore exponentiallysmall in the neighbourhood of the dividing streamline, from where it is presumablyslowly stripped to infinity (MK094).

In the present context of steady, or self-similar, solutions, the validity of theexpansion cann ot be extended farther from the core, since it was shown in Appendix Aof MK094 that no steady solution could be expected for the vorticity distribution inany part of the flow dominated by a strain with an extensional direction. It is onlywell inside the dividing streamline that the vortex shelters itself from the externalinfluence, and that a self-similar solution is possible.

The predictions of the asymptotic expansion on the shape and orientation of thevorticity distribution are tested against a vortex catalogue obtained from a numericalsimulation of two-dimensional turbulence. They are shown to agree well with thenumerical experiment, and the same is true of the spatial distribution of the totalstrain (o r dissipation) field.


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222 J. Jimbnez,H . K . MofSatt and C. Vasco

It is interesting that this isso even if the vortices in two-dimensional turbulenceare subject to strains which are not steady. The basic requirement for the asymptoticexpansion in this paper is that the timescale of the driving flow should be muchlonger than the turnover period of the core, which is equivalent toRer >> 1. Thiswas sharpened recently by Lingevitch& Bernoff (1995) who studied the initial-valueproblem of a viscous vortex subject to a sudden change in the external driving flow,and showed that the relaxation time isO(SZ-'Rey3),as conjectured in M K 09 4, 92. Ina two-dimensional turbulent field with vortices of circulationsO ( r and radii O(a) ,separated by distancesO ( D ) ,the evolution time for the strain isO ( D 2 / r ) , nd thecondition that it should be much longer than the relaxation time of the individualcores is that

(5.1)In our example this is marginally satisfied, since the left-hand side is approximately

100, while the right-hand side is of order15, but it is not clear whether the samewould be true for simulations at higher Reynolds numbers or at different times intheir evolution.

D2Q/r D 2 / a 2>> Re:/3.

This research w as partially supported by the Human Capital a nd Mobility programof the EU under contract CHRXCT920001.

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