Bulletin of the American Physical Society
2006 59th Annual Meeting of the APS Division of Fluid Dynamics
Sunday–Tuesday, November 19–21, 2006; Tampa Bay, Florida
Session GK: Chaos and Fractals |
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Chair: John Christos Vassilicos, Imperial College Room: Tampa Marriott Waterside Hotel and Marina Meeting Room 4 |
Monday, November 20, 2006 10:30AM - 10:43AM |
GK.00001: Topological Optimization of Rod Mixers Matthew D. Finn, Jean-Luc Thiffeault Stirring of fluid with moving rods is necessary in many practical applications to achieve homogeneity. These rods are topological obstacles that force stretching of fluid elements. The resulting stretching and folding is commonly observed as filaments and striations, and is a precursor to mixing. In a space-time diagram, the trajectories of the rods form a braid [1], and the properties of this braid impose a minimal complexity in the flow. We discuss how optimal mixing protocols can be obtained by a judicious choice of braid, and how these protocols can be implemented using simple gearing [2].\\[12pt] [1] P. L. Boyland, H. Aref, and M. A. Stremler, JFM 403, 277 (2000).\\[8pt] [2] J.-L. Thiffeault and M. D. Finn, http://arxiv.org/nlin/0603003 [Preview Abstract] |
Monday, November 20, 2006 10:43AM - 10:56AM |
GK.00002: Scalings and Decays of Fractal-generated turbulence John Christos Vassilicos, Daryl Hurst 21 planar fractal grids from 3 fractal families have been used in 2 wind tunnels to generate turbulence. This turbulence and its homogeneity, isotropy and decay properties are strongly dependent on the grid's fractal dimension $D_{f}\le 2$, the effective mesh size $M_{eff}$ (which we introduce and define) and the ratio $t_r$ of largest to smallest bar thicknesses, $t_{r}=t_{max}/t_{min}$. With blockage ratios as low as $\sigma = 25$\%, these grids generate turbulent flows with higher turbulence intensities and Reynolds numbers than higher blockadge ratio classical grids in similar wind tunnels and wind speeds $U$. The scalings and decay of the turbulence intensity $u'/U$ in the $x$-direction along the tunnel centre line are (in terms of the normalised pressure drop $C_{\Delta P}$ and with similar results for $v'/U$ and $w'/U$): (i) for fractal cross grids ($D_{f}=2$), $(u'/U)^{2}= t_{r}^{2}C_{\Delta P} fct (x/M_{eff})$; (ii) for fractal I grids, $(u'/U)^{2}= t_{r} (T/L_{max})^{2} C_{\Delta P} fct (x/M_{eff})$ where $T$ is the tunnel width and $L_{max}$ is the maximum bar length on the grid; (iii) for $D_{f}=2$ fractal square grids, the turbulence builds up till a distance $x_{peak}$ from the grid where the turbulence intensity peaks and then decays exponentially, $u'^{2}=u'^{2}_{peak} exp[-(x-x_{peak})/l_{turb}]$ where $u'^{2}_{peak}$ increases linearly with $t_r$, $x_{peak}\propto t_{min}T/L_{min}$ ($L_{min}$ being the minimum bar length on the grid) and $l_{turb}\propto \lambda^{2}U/\nu$ ($\nu$ being the air's kinematic viscosity and $\lambda$ being the Taylor microscale); $\lambda$ and the longitudinal/lateral length-scales remain approximately constant during decay at $x\gg x_{peak}$. [Preview Abstract] |
Monday, November 20, 2006 10:56AM - 11:09AM |
GK.00003: Topological chaos in cavities and channels Jie Chen, Mark A. Stremler Moving three or more stirrers around in a two-dimensional fluid domain can generate topological chaos, that is, chaos that cannot be removed by continuous deformation of the fluid with the boundaries and stirrers held fixed. Those stirrer motions that generate topological chaos are determined using the Thurston–Nielsen classification theorem, which also predicts a lower bound on the fluid stretching rate. Equivalent motions can be produced in a lid-driven cavity without stirrers by using periodic, piecewise constant motion of the top and/or bottom boundaries. We explore the properties of topological chaos in lid-driven cavities. Lid-driven cavity flow can also be combined with rectangular Poiseuille flow as a model of either pressure-driven flow in a channel with surface grooves or electro-osmotic flow in a channel with variations in surface potential. We demonstrate that this combination can be used to generate topological chaos in three-dimensional steady channel flow. [Preview Abstract] |
Monday, November 20, 2006 11:09AM - 11:22AM |
GK.00004: Transport in an oscillating/drifting vortex chain and L\'evy networks Lily Beauvilliers, Tom Solomon We present numerical studies of diffusive and superdiffusive transport in a chain of alternating vortices. If the chain oscillates laterally (with velocity amplitude $v_o$), mixing is chaotic and transport is diffusive with mixing predominately between adjacent vortices. If the vortices drift as well with drift velocity $v_d$, transport can be superdiffusive (if $v_d > v_o$), characterized by L{\'e}vy flights that allow fluid to travel several vortices in a short period of time. We investigate in detail the transition between normal and superdiffusive transport in this system, studying in particular the statistics of the flights. We also determine the relative coupling strengths (due to mixing) between vortices as time increases. Ultimately, our goal is to develop a model of a ``L\'evy network'' for oscillatory process occurring in extended fluid systems. [Preview Abstract] |
Monday, November 20, 2006 11:22AM - 11:35AM |
GK.00005: The effects of superdiffusive transport on front Mollie Schwartz, Tom Solomon We present experimental studies of the propagation of chemical fronts in an annular chain of alternating vortices. The vortex chain can be controlled to both drift (with velocity $v_d$) and oscillate (with velocity amplitude $v_o$) in the azimuthal direction. Transport in this flow is diffusive if $v_d < v_o$ and superdiffusive if $v_d > v_o$. The chemical front is produced using the excitable state of the Ruthenium-catalyzed Belousov-Zhabotinsky reaction. Previous experiments\footnote{Europhys. Lett. {\bf 69}, 819 (2005); Phys. Rev. E {\bf 72}, 046204 (2005).} have shown that the fronts often mode-lock to the external forcing for pure oscillatory time dependence ($v_d = 0$). We investigate the limits of this mode-locking behavior as the drift is increased, studying in particular any changes that occur when the transport becomes superdiffusive. An important parameter in these studies is the ratio $\eta = U/v_{rd}$ between the maximum flow velocity $U$ and the reaction-diffusion (no-flow) front velocity $v_{rd}$. We investigate changes in the observed behavior as $\eta$ is increased, increasing the relative importance of fluid advection in the advection-reaction-diffusion process. [Preview Abstract] |
Monday, November 20, 2006 11:35AM - 11:48AM |
GK.00006: Effect of chaos on transport to reactive boundaries from 3D flows in microfluidic systems Joseph D. Kirtland, Abraham D. Stroock Microfluidic systems are generally characterized by laminar, uniaxial flow and slow, purely diffusive mixing in the cross section, resulting in poor bulk mixing and limited mass transfer to reactive boundaries. It is known that the introduction of chaotic particle trajectories and large chaotic invariant sets results in improved mixing in the bulk, but the effect of chaos on transfer to boundaries is not well documented. We study three dimensional flows produced by grooves patterned along one or more of the microchannel walls, as in the Staggered Herringbone Mixer (SHM). Mass transfer to reactive boundaries can be substantially increased by the introduction of these 3D flows, and chaos can be crucial in ensuring sustained increases in mass transfer at high flow rates and large axial distances. We will present numerical and experimental results that demonstrate the requirements for producing this desirable behavior in terms of the chaotic characteristics of the flow and the connection of the principal chaotic invariant set to the reactive boundary. [Preview Abstract] |
Monday, November 20, 2006 11:48AM - 12:01PM |
GK.00007: Stability and Robustness against System Parameter Drift of Algorithms for the Control of Low Dimensional Chaos Thomas Olsen, Kjell Schroder, Katherine Carriker, Bonita Squires, Kara Yedinak, Richard Wiener Previously, we have demonstrated that the chaotic formation of Taylor-Vortex pairs in Modified Taylor-Couette flow with hourglass geometry may be controlled by the application of the Recursive Proportional Feedback algorithm\footnote{Rollins \textit{et al}, Phys. Rev. E \textbf{47}, R780 (1993).}$^{,}$\footnote{Wiener \textit{et al}, Phys. Rev. Lett. \textbf{83}, 2340 (1999).}. We have developed analogous algorithms that may be more effective in changing environments, where system parameters may drift. We present numerical simulations and analysis to determine the stability and robustness of these new algorithms against such drift. [Preview Abstract] |
Monday, November 20, 2006 12:01PM - 12:14PM |
GK.00008: Recurrence Analysis of Fluid Molecular Dynamics Simulation Theodoros Karakasidis, Athanasios Fragkou, Antonios Liakopoulos We present a Recurrence Quantification Analysis (RQA) of instantaneous temperature records obtained using molecular dynamics simulations of Lennard-Jones fluids. Simulations were performed at various system (temperature and density) states. The instantaneous temperature was recorded as function of time and consequently analyzed using Recurrence Quantification Analysis. We calculated several RQA variables such us determinism, maximum line, trapping time as functions of the system temperature and density. By comparison with other time series analysis methods it is demonstrated that RQA is useful in extracting significant characteristics of the system dynamics. The existence of vibrational and diffusional motion of the fluid atoms are reflected on the results of the recurrence analysis and related to physical quantities such as Mean Square Displacement (MSD) of the atoms. White bands represent diffusion events and thus are larger at low system densities. Determinism ({\%}deter) becomes smaller as temperature or density increases since collisions of atoms are then more frequent. [Preview Abstract] |
Monday, November 20, 2006 12:14PM - 12:27PM |
GK.00009: Estimating the State of Large Spatio-Temporally Chaotic Systems: Application to a Rayleigh-Benard Convection Experiment Matthew Cornick, Brian Hunt, Edward Ott, Michael Schatz, Huseyin Kurtuldu Data Assimilation (DA) refers to the estimation of a dynamical system's state from the combined knowledge of past observations (possibly incomplete and noisy) and knowledge of an approximate model for the systems time evolution. Here we consider DA for spatio-temporally chaotic systems, and, in particular, we study the Local Ensemble Kalman Filter DA technique. We have applied this technique to Rayleigh-Benard convection undergoing spiral defect chaos. Using a system model (Boussinesq equations) and time series of noisy shadowgraphs we obtain estimates of the temperature and velocity field everywhere in a convection cell. This technique provides us with an indirect measurement of quantities previously inaccessible such as mean flow. We use the method to form initial conditions which are then used to produce a model forecast and compared to experimental data. In addition, we show how the technique can be extended to estimate fluid parameters such as the Rayleigh number. [Preview Abstract] |
Monday, November 20, 2006 12:27PM - 12:40PM |
GK.00010: Test of the Fluctuation Relation in compressible turbulence on a free surface Mahesh Bandi, John Cressman, Walter Goldburg The statistics of lagrangian velocity divergence are studied for an assembly of particles in compressible turbulence on a free surface. Under an appropriate definition of entropy, the two-dimensional velocity divergence of a particle trajectory represents the local entropy rate, a random variable. The statistics of this rate are shown to be in agreement with the steady-state fluctuation relation of Gallavotti and Cohen over a limited range of averaging times. The probability distribution functions obtained in this analysis exhibit features different from those observed in previous experimental tests of the fluctuation relation. [Preview Abstract] |
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