k. bajer, h.k. moffatt and francis h. nex- steady confined stokes flows with chaotic streamlines

8/3/2019 K. Bajer, H.K. Moffatt and Francis H. Nex- Steady Confined Stokes Flows with Chaotic Streamlines

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Chaotic Streamlinest K. BAJER* , H.K. MOFFATT & FRANCES H. NEX

Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Silver Street, Cambridge CB3 9 E W , UK

A family of incompressible steady flows, called STF flows, is considered. These areStokes flows whose velocity is a quadratic function of cartesian coordinates satisfyinguSTF-n 0 on a sphere r = 1. When the flow is an O(e) perturbation of a flow utTFwith closed streamlines the particle paths are, for a long time, constrained near sometwo-dimensional surfaces (adiabatic drift) but they jump randomly from one suchsurface to another when they approach a stagnation point. Because of these randomjumps, which we call super-adiabatic drift, streamlines are apparently space-fillingthroughout entire domain. The Lyapunov exponent X is computed and it is found tobe positive on the space-filling streamlines. There is some evidence that X remainsbounded away from zero when e -+ 0. The separation of two distinct but initiallyclose fluid particles is also computed. It indicates efficient mixing of a passive scalarin the STF flows.1. INTRODUCTIONIt has been known for some years that even simple fluid flows may lead to exceed-ingly complicated particle paths. Three-dimensional steady flows can have chaoticstreamlines which densely fill some regions of space - a property relevant, for exam-ple, to the kinematic dynamo problem. In two-dimensional flows even weak timedependence may result in the chaotic motion of fluid particles, which can be usefulfor stirring and mixing.Several time-dependent, two-dimensional flows have been investigated so far: blink-ing point vortices (Aref 1984), Stokes flow between the alternately rotating, excen-tric circular cylinders (Aref & Balachandar 1986; Chaiken et al. 1986,1987; Ottinoet al. 1988); and also simple models of such physical flows as: weak waves in Taylorvortices (Broomhead & Ryrie 1988) or tidal flows in shallow seas (Pasmanter 1988).

* On leave of absence from the Institute of Geophysics, University of Warsaw, Poland

www.moffatt.tc


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460 K. Bajer, H.K. Moffatt& F.H. NexThe structure of streamlines has been studied in some space-periodic steady fl0ws.VThe best known are the so-called ABC flows (Henon 1966;Dombre e t al. 1986)and Q-flows (Beloshapkin e t al. 1989). In this paper we introduce a class of three-dimensional incompressible steady flows in a sphere which have a particularly richLagrangian structure. These flows have a very simple Eulerian representation: thecartesian components of the velocity are quadratic functions of the coordinates.Similar flows were considered before (Moffatt & Proctor 1985) n connection withStretch-Twist-Fold fast dynamo processes and for this reason we adopt the nameSTF flows for their modified version presented below.2. STF FLOWSWe consider a two-parameter family of quadratic flows:

uSTF = (az- 8sy, lls2+ 3y2+ z2 + P x z - 3, -as + 2yz - Pzy) , (1)where a, are positive real parameters. Clearly uSTF satisfies: V * uTSF = 0;U S T F . nIs = 0 where S is the surface of a unit sphere.*The flow (1) is a sum of the rigid rotation R around the y-axis with angular velocitya and the remaining part u tTF. Streamlines of u tTF are mostly closed loops withthe exception of some which are heteroclinic lines linking the hyperbolic stagnationpoints at (z,y,z) = (0,f1,0).Hence, for a


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Steady Stokes Flows with Chaotic Streamlines........... ................ .:...._I. . 1 . .. .

............ 4'>.....__._._ . '................. :....I

46 1

FIGURE 1. A single chaotic streamline of the STF flow: a ) The Poincarb section; b) Theprojective view.


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462 K. Bajer, H.K. M offatt & F.H. Nexrather orderly manner. Watching the PoincarC section being plotted one may noticethat for a long time points closely follow curves. These curves are sections oftwo-dimensional surfaces which meet at the stagnation points on the surface of aunit sphere and the streamline jumps from one such surface to another every timeit comes near the stagnation point. The set of these curves is traced out withapparently random spacing. It forms the distinct layered pattern seen in fig. l a ) .The motion along the curves can be explained in terms of an adiabatic invariant(Bajer 1989; Bajer & Moffatt 1989) and therefore we should call it adiabatic motion(or drift). The relatively large jumps mentioned above form what we term as super-adiabatic drift. These two kinds of drift persist as we decrease a . For CY = 0streamlines are closed loops, i.e. the PoincarC section of an orbit consists of onlytwo points. For an arbitrarily small but finite a orbits spread apparently over theentire domain and there are no visible islands of regularity. This is a very differentpicture from, for example, the ABC flow where, in the integrable case, streamlinesfill toroidal surfaces, most of them densely. Then the K A M tori surviving a smallperturbation ensure confinement of the chaotic streamlines. The KA M theory doesnot apply to flows with closed streamlines and hence the global chaos is possible forarbitrarily small perturbations.4. LYAPUNOV EXPONENT AND THE SEPARATION OF PARTI-CLESIn order to substantiate the above claim we compute the Lyapunov exponent asso-ciated with the STF flow. The Lyapunov exponent X is the asymptotic growth-rateof the length of an infinitesimal material line element ( advected by the flow:

1X = lim -log1(1 .t+oo tPositive X means exponential stretching - the signature of a chaotic flow. We solvenumerically the sixth order system of ODES for (:

We integrate (3 ) with a = 0.01 and plot in fig. 2a ) the function X ( t ) = $ log I((t)l.The corresponding PoincarC section of uSTF s shown in fig. 2b) . The value ofthe function X ( t ) was plotted (marked with a single dot in fig. 2a) every time thestreamline crosses the Poincark plane of section. Gaps in the plot of X ( t ) correspondto the long time that a particle spends near the stagnation points.


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Steady Stokes Flows w ith Chaotic Streamlines 463

FIGURE 2. U) Th e function X(t). Its asymptotic value for t -+ 00 is equal to the Lyapunovexponent of a streamline; b ) Corresponding Poinc arQ section.


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464 K. Bajer, H.K. Moffatt & F.H. Nex

Itime

FIGURE 3. Logarithm of the separation between two close particles as a function of time.The Lyapunov exponent has the dimension of (time)-', the same as the smallparameter a. The time-scale of the unperturbed flow utTF s not well defined.For the streamlines close to heteroclinic lines the turnover time is arbitrarily long.However, the perturbation R has a unique time-scale : In these units the valueof the Lyapunov exponent, as read from fig. 2a ) , is X R 1.2. Further computationsindicate that X remains O(1) as we increase a (up to a = 0.1).Finally in fig. 3 we plot the separation s between two distinct fluid particles whichinitially were very close (10-6 compared to the unit radius of the sphere). Herelog(s) is plotted as a function of time for a = 0.1. Jumps in log(s) correspond tothe jumps between different adiabatic surfaces.In a finite domain the separation, unlike the length of the infinitesimal line ele-ment, is bounded. Therefore a systematic increase in s saturates, and then s variesrandomly between s = 0 and its maximal value s = 2. The time needed for thesystematic increase to saturate could be regarded as the characteristic time-scalefor mixing of the spatial scales of the size of the initial separation (10-6 in thiscase).5 . CONCLUSIONSWe have shown that STF flows (1) in the limit of small a exhibit Lagrangianchaos of a kind so far unknown in the context of flow kinematics. Because of the


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Steady Stokes Flows with Ch aotic Streamlines 465degenerate character of the unper turbed flow ufTF which has closed streamlinesthere are n o K AM tor i , an d for a n a rbi t rar i ly small per turbat i on th e super-adiabat icdrif t sprea ds th e streamlines over the entire domain. T h e Lyapunov exponent ispositive, an d when measured o n th e time-scale of the s low per tu rbat io n i t appearsto b e bo und ed away from zero when E ( = ) + 0.This new typ e of chaos is likely to b e typical for pe rtu rb ed flows with closed stre am -lines. I ts tra ns po rt properties such as the chaotic advection of a passive scalar orthe possibility of a fast d yn am o deserve furt he r investigation.REFERENCES

A R EF , H. (1984) Stirr ing by chaotic advection. J . Fluid Mech. 143, -21.AR EF, H., & BALACHANDAR, S. (1986) Ch aotic advection in a Stokes flow. Phys.Fluids 29, 3515-3521.

BA JE R, K . (1989) Flow kinematics an d mag neto static equilibria. P h.D . Thesis.Cambridge University.BA JER , K. , MO FFAT T, H.K. (1989) On a class of steady confined Stokes flowswith chaotic streamlines. J. Fluid Mech. ( submit ted) .BELOSHAPKIN, V.V., CHERNIKOV, A.A., NATENZON, M.Ya. , PETRO-VICHEV , B.A., SA GD EE V, R.Z., & ZASLAVSKY, G.M. (1989) Chaoticstreamlines in pre-turbulent states. Nature 337, 33-137, J a n 1989.

linearity 1, 09-434.BRO OM HEA D, D.S ., & RY RIE, S.C. (1988) Par ticle p at h s in wavy vortices. Non-CHAIKEN, J., CHEVRAY, R . , TABO R, M. , & TAN, Q.M. (1986) Experimentals t u d y of Lagrangian turbu lence in a Stokes flow. Proc. Roy. Soc. A408, 165-

174.CHAIKEN, J., CHU, C .K. , TAB OR, M. , & TAN , Q.M. (1987) Lagrangian tu rbu -lence an d sp atia l complexity in Stokes flow. Phys. Fluids 30, , 687-694.

SOWARD, A.M. (1986) Chaotic streamlines in the ABC flow. J . Fluid Mech.HENON, M. (1966) Sur la topologie des lignes de courant dans un cas particulier.MO F F ATT, H . K ., & P R O C T O R , M.R.E . (1985) Topological c ons traints associated

D O M B R E , T., FRISCH, u., GREENE, J.M., HENON, M ., MEHR, A., &167, 53-391.C.R. Acad. Sci . 262 , 312-314.with fast d yna mo action. J . Fluid Mech. 154, 93-507.


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466 K. Bajer, H.K. Moffatt & F.H. NexMOFFATT, H.K. (1985) Magnetostatic equilibria and analogous Euler flows of ar-bitrarily complex topology. Pa rt 1.Fundamentals. J. Ffuid Mech. 159, 59-378.M OF FA TT, H.K. (1986) M agneto static equilibria and analogous Euler flows of arbi-

trarily complex topology. P ar t 2. Stability considerations. J. Fluid Mech. 166,OTT INO, J.M., LEONG, C.W., RISING, H., & SWANSON, P.D. (1988) Morpho-logical structures produced by mixing in chaotic flows. Nature 333, 19-425,Jun e 1988.PA SM AN TER , R.A. (1988) Anomalous diffusion an d anom alous stretching in vor-

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k. bajer, h.k. moffatt and francis h. nex- steady confined stokes flows with chaotic streamlines

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