Session LV: Chaos/Multifractality

Spatiotemporal chaos in the dynamics of buoyantly unstable chemical fronts

Nonlinear dynamics resulting from the interplay between diffusive and buoyancy-driven Rayleigh-Taylor (RT) instabilities of autocatalytic traveling fronts are analyzed numerically for fronts traveling in the gravity field and for various values of the relevant parameters. These are here the Rayleigh numbers of the reactant $A$ and autocatalytic product $B$ as well as the ratio $D=D_B/D_A$ between the diffusion coefficients of the two key chemical species. The interplay between the coarsening dynamics characteristic of the RT instability and the fixed short wavelength dynamics of the diffusive instability can lead in some regimes to complex new dynamics dominated by irregular succession of birth and death of fingers. By using spectral entropy measurements, we show the possibility of a transition between order and spatial disorder in this system. The analysis of the power spectrum further allows to identify similarities between the various spatial patterns while phase space representation is also discussed.   

Chaotic mixing in a curved pipe with periodic variations in curvature and torsion

The chaotic fluid mixing in a helix-like circular pipe with periodic variations in curvature and torsion caused by a steady viscous flow under an axial pressure gradient of relatively small Reynolds number is examined. An approximate equation obtained under the assumption of small and slowly-varying curvature and torsion is used to calculate the cross-sectional motion of fluid particles associated with their axial motion. We examine the dependences of mixing efficiency on a few geometrical parameters and on Reynolds number, and attempt to explain them by the variation in a characteristic ratio composed of curvature, torsion and Reynolds number.   

Topology of Chaotic Mixing Patterns

A stirring device consisting of a periodic motion of rods induces a mapping of the fluid domain to itself, which can be regarded as a continuous mapping of a punctured surface. Having the rods undergo a topologically-complex motion guarantees a minimal amount of stretching of material lines, which is important for chaotic mixing. We use topological considerations to describe the nature of the injection of unmixed material into a central mixing region, which takes place at injection cusps. A topological index formula allow us to predict the possible types of unstable foliations that can arise for a fixed number of rods. See http://arxiv.org/abs/0804.2520 (Chaos, in press).   

Sky Dancer: a chaotic system

We present the experimental study of a collapsible tube conveying an ascending air flow. An extreme of the membrane tube is mounted on the air blower exit, while the other extreme is free. The flow velocity can be varied. For low speeds -- and tubes short enough -- the cylinder stands up (stable state). As the velocity is increased, the system presents sporadic turbulent fluctuations, when the tube bends and rises again. As the air speed is increased again, the intermittent events become more and more frequent. Films realized in front of the system permit to observe waves that propagate in the tube. We measure that these waves have a sonic speed, confirming previous results. Moreover, films taken from the top of the system allow a quantitative characterization of the transition to chaos. By processing the images, we can reduce the evolution of the system to two states: stable (when it is raised) and chaotic (when the tube fluctuates). The histograms of unstable / stable states are coherent with an intermittent transition in the theory of chaos.   

Multifractal Analysis of Vortex Pair Formation of Modified Taylor-Couette Flow in Laminar and Turbulent Regimes

For sufficiently large effective Reynolds Numbers the formation of Taylor Vortex Pairs in Modified Taylor-Couette flow with hourglass geometry becomes irregular in time. At higher effective Reynolds Numbers the flow becomes turbulent, but Taylor Vortices may still be discerned. Again, for sufficiently high effective Reynolds Numbers, the formation of these vortex pairs becomes chaotic. Previously we have demonstrated that each process may be characterized as low dimensional chaos.\footnote{A. Kowalski, T. Olsen, \& R. Wiener, Bull. Am. Phys. Soc. \textbf{50-9}, P1.00030 (2006).} We now present a multifractal analysis\footnote{J. A. Glazier \& A. Libchaber, IEEE Trans. On Circuits and Systems \textbf{35-7}, 790 (1988).}$^,$\footnote{T. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, \& B. I. Shraiman, Phys. Rev. A \textbf{33}, 1141 (1986).} of these processes.   

Topological chaos in wide lid-driven channels

Rapid fluid mixing can be produced in laminar flows through a high-aspect-ratio microchannel by means such as pressure-driven flow with staggered surface groove patterns or electro-osmotic flow with potential differences between the upper and lower boundaries. Under certain conditions, passive fluid particles or groups of particles can act as ``rods'' that stir the surrounding fluid and produce exponential stretching. The occurrence of ``topological chaos'' guarantees rapid mixing in these flows, and the Thurston-Nielsen theorem predicts a quantitative lower bound on complexity in the dynamics of the flow. We will present an exact solution for two-dimensional Stokes flow in a lid-driven cavity with periodic side wall boundary conditions and extend this model to approximate three-dimensional channel flow. We will examine the occurrence of topological chaos in these flows and discuss the mixing efficiency.   

Chaotic advection in pulsed source-sink systems

Pulsed operation of a source and a sink is a classic approach to generating chaotic advection in the unbounded plane. This approach provides motivation for mixing laminar flows in high-aspect-ratio volumes using an arrangement of sources and sinks.~ In bounded systems the sources and sinks must operate in pairs in order to conserve volume.~ When the sources and sinks are arranged on the boundary of a circular domain, particle motions are given explicitly by a discrete mapping. This mapping is used to explore the optimal operating parameters for producing chaos with three different source-sink configurations in a circular domain.   

Topological chaos in flows on surfaces of arbitrary genus

The emerging field of topological fluid kinematics is concerned with design and analysis of effective fluid mixers based on the topology of the motion of stirring apparatus and other periodic flow structures. Knowing even a small amount of flow topology often permits very powerful diagnoses, such as proving existence of chaotic dynamics and a lower bound on mixing measures based on material stretching. In this paper we present a canonical method for examining flows on surfaces of arbitrary genus given the flow topology encoded as a braid. The method may be used to study fluid mixing driven by an arbitrary number of stirrers in either bounded or spatially periodic fluid domains. Additionally, and unlike previous techniques, the current work may also be applied to flows on manifolds of higher genus.   

Chaotic mixing and superdiffusion in a two-dimensional array of vortices

We present experimental and numerical studies of mixing and long-range transport in an array of vortices forced by a magnetohydrodynamic technique. A current passing horizontally through a thin electrolytic solution interacts with a magnetic field produced by an array of magnets below the fluid. If the current is parallel to one of the primary directions of the magnet array, a square array of vortices is produced. If the current is tilted with respect to the magnet array, however, wavy channels form diagonally through the vortex pattern, allowing tracers in the flow to travel long distances in a short period of time. The addition of a time-dependent current results in a combination of chaotic and ordered vortex/jet regions that produces Levy flights and superdiffusive transport. If an AC current is applied in both cardinal directions, the resulting chaotic mixing is typically barrier-free.   

Unmixed islands in quasi-periodically-driven flows

Nested invariant 3-tori surrounding a torus braid of elliptic type are found to exist in a quasi-periodically forced fluid flow. The Hamiltonian describing this system is given by the superposition of two steady stream functions, one with an elliptic fixed point and the other with a coincident hyperbolic fixed point. The superposition, modulated by two incommensurate frequencies, yields an elliptic torus braid at the location of the fixed point. The system is suspended in a four-dimensional phase space (two space and two phase directions). To analyze this system we define two three-dimensional, global, Poincar\'{e} sections of the flow. The coherent structures (cross-sections of nested 2-tori) are found to each have a fractal dimension of two, in each Poincar\'{e} cross-section. This framework has applications to tidal and other mixing problems of geophysical interest.