H.K. Moffatt- Spiral structures in turbulent flow

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  • 8/3/2019 H.K. Moffatt- Spiral structures in turbulent flow


    New Approaches and Concepts in lhrbulence, Monte Verith,0 Birkhauser Verlag Base1 121

    Spiral structures in turbulent flowB y H . K . Moffatt

    Department of Applied Mathematics and Theoretical Physics,Silver Street, Cam bridge, CB3 9EW.1 IntroductionSpiral structures are natural candidates for the role of the generic struc-tures of turb ulen t flow, because they are the eventual o utco m e of Kelvin-Helmholtz instability, an all-pervasive phenomenon associated with nearlyall shear flows at very high Reynolds number. Such structures were pro-posed by Lundgren (1982) in a model of the f ine structure of turbulencein which axial stretching of rolled-up spiral vortices played an essentialrole. Th is model could be viewed as a na tur al development of Townsends(1951) mod el of th e dissipative structu res of turbulence in terms of a ran-dom d istr ibution of vortex sheets an d/o r tubes, each such stru ctur e beingsubjected to the local rate of strain (assumed uniform and constant) as-sociated with all other vortex structures (see Batchelor 1982, $7.4). I t isknown t h a t compressive stra in (with two positive principle rates of stra in)is more likely in isotropic turbulence than extensive strain, so tha t sheetsform with higher probabili ty tha n tub es. However these are im m ediatelysubject to the Kelvin-Helmholtz instability which may b e imp ede d, bu t notentirely suppressed, by the stretching process.A rand om superposit ion of vortex sheets has th e property th at the re-sulting velocity field has a finite number of discontinuities per unit lengthon any straight l ine transversal. Th is leads to an energy s pec trum functionE ( k ) proport ional to k - 2 (with an exponential cut-off at wave-numbersof th e order of th e reciprocal of th e typical vortex sheet thickness). T h eKolmogorov cascade theory, by con trast, gives E ( k ) proport ional to k - 5 / 3(with a n exponential cut-off a t the inner Kolmogorov scale). It is natu ralthen to enquire as to w hat structures in x-space can give rise t o such apower-law. Th is question was raised by Moffatt (1984) where it was ar-gued th a t th e aCcumulatiQn po ints of discontinuities associated w ith spira lstructures could give rise to fractional power laws k - x with 1 < A < 2. Weshall pursue this argu me nt a little further in the present note.structures in two-dimensional turbulence hasbeen considered b (1988) who considered the wind-up of a weakThe inflhence


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    vortex patch by a strong concentrated vortex in its vicinity, and showedthat fractional power laws E ( k )- - with 3 < A < 4 could be obtainedthrough this mechanism. This provides a convenient bridge (and reconcilia-tion) between the vortex patch theory of Saffman (1971) and the enstrophycascade theory of Kraichnan (1967) and Batchelor (1969) which give L-4and k - 3 respectively. The methods tha t we adopt in the present paper aresimilar to those employed by Gilbert.

    2 The Kelvin-Helmholtz instabilityA vortex sheet, i.e. a tangential discontinuity (of magnitude U) in tangen-tial velocity, is well known to be unstable, on linear analysis, to perturba-tions of wavenumber k, he growth rate being of order k U . The fact tha tthe shortest wavelengths grow most rapidly means that the linear problemis ill-posed, and, as shown by Moore (1979), the nonlinear development ofan initially sinusoidal perturbation develops a mild singularity at a finitetime t , of order (LU)-,argely as a result of the singular character ofthe dispersion relation. The mechanism of the instability, as described byBatchelor (1967), is that the strength of the vortex sheet increases nearthe inflexion points where the surface is tilted in the sense favoured by thevelocity difference; this tendency persists in the nonlinear regime, and attime t = t,, the strength of the vortex sheet is cusp-shaped at the inflexionpoint (see figure 1).

    Numerical procedures based on the Euler equations cannot be continuedbeyond this finite time singularity. If however the equations governing theevolution of the sheet are modified slightly, in a manner that mimics the factthat any vortex sheet in a real fluid would not be a perfect discontinuity,but would have finite thickness, then the integration can be pursued beyondt = t,. Such a desingularisation procedure has been adopted by Krasny(1986) who integrated the equations for vortex sheet evolution for varioussmall values of a desingularisation parameter 6. When 6 = 0, the evolutionbased on the Euler equations is recovered. Figure 2 shows a reproductionof Krasnys solution for the vortex sheet, for 6 = 0.05, and for t = 8tJ3. Atthis stage, five complete turns have appeared in the rolled-up vortex sheet.As 6 increases, the number of turns (at fixed t ) decreases; conversely, itmay be supposed that, as 6 decreases towards 0, the number of turns atfixed t (> t , ) increases without limit.

    A straight line transversal passing through the centre of this spiral nowsamples an accumulation point of discontinuities of the velocity field. It isthis property that can lead to a fractional power law in the energy spec-trum. A random distribution of spiral vortices of this kind can therefore, inprinciple, yield a Kolmogorov spectrum (or any other power law spectrum)if the structure is right. We investigate this by an even more simplified

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    Figure 1. The mechanism of Kelvin-Helmholtz instability. (a) The unper-turbed state; (b) the instability mechanism associated with accumulationof vorticity at the inflexion point; (c) vortex sheet strength, showing thecusp at the inflexion point at the instant t = t , when the curvature becomesdiscontinuous (Moore 1979); (d) development of the spiral singularity fort > t , .model in the following section.

    3 A simple modelConsider a set of concentric circular vortex sheets of radius x, = rm-a(m = 1 , 2 , 3 etc.). Let 62, = 2, - xm+l(- m-O+' for large m), andsuppose that the velocity in the annulus 6xm is um - mp - x m a . Theand the condition that the total energy be finite then implies that /? < a.The situation is shown schematically in figure 3.

    The spectrum of the velocity field may be calculated by the method

    - eenergy in the annulus 62 is then E, - Jz,+,.. &xmdxm - m2@-2a-1,

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    . 4 X . 6

    Figure 2. (From Krasny 1986) Structure of the spiral vortex computedby Krasny according to a desingularised prescription.

    developed by Gilbert (1988). The Fourier coefficients of U(.) are given bya, = Z l ' u ( z ) s i n n z c i zr (3.1)

    T he factor s in( $n6zrn+1) s effectively zero if r d a + l ) >> n; hence, providedUrn does not increase too rapidly with m,' the number of terms makingsignificant contributions to this sum is M = O(n*). T h e t e rm R, =sin fn (+m+l + 2 is effectively a random number in the interval [-1,1],and the sum can be thought of as the result of a rand om walk in which th estep-length increases like rnp (where m labels the step). It is not difficult

    J.C.Vassilicos (privatecomunication) has calculated that the conditionfor validityof this argument is p 5 $0.

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    Figure 3. (a) Simplified m odel involving concentric v ortex sheets; (b ) thecorrespo nding velocity profile on a transversal through the centre.to deduce tha t 2 2p+ 1

    CY+l'on - -', where X = 2 --If there is a random superposition of such structures, then a straightline transversal will no t pass thro ugh the centre of all of the m , but rathe rwill pass at dis tance y from the centre where all values of y between -1a n d +1 are sampled with eq ua l probability. Clearly, when y is small butnon-zero, th e infinite s u m in th e above equation is cut-off at a large butfinite value of m. T h e resulting value of a: can then be averaged over allvalues of y, giving the modified result

    Q(ai(y)) - -" where p = X +-1 + Q 'We may note here in parenthesis the special values ,O = 0,cr = 3, fo rwhich X = $ , p = 3. These values were relevant in the work of Gilbert(1988),who considered however not the wind-up of a vortex sheet, butthe wind-up of a discontinuity of vor t i c i f y (rather than of velocity) by

    a conce ntrated point-vortex. Consequently, in th e simp lest configurationconsidered, Gilb ert found a n enstrophy spectru m proportional to k-z, anda corresponding energy spectrum proportional to k- S .


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    1- 5Xm



    m1Figure 4. Plot of t m T against m for the Krasny vortex showing a linearrange for 1 < rn < 8. This implies that the spiral can be representedreasonably well by the curve r (0 - 0)2 = cst., and that the circulating

    velocity behaves like r - 2 , except in the very central region.1

    It is interesting to re-examine the structure of the Krasny vortex inrelation to this analysis. Figure 4 shows a plot of zmT, gainst m, wherern labels the intersections of the spiral with the z-axis, as before. Thereis a good straight-line relationship for 1 < rn < 8, consistent with fittingto a spiral of the form r (0 - 6 0 ) ~ cst.. The points for m = 9,10,11,depart from this straight-line behaviour, and this is presumably due to thefact that the vortex does not achieve an asymptotic structure in the centralregion (or perhaps the desingularisation procedure is at its worst in thisregion).

    With (Y = 2, as suggested by this result, a Kolmogorov exponent p =is obtained for the choice ,i3 = 1. For this choice, U, N x m 2 , andcorrespondingly the circulating velocity associated with the spiral is u(r) -

    r'T. This slower fall-off than the velocity (- r'l) associated with a point-



    'R. Krssny (privatecommunication)confirms this inference in further computationswith desingularisationparameter 6 = 0.03 and a spiral with 21 turns.

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    vortex is of course related to the more diffuse distribution of vorticity inthe spiral region. The value p = 1 means that the jump in velocity is thesame across each discontinuity, i.e. that the strength of the vortex sheet iseffectively constant except perhaps in the very central region. The evidenceavailable from Krasnys computations is not conclusive on this point, butis certainly not inconsistent with this conclusion.4 DiscussionVisual observations of turbulent flow, and equally, direct numerical simu-lations (see for example Kerr 1985), provide ample evidence for the exis-tence of vortex tube structures, frequently exhibiting a spiral cross-sectionalstructure. The suggestion of this paper, which is a development of the ar-gument advanced in Moffatt (1984, 1990), is that a random distributionof vortices of this kind is all that is needed to provide an inertial rangespectrum proportional to k-5. There is certainly no need to consider ahierarchy of structures on different length-scales, with a cascade of energyfrom one scale to the next. A single generic structure, which evolves in timedue to its internal dynamics, and possibly (as in Lundgrens 1982 model)through additional straining associated with other structures, is sufficientto provide an inertial range spectrum of Kolmogorov type, although forvery different reasons from those first advanced by Kolmogorov. This doesnot of course mean that other physical processes (e.g. vortex break-down)may not be important also in turbulent flow. But it does mean that thesearch for characteristic vortical structures is well worthwhile, not only toprovide a better understanding of the dynamics of the energy-containingeddies, but also in understanding the process of transfer of energy to thesmall scales where viscous dissipation is operative.


    5 AcknowledgmentsI am grateful to Christos Vassilicos and Andrew Gilbert, who have madesignificant input in recent discussions of the problem of analysing spiralstructure.

    This paper was originally presented at the Conference Wavelets, Frac-tals and Fourier Transforms of the Institute of Mathematics and its Ap-plications in December 1990, and is published with the agreement of theIMA.6 AppendixEverson and Sreenivasan (1992) have recently investigated spiral structuresthat develop in the field of a passive scalar contaminant, and have found

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    that these spirals are better modelled by a logarithmic spiral rather than byan algebraic spiral of the type discussed above. The passive scalar problemis however rather different from the spiral wind-up associated with Kelvin-Helmholtz instability, for which the nonlinear dynamics associated with theEuler equation plays an essential role.An important dynamical mechanism for the formation of vortex sheetshas recently been identified by Pumir and Siggia (1992a,b). With anyswirling flow, there is an associated unbalanced centrifugal force, whichaccelerates the flow outwards, and which intensifies the vorticity associatedwith the swirl in the accelerating region of the flow. Pumir and Siggia havesuggested that shear instabilitiesof the developing front may lead, withinthe framework of the Euler equations, to a finite time singularity. Whetherthis is true or not, the mechanism is undoubtedly a potent one for thegeneration of vortex sheets and spiral structures, and it would seem to be anatural process in any region of the flow where the streamlines are curved(and that of course means virtually everywhere).7 References

    1 . Batchelor, G.K. (1967). A n Introduction to Fluid Dynamics. Cam-bridge University Press.

    2. Batchelor, G.K. (1969). Computation of the energy spectrum in ho-mogeneous two-dimensional turbulence. Phys. Fluids Suppl. 11, 233.3. Batchelor, G.K. (1982). The Theory of Homogeneous Turbulence

    (Cambridge Science Classics, CUP).4. Everson, R.M. & Sreenivasan, K.R. 1992). Accumulation rates ofspiral-like structures in fluid flows. Proc. Roy. Soc. Lond. A 437,39 1-40 1.5 . Gilbert, A.D . (1988) . Spiral structures and spectra in two-dimensionalturbulence, J. Fluid Mech. 193, 75-497.6. Kerr, R.M.1985). Higher-order derivative correlations and the align-

    ment of small-scale structures in isotropic numerical turbulence. J .Fluid Mech. 153, 1-58.7 . Kraichnan, R.H. (1967). Inertial ranges in two-dimensional turbu-

    lence. Phys. Fluids 10, 417-1423.8. Krasny, R. (1986). Desingularisation of periodic vortex sheet roll-up,J . Computational Physics 65, 292-313.9 . Lundgren, T.S. (1982). Strained spiral vortex model for turbulent

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    10. Moffatt, H.K. (1984). Simple topological aspects of turbulent vortic-ity dynamics. In Turbulence and Chaotic Phenomena in Fluids, ed.T. Tatsumi (Elsevier) 223-230.

    11. Moffatt, H.K. (1990). Fixed points of turbulent dynamical systemsand suppression of nonlinearity. In Whither Turbulence? - Turbu-lence at the Crossroads, ed. J.M. Lumley. Lecture Note s in Ph ysics357 (Springer-Verlag) 250-257.

    12. Moore, D.W. (1979). The spontaneous appearance of a singularity inthe shape of an evolving vortex sheet. Proc. Roy. Soc. A365, 105.

    13. Pumir, A. & Siggia, E. (1992a). Blow up in axisymmetric Euler flows.In Topological Aspects of the Dynamics of Fluids and Plasmas, ed.H.K. Moffatt et al. Kluwer Acad. Publ. Series E: Applied Sciences218 293-302.

    14. Pumir, A. & Siggia, E. (199213). Development of singular solutions tothe axisymmetric Euler equations. Phys . of Fluids A 4, 1472-1491.15. Saffman, P.G. (1971). On the spectrum and decay of random two-

    dimensional vorticity distributions at large Reynolds number. Stud.,Appl . Maths 50, 377.16. Townsend, A.A. (1951). On the fine-scale structure of turbulence.Proc. Roy. Soc. A209, 418.