H.K. Moffatt, S.Kida and K. Ohkitani- Stretched vortices - the sinews of turbulence;...

8/3/2019 H.K. Moffatt, S.Kida and K. Ohkitani- Stretched vortices - the sinews of turbulence; large-Reynolds-number asympt

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J . Fluid Mech. (1994), vol. 259, p p . 241-264Copyright 0 1994 Cam bridge University Press

24 1

Stretched vortices- he sinews of turbulence;large-Reynolds-number asymptoticsBy H. K. MOFFATT, S.KIDA A N D K. OHKITANI

Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Silver Street, Cambridge CB3 9EW , U KResearch Institute for Mathematical Sciences, Kyoto University, Kyoto 606-01, Japan

(Received 22 April 1993)A large-Reynolds-number asymptotic theory is presented for the problem of a vortextube of finite circulation r subjected to uniform non-axisymmetric irrotational strain,and aligned along an axis of positive rate of strain. It is shown that at leading orderthe vorticity field is determined by a solvability condition at first-order in E = 1/R,where R, = T/v. he first-order problem is solved completely, and contours ofconstant rate of energy dissipation are obtained and compared with the family ofcontour m aps obtained in a previous numerical study of the problem. It is found th atthe region of large dissipation does not overlap the region of large enstrophy; in fact,the dissipation rate is maximal at a distance from the vortex axis at which theenstrophy has fallen to only 2.8% of its maximum value. The correlation betweenenstrophy and dissipation fields is found to be 0.19 +O ( e 2 ) .The solution reveals thatthe stretched vortex can survive for a long time even when two of the principal ratesof strain are positive, provided R, is large enough. The mann er in which the theorymay be extended to higher orders in E is indicated. T he results a re discussed in relationto the high-vorticity regions (here described as sinews) observed in many directnumerical simulations of turbulence.

1. IntroductionOne of the most striking features of the structure of turbulence observed in many

numerical simulations (Siggia 1981; Kerr 1985; Hosokaw a & Yam am oto 1989, 1990;She, Jackson & Orszag 1990; Ruetsch & Maxey 1991; Vincent & Meneguzzi 1991;Douady, Couder & Brachet 1991; Kida & Oh kitan i 1992; JimCnez et al. 1993; Kid a1993) is the emergence of high-vorticity regions concentrated in tube-like structureswhich occupy a relatively small fraction (N 1%) of the total volume, but whichaccount for a much larger fraction (typically 1&20%) of the viscous dissipation ofturbulent energy (Hosokawa & Yamamoto 1989).7 The tubes have a length of theorder of the integral scale of the turbulence, and a cross-sectional radius somewherebetween the inner Kolmogorov scale and the Taylor microscale (Tennekes 1968 ;Vincent & Meneguzzi 1991), and ar e generally interpreted as vortex tubes which arestretched an d concentrated, in a m anner analogo us to th e familiar Burgers vortex, bythe local straining associated w ith the turbu lence field. The radius of the Burgers vortexsubjected to a stretching rate y is 6 - v / y ) i ,and if y is a typical turbulence strain rate,then 6 is of the order of the inner Kolmogorov scale.is lowered, then both the volume fraction and the fraction of dissipation increase.t These percentages depend o n the thresho ld th at is used to define high vorticity. If this threshold

www.moffatt.tc


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242Vortices have been described as the sinews of fluid motion (Kiichemann 1965;Saffman & Baker 1979), a term that now seems particularly apt for the aboveconcentrated vortex tubes. Just as sinews serve to connect a muscle with a bone orother structure, so the con centra ted vortices of turbulence serve to connect large eddiesof much w eaker vorticity; an d jus t as sinews can take the stress and strain of musculareffort, so the concentrated vortices can accom mod ate the stress associated w ith the lowpressure in their cores and the strain imposed by relative motion of the eddies intowhich they must merge a t their ends. Thu s the term sinews of turbulence seems well-chosen to describe these high vorticity regions;? hence the title of this paper.The idea that the dissipative structures of turbulence may be well represented by arandom distribution of strained vortex sheets or tubes goes back to Townsend (1951)(see the discussion in Batchelor 1953, 57.4). Suppose that the background strainingflow has the form

H . K . MofSatt, S . Kida and K. Ohkitani

U = ( E X ,Py, YZ), (1 1)where a + p + y = o , a < O < y , p 2 a . (1.2)If /3 < 0, then there is one positive principal rate of strain ( y ) , and there is a tendencyto form vortex tubes aligned with the corresponding axis of strain (Oz); we shalldescribe this type of strain as axial stvain. If /3 > 0, then there are two positive principalrates of strain (p and r), nd there is a tendency to f orm vortex sheets in the plane ofthe corresponding axes of strain ( O y ,O z ) ;we shall describe this type of stra in as biaxialstrain. If = 0 and a = - , then of course we have the case of plane strain. We maycharacterize the different types of strain by the single strain parameter

as follows: h = 0, axisymmetric axial strain ;0 < h < 1, axial stra in;A = 1, plane strain;1 < h < 3, biaxial strain with 0 < ,8 < y ;h = 3, axisymmetric biaxial strain (P = y ) ;h > 3, biaxial s train with p > y .Note that the determinant of the rate-of-strain matrix is given by

.pY = ay3(1 - A * ) . (1.4)It is well known that < 0, the overbar denoting the ensemble average, inhomogeneous isotropic turbulence, so that there would appear to be a statistical biastow ards regions of biaxial strain. H owev er, as pointed ou t by JimCnez (1992) and Kida(1993), the presence of a s trained v ortex itself modifies the local strain field, tending toconvert ax ial strain to biaxial strain, so that care is needed in this inference. Moreover,vortex sheets are in any case subject to Kelvin-Helmholtz instability, an instability thatis modified but not completely suppressed by strain (Lin & Corcos 1984); thisinstability leads to spiral wind-up of a vortex sheet and to the formation of a row ofparallel vortex tubes of spiral structure. Spiral structures have been considered as

t The alternative term w orm s has been suggested by Y am am oto & Hoso kaw a (1988). We preferthe term sinews for the reason given above, an d also because vortex tubes, unlike worms, d o nothave clearly defined end -poin ts.


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Stretched vortices - he sinews of turbulence 243candidates for the typical dissipative structu res of turbulence (Lundgre n 1982;Moffatt1984, 1993) but clear evidence for spiral structures fro m direct num erical simulations(DNS) of turbulence is not yet available.Vortex tubes form ed either in regions of axial strain or (via the K elvin-Helmholtzmechanism) in regions of biaxial strain are in general represented by fully nonlinearsolutions of the Navier-Stokes equa tions, the nonlinearity arising thro ugh convectionof the (generally non-axisym metric) vorticity field o y the rota tion al velocity U (whereo = V x U). Only in the axisymmetric case (J= a ) is an exact solution of theNavier-Stokes equa tion of finite circulation available ; this is the Burgers vortex forwhich o = (0, 0, w ( r ) ) with

Here, r is the total circulation associated with the vortex. If a d 8 < 0, and if theReynolds numberis very small, so that self-induced convection of vorticity is negligible, then thelinearized vorticity equation admits a unique solution similar to (1.5), namely

R, = rIv (14

so that iso-vorticity contours w = const. are ellipses with principal axes aligned w iththe principal axes of strain (see Appendix A).Study of the effects of nonlinearity for larger R, was initiated by Robinson &Saffman (1984) (see also Saffman 19 92 ,s13.3), wh o developed a double series solutionin terms of the Reynolds number (still assumed small) and the strain parameter A ,representing the departure from axisymmetry in the imposed strain field (1. l) , an d alsoassumed small ; they also provided numerical evidence for the continued existence ofsuch vortices for Reynolds num ber u p t o 100 an d values of h up to 0.75. In turbulence,we are particularly interested in large values of R, (Jimenez et al. 1993 provide evidencefor a scaling law R, - R e f where Re is the turbulent Reynolds number based on theTaylor microscale) and also in the full range of values of A , not only the range 0 1 (i.e. ,4 > 0) since, as will emerge, stro ng vortex tubes

can survive for a very long time even in regions of biaxial strain (see $ 5 ) .Renewed interest in this problem has been kindled by two numerical studies: firstthat of Buntine & Pullin (1989) who studied the time-dependent m erger of two vorticesin a strain field with h = i an d Rr/27c in the range 10 to 1280; an d second, tha t of Kida& Ohk itani (1992, hereafter referred t o as K 0 9 2 ) who (in the Appendix t o their paper)analysed the stretched vortex problem numerically an d comp uted n ot only steady iso-vorticity con tours fo r h = i an d a range of values of R r up t o 500, but also the contou rsof constant energy dissipation rate @ = 2vsijsij, where s i j is the rate-of-strain tensor(including both the background strain field and the contribution from the vortex).KO 92 showed tha t (i) the principal axes of the contou rs w = const. ro tate anticlockwisetowa rds the lines x = k y a t 45" o the axes of strain Ox, O y as R r increases, and at thesame time tend t o become m ore circular in f or m ; (ii) the conto urs of the dissipationfunction @ evolve in a rather complex manne r as R, increases, but the function alwaysexhibits two maxima on or near the line x = - y an d symmetrically displaced from thevortex centre. A similar double-pe aked structure of the dissipation func tion was notedin the high-vorticity regions (sinews) of a DNS at Reynolds number (based on the


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246and for the vorticity

H . K . MoJffatt, S . Kida and K. Ohkitani

At order E, we obtain the Euler equation(2.13)

(2.14)so that wo = F(yk0)or some function B.f we assume that the flow $o has a singlestagnation point at r = 0, the streamlines being closed curves around this stagnationpoint, then it seems highly probable tha t the only solutions of (2.14) in an u nboundedfluid and with localized vorticity are of the form $o = $o(r), i.e. the streamlines (andso the iso-vortic ity lines also) are circles r = const. Certainly this is consistent with thetrend noted by K092. We note moreover that it has been recently shown byLinardatos (1993) that in the closely related problem of finding magnetic equilibria ina perfectly conducting fluid by the m ethod of magnetic relaxation (Moffatt 1985), theminimum-energy states with a single null-point do indeed have circular field lines.At any rate, we shall restrict attention to this class of solutions of (2.14), i.e. weassume $o = $o(r) so thatand the associated velocity com ponents a re

0 y ) = - $-- -1$ 0 , (2.15)uo = 0 , v, = -+;. (2.16)

Here, an d subsequently, the prime is used to denote differentiation with respect to r .The function $o(r) is not determined at this level of approximation.Now equating terms of order E in (2 .7), we obta in

Averaging over 8, we obtain the solvability conditionLowo= 0,

and this has solution, finite at r = 0 and vanishing at r = CO, (2.18)(2.19)

where we have normalized so that the total (dimensionless) vortex strength is unity(consistent with the dimensional co nstrain t (2.2)). Th us, despite the non-axisymmetryof the strain ( A > 0), the vorticity field at leading order is precisely that of theaxisymmetric Burgers vortex and is independent of the strain parameter A . Thisremarkable result has already been obtained by Neu (1984) for the particular case ofplane strain ( A = 1). Th e result is quite subtle because it means tha t viscosity has a nim por tan t residual effect even in the limit of infinite Reynolds num ber. I n this respect,the result (2.19) is reminiscent of the Prandtl-Batchelor theorem (Batchelor 1956) fortwo-dimensional steady flow with closed streamlines, which establishes a similarresidual effect of viscosity (in establishing a uniform vorticity distribution outsideboun dary layers) in the limit v +0. By an alogy with th at situatio n, we may conjecturetha t it will take a time of order Rri to establish the steady state (2.19) starting from aninitial condition in which w is localized within an area of order unity but otherwise


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Stretched vortices - he sinews of turbulence 247

rFIGURE. The functionf(r) determined by (2.25) , (2.28) and (2.29). The dashed curve shows therequired convergence of rzf(r) to a constant as r --f CO.

arbitrary (Rhines & Young 1982). (A residual disturbance approximately constant onstreamlines $ = const. may persist on the longer timescale O(R,); he time-dependentproblem (cf. Buntine & Pullin 1989) deserves furthe r study in this respect.) Note th atthe circulatory velocity corresponding to (2.19) is127cr0(r)=- 1- -p/4),and the streamfunction is

Returning now to (2.17), and using (2.18), we have

Hence,i a- - (U ; $l +V , w l ) = AL, U , = !+o COS 28.r ae

uoV2@, - U ; @ l = -+Ar2wl, sin 28 +q ( r ) ,for some function q(r ) . Hence we may setwherewith

$l = AAr)sin 28 + g ( r ) ,f" +r - ' f ' - 4 r -2 f = ( f - i r 2 ) r ( r )

(2.20)

(2.21)

(2.22)(2.23)(2.24)(2.25)(2.26)

The function g ( r ) is not determined at this level, but will be shown to be zero (seethe final paragraph of this section) on the basis of a solvability condition at order e2.r ( r ) = - + g 2 - r92r+ . ., (2.27)

and, by elementary techniques, the expansion forflr) may be found in the formf ( r ) = a r 2 + & ( ~ - a ) ( r 4 - & r 6 + ...), (2.28)

where the constant a is to be determined in such a way that the outer boundary(2.29)ondition r 2 f ( r ) + C as r+Oo

is satisfied (ensuring tha t the flow is irrotation al as r +.CO).Num erical solution of (2.25)determinesf(r) (see figure 2), an d the constan ts a and C:(2.30)

For small r , q(r ) has the Taylor expansion

U = -0.381475. .., C = - 7.4723... .


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248 H . K. MofSatt, S. Kida a nd K. Ohkitani1.0

0.5

0 rFIGURE. The vorticity functions 47co,(r) (solid line), -D(r) (dashed line) defined by(2.19) and (2.34).

Note that f ( r ) < 0 for all r > 0, and that f ( r ) has a single well-defined minimum atr = r1 z .94 whereflr,) z - .10. Th e convergence of rffor large r is rapid; in fact,it is easily shown that rzf- c - y4 e - r 2 ~ 4 as r + m . (2.31)Assuming for the moment that g( r ) = 0, we now have $ to order e in the form

$ = $o( r ) + e1f r )sin 28, (2.32)where el = eh. The corresponding vorticity is

o = wo(r)+e1Q(r) sin 28, (2.33)where Q(r) = - -l(rf) +4rP2f = ($2 f) 1 (2.34)using (2.25). Clearly Q(r) < 0 for all r (figure 3). No te the asymptotic behaviour

O(r) -+(1-4a) ( r2-&r4+ . . .) fo r r < 1, (2.35)and O(r>- ~ y 4 -r4/4 as r - t c o . (2.36)We note a potential difficulty here, in tha t

(2.37)so that the leading term wo(r) in (2.33) is not dominant for large r . However, it isdom inant provided r2 < 2(ne1)4, (2.38)and when r2 - 2(ne1)-i, both wo(r)and e1 Q(r) are of order (4n)-lexp [ -:(ne1)-i], i.e.transcendentally small. We discuss this point further in 6 5 .

The streamlines are now given, to order el , byr = ro+ e1 r,(8), (2.39)

where, by substituting in (2.32) and equating terms of order e1,(2.40)


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Stretched vortices - he sinews of turbulence 249

3

2

1

Y O

1

2

3

I I I I I II

I I I I I I I-3 -2 -1 0 1 2 3

X

FIGURE. Contours i = const. (thick) and w = const. (thin) obtained from (2.32) nd (2.33) ithe1 = 0.005. Note that the streamlines are quite nearly circular, whereas the iso-vorticity lines aremore strongly flattened. (Contour levels : $ = -0.01 n (n = 1,2,. .,7); = w,,, + w,,, - ,,,) n/7(n = 1,2,..,6);U,,, = -4.737 x 10-4, w,,, = 7.938 lO-'.)Thu s, the streamlines are ellipses with m ajor axes o n 8 = 3 n /4 , 7 n / 4 and m inor axeson 0 = n / 4 , 5 n /4 (because ( ro) /vo(ro) 0 ) he eccentricity of these ellipses is maxim ala t ro z .9 .Similarly provided e1 4 and r2 < 2(ns,)-f, the lines of constant vorticity are given

(2.41)by

r = ro+ e1 rw(0) ,where (2.42)Again, these are ellipses with the same orientation of majo r an d minor axes as above,a property that is evident in the iso-vorticity plots of K O 9 2 (figure 1 3 d - f ) for R, 2100. The tendency of the principal axes of the iso-vorticity ellipses to rotate in acounterclockwise sense from the principal axes of strain (x = 0 , y = 0) was noted byRobinson & Saffman (1984); he present analysis shows tha t the total angle of rotationasymptotes to i n as Rr+ 00.Although the princ ipal axes of streamline ellipses and iso-vorticity ellipses coincide,the curves d o n ot themselves coincide because the eccentricities at given ro are clearlydifferent in the two cases. At order E , vorticity is no longer con stant o n streamlines; theslight mismatch is accommodated by the strain field and by viscous diffusion. Thesituation is illustrated for the case e1 = 0.005 in figure 4 , which shows contours of @and w based on (2.32) and (2.33) .N ote tha t, despite the qu ite strong ellipticity of the


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250 H . K. M oflat t, S . Kida and K . Ohkitaniiso-vorticity lines, the streamlines settle down rapidly to circular form outside thevortex core.Proof that g(r) 0

Consider now the terms of order e2 in (2.7); these give

All the terms in hL, w1 and r- l a($,, w , ) / a ( r ,8) are propo rtional to sin 28 or cos 28 (fordetails see $ 4 below), and so give zero when integrated from 8 = 0 to 8 = 2n. Hence,integrating (2.43) over 8 gives the solvability cond itionL,( -V2g) = 0 (2.44)

using the vorticity w1 = -V2$, with $l given by (2.24). It follows thatVZg = k e-rz14 (2.45)for some constan t k . The no rmalization condition coupled w ith the zero-order solution(2.19) now implies that k = 0. Hence V2g(r)= 0 an d the only solution finite at r = 0 isg = const. ; since an additive constant in the streamfunction does not affect the flow,we may clearly take g = 0, as anticipated above.3. Spatial distribution of viscous dissipation

Let sij be the (dimensionless) rate-of-strain tensor, with elements

Then the rate of viscous dissipation is given by@ = 2sijsij = 2[D+e2(;h2 ;)I,

D = 2 $ i y + ( $ X X - yy)2 - % $xy,(3.2)

where (3.3)and, as before, E, = he. (The dimensional dissipation rate is given by ~(r/6)~( y 2 P / v ) .) The term e2(;h2+I)n (3.2) comes from the background strain field and isthe uniform value of +@ far from the vortex; hence D represen ts the excess of @ overthis background values due to the presence of the vortex. Converting to polarcoordinates, (3.3) becomes

D ( r , 8) = a($ - -'$, - -2$,e)2 +2r-'($,, - - l $ J 2-el($rr-r-1$'r-~-2$,e)sin28-2e, r-1($r,-r-1$g)cos28. (3.4)

It proves necessary to consider terms up to order e2 in this expression, in order to geta uniformly valid first approximation to D . Substituting $ = $,(r) + , f l y ) sin 28 +e2$2(r,8), we obtainD(r , 8) = ;(H(r))' - 1H ( r ) 1-G ( r ) ) in 28 + e9[(;G2- G) sin228 + aF2- )cos2281

+ e 2 H ( r )G 2 ( r ,8)+ O(e3) , (3.5)


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Stretch ed vortices - he sinews of turbulence1 1 1 1 1 1 1 1 I I I U 1 1 1 1

25 1

FIGURE. The functions F(r) (chain-dashed), G ( r ) (solid), 32nH(r) (dashed)given by (3.6), (3.7) and (3.8).where

and Gz ( r , e, = $zrr - -$rZr - -2$266* (3.9)The functions F, G, H a r e shown in figure 5 . Note the following asymptotic behaviou r:for small r ,F(r) - a + ( ~ - a ) r 2 - & ( + - a ) r 4 + . . . , (3.10a)G ( r )- a + ( a - a ) r 2 - g ( a - a ) r 4 + . . . , (3.10b)132n( r ) - - ( r 2 - + r 4 + ...); ( 3 . 1 0 ~ )and as r + CO,

F(r)---2,C+ O(r2 e-r2/4), (3.11a)G ( r )-y+O(r4 e-ra/4), (3.1 1b)

r

H(r) -- 0(e-r2/4). ( 3 . 1 1 ~ )The presence of the small factor (32n)-l z .01 in ( 3 . 1 0 ~ )s rather striking; looking atthe structure of (3.5) suggests that, for numerical purpose, 32x6, might be a moreappropriate expansion parameter than e1 (see the asymptotic behaviour of D( r ,8 ) asr+O, obtained below (3.15)). Note also that

H(r) = 0 and H ( r ) = H,,, x 0.0237 at r = r , z .67, (3.12)and that G ( r )< 0 for all r . Note finally that the functions F(r) and G ( r ) are almostidentical for r 5 1, where the first two terms of (3.10~2, ) give an excellentapproximation to both functions.

nr2

The function $z ( r , 0) is determined in $4. All we need note here is thatas r + O , $, = O ( r 2 ) and so G, = O( l ) ,and as r + CO, $, = O(r-) and so G, = O(r-4).

( 3 . 1 3 ~ )(3.13b)


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252 H . K . Moffatt, S. Kida and K . Ohkitani

FIGURE. Contour plots (a) of the inner dissipation fuzction e ; z f i ( r ^ , O ) defined by (3.15) an d( b ) of the outer dissipation function q z D ( F ,0) defined by (3.19).The structure of the dissipation field (3.5) may be best understood by consideringthree regions.

(i) r = ~ ( e i )Defining a n ' nner variable '

r^ = (32ne1)-b, (3.14)the leading-order contribution to D in the region where r^ = 0(1 ) is6(?,) - e:[;t4 - 1- a) r^' sin 28 +4a(2a - )] , (3.15)

the final term coming from the third term of (3.5). Th e need to retain terms of orderez in (3.5) should now be clear; note however that the final term of (3.5) makes nocon tributi on t o (3.15) sincee2H(r )G z ( r , ) = O ( e z r 2 )= O(e4i") (3.16)

in this region. The function ~ ; ~ d ( i ,) defined by (3.15) has universal fo rm independentof both h and e = l / R r ; ts contours are shown in figure 6 (a ) . It is minimal at r^ =(1 -4a)i z 1.59, 6 = n/4, 5n/4, and thereDmin = - 0 . 5 ~ ; . (3.17)a

(ii) r = o(E+)Similarly, defining an 'oute r variable '

i = (nE1)b, (3.18)we find that in the region where i = 0(1), the leading contribution to (3.5) is

b(i,1 - : [ ! ~ J - ~- -* in 281. (3.19)Again, e ; 'b ( r , 6) has universal form; its contours are shown in figure 6(b). It isminimal at i = 1, 6 = n/4, 57c/4, and there we again find

Dmin = -0.56; . (3.20)w


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Stretched vortices - he sinews of turbulence 253(Actually, as noted below (2.37), the asymptotic solution is suspect for r E;$, and iscertainly invalid in the outerm ost region r 9 e;:; this is discussed further in $ 5 and inAppendix A .) The results (3.17) and (3.20) imply that the correspond ing minim a of @are given by

= $2, (3.21)independent of A .(iii) r = O(1)

When r = O(1), it is sufficient to retain the first two terms of (3 .9 , i.e.D(r , 6 ) - D l ( r ,0) = ; (H(r))2- 1 H(r) (1- G ( r ) ) in 26. (3.22)

This function is symm etric about the diagonal lines x = k y , and s ince H (r )(l - G ( r ) )> 0 for all r > 0, D is maximal as a function of 6 on x = - y , minimal o n x = y . In thelimit e1 = 0, the contours D l ( r ,6 ) = const. become circles r = const. (a behaviour th atmay be seen bo th in the outer limit t+ 00 of (3.15) and in the inner limit i + O of (3.19))and D , is maximal where H (r) is maximal, i.e. at r = re x .67. This maximum valueis thus, asymptotically, D l m a x- ;Pm,,x .808 x 10P4. (3.23)

The perturbation term in (3.22) breaks this circular symmetry, and, provided we(3.24)

avoid the critical circle r = r, , the contours are given to first order in e1 byr = ro+ e1 rD(6),

where, by substituting in (3.22) and linearizing,(3.25)

These contou rs are ellipses, with m ajor axis x = - y for f l ( r o ) < 0, i.e. ro > rc , and withmajor axis x = y fo r f l ( r o ) > 0, i.e. r,, < r,.This approxim ation is clearly invalid near ro = r , where H(ro) = 0. Ne ar this value,H ( r )x HC+f(r-rJ2H:, (3.26)

where H , = H(r, ) ,H: = H(r,) , so tha t (3.22) becomesD,(r, 6 )xge+ ( r- J 2H , H i - 1 H,( 1- G,) sin 26, (3.27)and the contour

D = - 1H e(1-G,) = (0.028- . 6 ~ ~ )10-2 = D,, (3.28)say, is given by

1-G ,( r - r , ) = 2el-(1 -sin26).IH, I (3.29)This is a separatrix joining saddle points a t Y = r,, 6 = x/4, 5n/4. Maxima of D occura t r = rc, 6 = 3x/4, 7n/4, and thereD,,, =;e 6 , H,( 1- G,) = (0.028 +2 . 6 ~ ~ )10P2. (3.30)There is a cats eye structure with width (a t 6 = 3x/4 , 7x /4) given from (3.29) by

9

(3.31)F L M 259


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254 H . K . Mofa t t , S . Kida and K . Ohkitani

5

0

-5

-5 0 5 -5 0 5FIGURE. Con tour plots of the dissipation function D(r, 0) defined by (3.32). The maxima of D aremarked with and the (global) minima with x , The co nto ur levels are equally spaced at one-seven thof the difference Dm,,-Dmin. The separatrices D = D o nd D = D, re also included (thick lines),and the plots are shaded light grey where min(D,,D,) < D < max (Do,D ,) and dark grey whereD > max(D,, D,).a)e1 = h/R, = 0.01 (Do D,),b )e1 = 0.005 (D, Do),c) e1 = 0.0025 (D, Do),(d) 1 = 0.001 (D, Do).ompare with the computed contours in figure 14(c-f) of K 0 9 2 .All of these features may be observed in figure 7 ( d ) which shows contours of thefunction (3.32) below for e1 = 0.001.The above discussion makes it clear that a uniformly valid approximation toD ( r ,0) for small e is given by simply omitting the term e2H(r )G 2 ( r , ) in (3.5) which isalways dominated by other terms. Thus

D ( r ,0) - (H(r))' - 1 H ( r ) 1- G ( r ) ) in 20+ ~ ~ [ ( ; G ~ - G ) s i n ~ 2 8 + ( ~ F ~ - F ) c o s ~ 2 8 ] .3.32)

Contours of this function are shown in figure 7 fo r e1 = 0.01, 0.005, 0.0025, 0.001.These values are chosen to enable comparison w ith the contours obtained numerically


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Stretched vortices- he sinews of turbulence 2553

(x 10-4)2

D(r , x/4)I

0

0.00656

0 2 6 84 r-11 I I I I " ' I I " I I I I ' I IFIGURE . The function D(r,x/4) defined by (3.34) for various values of sl. A saddle point ofD( r ,0) occurs very near the point where D ( r ,n/4) is maximal. When e1 x 0.00656, the value D, tthis saddle equals D o ; 2.69s;.and presented in figure 14(c-f) of K 0 9 2 . The value E , = 0.01 (for which 32x6, x ) isthe largest for which the present asymptotic theory can reasonably be applied. Theseparatricesthrough the origin and D = D, hrough the other saddle points on x = y , are includedin each case, and the contour plots are shaded light grey in the region where min (Do,D,)< D < max (Do,,),nd dark grey where D > max (Do,,).he following pointsshould be no ted: (i) the inner region (r" = O(1)) is best seen in figure 7(d) (el = 0.001)(compare w ith figure 6a) and the outer region is best seen in figure 7(a) (el = 0.01)(compare with figure 6b) ; (ii) the cat's eye pattern and the ellipses both inside andoutside are best seen in figure 7 ( 4 ; (iii) in all cases, the regions (dark grey) of strongdissipation are set distinctly off-centre on the diagonal x = - y , very much as revealedin the corresponding computations of K 0 9 2 .There is a change of topology of these contou rs as e1 increases from very sm all levels.This may be understood from consideration of the function

(3.34)

D = D o 4 ~ ( 2 ~ -)e; x .69s; (3.33)

D(r , n/4) = ;(H(r))' - 1 H(r) 1-G ( r ) )+e;(;G2-G)shown in figure 8 for various values of e1. The change of topology occurs when e1 =elCx .00656, at which value D o D,. he corresponding contour plot is shown infigure 9. There are further interesting changes in the topology of D(r ,O) for largervalues of e but these lie beyond the range of validity of the asymptotic analysis, andare therefore not considered here.The most striking property of the field of dissipation (3.32) is that, as el+O, thedissipation becomes quite sharply concentrated near the radius r = 2.67 at which H ( r )is maximal. At this radius,

so that the enstrophy (w0(r))' is only 0.028(~(0))' , .e . 2.8 % of its maximum value. Thismeans that the region of large enstrophy and large dissipation are effectively non-overlapping, an im portant conclusion in the turbulence context (see $ 6 below). The 6-averaged enstrophy and dissipation, correct to order e, are given by( U ' ) = ( W 0 ( r ) ) ' = (167c2)-l e-r2/2, 2(D ) = (H(r))', (3.35)

9-2


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256 H . K . MofSatt, S. Kida and K . Ohkitani

t I y

t 1t I 1l i l t I I l l l l l l l l l l l l l-5 0 5

FIGURE. Contour plot of D ( r ,8 ) for the critical case e1 = el e z .00656 (Do DJ.Shaded area as in figure I .and it is not hard to verify that

JOm (H(r))%dr = JOz (w,(r))'rdr = (167P-l.It is also easy to verify that at r = 2/2,

and thatand

1; (w,(r))'rdr = 0.632 (w0(r))'rdr,11: ' H(r)) 'rd r = 0.0128 (H(r))'rd rJOX

(3.36)

(3.38)(3.39)

Hence 63.2 YO f the total e nstrop hy associated with the vortex lies within the radiusr = 2/2, whereas 98 .7% of the viscous dissipation occurs outside this radius.The correlation between the fields w&r) and H (r) is given by

This integral can be evaluated, with the result

(3.40)

(3.41)1 = - 1 + 2 1 n 2 = 0.386.


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Stretched vortices- he sinews of turbulence 257( 2 y @ m a z ( 2 2 - Q m t n

1 Fro m (3.32) Fro m (3.30) Numerical Fro m (3.32) Num erical(a) 0.01 3.00 2.98 2.90 1.50 1.50(b ) 0.005 5.76 5.13 5.57 1.50 1.50(c) 0.0025 15.5 15.4 15.1 1.50 1.50(d)0.001 79.1 78.1 77.3 1.50 1.50TABLE . Maximum and minimum dissipation rate for the case h = t. The numerical values wereobtained by integrating (A 3) by the finite difference method as in KO92 with a larger domain(- 15 < x,y < 15) and a finer mesh (Ax = Ay = 0.25).The correlation between the enstrophy field ( w ( r ,8))' and the dissipation field D ( r ,e )is given, correct to order E, by lomr d r

1+ O(e2) .pz = (1: w i r d r1 H4rdr)i

The integrals have been evaluated numerically, with the result1.157 x 10-6 + O(E')= [(2.005x lOP5) (1.882 x 10-')>1:

(3.42)

= 0.19 + O(2). (3.43)The results (3.41), (3.43) shou ld be testable throu gh a pp rop riate m anipu lation of DNSdata .?Finally, we note some numerical comparisons. Table 1 shows the maximum andminimum values of @ / 2 2 obtained from the function (3.32) for the case h = f, ogetherwith the corresponding figures comp uted for this case by K 0 9 2 . The minimu m valuesare in perfect agreement. There is however a systematic difference of about 10%between the maxima given by the asymptotic theory and the maxima previouslycomputed. The reason for this difference is that, for the larger values of e, theasymptotic theory is perha ps no t sufficiently accura te, while for the smaller values ofel, the numerical procedure of KO92 is slow to converge. We have repeated thecomputation of KO92 for the case h = i, R, = 500 (i.e. e1 = 0.001) using a largerdomain and a finer mesh, and the discrepancy between the num erical solution an d theasymptotic theory is indeed reduced (see Table 1).4. Higher-order asymptotics

Let us now briefly consider the prob lem posed by (2.43) at o rder E'. As shown at theend of 52, the solvability condition for this equa tion yields the result g ( r ) = 0. Then,(2.43) simplifies to the form

c?-e (wl, $2 +v,, 02) 2hv,(r) M ( r )sin 28 + 4A2v0(r)N ( r )sin 48, (4.1)t As pointed out by a referee, pz is zero for a Rank ine vortex (for which the enstrophy is non-zeroonly for r < 6 and the dissipation is non-zero only fo r r > 6) . On the other hand, pz = 1 for a vortexlayer in which w = (0 ,0, w ( y ) ) and dissipation is proportion al to enstrophy (T anak a & Kida 1993).


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H . K . MofSatt, S. Kida and K. Ohkitani58where

N ( r ) = (r2Q- rQ - ( sZ -fsZ)).16v,(r) (4.3)Integra ting (4.1) with respect to 8 an d dividing by vo(r), we ob tain

V2$, - ( r )$, = hM(r) os 28 + h2N(r ) os 48 + g , ( r ) , (4 4)for some function g 2 ( r ) .The solution of (4.4) may be sought in the form

$,(r , 8 ) = h2$, ,(r)+ hf,(r)cos 28 + h2f,(r) os 48, (4.5)where L, f , 7;+ -lf, - rP2f2= 7 , +M ( r ) , (4.6)and L,f, 7;+r-y,- 6r-2f4 = 7 , +N ( r ) . (4.7)The function $,, is determined by a solvability condition at O ( e 3 ) see Appendix B).Note that the equation L2f= 0 has linearly independent solutions r2 and rP2,while theequation L, f = 0 has linearly independent solutions r4 and r-*.Now, using (2.28) and (2.35), the behaviour of M and N for small r may becalculated, and isSimilarly, from (2.31) and (2.36), we find that

M ( r )- n(1 -4a) r2 , N ( r )- &n(l -4a)r4. (4.8)M ( r ) N --nr~4 e-r2/4 N ( r )- -Am*-r24 as r + co. (4.9)These properties guarantee that solutions of (4.6) and (4.7) may be found with theproperties f , - a 2 r 2 as r + O , f , - C,r + as r + co, (4.10)

f , -a4r4 as r + O , f , - C,rP4 as r + c o . (4.1 1)Hence the solution (4.5) is uniquely determined. Again it has the property that theassociated vorticity w2 = - V2$, is exponentially small as r + CO.

Th e streamfunction $ = $ , + ~ $ ~ + e ~ $ , now has the form$ = $,(r) + ehf(r)sin 28 + e2(h2$, ,(r)+hf2(r) os 28+ h2f,(r) os 48) + O(e3) . (4.12)

Note that $ now involves e and h independently, and no t just through the combinatione1 = eh. Mo reover, the symmetry abo ut the diagonals x = k y s clearly broken a t ordere2, and this will carry over to the dissipation function also.This procedure m ay now be carried to higher orders in e if required. The main pointto note here is that a w ell-defined procedure exists. This lends confidence to o ur claimthat the essential features of the flow are well captured by the first two terms of theexpansion (2.11) when e is sufficiently small.5. The case h > 1 (biaxial strain)

Th e asymptotic solution th at we have obtain ed in $2 appe ars to be valid whetherh c 1, or h > 1, provided merely that e1 = eh 4 . This suggest that concentratedvortices should be able to survive even in a biaxial strain field where both /3 and y arepositive.


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Stretched vortices - he sinews of turbulence 259There are two difficulties here, the first of which we have already no ted in relationto the behaviour (2.37) of the ratio Q ( r ) / w , ( r ) for large r . The fact tha t EIIQ(r)I eventuallybecomes of the same order of magnitude as wo(r) mean s that the expansion (2.1 1)cannot be uniformly valid all the way to r = CO. How ever, at the large values of r (of

order e;:) at which it breaks down , the vorticity is already of order exp [ - i ( n ~ ~ ) - i ] ,.e.so small that the non-uniformity is of no consequence. (The situation is a littlereminiscent of the breakdow n of the inner solution for Stokes flow past a sphere, abreakdown th at can be rectified by the use of matched inner a nd o uter expansions.) A tvery great distances ( r % e;;) the solution should match to a solution of the linearizedform of (2.4). We show, in Appendix A , tha t the unique stable steady solution of thislinearized equatio n is the solution (1.7). Hence in the oute rmost regions, the principalaxes of the iso-vorticity ellipses must r ota te back tow ards th e principal axes of strain(0x2O y ) .The second difficulty arises specifically for h > 1. Far from the vortex core, thevorticity is exponentially small; moreover the imposed strain field (ax,y , yz )dominates (at sufficient distance) over the velocity vo(r)- /2nr associated with thevortex. I n this far field, the vorticity behaves like a passive scalar an d when ,!? > 0, itis convected t o y = f O (the linearized solution (1.7) is no t available when ,!? > 0). Thestrain dom inates over vo(r )a t a (dimensional) distance

r - % (5.1)from the vortex, and at distances greater than this a strictly steady solution with,!? > 0 is apparently not possible.However, the vorticity is transcendentally small in this reeion so that the rate ofreduction of the circulation T ( r ) nside a circle of radius r - -pS due to this strippingmechanism? is likewise transcendentally small. In order of magnitude, when(r /6) - e-,say, so that the timescale t , of this process is

which is effectively infinite when E 4 . Thus, although in a strict sense equa tion (2.4)has no solution vanishing a t infinity when ,!? > 0, the so lution th at we have obtainedin $ 2 is quasi-steady when /3 > 0, with a rate of loss of circulation (by stripping) thatis so small as to be effectively negligible. Ou r conclusion is that the stretched vo rtex ca nsurvive for a time that is effectively infinite even when h > 1 (i.e. ,!? > 0).A fortiori, the same remarks apply to the plane strain situation h = 1 (i.e. ,!?= 0). Inthis case, vorticity can be lost to y = f O by viscous diffusion which (a t large distancesfrom the core) is not opposed by inward convection. The rate of loss of vorticity islikely to be even smaller than estimated above fo r the case ,!? > 0, provided always th at 4 .t The mechanism should not be confused with the inviscid stripping mechanism that ca n act o nelliptical vortices in two-dimensional s training flow (Dritschel 1989, 1990; Legras & Dritschel 1993).


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260 H . K . M o f a t t , S. Kida and K . Ohkitani6. Discussion

In this paper we have obtained an asymptotic solution of the nonlinear equationdescribing a vortex of finite circulation r subject to a non-axisymmetric strain field.This solution is applicable in the limit R, = r/u+ cc, nd is valid for all values of theparameter h = ( a- ) / (a+p) representing the departure from axisymmetry in thestrain field.F o r 0 < h < 1, the strain field is axial, with a < p < 0. Previous numerical evidence(Robinson & Saffman 1984; K 0 9 2 ) indicated the existence and structure of solutionsof the governing equation (2.4) for particular values of the param eters R, and A . Thegreat adv antage of the asym ptotic solution is that it is valid for a rbitr ary values of theseparameters provided eh 4 1 where e = l / R p M oreov er, the solution may be analysedin detail, to explain the rather complex structure of the field of viscous dissipationrevealed by the com puted solution of K 0 9 2 , an d the manner in which this changes asR, increases. The dissipation has been fo und t o be located predominantly in the regionof low enstrophy near the radius r / 6 = 2.67, and the correlation between dissipationand enstrophy fields has been found to be 0.19 +O ( 2 ) for e 4 1.Th e case h = 1 (p= 0 orplane strain) is of particular interest, as noted by N eu (1984)who developed a theory of unsteady vortex dev elopment for this case, with a view toexplaining vortical structures (e.g. braids) in turbulent mixing layers (Lin & Corcos1984). T he theory of the present paper is clearly applicable when h = 1, an d providesdetailed steady solutions for the persistent stretched braid-type vortices that are sucha characteristic feature of free-shear-layer flows.

The case h > 1 (p> 0 or biaxial strain) is also of great interest. A weak vortexaligned with the z-axis cann ot possibly su rvive in the presence of such a strain field: thelinearized solution (1.7) is available only when bot h a and ,8 are negative. H owever, astrong enough vortex may well survive because its rapid rotation always tends tore-establish the circular cross-section, thus thwarting the disruptive tendency of thepositive strain rate p, although as explained in $ 5 an exceedingly slow decay ofcirculation is inevitable. With this reservation, the asymptotic solution that we haveobtained is equally valid in the range 1 < h < 3 (corresponding to 0 > p < y ) . Thisma y help to explain why vortices, once formed, ar e remarkably p ersistent ; even if theymove into a n environment in which ,8 > 0, they can continu e to survive with a m inoradjustment of structure, provided simply that e1 = h/R, remains very small.Even if we move into the regime h > 3 (i.e. /3 > y), the asym ptotic solution, with thevortex still aligned along the y-axis of strain, apparently still exits. However, if thevortex is disturbed away from this axis of strain, it will tend t o rota te tow ards the largerc(3) axis of strain so that we revert to a situation 1 > A' > 3 with a revised ordering of(a,p,y) and A' = (a -y ) / (a+y ) . We speculate that the vortex can still persist, butrotated throug h n /2 from its initial direction. This argumen t suggests moreover th atif a,p and y vary continuously and not too rapidly with time, then the vortex mustalways tend t o line up with a n axis of positive rate of strain, this being the only stableconfigu ration; only in exceptional circumstances therefore could it find itself alignedwith an axis of negative rate of strain (which could lead to its extinction).Th us it appe ars tha t, if an d when strong vortices form thro ugh the well-understoodvortex stretching mechanism, they have a goo d chance of surviving for a long time. Anetwork of such sinews connecting a nd m erging with regions of mu ch weaker vorticitytherefore provides an attractive starting point for the understanding of the fine-scalestructure of turbulence. Further DNS investigations varying the threshold level ofvorticity as well as rate of strain (see Tanaka & Kida 1993) for detection of vortex


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Stretched vortices - he sinews of turbulence 26 1structures is desirable to identify any competing structures that may be responsible forthe balance of viscous dissipation.

This work was carried out while H. K . M. held a visiting appointment at theResearch Institute for Mathematical Sciences (RIMS), Kyoto University. The kindhospitality of all at RIMS is gratefully acknowledged.Appendix A. The far fieldthe strain field (ax,/3y),nd we may therefore consider the linearized form of (2.6),As noted in 0 , fo r r % sdS, the induced velocity (U%, y ) s negligible compared with

which must hold asymptotically, irrespective of the value of e (= l/&). We havealready noted the solution

when a < 0 ,P < 0 (i.e. (1.7) in dimensionless form). We now show that this is theunique solution of (A 1) of finite circulation, a nd tha t n o such solution exists if /3 > 0.Consider the initial-value problem for w(x, ; ),ao aw aw-+ax-++y- = w + v 2 w ,a t ax ay

W(X,Y; ) = w0(x,), w,(x, )dx dy = 1. (A 4)ssithEquation (A 3) adm its Kelvin mo de solutions (familiar from rapid d istortion theory)a Of the form

w(x, ; ) = w(t) exp [ik (t) XI, (A 5 )whereandi.e. (eWzat ) +-O e-zat- 1)2PThe solution of the problem (A 3), (A 4) is given by a superposition of such Kelvinmodes. Let

O,(X,Y)= ~ ~ ~ ~ ~ ~ o ~ , ~ 0 2 ~ ~ ~ ~ a ~ x ~ ~ o lA 9)The condition of finite circulation ensures that this Fou rier tran sform exists; and (A 4)implies tha t dO(O,) = (2n)-2. (A 10)Noting that


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262the required solution of (A 3), (A 4) is given by

H . K . MofSatt, S. Kida an d K. Ohkitani

w ( x , y ; ) = do( ko l, ,,)exp 1)+-(eC2Pt-:2 1)Is 2P= ss&.uleat,k, ePt) ex p (1 -east)+q(1- 2Pi)]e ik .xd k , k,. (A 12)

I f c r < O a n d / ? < O , th en as t + c o ,w ( x , y ; ) - ~ d , ( 0 , 0 ) e x p [ ~ + ~ ] e z k ~ x d k , d k 22 k2 .

= exp [ ; (Ex2 +Py2)>I27cusing (A 10). Hence (A 2) is a global attractor for solutions of (A 3) and is thereforethe unique stable steady solution in this case. (There may conceivably exist unstablesteady solutions that cannot be determined by this method.)If /3 > 0, then the form of the exponential factor in (A 12) makes it clear that thereis no stable steady solution (with finite non-zero circulation).Appendix B.Determination of $02(r )from solvability condition at 0 ( e 3 )

Consideration of terms of order e3 in (2.7) gives the equa tion

Here, $1, w1 and $, are given from (2.32), (2.33), (4.5) by$1 = h f ( r ) in 28, w1 = hSZ(r) in 28, (B 2)

$, = hz$02(r) hf2(r) os 28 + h2f,(r)cos 48, (B 3)w2 = h 2 9 , , ( r )+hSZ,(r)cos 28 +h2SZ4(r)os 48, (B 4)

where QO2W - -Yr$h2)', (B 5 )Q 2 @ ) = -L,f2, Q , ( r ) = -U. (B 6)

and we have then also

Now the left-hand side of (B 1) (cf. (2.17), (4.1)) may be written as1 8r a8 (B 7)- - ( U ; $3 + w3) .

Each term on the right-hand side of (B 1) may be calculated explicitly. Th e solvabilitycondition is obtained by integrating the equation from 8 = 0 to 8 = 2n. The left-hand side integrates to zero, and on the right-hand side we get contributions from- , SZO2(r) and also from terms involving the factor cos228 or sin228. The resultingequation for SZ,, is L, SZo2 = - - l Y ( r ) , (B 8)where S(r ) = +r2SZ2 f 2 Z-fa,. (B 9)


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Stretched vortices - he sinews of turbulence 263The right-hand side of (B 8) is O ( r 2 ) s r + O , an d exponentially small as r + CO . Thefunction Go, inite a t r = 0, exponen tially small as r + CO , an d satisfying the norm alizingcondition

1; rGo2(r ) r = 0 (B 10)is then given byThen, $02(r) is given from (B 5 ) by

J o J oWe see therefore that there is a small modification to the &averaged structure of thevortex at O(s2h2) .

R E F E R E N C E SBATCHELOR,. K. 1953 Homogeneous Turbulence. Cambridge University Press.BATCHELOR,. K . 1956 O n steady laminar flow with closed streamlines at large Reynolds number.J. Fluid M ech. 1, 177-190.BUNTINE,. D . & PU LL IN , . I. 1989 Merger an d cancellation of strained vortices. J . Fluid M ech.DOUADY,. , COUDER, .& BRA CHE T, . E. 1991 D irect observation of the intermittency of intenseDRITSCHEL, . G. 1989 Strain induced vortex stripping. In Mathematical Aspects of VortexDRITSCHEL,. G. 1990 The stability of elliptical vortices in an extern al straining flow. J . FluidMech.HOSOKAWA,. & YAMAMOTO,. 1989 F ine structure of a directly simulated isotropic turbulence. J .HOSOKAWA,. & YAMAMOTO,. 1990 Intermittency of dissipation in directly simulated fullyJIMENEZ,. 1992 Kinematic alignment effects in turbulent flows. Phys. Fluids A 4, 652-654.J I ~ N E Z ,., W RA Y, . A., SAFFMAN,. G. & ROGALLO, . S. 1993 The structure of intense vorticityKERR,R . M . 1985 Higher-order derivative correlation and the alignment of small-scale structures

205, 263-295.vorticity filaments in turbulence. Phys. Rev. Let t. 67 , 983-986.Dynamics (ed. R. E. Caflisch), pp. 107-1 19. SIAM.210, 223-261.Phys. Soc. Japan 58, 2G23 .developed turbulence. J . Phys. Soc. Japan 59, 401-404.

in homo geneous isotropic turbulence. J . Fluid M ech. 255, 65-90.in isotropic turbulence. J. Fluid Mech. 153, 31-58.KIDA,S. 1993 Tube-like structures in turbulence. Lecture Notes in Numerical Applied Analysis 12,137-159.

KIDA,S. & OHKITANI,. 1992 Spatio-temporal intermittency a nd instability of a forced turbulence.Phys . Fluids A 4, 1018-1027 (referred to herein as K 09 2) .KUCHEMANN,. 1965 Report on the IUT A M Symposium on concentrated vortex motions in fluids.J. Fluid M ech. 21 , 1-20.LEGRAS,. & DRITSCHEL,. G. 1993 Vortex stripping and the g eneration of high vorticity gradientsin two-dimensional flows. Appl. Sci. Res. 51, 445-455.LIN,S. J. & CORCOS, . 1984 The m ixing layer: deterministic models of a turbulent flow. Par t 3. Theeffect of plane strain on the dynamics of streamwise vortices. J. Fluid Mec h. 141, 139-178.LINARDATOS,. 1993 Determination of two-dimensional magnetostatic equilibria and analogousEuler flows. J. Fluid Mech. 246, 569-591.LUNDGREN,. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25 ,MO FFATT , . K . 1984 Simple topological aspects of turbulent vorticity dynamics. In Turbulence and

2 193-2203.Chaotic Phenomena in Fluids (ed. T. Tatsumi) , pp . 223-230. Elsevier.


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264MOFFATT, . K. 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complextopology. Pa rt 1 . Fundam entals. J. Fluid M ech. 159, 359-378.MOFFATT,H . K . 1993 Spiral structures in turbulent flow. In Fractals, Wavelets and FourierTransforms: New Developments and New Applications (ed. M . Farge, J . C . R. Hunt & J . C.Vassilicos), pp. 317-324. Cla ren don .

H . K . Moffat t , S. Kida and K. Ohkitani

N E U ,J. C . 1984 The dynamics of stretched vortices. J. Fluid Mech . 143, 253-276.RHINES, . B. & YOUNG,W. R . 1982 H omo genization of potential vorticity in planetary gyres. J .Fluid Mech. 122 , 347-367.ROBINSON,. C. & SAFFMAN, . G. 1984 Stability and structure of stretched vortices. Stud. Appl.Maths. 70, 163-181.RUETSCH, . R. & MAXEY,M . R . 1991 Small scale features of vorticity and passive scalar fields inhomogeneous isotropic turbulence. Phys. Fluids A 3, 1587-1 597.SAFFMAN,. G. 1992 Vortex Dynamics. Cambridge University Press.SAFFMAN,. G. & BAKER,G. R. 1979 Vortex interactions. Ann. Rev. Fluid Mech. 11, 95-122.SHE, Z.-S., JACKSON, . & ORSZAG, . A. 1990 Intermittent vortex structures in homogeneousSIGGIA, . D. 1981 Nu me rical study of small scale intermittency in three-dimensional turbulence. J .T A N A K A ,M. & KIDA, S . 1993 Characterisation of vortex tubes and sheets. Phys. Fluids A 5 ,TENN EKES, . 1968 Simple model fo r the sm all-scale structure of turbulence. Phys. Fluids. 11,TOWNSEND,. A . 1951 O n the fine scale structure of turbulence. Proc. R. Soc. Lond. A 208 ,536542 .VINCENT, . & ME NEGU ZZI, . 1991 T he spatial structure and statistical properties of homogeneousYAMAMOTO,. & HOSOKAWA,. 1988 A decaying isotropic turbulence pursued by the spectral

isotropic turbulence. Nature 344, 226-228.Fluid Mech. 107, 375-406.2079-2082.669-671.

turbulence. J. Fluid M ech. 225, 1-25.method. J. Ph ys. Soc. Japan 57, 1532-1535.


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J . Fluid Mech. (1994), vol. 266, p . 371Copyright 0 1994 Cambridge University Press

CORRIGENDUMStretched vortices - he sinews of turbulence;

large-Reynolds-number asymptoticsBY H. K . MOFFATT,. KIDA n d K. OHKITANIJournal of Fluid Mechanics, vol. 259 (1994), pp. 241-264

Figure 7 (page 254) as printed was so dark that some of the contours and the starscould not be seen. A clearer version is reproduced below.

L -+

I " ' " ' ' I " 'I-5

0

-5

-5 0 5FIGURE(corrected). Co ntou r plots of the dissipation function D( r ,0) defined by (3.32). The maximaof D are marked with * an d the (global) minima with x .The con tou r levels are equally spaced at one-seventh of the difference DmaZ-Dmzn.he separatrices D = D o and D = D, re also included (thicklines), and the plots a re shaded light grey where min ( D o ,D,) D < max ( D o ,D,) nd d ark grey where

H.K. Moffatt, S.Kida and K. Ohkitani- Stretched vortices - the sinews of turbulence;...

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