Lagrangian Chaos Research Articles

Transport control in fusion plasmas by changing electric and magnetic field spatial profiles

For magnetically confined plasmas in tokamaks, we have numerically investigated how Lagrangian chaos at the plasma edge affects the plasma confinement. Initially, we have considered the chaotic motion of particles in an equilibrium electric field with a monotonic radial profile perturbed by drift waves. We have showed that an effective transport barrier may be created at the plasma edge by modifying the electric field radial profile. In the second place, we have obtained escape patterns and magnetic footprints of chaotic magnetic field lines in the region near a tokamak wall with resonant modes due to the action of an ergodic magnetic limiter. For monotonic plasma current density profiles we have obtained distributions of field line connections to the wall and line escape channels with the same spatial pattern as the magnetic footprints on the tokamak walls.

Low Reynolds number scalar transport enhancement in viscous and non-Newtonian fluids

Enhancement of heat and/or mass transfer via turbulence is often not feasible for highly viscous, non-Newtonian or shear sensitive fluids. One alternative to improve transport within such materials is chaotic advection, whereby Lagrangian chaos occurs within regular (non-turbulent) flows [J.M. Ottino, The Kinematics of Mixing: Stretching, Chaos and Transport, Cambridge University Press, Cambridge, 1989]. Complex interactions between chaotic advection and diffusion yields enhanced dispersion, and the topology of the Lagrangian dynamics is governed by the set of control parameters for the flow device. What parameter set maximises scalar dispersion for a given fluid rheology and diffusivity? Most studies to date have only considered a handful of points in the parameter space Q , but as this space may be large and the solution distribution complex (fractal), robust optimisation requires detailed global resolution of Q . By utilising a novel spectral method [D.R. Lester, G. Metcalfe, M. Rudman, H. Blackburn, Global parametric solutions of scalar transport, J. Comput. Phys. (2007). doi: doi:10.1016/j.jcp.2007.10.015] which exploits the symmetries often present in chaotic flows, we can resolve the asymptotic transport dynamics over Q , facilitating the identification of optima and elucidating the global structure of transport. We employ this method to optimize scalar transport for both Newtonian and non-Newtonian fluids in a chaotic mixing device, the Rotated Arc Mixer (RAM). Significant (up to sixfold) acceleration of scalar transfer is observed at Peclét number Pe = 10 3, which furthermore increases with Pe.

Stretching fields and mixing near the transition to nonperiodic two-dimensional flow

Although time-periodic fluid flows sometimes produce mixing via Lagrangian chaos, the additional contribution to mixing caused by nonperiodicity has not been quantified experimentally. Here, we do so for a quasi-two-dimensional flow generated by electromagnetic forcing. Several distinct measures of mixing are found to vary continuously with the Reynolds number, with no evident change in magnitude or slope at the onset of nonperiodicity. Furthermore, the scaled probability distributions of the mean Lyapunov exponent have the same form in the periodic and nonperiodic flow states.

Chaotic transport in reversed shear tokamaks

For tokamak models using simplified geometries and reversed shear plasma profiles, we have numerically investigated how the onset of Lagrangian chaos at the plasma edge may affect the plasma confinement in two distinct but closely related problems. Firstly, we have considered the motion of particles in drift waves in the presence of an equilibrium radial electric field with shear. We have shown that the radial particle transport caused by this motion is selective in phase space, being determined by the resonant drift waves and depending on the parameters of both the resonant waves and the electric field profile. Moreover, we have shown that an additional transport barrier may be created at the plasma edge by increasing the electric field. In the second place, we have studied escape patterns and magnetic footprints of chaotic magnetic field lines in the region near a tokamak wall, when there are resonant modes due to the action of an ergodic magnetic limiter. A non-monotonic safety factor profile has been used in the analysis of field line topology in a region of negative magnetic shear. We have observed that, if internal modes are perturbed, the distributions of field line connection lengths and magnetic footprints exhibit spatially localized escape channels. For typical physical parameters of a fusion plasma, the two Lagrangian chaotic processes considered in this work can be effective in usual conditions so as to influence plasma confinement. The reversed shear effects discussed in this work may also contribute to evaluate the transport barrier relevance in advanced confinement scenarios in future tokamak experiments.

Chaotic mixing and mixing efficiency in a short time

Several studies of the chaotic motion of fluid particles by two-dimensional time-periodic flows or three-dimensional steady flows, called Lagrangian chaos, are first introduced. Secondly, some of the studies on efficient mixing caused by Lagrangian chaos, called chaotic mixing, are reviewed with discussion of several indices for the estimation of mixing efficiency. Finally, several indices to estimate the efficiency of mixing in a short time, such as those related to transport matrices, stable and unstable manifolds of hyperbolic periodic points of Poincaré maps, and lines of separation, are explained by showing examples of mixing by two-dimensional time-periodic flows between eccentric rotating cylinders and mixing by three-dimensional steady flows in a model of static mixers.

The role of helicity and stretching in forced kinematic dynamos in a spherical shell

Considerations of mean-field theory suggest that small-scale helical flows are an effective means of generating large-scale (mean) magnetic fields, whereas fast dynamo considerations reveal the importance of Lagrangian chaos in the flow for generating small-scale magnetic fields in the limit of high magnetic Reynolds number. We explore these ideas further by considering the kinematic magnetic fields generated by three forced steady flows in a spherical shell that differ both in their helicity and in their stretching properties. The full magnetic induction equation is solved numerically, with no a priori assumptions about the nature of the generated magnetic field. There are two surprising aspects to our results. One is that the most significant mean field is generated by a flow with zero net helicity; the other is that the flow with the “best” stretching properties turns out to be the most inefficient dynamo. Our results, therefore, suggest that it may not be possible to determine the nature of a kinemati...

Amplification of Interstellar Magnetic Fields and Turbulent Mixing by Supernova‐driven Turbulence. II. The Role of Dynamical Chaos

In this paper we further advance the study of magnetic field amplification in the interstellar medium that we started in 2004. We show that the flux growth rate is comparable to the rate of magnetic energy growth found in the earlier paper. We also demonstrate the role of intermittency in field amplification. The density shows a double-peaked PDF, consistent with the cooling curve that was used. The PDF of the magnetic field shows a high-end tail, providing a telltale signature of the operation of the small-scale dynamo. The magnetic field strength correlates with the density as ~ ρα, with α = 0.386. As a result, the field amplification takes place more vigorously in the lower temperature, denser gas. The Lagrangian chaos in the simulated turbulent flows is studied in substantial detail. It is shown that the stretching rate of material lines, as well as the Lyapunov exponents, can be used to gain important insights into the growth of magnetic field. The cancellation exponent for the small-scale supernova-driven dynamo is derived, and it is shown that constructive folding of field lines in the dynamo is very inefficient. We also show that our Lagrangian approach can yield actual measures of the turbulent diffusivity in the simulated ISM. The turbulent diffusivity provides insights into the mixing of elements from supernova ejecta on macroscopic scales. The high rates of line stretching in interstellar turbulence suggests that the eventual diffusion of elements at the molecular level is very efficient. Many of the diagnostics of turbulence that are presented here can be used to make direct connections between simulations and observations.

Intersections of stable and unstable manifolds: the skeleton of Lagrangian chaos

Abstract We study Hamiltonian chaos generated by the dynamics of passive tracers moving in a two-dimensional fluid flow and describe the complex structure formed in a chaotic layer that separates a vortex region from the shear flow. The stable and unstable manifolds of unstable periodic orbits are computed. It is shown that their intersections in the Poincare map as an invariant set of homoclinic points constitute the backbone of the chaotic layer. Special attention is paid to the finite time properties of the chaotic layer. In particular, finite time Lyapunov exponents are computed and a scaling law of the variance of their distribution is derived. Additionally, the box counting dimension as an effective dimension to characterize the fractal properties of the layer is estimated for different duration times of simulation. Its behavior in the asymptotic time limit is discussed. By computing the Lyapunov exponents and by applying methods of symbolic dynamics, the formation of the layer as a function of the external forcing strength, which in turn represents the perturbation of the originally integrable system, is characterized. In particular, it is shown that the capture of KAM tori by the layer has a remarkable influence on the averaged Lyapunov exponents.

Complex magnetohydrodynamic low-Reynolds-number flows.

The interaction between electric currents and a magnetic field is used to produce body (Lorentz) forces in electrolyte solutions. By appropriate patterning of the electrodes, one can conveniently control the direction and magnitude of the electric currents and induce spatially and temporally complicated flow patterns. This capability is useful, not only for fundamental flow studies, but also for inducing fluid flow and stirring in minute devices in which the incorporation of moving components may be difficult. This paper focuses on a theoretical and experimental study of magnetohydrodynamic flows in a conduit with a rectangular cross section. The conduit is equipped with individually controlled electrodes uniformly spaced at a pitch L. The electrodes are aligned transversely to the conduit's axis. The entire device is subjected to a uniform magnetic field. The electrodes are divided into two groups A and C in such a way that there is an electrode of group C between any two electrodes of group A. We denote the various A and C electrodes with subscripts, i.e., A(i) and C(i), where i=0,+/-1,+/-2, .... When positive and negative potentials are, respectively, applied to the even and odd numbered A electrodes, opposing electric currents are induced on the right and left hand sides of each A electrode. These currents generate transverse forces that drive cellular convection in the conduit. We refer to the resulting flow pattern as A. When electrodes of group C are activated, a similar flow pattern results, albeit shifted in space. We refer to this flow pattern as C. By alternating periodically between patterns A and C, one induces Lagrangian chaos. Such chaotic advection may be beneficial for stirring fluids, particularly in microfluidic devices. Since the flow patterns A and C are shifted in space, they also provide a mechanism for Lagrangian drift that allows net migration of passive tracers along the conduit's length.

Two-dimensional turbulence of dilute polymer solutions.

We investigate theoretically and numerically the effect of polymer additives on two-dimensional turbulence by means of a viscoelastic model. We provide compelling evidence that, at vanishingly small concentrations, such that the polymers are passively transported, the probability distribution of polymer elongation has a power law tail: Its slope is related to the statistics of finite-time Lyapunov exponents of the flow, in quantitative agreement with theoretical predictions. We show that at finite concentrations and sufficiently large elasticity the polymers react on the flow with manifold consequences: Velocity fluctuations are drastically depleted, as observed in soap film experiments; the velocity statistics becomes strongly intermittent; the distribution of finite-time Lyapunov exponents shifts to lower values, signaling the reduction of Lagrangian chaos.

Lagrangian chaos and multiphase processes in vortex flows

We discuss experimental and numerical studies of the effects of Lagrangian chaos (chaotic advection) on the stretching of a drop of an immiscible impurity in a flow. We argue that the standard capillary number used to describe this process is inadequate since it does not account for advection of a drop between regions of the flow with varying velocity gradient. Consequently, we propose a Lagrangian-generalized capillary number C L number based on finite-time Lyapunov exponents. We present preliminary tests of this formalism for the stretching of a single drop of oil in an oscillating vortex flow, which has been shown previously to exhibit Lagrangian chaos. Probability distribution functions (PDFs) of the stretching of this drop have features that are similar to PDFs of C L. We also discuss on-going experiments that we have begun on drop stretching in a blinking vortex flow.

Chaotic advection of finite-size bodies in a cavity flow

We considered advection of neutrally buoyant discs in two-dimensional chaotic Stokes flow. The goal of the study is to explore a possibility to enhance laminar mixing in batch-flow mixers. Addition of freely moving bodies to periodically driven chaotic flow renders the flowfield nonperiodic [D. F. Zhang and D. A. Zumbrunnen, AIChE J. 42, 3301 (1996)], i.e., the Lagrangian chaos of the bodies motion induces Eulerian chaos of the flow that makes mixing more intensive. The presence of three bodies creates new topological features that do not exist in “pure” fluid. The trajectories of the discs in the augmented phase space tangle and form a braid that leads to so-called topological chaos [P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)]. Simulations were performed using a new variant of the immersed boundaries method that allows the direct numerical simulation of fluid–solid flows on a regular rectangular grid without explicit calculation of the forces that the particles exert on the fluid.

Bifurcations for fluid stirring in circular cavity

Lagrangian chaos associated with Stokes flows in a circular cavity is addressed in the present study. Two belts arranged on the boundary to be rotated at various speeds are employed to drive this flow motion. The evolutions of several tori are examined when the periods are increased. Then the types of transitions through which the flows change from regular motions to chaotic ones are clearly identified. The routes to chaos for the present studied flows are subject to some successive bifurcations of island‐chains. Those include period‐5, 4, 3, 5, 7 along the direction with increased periods and finally the flows arrive at random motion. The detailed processes of the fluid stirring were numerically visualized and some animations were produced for instructional uses.

Lagrangian structures, integrability and chaos for 3D dynamical equations

In this paper we consider the general setting for constructing Action Principles for three-dimensional first order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and we show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behavior or homoclinic orbits have not been verified up to now. The Euler-Lagrange equations we get for these systems usually present time reparameterization symmetry, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrability. In particular, we show that the so-called ABC system is completely integrable if it possesses one constant of the motion. Pacs numbers: 45.20.Jj, 02.30.Ik, 05.45.Ac Keywords: Lagrangian, integrable system, chaos, Lotka Volterra

Note on chaotic advection in an oscillating drop

Mixing by chaotic advection within an oscillating drop is investigated by means of an elementary model (Hill’s spherical vortex). The drop remains spherical and the velocity field within the drop is assumed to rotate around a radial axis, in a periodical manner. Poincaré sections in the vicinity of fixed points are obtained asymptotically in the limit of small amplitude oscillations, and show that the trajectory of elliptic points undergoes a resonant mechanism. The flow inside the drop can display Lagrangian chaos when the drop oscillates. In particular, tracers located near the boundary of the drop are observed to contaminate the whole drop within a few periods.

A thermal model for prediction of the Nusselt number in a pipe with chaotic flow

It is currently well established that Lagrangian chaos intensifies heat transfer significantly [J. Fluid Mech. 209 (1989) 335]. It thus appears to be a promising technique for the design of compact, high-performance heat exchangers and heat exchanger-reactors. However, the design of such apparatus requires extensive calculations. The objective of this work is to implement a simplified thermal model with which to simulate heat transfer in a twisted pipe (of a shell and tube heat exchanger) of two tube configurations, helically coiled or chaotic, without requiring the heavy calculations needed in the numerical resolution of the Navier–Stokes and energy equations. The large database obtained from the parametric study of the variation of the Nusselt number using the heat transfer model developed here, allows one to correlate Nu with Re, Pr, N bends: Nu=1.045 Re 0.303 Pr 0.287 N bends −0.033. This correlation is valid for coil geometry with alternating planes of curvature, i.e. chaotic configuration and the range of validity of the correlation is Re∈[100;300], Pr∈[30;100] and N bends∈[3;13].

The effect of Lagrangian chaos on locking bifurcations in shear flows.

The effect of an externally imposed perturbation on an unstable or weakly stable shear flow is investigated, with a focus on the role of Lagrangian chaos in the bifurcations that occur. The external perturbation is at rest in the laboratory frame and can form a chain of resonances or cat's eyes where the initial velocity v(x0)(y) vanishes. If in addition the shear profile is unstable or weakly stable to a Kelvin-Helmholtz instability, for a certain amplitude of the external perturbation there can be an unlocking bifurcation to a nonlinear wave resonant around a different value of y, with nonzero phase velocity. The interaction of the propagating nonlinear wave with the external perturbation leads to Lagrangian chaos. We discuss results based on numerical simulations for different amplitudes of the external perturbation. The response to the external perturbation is strong, apparently because of non-normality of the linear operator, and the unlocking bifurcation is hysteretic. The results indicate that the observed Lagrangian chaos is responsible for a second bifurcation occurring at larger external perturbation, locking the wave to the wall. This bifurcation is nonhysteretic. The mechanism by which the chaos leads to locking in this second bifurcation is by means of chaotic advective transport of momentum from one chain of resonances to the other (Reynolds stress) and momentum transport to the vicinity of the wall via chaotic scattering. These results suggest that locking of waves in rotating tank experiments in the presence of two unstable modes is due to a similar process. (c) 2002 American Institute of Physics.

Noise-induced enhancement of chemical reactions in nonlinear flows

Motivated by the problem of ozone production in atmospheres of urban areas, we consider chemical reactions of the general type: A+B-->2C, in idealized two-dimensional nonlinear flows that can generate Lagrangian chaos. Our aims differ from those in the existing work in that we address the role of transient chaos versus sustained chaos and, more importantly, we investigate the influence of noise. We find that noise can significantly enhance the chemical reaction in a resonancelike manner where the product of the reaction becomes maximum at some optimal noise level. We also argue that chaos may not be a necessary condition for the observed resonances. A physical theory is formulated to understand the resonant behavior. (c) 2002 American Institute of Physics.

C∞-Interpolation of Discrete Fields on Regular and Irregular Grids

The article proposes a simple C∞ interpolation of discretized lattice fields on regular and irregular grids. The method is based on localized C∞ but not analytic basis functions, which vanish outside an open set (region of influence). As a result, the interpolating fields at a point depend exclusively on the nodal values within the region of influence. The method can be applied to generic fields whose support is a limited set of n-dimensional space, starting from discretized values given on regular or irregular grids. Particular attention is focused on the interpolation of CFD-computed velocity fields that give rise to Lagrangian chaos.

Electrohydrodynamically driven chaotic mixing in a translating drop

When a liquid drop of given dielectric constant, resistivity and viscosity is translating in the presence of an electric field in a liquid of different dielectric constant, resistivity and viscosity, the resulting internal circulation is a superposition of the Hadamard–Rybcynski and Taylor (quadrapole) circulations. For sufficiently large field strengths, the quadrapole structure of the Taylor flow causes an internal stagnation disk to occur. When this flow field is modulated by an ac component of the electric field, at low Stokes numbers, the resulting Lagrangian particle trajectories can exhibit chaotic behavior. This Lagrangian chaos is examined via Poincaré maps and both the degree and rates of mixing are studied for a wide range of field strengths and modulation frequencies.