H. K. Moffatt-Topological Approach to Problems of Vortex Dynamics and Turbulence

8/3/2019 H. K. Moffatt-Topological Approach to Problems of Vortex Dynamics and Turbulence

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Current Trends inTurbulence Research

m E : 2 5 r a n overMichael MondYeshajshu UngerBen-Gurion University of the NegevBeer-Sheva, Israel

Volume 112PROG RESS INASTRONAUTICS AND AERONAUTICSMartin Summerfield, Series Editor-in-ChiefPrinceton Combustion Research Laboratories, Inc.M onm outh Junction, New Jersey

Technical papers from the Proceedings o f the Fifth Beer-Sheva International eminar on Magnetohydrodynamics Flow and Turbulence, Ben-Gurion Uni-versity of Negev, Beer-Sheva, Israel, March 2-6, 1987, and subsequentlyrevised for this volume.Published by the American Institute of Aeronautics and Astronautics, Inc.,370 LEnfant Promenade, SW, W a ngton, DC 20024-2518.h

www.moffatt.tc


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142 H. K. MOFFATT.eddies of the turbulence (for a recent review, see Ref. 1); it is necessaryto understand why these structures form and how they persist inthe presence of the random strong perturbations associated with thebackground turbulent velocity field U (x, ) .B. ntermittency of DissipationE = V ( L h i / a z j ) 2 , s intermittent in the following sense: if E , , is arbi-trarily small compared with the mean 3 , then the fraction V, /V of anyvolume V of fluid within which E > E , , tends to zero as the Reynoldsnumber Re tends to infinity. This means that as Re -+ 00, thenE(X, ) becomes increasingly spiky, being concentrated almost entirelyin this vanishingly small proportion V,, of the volume V .

The local rate of dissipation of energy per unit mass,

C. Inertial Range Spectrummensional grounds, an inertial range energy spectrum

The universal equilibrium theory of Kolmogorov predicts, on di-

for k , ), < w 2 >+ 00 as U + 0. The


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TOPOLOGICAL APPROACH TO VORTEX DYNAMICS 143equation for enstrophy4 is

and the production term < w ; w j a u i l a x j > is associated with stretch-ing of vortex lines as these are distorted by the flow. This stretchingof vortex lines is presumably associated with the tendency of lineelements, on average, to increase in length;6 but this type of argu-ment is, in itself, too simplistic (it does not apply, for example, to) two-dimensional turbulence, for which line elements parallel to thevorticity field do not increase in length); and what is needed is adynamical model in which the increase of enstrophy is (by some ex-plicitly three-dimensional mechanism) an essential and natural ingre-dient. Lundgrens model, referred to above, does incorporate vortexstretching, and it is interesting that this requirement appears to beintimately connected with the extraction of a Kolmogorov spectrum.lence6-* that provides a possibility of comprehending all four aspectsof the problem, as described above, within a single coherent frame-work. We visualize a turbulent velocity field u(x, ) as evolving in thefunction space of all solenoidal vector fields of finite energy density. Ast increases, the turbulent state is represented by a point that followsa trajectory in this function space. Of particular interest are the fixedpoints of the associated dynamical system, and (in the limit Re -+ m)these are the steady solutions of the Euler equations, or Euler flowsuE(x),which may have embedded tangential discontinuities (i.e., vor-tex sheets) or other singularities, provided these satisfy the constraintDof finite energy density (e.g., concentrated line vortices are excluded).

The existence of vortex sheets in nearly all Euler flows of anystructural complexity indicates that these flows will generally be un-stable to local instabilities of the Kelvin-Helmholtz type (e.g., Ref. 9,Chapter 7) i.e., the fixed points in the function space are generallyunstable; we should hardly expect otherwise because unsteadiness isan essential and inescapable feature of the turbulence problem. Thatfixed points are unstable does not, however, make them any less inter-esting from the point of view of understanding the global structure oftrajectories in the function space. For many low-order dynamical sys-tems having only unstable fixed points, location of these fixed pointsis nonetheless an essential preliminary to mapping out the strange at-tractor (or other limit set) to which trajectories asymptote. To take

In this paper, we review an approach to the problem of turbu-


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144 H. K.MOFFATTan even simpler example, a compound pendulum moving with enoughenergy for its center of gravity to describe complete circles in a verti-cal plane will spend more time in the neighbourhood of its position ofunstable equilibrium (vertically upward) than it spends near its posi-tion of stable equiliibrium (vertically downward); and a time averagewill reveal the characteristics of the unstable (rather than the stable)equilibrium! It would therefore be foolish to ignore the unstable fixedpoints, particularly when the energy level of the system is high.

11. The Helicitv InvariantIt is important for what follows that the Euler equations of in-viscid flow

a U 1- + u - v u = - - v pat P ' (4)with v - U = 0 and p = const (density) admit an integral invariantfor every volume V bounded by a surface S moving with the fluid onwhich w n = 0, namely,

X = l u . w d V ( 5 )This invariant, the helicity of the flow within V , s associated with thedegree of linkage, or knottedness, of the vortex lines in V andtherefore has topological significance: it provides a bridge betweenfluid mechanics and isotopy theory in topology.It is equally important that a similar invariant exists for anysolenoidal vector field that is frozen in the fluid; the prototype examplehere is the magnetic field B ( x , t ) in a perfectly conducting fluid, forwhich the invariant analogous to )I is the magnetic helicity l3

1

P

XM = L A - B d Vwhere B = v x A, nd B .n= 0 on S.The helicity satisfies the Schwarz inequality


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TOPOLOGICAL APPROACH TO VORTEX DYNAMICSMoreover, if the flow is confined to a fixed bounded domain D, thenwe have a Poincarb inequality,

145

where go depends only on the geometry of D.

and, by the same token, in the magnetic context,

provided B .n = 0 on aD. The significance of this lower bound onthe energy associated with a frozen-in field w as noted by Arnold12: ithas an important bearing on the question of existence of Euler flows(i.e., fixed points of the Euler system), as will now be described.

111. Fixed Points of the Euler EauationsRegarded as a Dvnamical System

The Euler equations (4 ) may be written in the equivalent form

where h = p / p + i u2 .Thus, Euler flows u(x) satisfy

There is a very well-known exact analogy between this equation andthe equation ofmagnetostatic equilibrium in a perfectly conductingfluid, namely,

j x B = V p (13)The great significance of this analogy is that it provides a meansof constructing a wide class of solutions of Eq. (12) of arbitrarily

complex topological structure. The details are given in Moffatt,' andonly the briefest outline will be given here. Let B,(x) be a solenoidalfield of arbitrarily complex structure and generally not satisfying thecondition of magnetostatic eq~i1ibr ium. l~uppose that this field is


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146 H. K.MOFFATT ,embedded in a viscous but perfectly conducting fluid (a mathematicalidealization, which is nonetheless appropriate to the purpose) and thatthe fluid is initially at rest. Obviously the Lorentz forcej xB will causethe fluid to move, with velocity v(x, ) , say, for t > 0, nd energy willbe dissipated by viscosity for as long as this motion continues. Thetotal energy therefore decreases monotonically, but is bounded belowby virtue of the inequality (10) associated with nontrivial (conserved)topology of the frozen-in field. Hence, ultimately, as t --t 00, v(x,t)must tend to zero everywhere, and

B(x,t) --t BE(.) (14)where BE(x) s a magnetostatic equilibrium field that is topologicallyaccessible from the initial field B,(x). The analogy between Eqs. (12)and (1 3 ) , and specifically between B in the magnetic problem and Uin the Euler flow problem, permits us to conclude that if U(x) is akinematically possible flow of arbitrarily complex structure, then thereexists at least one Euler flow uE(x) hat is topologically accessiblefrom U(x), n the sense that the streamlines of uE may be obtainedfrom the streamlines of U by a frozen-field distortion associated with asubsidiary velocity field v(x, ) (0 < t < 00) of finite total dissipation.

The process of magnetic relaxation (14) may, and in general does,involve the appearance of tangential discontinuities where magneticsurfaces are squeezed together by opposing Lorentz forces. The re-sulting magnetostatic equilibria therefore involve current sheets em-bedded within them but are nevertheless stable within the perfectlyconducting framework because the magnetic energy will obviouslybe minimal with respect to frozen-field perturbations about the stateBE(x).

The analogous Euler flows uE(x)have vortex sheets embeddedwithin them, which are presumably subject to the Kelvin-Helmholtzinstability. There is no conflict here with the analogy between Eqs.(12) and (13) because this analogy applies strictly to the equilibriumstates but not to questions of stability about these states.

IV. Structure and Stability of Euler FlowsThe argument in sec. I11 establishes the existence of a wide class

of fully three-dimensional solutions of the steady Euler equation (12)that are all topologically distinct if the reference fields U(x) fromwhich they are derived are topologically distinct. Each flow, nev-


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TOPOLOGICAL APPROACH TO VORTEX DYNAMICS 147

I

b CFig. 1 a) Typical structure of Euler flow; within the toroida D1 and D2,the streamlines are ergodic and the flow has maximal helicity; b) and c)Poincar6 sections Ill , l l of the flow indicating the ergodic blobs (or coher-ent structures) separated by surfaces h = conat on which vortex sheets maybe located.

ertheless, has certain structural properties that can be very simplydescribed. First, note from Eq. (12) that since U vh = 0, thestreamlines generally lie on surfaces h = const. The only situationthat permits streamlines to escape this constraint occurs when, insome subdomain, D1 say, g h is zero and

w = a1u (15)where a1 is constant. This is a Beltrami flow (in D1) f maximalhelicity in the sense that the upper bound on 1x1 permitted by theSchwarz inequality (7) is attained (note that w 0 n = 0 on BD1, sothat the helicity in D1 is well defined). Within D1 or at least withinparts of D1, treamlines may be space filling (i.e., ergodic in the three-


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148 H. K. MOFFATTdimensional subspace) in the manner identified for a particular space-periodic Beltrami flow by HQnon(I4) and Dombr6 et al.

It is easy to visualize such regions as of toroidal topology as in Fig.l a (although they may, in general, be much more complex). Differentergodic toroids of maximal helicity D1,D2,..may be linked andknotted with arbitrary complexity (determined by the topology of thereference field U(x) and different Poincare sections of the flow Ill,Il2then have the structures indicated in Figs. l b and lc. The spaceseparating the toroids is filled with a family of surfaces h = const(or possibly (IY: = const , where w = cy(x)u,U var = 0), and it is onthese surfaces that tangential discontinuities of U may appear. It isimportant to note that vortex sheets cannot be present in the ergodicregions since U = a - l w would then be singular also, with infiniteenergy density, and this is excluded. So the vortex sheets, if present,are necessarily separate from the ergodic structures and occur throughconfluence of surfaces h = const with discontinuities related by

across the sheet (i.e., [ p ]= 0).The relevance of such solutions to the problem of turbulence

should now be clear: we interpret the toroidal ergodic blobs as thecoherent structures of the flow; these are bounded by a stream sur-face and would therefore be detectable in a vhal ization process; in-deed, if a dye is injected at any point in such a region, it will rapidlyspread to fill the region.

The vortex sheets, on the other hand, are the site of strong vis-cous dissipation (when viscosity is restored to the equations). Thesevortex sheets, as has already been indicated, are subject to instabili-ties of the Kelvin-Helmholtz type, so that characteristic double-spiralstructures may be expected to develop and to induce unsteadiness inotherwise stable parts of the flow field. Under such perturbations,the exact alignment of w and U within a coherent structure will bedisturbed, but nonlinear energy transfer associated with the U x wterm in the Navier-Stokes equation may be expected to remain small,which is why these structures may persist over long time scales.

V. Spiral Structures and the Kolmogorov SpectrumIt has been argued previously l6 that the development of double-

spiral structures (see Fig. 2) can be associated with a spectral power


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TOPOLOGICAL APPROACH TO VORTEX DYNAMICS 149

aFig. 2 Growth of double spirals associated with Kelvin-Helmholtz instability;a new vortex sheet can form due to stretching action associated with growthof adjacent spirals.

law E(k) - -' with fractional exponent 1< X < 2. If the structureof the spiral is known, with a (near) accumulation point of velocitydiscontinuities on any straight-line transversal passing near the 'eye'of a spiral, then the value of X may be calculated. There is here animportant difference between two-dimensional Kelvin-Helmholtz in-stability (for which the wave-crests are parallel to the vorticity vectorwithin the sheet) and three-dimensional modes for which the wave-crests are inclined to the vorticity vector, as must occur, for example,if the vortex sheet is wrapped on a torus, the vorticity being ergodicon the torus. In the latter case, the component of vorticity perpendic-ular to the wave-crests is stretched by the instability, the associatedvelocity jump remaining constant across the sheet.parallel to the spiral axis becomes more and more concentrated nearthis axis (the beginnings of this process being seen in the analysis ofMoore"), so that the winding process becomes similar to that dueto a concentrated line vortex superposed on the sheet. Here the resultsof Gilbert" on a related two-dimensional problem are illuminating:Gilbert considered the wind-up of a weak vorticity discontinuity by aneighboring concentrated line vortex and found that the energy spec-

0

In the process of spiral winding, the component of vorticity


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' 150 H.K. MOFFATTtrum associated with the developing spiral structure has a k-"j3fall-off (a result that receives some support from numerical simula-tions of vortex interactions in a mixing layerIg). The correspondingresult for the wind-up of a vortex sheet by a superposed line vortexperpendicular to the vorticity in the sheet would be a k-'j3 spec-trum; but the three-dimensional problem is complicated by the factthat, for the Kelvin-Helmholtz instability, the strength of the vortexsheet (i.e., of the surface vorticity) becomes non-uniform, as indicatedabove, and the winding is not precisely that due to a concentrated vor-tex. Lundgren's3 model of a strained spiral vortex suggests that thek-'I3 spectrum may not, in fact, be too sensitive to the details of thewinding process, but this is an area that clearly merits further study.

The picture that then emerges is that the inertial range spectrumis associated with a particular type of developing singularity, namely,a Kelvin-Helmholtz double spiral, distributed randomly throughoutthe flow field. Each spiral will grow until it interacts with neighboringspirals, when pairing of spirals may occur, the 'mature' spirals actingas possible foci for new coherent structures.

At the same time, we see a mechanism for the formation of newvortex sheets (see Fig. 2), through stretching of the vorticity com-ponent in the plane of the straining motion associated with spiralexpansion. The vorticity in the new sheet is perpendicular to theaxes of the 'parent' spirals, and the process (related to the processof production of enstrophy) is essentially three-dimensional in charac-ter. (There is no comparable mechanism for the production of vortexsheets in two-dimensional turbulence.)

VI. SummaryThe aim of this paper has been to provide a global description of

turbulence that incorporates in a natural way the four features speci-fied in the introduction, namely, coherent structures, intermittency ofdissipation, an inertial range spectrum, and a three-dimensional pro-cess of enstrophy production. We achieve this by focusing attentionon the fixed points of the Euler equations (i.e., Euler flows of generalthree-dimensional structure) and the instabilities to which these flowsmay be subject. The coherent structures emerge as regions in whichstreamlines may be ergodic and helicity density is high. Dissipationis concentrated in the vortex sheets located in regions separating thecoherent structures and is thus spatially intermittent. The vortexsheets are unstable by the Kelvin-Helmholtz mechanism and form

'


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TOPOLOGICAL APPROACH TO VORTEX DYNAMICS 151spiral structures that we conjecture are intimately bound up with theappearance of an inertial range spectrum with fractional power lawk - A . Finally, this same instability is responsible for stretching of vor-tex lines inclined to the wave-crests and thus for enstrophy productionand formation of new vortex sheets.Development of a detailed model of this kind is important be-cause, if the characteristic structures and instabilities are understood,then it should be possible to derive not only second-order statisticssuch as the energy spectrum but also higher-order statistics withoutb appeal to closure assumptions, which are hard, if not impossible, tojustify.through the helicity invariant ( 5 ) , which is maximal in the ergodicblobs of Euler flows. There is some evidence from direct num-erical simulation of space-periodic flows such as the Taylor-Greenvortex20*21hat coherent structures of nonzero helicity where dissi-pation is low do tend to emerge as the flow develops. Further workalong these lines, as well as direct experimental measurement of helic-ity fluctuations in turbulent flow, is urgently needed to complementthe theoretical developments.

Topological (as opposed to analytical) considerations enter

AcknowledgmentsI am grateful to D r. M. Garnier and Prof. R. Moreau of MADY-

LAM for hospitality at the Institut de MQcanique, Grenoble, and toCNRS for financial support during the writing of this paper.

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152 H.K. MOFFATTMechanics, Vol. 159,1985,pp. 359-378.'Moffatt, H. K., Magnetostatic Equilibra and Analogous Euler Flows of Ar-bitarily Complex Topology. Part 2: Stability Considerations, Journal ofFl uid M echanics, Vol. 166,1986,pp. 359-378.*M offa tt, H. K., On th e Existence of Localized Rotatio nal DisturbancesWhich Prop agate W ithout Chang e of Struc ture in an Inviscid Fluid, Journa lof Fluid Mechanics, Vol. 173, p. 289-302.'Batchelor, G. ., Introdu ction to Fluid Dynamics, Cambridge Univ. Press,Ca m bridge , U.K., 1967."Moreau J.-J., Co nsta nts d'un ilat tourbillonaire en fluide parfait b arotrop e,ComDtes Rendus Hebdomadaires des ScQancesde I'Acadkmie des Sciences,Vol. 252, 1961, . 2810."Moffatt, H. K., O n th e Degree of Knottedness of Tangled Vortex Lines,Journal of Fluid Mechanics, Vol. 35,pp. 117-129.'2Arn old, V., T h e Asym ptotic Hopf In variant an d Its A pplications,Proceedinns of t h e Summer School in Differential E quation s, ArmenianSSR Academy of Science, 1974 (in Russian).laWoltjer, L., A Th eorem on Force-Free Magnetic Fields, Proceedings of th eNational Academy of Sciences of the United States ofAmerica, Vol. 44, 958, p. 489-491.14HQnonM., Com ptes Rend us Hebdomadaires des ScQancesde 1'Acadkmiedes Sciences, Vol. 262,p. 312.15Dombr6,T., Frisch, U., Greene, J. M., HQnon,M., Mehr, A., a n d Soward,A. M., Ch aotic S tream lines in th e ABC Flows, Jo urna l of F luid Mechanics,16Moffatt H. K., Simple Topological Aspects of Turbulent Vorticity Dynam-ics, Turbulence a nd Chaotic P henom ena in Fluids, edited by T. Tatsu mi, El-sevier, New York, 1984,pp. 223-230.l 'Moore, D.W., T h e Spo ntaneo us Ap peara nce of a Singularity in th e Shapeof an Evolving Vortex Sheet, Proceedinns o >fthe Royal Society of London,Ser. A, Vol. 365, 1979,p. 105.'*Gilbert, A. D., Spiral Structures a nd Spe ctra in Two-Dimensional Turbu-lence, Journal of Fluid Mechanics (to appear), 1988."Lesieur, M., Sta qu et, C., and Le Roy, P., The Mixing Layer and Its Co-herence Examined from th e Point of View of Two-Dimensional Turbulence,Journal of Fluid Mechanics (to appear), 1988.2o Sh tilm an , L., Levich, E., O rszag, S.A., Pelz, R.B., a nd Ts inober, A., O nth e Role of Helicity in Com plex Fluid Flow, Physics Le tters A, Vol. 113, p.21Pelz, R. B., Yakhot, V., Orszag, S.A., Shtilman, L., and Levich, E.,

Vol. 167,pp. 353-391.

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H. K. Moffatt-Topological Approach to Problems of Vortex Dynamics and Turbulence