H.K. Moffatt and B.R. Duffy Local similarity solutions and their limitations
H.K. Moffatt Formation and disruption of concentrated vortices in turbulence

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Transcript of H.K. Moffatt Formation and disruption of concentrated vortices in turbulence

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FORMATION AND DISRUPTION OF CONCENTRATEDVORTICES IN TURBULENCE
H.K. MOFFATTIsaac Newton Institute for Mathematical Sciences20 Clarkson RoadCambridge CB3 OEH, U K
T h e Burgers vortex (Burgers 1948) is described by a n exact solution ofthe NavierStokes equations in which the effects of uniform axisymmetricstretching are in equilibrium with viscous diffusion. Burgers introducedthis vortex as a math em atica l model i l lus trat ing the theory of turbulence ,an d he noted par t icular ly th at the vortex h ad the property th at the ra te ofviscous dissipation p er u nit leng th of vortex w as indepe nde nt of viscosity inth e lim it of vanishing viscosity (i.e. high Reyn olds nu m be r). T hi s is of coursea n a ttr ac tiv e fea tur e in th e light of Kolmogorovs theory of turbulence , inwhich the rate of dissipation of energy per unit volume E an d the kinematicviscosity U are regarded as independent variables.T h e idea th at the smallscale s tru cture s of turbulence might be representable in term s of a rand om dis tr ib ution of vortex sheets or tub es wastaken u p by Tow nsend (1951). Townsend showed tha t a ran do m dis tribut ion of vortex sheets would give rise to an energy spectrum proportionalto k 2 (multiplied by an exponential viscous cutoff factor) this powerlawreflecting th e fact t h a t o n any straight l ine through th e field of turbulence,intersecting a finite number of vortex sheets, there will be (in the limit ofvanishing viscosiby) a finite number of discontinuities of velocity per unitleng th. A rand om dis tr ib ution of vortex tube s gave rise to a powerlaw k(aga in m odified by a n ex pon entia l cuto ff), th is slower falloff w ith k beingassociated with the more singular behaviour in physical space associatedwi th a l ine vo rtex.T h e Kolmogorov sp ect rum k  5 / 3 lies tantalisingly betwee n k  l a n d k  2 ,suggesting t h a t th e typical (or generic) s truc tures in xspace which may beresponsible for such a s pe ctru m should involve some com promise betweentubes and sheets , for example spiral s tructures (Lundgren 1982, Gilbert1988 ), these possibly arising thro ug h th e interaction of tubes and sheets(K ras ny 1986, Moffatt 1993).
32 1U. Frisch (ed .),Advances in Turbulence VII, 321330.@ 1998 Kluwer Academic Publishers. Printed in the Netherlands.
www.moffatt.tc

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328 H.K. MOFFATTDuring the last fifteen years, evidence from direct numerical simulation
(DNS) of turbulence has accumulated indicating the presence of concentrated tubelike structures in the vorticity field (Siggia 1981, Kerr 1985,Yamamoto & Hosokawa 1988, Vincent & Meneguzzi 1991, and others).This has led to a great revival of interest in the primitive theories of Burgers and Townsend, and a reevaluation of possible models of turbulence interms of simple vortex structures.
It was noticed by Kida & Ohkitani (1992) that the dissipation structure in the concentrated vortices of 3D turbulence exhibit two maxima,offset from the centre of the vortex, and they suggested that this might beexplained in terms of the action of nonaxisymmetric strain acting on thevortex. This suggestion was taken up by Moffatt, Kida & Ohkitani (1994)who developed a high Reynolds number asymptotic theory of a vortex subjected to nonaxisymmetric strain. Determination of the dissipation structure involved pursuing the analysis to third order in the small parameterRe', and this analysis did indeed reveal the two peaks in the dissipation structure, arising from a symmetrybreaking splitting of the circle ofmaximum dissipation that occurs for the Burgers vortex. Remarkably, atleading order in the asymptotic analysis, the axisymmetric Burgers vortexemerges in spite of the nonaxisymmetric character of the strain. This isbecause, at high vortex Reynolds number, the vortex spins rapidly in thestrain field, and experiences the &averaged strain, which is axisymmetric.A similar behaviour had been previously recognised by Ting & Tung (1965),and by Neu (1984). Even more remarkably, the solution of Moffatt, Kida& Ohkitani indicates that the vortex can survive for an exponentially longtime even when one of the rates of strain in the plane of crosssection ofthe vortex is positive. Again, this is because it is only the &averaged strainthat is relevant at leading order.
The characteristic dissipation structures identified by Moffatt, Kida &Ohkitani are present also, although for rather different reasons, in twodimensional freely decaying turbulence, at the stage when identifiable vortices emerge from a random initial s ta te (McWilliams 1984, 1990, Jimknez,Moffatt & Vasco 1996). Each vortex in such a field moves with the local velocity induced by all the other vortices, and is also subject to thetwodimensional strain field associated with the presence of all the othervortices. At high Reynolds number, the effect of this strain field is to distorteach vortex crosssection to slightly elliptical form; the associated dissipation field has precisely the same structure as that determined in the earlierwork of Moffatt, Kida & Ohkitani. This remarkable result is a consequenceof the analogy between steady stretched threedimensional vortices andunsteady unstretched twodimensional vortices, as described by Lundgren(1982).

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FORMATION AND D ISRUPTION O F VORTICES IN TURBULENCE 329The elliptic deformation of vortices in both two and threedimensional
turbulence makes them prone to the type of threedimensional resonantinstability identified by Bayly (1986) and Pierrehumbert (1986). As shown,however, by Le Dizks, Rossi & Moffatt (1996), stretching carries the wavenumber of sinusoidal perturbations through the unstable wavenumber bandin a finite time, so that infinitesimal disturbances are always asymptoticallystable. This mechanism is not present for twodimensional (unstretched)vortices, and one may reasonably conjecture that 2D turbulence is alwaysunstable to 3D disturbances (in the absence of stabilising mechanisms suchas stratification or magnetic field).
A serious limitation of the Burgers model in the context of threedimensional turbulence lies in the assumption of the uniformity of the strainfield, and the associated infinite length of the stretched vortices. In fact, theregion of concentration is of finite length, the vortex lines diverging more orless rapidly at the ends of these regions of concentration. This finite lengthis apparent also in experiments (Douady, Couder & Brachet 1991) designedto detect intense vortex filaments in turbulent flow of liquids seeded withsmall gas bubbles.
Variation of the strain field arises through the nonuniform action ofthe other vortices near to the parent vortex whose structure is considered.In so far as these other vortices may be treated as point vortices, the nonuniform strain field acting on the parent vortex is a strain field associatedwith a nonuniform potential flow. In this lecture, a simple model will bedeveloped involving the action of distributed vortices on a twodimensionalstretched vortex sheet (of Burgers type). The problem is treated by usingthe potential 6 nd stream function ?J of the nonuniform straining flow asindependent coordinates. The advectiondiffusion equation for the parentvortex sheet has universal form in terms of these coordinates, and a widefamily of exact solutions of the NavierStokes equations is thus generated.A variety of solutions will be described, which provide a good indication ofthe manner in which such vortex sheets may disrupt in regions of strongnonuniformity of strain.
A similar technique runs into difficulties for the analogous axisymmetricproblem, because in this case the advectiondiffusion equation, expressed interms of the relevant 4 and $J,is not universal. Nevertheless, the approachdoes indicate one mechanism by which vortex disruption can occur.ReferencesBayly, B. (1986) Threedimensional instability of elliptical flow, Phys. Rev . Lett . 5 7 ,Burgers, J . M . (1948) A mathematical model illustrating the theory of turbulence, A d v .p. 2160.A p p l . Mech. 1,pp. 171199.

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330 H.K. MOFFATTDouady, S . , Couder, Y. & Brachet, M.E. (1991) Direct observation of the intermittencyGilbert, A.D.1988) Spiral structures and spectra in twodimensional turbulence, J. FluidJimCnez, J . , Moffatt, H.K. and Vasco, C. (1996) The structure of the vortices in freelyKerr, R.M. (1985) Higherorder derivative correlations and the alignment of smallscaleKida, S . and Ohkitani, K. (1992) Spatiotemporal intermittency and instability of a forcedKrasny, R . (1986) Desingularisation of periodic vortex sheet rollup, J . Computa t iona lLe DizBs, S . , Rossi, M. & Moffatt, H.K. (1996) On the threedimensional instability ofLundgren, T. (1982) Strained spiral vortex model for turbulent fine structure, P h y s .McWilliams, J. C. (1984) The emergence of isolated coherent vortices in turbulent flow,McWilliams, J . C . (1990) The vortices of twodimensional turbulence, J . Flu id Mech. 219,Moffatt, H.K. (1993) Spiral structures in turbulent flow, New Approaches and Concepts
in Turbulence , Mo nte Veritd, Birkhauser Verlag Basel, pp. 121129.Moffatt, H.K. , Kida, S . and Ohkitani, K. (1994) Stretched vortices  the sinews of turbulence; largeReynoldsnumber asymptotics, J . Fluid Mech. 259, p. 241264.Neu, J.C. (1984) The dynamics of stretched vortices, J. Fluid Mech. 143, p. 253276.Pierrehumbert, R .T . (1986) Universal shortwave instability of twodimensional eddies inSiggia, E.D. (1981) Numerical study of small scale intermittency in threedimensionalTing, L. and Tung, C. (1965) Motion and decay of a vortex in a nonuniform stream,Townsend, A.A. 1951) On the fine scale structure of turbulence, Proc. R. Soc. Lond. AVincent, A. nd Meneguzzi, M. (1991) The spatial structure and statistical properties ofYamamoto, K. and Hosokawa, I. (1988) A decaying isotropic turbulence pursued by the
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an inviscid fluid, Phys . Rev. et t . 57, . 2157.turbulence, J. Fluid Mech. 107, p. 375406.Phys. Flu ids 8, pp. 10391051.208, p. 534542.homogeneous turbulence, J . Fluid Mech. 225, pp. 125.spectral method, J . Phys. Soc . Japan 57, p. 15321535.