Session MM - Chaos and Fractals.
MIXED session, Tuesday afternoon, November 23
Whidbey, Westin Seattle

[MM.001] Two-regime power-law behavior in a Lennard-Jones fluid

We study the dynamical behavior of a Lennard-Jones fluid using constant-energy molecular dynamics simulations. The system is studied at several points of the density-temperature phase diagram. Power spectra of instantaneous temperature and pressure show a two-regime power-law 1/f^a behavior. Rescaled range analysis (Hurst test) reveals also this two-regime behavior with strong memory effects for short times and gradual loss of memory at longer times. The dependence of memory on system density and temperature is discussed. By analyzing the Mean Square Displacement (MSD) of the atoms, the two-regime behavior is found to be associated with two kinds of atomic motion: vibrational (high frequency) and diffusional (low frequency). Characteristic times extracted from MSD calculations are in agreement with times deduced from spectral analysis. The system exhibits high-dimensional chaotic dynamics which is difficult to distinguish from stochastic behavior. We calculate the largest Lyapunov exponent of the system and discuss algorithms for the computation of the full Lyapunov spectrum.

[MM.002] Non-reversiblity in periodically driven flows by viscous dephasing

We show that in generic situations a reversible periodic driving of a low-Reynolds number flow will not result in a reversible flow field. In the famous experiment of GI Taylor with a blob of dye between concentric cylinders only a single eigenmode of the Stokes operator is involved and reversibility is assured. In generic situations several modes are present and since each eigenmode responds with a phase delay that depends on its eigenvalue and the driving frequency, not all modes will be in phase anymore. Such a viscous dephasing then breaks time reversal symmetry. The general theory is illustrated for 2-d vortex patterns that can be generated in current driven flows in a magnetic field.

[MM.003] Lagrangian chaos and resonance phenomena in Stokes flows

We discuss the Lagrangian chaos in a Stokes flow between two counter-rotating coaxial cylinders with the angular velocities depending on the coordinate along the axis. The streamlines of the unperturbed system are closed and the flow is regular. However, if an additional weak flow parallel to the axis is applied, Lagrangian chaos develops. We show that this phenomenon is caused by quasi-random changes (jumps) in the adiabatic invariant of the flow, which occur when streamlines cross the resonance surface. We demonstrate that the accumulation of the jumps leads to the diffusion of adiabatic invariant. For multiple resonance crossings. We discuss the statistical properties of the jumps. We study the long-time mixing properties of the flow and compare our results with numerical simulations.

[MM.004] Mode-locking of chemical pulses in an advection-reaction-diffusion system

We present results of experimental studies of the propagation of chemical fronts in a chain of oscillating vortices, a system in which mixing of passive impurities is chaotic. A variation of the photosensitive Belousov-Zhabotinsky chemical reaction is used in which a reaction pulse can be triggered with a silver wire. The speed at which this pulse propagates through the vortex chain is measured and compared with recent theories(M. Cencini, A. Torcini, D. Vergni and A. Vulpiani, Phys. Fluids 15, 679 (2003)) for front propagation in a similar flow. Several mode-locked regimes are observed in which the pulse propagates an integer number of vortex widths in an integer number of oscillation periods. Measurements of the widths of these locking regimes are made as a function of the oscillation amplitude, revealing classic Arnol’d tongue behavior.

[MM.005] Chemical pulses and superdiffusive transport in an oscillating-drifting vortex chain

We discuss experimental and numerical studies of transport and pulse propagation in an alternating vortex chain that can both oscillate and drift. Previous studies have shown that transport in the oscillating vortex is chaotic, with a variance that grows linearly with time; i.e., the transport is diffusive in nature. The addition of drifting to the vortex chain allows for superdiffusive transport with a variance that grows faster than linearly with time. We present particle-tracking data for transport in this system, along with experimental studies of the propagation of chemical pulses for the oscillating/drifting vortex chain.

[MM.006] Self-consistent chaotic mixing of active impurities in a vortex chain

Experiments are performed on the effects of active mixing on the dynamics of a vortex chain. The process is called “self-consistent” because of its circular nature: the mixing affects the time-dependence of the flow which, in turn, affects the mixing. The flow is forced magnetohydrodynamically -- an electrical current passes through a salt-water solution and interacts with an alternating magnetic field to generate a chain of vortices. The system is forced with a current below that for time-dependent oscillations. A more concentrated salt solution is injected as the active impurity. An instability arises in which blobs of more- and less-concentrated salt solution are advected around the vortices, causing oscillations in the vortex boundaries and leading to chaotic mixing of impurities between vortices.

[MM.007] Simulations of the Reaction-Diffusion System demonstrating the Transition from Purely Temporal to Spatio-Temporal Chaos

The Reaction-Diffusion model (H. Riecke and H.-G. Paap, Europhys. Lett. \textbf14), 1235 (1991).has been applied to a wide variety of pattern forming systems. It correctly predicted a period doubling cascade to chaos in Taylor-Couette flow with hourglass geometry(Richard J. Wiener \textitet al), Phys. Rev. E \textbf55, 5489 (1997).. We have conducted a series of such simulations, varying the length of the system. This has enabled us to study the transition from a purely temporal chaos of the formation of new pairs of Taylor Vortices at a single location, to a spatio-temporal chaos of formation across a range of locations. Application to anticipated experiments will be discussed.

[MM.008] Ruelle-Takens scenario in a confined two-dimensional flow

Ruelle, Takens and Newhouse (Commun. Math. Phys. 20, 1971; 64, 1978) argued that, as a function of an external parameter, the route to chaos in a fluid flow is a transition sequence leading from stationary (S) to single periodic (P), double periodic (QP_2), triple periodic (QP_3) and, possibly, quadruply periodic (QP_4) motions, before the flow becomes chaotic (C). However, the quasi-periodic motions with 3 or 4 incommensurate frequencies were shown to be unstable with respect to small perturbations and may, hence, not be observed at all. Using direct numerical simulations, we show that the route to chaos in a forced two-dimensional flow, confined to a square domain with two no-slip boundaries, essentially follows a Ruelle-Takens scenario. In this flow the kinematic viscosity \nu serves as the control parameter. Its value is decreased in discrete steps, whilst keeping the amplitude of a time-independent external forcing fixed. As such, the sequence encountered in our simulations, within the interval 1/700\geq\nu\geq 1/1700, is S\rightarrow P\rightarrow QP_2\rightarrow QP_3\rightarrow C, where each transition corresponds to a Hopf bifurcation. These motions are analyzed in terms of their fundamental frequencies and a phase-space reconstruction.

[MM.009] Reynolds number dependence of fluid mixing in the partitioned-pipe mixer

Reynolds number dependence of chaotic mixing of fluids in the partitioned-pipe mixer (PPM) is studied by using numerically obtained velocity fields. We found that transport barriers once expand as the Reynolds number, R_e, is increased. Although they tend to disappear with increase in R_e, some of them remain for relatively large R_e in some cases in which a geometrical condition of the system is changed. Existence of the barriers can be related to dimensionality of the velocity field. On mixing in the chaotic region, we found that a high shear region apart from the rigid wall appears as R_e is increased enough. This appearance of a high shear region is expected to enhance mixing efficiency in the chaotic region.

[MM.010] Dynamic Subgrid-Scale Drag Modeling for Turbulent Flow Over Fractal Trees

LES ideas, in particular the dynamic model (Germano et al. 1991), are extended and applied to the problem of SGS drag modeling for flow over trees. The SGS branch drag coefficient is determined as in dynamic LES by enforcing model consistency amongst SGS branches and the smallest resolved branches, and assuming scale invariance of the drag coefficient. The formulation allows for dependence of the drag coefficient on the geometrical configuration of a given branch (e.g. orientation relative to the incoming flow), which results in a linear system for the drag coefficients. Results are obtained by performing LES of high-Re turbulent flow over idealized leafless, self-similar fractal trees as a test case. A posteriori tests show that the dynamic drag coefficients are stable in many cases, although less robust coefficients are associated with trees with a high degree of branching. The total SGS drag force is observed to be a significant fraction of the total drag force on the tree, highlighting the importance of the drag model. Observed nontrivial dependence of the dynamically obtained drag coefficients on branch orientation will also be presented.

Part M of program listing