yasuhide fukumoto and h.k. moffatt- motion of a thin vortex ring in a viscous fluid: higher order...

8/3/2019 Yasuhide Fukumoto and H.K. Moffatt- Motion of a thin vortex ring in a viscous fluid: higher order asymptotics

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MOTION OF A THIN VORTEX RING IN A VISCOUS FLUID:HIGHER-ORDER ASYMPTOTICS

YASUHIDE FUKUMOTOGradua te School of Mathematics,Kyushu University 33, Fukuoka 812-81, JapanANDH. K . MOFFAT TIsaac N ewton Institute for Mathematical Sciences,20 Clarkson Road, Cambridge CB3 OEH,UK

Abstract. The motion of an axisymmetric vortex ring of small cross-section in aviscous incompressible fluid is investigated using the method of matched asymp-totic expansio ns. A general formula for the ring speed is obtaine d up t o third orderin E = SIR0 (= (v /I?) l l2) ,he ratio of core to curv ature radii, which takes accountof th e influence of th e self-induced st ra in . Here is th e circulation and v is thekinematic viscosity of fluid. It is pointed o ut th at the dipole distributed along thecenterline of the ring plays a vital role in its movement. Its strength needs bespecified at the initial instant in order to remove the indeterminacy of the the-ory. A new asymptotic development of the Biot-Savart law enables us to calculatethe non-local induction velocity at O( c3) from th e dipole. In a special case, werecover D ysons inviscid form ula (1893) . It is dem onstrated t ha t th e viscosity acts,a t O ( e 3 ) , o expand the radius of the loop consisting of the stagnation points inthe core, when viewed from a certain comoving frame.1 . IntroductionThe motion of a vor tex ring is one of the m ost classical an d fund am enta l problemsof vortex dynamics. Extending Kelvins result, Dyson (1893) (see also Fraenkel1972) obtained th e speed U of an axisymmetric vortex ring , embedded in an invis-cid incompressible fluid, up t o third (virtually fou rth) order in a small parameter:

where r is the circulation, Ro is the ring radius and E = SIR, is the ratio ofcore radius 6 t o Ro. The vorticity distribution in the core is proportional to the21

E. Krause atid K. Gersteri (eds.), IUTAM Symposium on Dynamics of Slender Vortices, 21-34.0 1998Kluwer Academic Publishers. Priilted in the Netherlands.

www.moffatt.tc


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22 YASUHIDE FU KUM OTO AND H. K. MOFFATTdistance from the sym m etry axis. We consider Kelvins formula as the first orderand E2-term as the third . Th e vortex ring induces a local straining field on itselfwhich deforms the core into an ellipse at second order:

{ [:log(:) -I c 0 s 2 e + . . . ,1= 6 1 + E 2where ( r , e ) are local cylindrical coordinates about the core center which will beintroduced in $2. It is remarkable t ha t the inclusion of the third -order ter m in th epropagation velocity achieves a great improvement in approximation: equation(1) exhibits fair agreement even with the exact value for the fat limit of Hillsspherical vortex ( E = a).

The viscosity acts to diffuse the vorticity. Its influence on the propagationspeed, at large Reynolds number, was calculated by Tung and Ting (1967) (Cal-legari and Tin g 1978) and Saffman (1970), up t o O( (v/17)1/z ),as

.=-{log(*) - - ( l - y + l o g 2 ) + . . .4.irRo 2 f i 2 (3 )

where t is the time measured from a virtual instant at which the vorticity is 6-function concentrated, and y = 0.57721566. . . is Eulers constant. The vorticitydistribution ha s a diffusing Gaussian profile with circular symmetry.Recent dire ct numerical simulations of fully developed turb ulen ce have revealedth at the small-scale structu re is dom inated by high-vorticity regions conc entratedin tub es (see, for ex am ple, Siggia 1981; Kerr 1985; Hosokawa a nd Yam am oto 1989 ).These occupy a small fraction of the total volume, but are responsible for a muchlarger fraction of viscous dissipation. This observation led Moffatt e t al. (1994,1996) to develop a large-Reynolds-number asym ptotic theor y t o solve the Navier-Stokes equations for a vortex subjected t o uniform non-axisymmetric irro tatio nalstr ain . T he solution satisfactorily a cco unts for th e stru ctu re of the dissipation fieldpreviously obtaine d by numerical com putation (K ida an d Oh kitani 19 92). T heviscosity is a n age nt to pick ou t vorticity distribution. At leading order, the Burg-ers vortex is obtain ed, and at the next order ( O ( v / l ? ) ) , quadrupole componentemerges, reflecting an elliptical vorticity distribution. The salient feature is thatthe major axis of the ellipse is aligned at 45 to the principal axis of the externalstrain. This fact leads us to the belief that the strained crosssection of a propa-gating v ortex ring, comm only observed in natu re, is established as a n equilibriumbetween self-induced strain and viscous diffusion.

T he a im of our s tud y is to elucidate th e struc ture of this s trained core an d i tsinfluence on the translation speed of an axisymmetric vortex ring. As a first step,we present, in this paper, a general framework to addre ss this problem. A part ia lanswer is given as to how viscosity affects the radial drift of vorticity.T he m etho d of ma tched asym ptotic expansions has been previously developedto derive the velocity of a slender curved vortex tube in a fluid both with and


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M O T I O N O F A VISCOUS VO RTEX RING 23without viscosity (Tung and Ting 1967; Widnall et al. 1971; Callegari and Ting1978; Klein and Majda 1991). However these studies have been limited to thesecond-order curvature effect (Moore and Saffman 1972; Fbkum oto and Miyazaki1991). The self-induced straining field of a vortex ring makes its appearance atsecond order in E = ( v / J ? ) l I 2 ,nd the translation speed is affected at the nextorder . We make an attempt to extend asymptotic expansions to a higher orderand to calculate the speed of a vortex ring up t o U ( t 3 ) .The existing asymptotic formula for the potential flow caused by a circularvortex loop is not sufficient t o ca rry through this p rogram . After a brief statementabout the general sett ing of asym ptotic expansions in $2, we obtain an asy mptoticexpression of the Biot-Savart integra l acco mm odatin g an arb itrar y vorticity dis-tribution in $ 3 . In $4, th e inner expansions are recalled a nd ex tende d to secondorder. Based on th eses, we establish , in 5 , a general formula for th e tran slationvelocity of a vortex ring , valid up t o third order. Dyson's formula (1) is recoveredin a special case. Moreover, it is revealed that the radius of the loop consisting ofthe stagnation points in the core, when viewed from the frame moving with thecore, ex pan ds linearly in tim e owing t o the action of viscosity. O ur pro cedu re pur-suing higher-order asy m pto tics highlights the significance of the dipole distr ibu tedalong the r ing. I ts s trength must be prescribed a t the initial insta nt an d therebythe problem of undetermined constants at O ( E )s remedied.2. Formulation of the matched asymptotic expansionsTwo length scales are available, namely, the (typical) core radius 6 nd th e r ingradius Ro. We ass um e t h a t the ir rat io is very small. We reta in only the slow modeof core dynamics, suppressing fast waves on the core. Then, in view of ( l ) , hetime-scale is of order Ro/(I'/Ro)= Ri/J?. n the presence of viscosity, the coreradius grows as b N ( ~ t ) ' / ~(v/l?)1/2Ro uring this t ime. Thu s our assum ptionrequires that

t = SIR0 =


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24 YASUHIDE FUKUMOTO AND H. K . M OFFAT Tflow which decays rapidly with distance from th e core center. T hu s we ar e led t oinner and outer expansions (Tung and Ting 1967). The inner region has lengthscale of ord er th e core radiu s 6 an d the re we seek th e solution of th e Navier-Stokesequations matched t o th e outer solut ion given by (6).

It is expedient to choose a coordinate frame moving with the core center( R ( t ) ,Z ( t ) ) in which we intro du ce local cylindrical co ordinates (r,8 ) such tha tp = R ( t ) + r c o s e , z = Z ( t ) + r s i n e . (7)

Introduce dimensionless variables:

Here 2) is the velocity relative to the moving coordinates and the difference innormalization between the last two of (8) should be noted. Th e equation handledin the inner region are the vorticity equation and the relation between C and $.Dropp ing the star s, these take the following form:

wherep = R + er c o s e ,

A is th e two-dimensional L aplacian , an d U and v are the r- and &com ponents ofth e relative velocity U :


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M O T IO N O F A VISCOUS VO RTEX RING 25Th e permissible solution must satisfy th e condition1,

U an d v are finite at T = 0 ; (14)the requirement that it smoothly matches the outer solution will determine thevalues of and Z ( i ) ( i = 0 , 1 , 2 , . .).3. Out e r solutionThe streamfunction I+J~ for the flow induced by a circular vortex loop C = S (p -R)S (z) of unit s trength is obtainable from (6):

We call (15 ) th e mono pole field. So fa r, this has been exclusively employed as theouter solution.It turns out however that, when going into higher orders, (15) is not enough

to be qualified as th e o uter solution. T he elab oration of th e detailed struc ture of(6) is unavoidable. To this a im , it is advantageous t o ad ap t D ysons technique toan arbitrary distribution of vorticity:

Th e exp ected s patia l dependence of vorticity distribution isp ) = 0) , (1 7 4c ( l ) = &) case, (17b)~ ( 2 ) = ci2)+ (4;) c o s 2 e , (17c)~ ( 3 ) = &) cos e + &) sin e + &) cos 30 . ( 1 7 4

For


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26 YASUHIDE FUKUMOTO AND H. K. M OFFAT Tintegrating with respect to x' a nd z ' , and taking th e derivatives of $m , subst i tu tedfrom (15), with respect to R and z , we obtain the asymptotic form of the outersolution valid at T


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M O T I O N O F A VISCOUS VORTEX RING 27unknown, but will be fixed by the inner expansions and the matching procedure.It will be clarified t h a t th e dipole com pon ents ( i t ) , (ii),&) are distinctive. In th esub seq uen t sections we investigate th e flow field inside t he core.4. Inner expansions up to second orderBefore going t o th ird ord er, we give a brief outline of the inner perturbat ions upto second order.

Collecting like powers of E in (9) and ( l O ) , along with (11)-(12b), subs t i tutedfrom (13a)-( 13 d) , th e Navier-Stokes equa tions at each order are deduced succes-sively.

At O(EO), e obtain the Jacobian form of the Euler equation:

where we define [ ( (O ) , $ ( 0 ) ] = a((('), ( ' ) ) / d ( r ,B ) / T . Hence [ (O ) = 3($('))or somefunction 3.Suppose th a t the flow $ (O ) has a single stag natio n point a t T = 0, thestream lines being all closed ar ou nd th at point. T hen it is probable th at the solutionof (2 2) , coupled w ith = $ ( ' ) ( r ) , t ha tis, the stream lines are circles (M offatt e t ad. 1994)2.T he functional form of $ ( O ) ( r )and


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28 YASUHIDE F U K U M O T O AND H. K . M OFFAT Twhere Ro = R(O)(with some abuse of nota tion ) and

Here we have used the fact that the axisymmetric part of C ( l ) is suppressed fromthe resul t of (31) an d the analysis of the vorticity equ ation at O ( c 3 ) . he solutionmeeting the cond ition th a t th e relative velocity ( ~ ( ~ 1 ,l ) )s finite at T = 0 is= cos 8 + $!a) sin 8 ; ( 2 8 4

(Widnall e t al . 1971; Callegari an d T ing 197 8). (1 )Irrespective of any choice of the parameter values cl: and c12 , the matching

resul ts in (3) and lp) 0 .To have an idea on the constants, we revisit the discrete model in an inviscid

flow studied by Dyson. At leading order, i t is the Rankine vortex, that is, thevorticity is con stan t in th e circular core of unit rad ius surrou nded by an ir rota tiona lflow:

Continuity of velocity across the core boundary T = 1 gives

(Widnall e t al. 197 1). However a difficulty arises when the discrete distributionis replaced by a continuous one, because the continuity condition is no longer ofhelp. To make matters worse, both c and c admit arbitrary time dependenceas long a s we stick to the ma tching condition (30 ). Th is is tru e also for the discretemodel, and therefore (33) is merely one possibility.


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MOTION O F A VISCOUS VORTEX RING 29We can show that ci i ) and cri) serve as the parameters placing the circular core

in the moving frame, to an accuracy of O ( E )n terms of the inner spatial scale.Increase of ci i ) and c i i ) by c amounts to the shift of the core-center by EC/ROnthe p- and z-directions respectively. Without loss of generality, we may assumethat c i i ) = 0. Still, a freedom of the choice of the location of the center in theradial direction is at our disposal. We realise thEtt;fixing the initial location of thecore is equivalent to giving the strength of dipoleat t = 0 , and (30) is supersededbv

Comparison of (34) with (18) gives rise to the following identity:

With the specification of do(O), a proper formulation of the initial-value problemis completed. Yet, we suffer from arbitrariness of the temporal evolution of do ( t ) .We can verify that this is consistently absorbed into the third-order radial velocity

as exemplified at the end of $5 . It implies that the perturbation solution isunique, while it has an infinite variety of representations.

Next, we proceed to the second-order perturbation + ( 2 ) . It is shown to havethe following &dependence:$(2) = ?&' $4;) cos 2 8 , (36)

meaning that a quadrupole is produced in conjunction with the elliptical coredeformation. The governing equations and matching conditions are

with

and


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30 YASUHIDE F UK UM OT O AND H. K . M O F F A TT

where CA2) is the axisymm etric p art of th e second-order vorticity p ertu rba tion and1 aab = ---d o ) d r

Finding Cf) requires us to analyse the vorticity equation at O(e4).5. Third-order velocity of a vortex ringWe are now ready to tackle the third-order problem. The dipole field again showsup as the result of nonlinear interactions among the mono-, di- and quadru-polesu p to 0 ( e 2 ) . It is this field that takes part in the correction to the ring speed atO ( c 3 ) .Th e s t reamfunct ion at 0 ( e 3 ) consists of three terms:

+(3) = + cos B + sin B + +$;) cos 38 , (42)only cos 8 nd sin 8 components being relevant t o the speed.After lengthy but tedious algebra, the Navier-Stokes equations collapse to thefollowing e qua tion for gl13):

where

and


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MOTI ON O F A VISCOUS VO RTEX RING 31Th e bounda ry conditions are

and , from (18),

with d3 ) eing the s tre ng th of th e third-orde r dipole. Th e last term of (47) pertainsto fixing the location of the core center with an accuracy of O ( e 3 ) , ut may beignored for determining the speed at the present order. To deduce Z(),e canskip th e full solution of (43)-(47). It suffices to mu ltiply (43) by r 2 and t o in tegratefrom 0 t o some large value w ith respect t o r . To simplify the expression, (37)-(41)is invoked. Tak ing the limit r -+ CO, we eventually arriv e at the desired formula:

where definitions (27) and (41) of a and b should be remembered, and

(48c)The fact that (48c) includes the parameter do brings out the contribution of thedipole distributed along the core-centerline to the induction velocity at O ( e 3 ) )which has so far gone unnoticed. This is traced back to the matching condition(47 ), essentially non-local in its na tur e. It must be emphasised that the asymp-totic formula (18) of th e B iot-Savart integral is essential to make th e system aticevaluation of multi-pole induction feasible.


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32 YASUHIDE F UKU MO TO AND H. K . MOFFATTIn o rder t o f ind Z(2) ,t remains to numerically calculate an d $!j;). Fortu-nately, the explicit solution is at hand for the Ran kine vortex (32 ) . In this case,

B = = l o g ( T ) - 712 7 r 1 5 . 2 5 ~ 2 (49)

Noting that a = -26(r - 1) an d (4 1), th e la st four integrals of (48a) vanish a nd

in accordance w ith (1).Otherwise sta te d, (48a) is a gene ralisation of Dysons resultto an arbitrary distribution of leading-order vorticity in the presence or absenceof viscosity.The r est of this section concerns th e th ird-orde r radial velocity k(2 ) .he equa-tion for $!;) is reducible to

subject to the matching condi t ion that c( 1/r as r + CO. As before, weimplement th e integratio n of (51) with respect t o r after multiplication by r 2 . T hediffusion equation (23) of


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MOTION O F A VISCOUS VORTEX RING 33Finally we illustrate how the vorticity distribution radially evolves starting

from a delta-function core (24) . In this case, P, = 7rRi identically with the 0 ( c 2 )correction term being absent. The particular solution Qii)given by (29) corre-sponds to th e dipole field whose stagnati on point is permanent ly sitting at T = 0(Klein an d Knio 1995). Th e evaluation of th e behaviour of Qii),t large values ofT , is carried ou t w ith ease to yield

D o % 0.41225489 x 4 .27rComparing with (52) , we reach the conclusion th a t , given initially a circular linevortex of radius Ro, th e stagnation point ps( t ) in the core drifts outward linearlyin time owing t o t he action of viscosity:

ps 2 Ro + 0 . 4 1 2 2 5 4 8 9 ~ t / R o . (55)ReferencesCallegari, A . J. and Ting, L. (1978) Motion of a curved vortex filament with decayingvortical core and axial velocity, S I A M J . A p p l . Maths 35, p. 148-175.Dyson, F. W. (1893) The potential of an anchor ring - part 11, P hi l . P ans . R oy . Soc .Lond . A 184, p. 1041-1106.Fraenkel, L. E. (1972) Examples of steady vortex rings of small cross-section in an idealfluid, J . Fluid Mech. 51, p. 119-135.Fukumoto,Y . nd Miyazaki, T. (1991) Three-dimensional distortions of a vortex filament

with axial velocity, J . Fluid Mech. 222, p. 369-416.Gidas, B., Ni, W.-M., and Nirenberg, L. (1979) Symmetry and related properties via themaximum principle, Commun. Math . Phys . 68, pp. 209-243.Hosokawa, I. and Yamamoto, K. (1989) Fine structure of a directly simulated isotropicturbulence, J . Phy s . Soc . Japan 59, p. 401-404.Jimknez, J., Moffatt, H. K., and Vasco, C. (1996) The structure of vortices in freelydecaying two-dimensional turbulence, J . Fluid Mech. 313, p. 209-222.Kerr, R. M. (1985) Higher-order derivative correlation and the alignment of small-scalestructure in isotropic turbulence, J . Fluid Mech. 153, p. 31-58.Kida, S. and Ohkitani, K. (1992) Spatiotemporal intermittency and instability of a forcedturbulence, Phys . Fluids A 4, p. 1018-1027.Klein, R . and Knio, 0 .M. (1995) Asymptotic vorticity structure and numerical simulationof slender vortex filaments, J . Fluid Mech. 284, p. 275-321.Klein, R and Majda, A. J. (1991) Self-stretching of a perturbed vortex filament. I. Theasymptotic equation for deviations from a straight line, Physzca D 49, p. 323-352.Moffatt, H. K., Kida, S . , and Ohkitani, K. (1994) Stretched vortices - the sinews ofturbulence; large-Reynolds-number asymptotics, J . Fluid Mech. 259, p. 241-264.

Moore, D. W. and Saffman, P. G . (1972) The motion of a vortex filament with axial flow,P h i l . P a n s . R . Soc . Lond. A 272, p. 403-429.Saffman, P. G . (1970) The velocity of viscous vortex rings, Stud . A p p l . Math . 49, p. 371-380.Siggia, E. D . (1981) Numerical study of small scale intermittency in three-dimensionalturbulence, J . Fluid Mech. 107, p. 375-406.


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34 YASUHIDE FUK UM O T O AND H. K. MOFFATTTung, C. and Ting, L. (1967) Motion and decay of a vortex ring, Phys. Fluids 10, p. 901-

910.Widnall, S. E., Bliss, D. B ., and Zalay, A. (1971) Theoretical and experimental studyof the stability of a vortex pair, In Aircraft W a k e Turbulence and i ts Detection (eds,Olsen, Goldberg, Rogers), Plenum, pp. 305-338.

yasuhide fukumoto and h.k. moffatt- motion of a thin vortex ring in a viscous fluid: higher order...

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