Bulletin of the American Physical Society
63rd Annual Meeting of the APS Division of Fluid Dynamics
Volume 55, Number 16
Sunday–Tuesday, November 21–23, 2010; Long Beach, California
Session LF: Chaos and Fractals |
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Chair: Jean-Luc Thiffeault, University of Wisconsin-Madison Room: Long Beach Convention Center 103A |
Monday, November 22, 2010 3:35PM - 3:48PM |
LF.00001: On the atmosphere of a moving body Johan Roenby, Hassan Aref We have explored whether a rigid body moving freely with no circulation around it in a two-dimensional ideal fluid can carry a fluid ``atmosphere'' with it in its motion. Somewhat surprisingly, the answer appears to be ``yes''. When the body is elongated and the motion is dominated by rotation, we demonstrate numerically that, indeed, regions of fluid follow the body in its motion. Since there is a double-island structure for the case of pure rotation, as already found by Morton and Darwin many years ago, we see the existence of an atmosphere for the moving body as an example of the stability of Kolmogorov-Arnold-Moser tori. Our observations were reported in {\it Physics of Fluids} {\bf 22} (2010) 057103. The presentation will include animations not published with the paper and some indications of further work. [Preview Abstract] |
Monday, November 22, 2010 3:48PM - 4:01PM |
LF.00002: The phase portrait of aperiodic non-autonomous dynamical systems Ana M. Mancho, Carolina Mendoza Geometry has been a very useful approach for studying dynamical systems. At the basis are Poincar\'e ideas of seeking structures on the phase space that divide it into regions corresponding to trajectories with different dynamical fates. We present a methodology to build global Lagrangian descriptors for arbitrary time dependent flows based on the intrinsic geometrical and physical properties of trajectories. Our new Lagrangian descriptors are applied to flows with general time dependence as those in geophysics. They succeed in detecting simultaneously, with great accuracy, invariant manifolds, hyperbolic and non-hyperbolic flow regions. We analyze convenience of different descriptors from several points of view: regularity conditions requested on the vector field, rate at which the Lagrangian information is achieved and computational performance. Comparisons with other traditional methods such as Finite Time Lyapunov Exponents (FTLE) will be also discussed. [Preview Abstract] |
Monday, November 22, 2010 4:01PM - 4:14PM |
LF.00003: On the cost-effectiveness of mixing optimization Oleg Gubanov, Luca Cortelezzi We consider the problem of estimating the cost-effectiveness of an optimal mixer (Gubanov \& Cortelezzi, {\em J. Fluid Mech.}, vol. 651, 2010), a mixer able to generate a mixture with a desired level of homogenization over a wide range of operating conditions while minimizing the homogenzation time and cost. We generate a family of optimal mixers by extending the formulation of the sine flow (Liu {et al.}, {\em Chaos, Solitons and Fractals}, vol. 4, 1994). We derive the Fourier sine flow, an egg-beater type of flow, which stirs a mixture by blinking velocity fields whose profile is defined as a Fourier sine series. We generate the four lower-level mixers by truncating the Fourier representation of the velocity profile to one, two, three and four modes, respectively. We formulate a constrained optimization problem for the velocity profiles. We use the mix-norm (Mathew {\em et. al}, {\em Physica D}, vol. 211, 2005) as a cost function. We couple profile and protocol optimizations and solve the problem every time the velocity fields are blinked. We compare the homogenization times achieved by the mixers. We show that, unexpectedly, the homogenization time does not decrease monotonically with increasing power input. Our results indicate that mixing optimization is most cost-effective at lower power inputs, it should be avoided in the low-middle range and becomes less attractive for higher power inputs. [Preview Abstract] |
Monday, November 22, 2010 4:14PM - 4:27PM |
LF.00004: Coupled Droplet Oscillators and Symmetry Detectives David Slater, Paul Steen Symmetry detectives offer an automated method to classify the symmetries of solutions to dynamical systems -- we shall illustrate on a coupled-droplet oscillator problem. The primary application of detectives has been the determination of symmetries of attractors as well as the detection of symmetry-changing bifurcations. This is achieved by mapping a trajectory into an appropriate representation space and detecting symmetry by computing distances to fixed point subspaces of subgroups. We utilize symmetry detectives in our analysis of the trajectories of a fourth-order $S_3$ symmetric model of three coupled inviscid liquid droplets. Since there is no dissipation in the model, there are no asymptotically stable attractors. Hence, solutions away from equilibrium are the focus. In particular, we examine trajectories with no initial velocity. Results of the symmetry detective approach are contrasted to a computation of the largest Lyapunov exponent, which indicates whether dynamics are chaotic or quasiperiodic. Both methods can be applied to a grid of initial conditions in an automated fashion. Our results demonstrate a strong correlation between symmetries and nonlinear dynamics. [Preview Abstract] |
Monday, November 22, 2010 4:27PM - 4:40PM |
LF.00005: Trapping of Swimming Particles in Chaotic Fluid Flow Nidhi Khurana, Jerzy Blawzdziewicz, Nicholas T. Ouellette We computationally study the dynamics of active particles suspended in a two-dimensional chaotic flow. The point-like, spherical particles have their own intrinsic velocity, and can therefore break transport barriers (KAM) tori) in the flow. Even a small amount of swimming significantly affects the mixing. However, small but finite values of the swimming speed can lead to a decrease in mixing efficiency, as swimmers can get stuck in traps that form near elliptic islands in the flow field. We study the statistics of trapping times and its effect on transport dynamics. [Preview Abstract] |
Monday, November 22, 2010 4:40PM - 4:53PM |
LF.00006: Rayleigh-Taylor unstable, premixed flames: the transition to turbulence Elizabeth Hicks, Robert Rosner A premixed flame moving against a sufficiently strong gravitational field becomes deformed and creates vorticity. If gravity is strong enough, this vorticity is shed and deposited behind the flame front. We present two-dimensional direct numerical simulations of this vortex shedding process and its effect on the flame front for various values of the gravitational force. The flame and its shed vortices go through the following stages as gravity is increased: no vorticity and a flat flame front; long vortices attached to a cusped flame front; instability of the attached vortices and vortex shedding (Hopf bifurcation); disruption of the flame front by the shed vortices, causing the flame to pulsate; loss of left/right symmetry (period doubling); dominance of Rayleigh-Taylor instability over burning (torus bifurcation); and, finally, complex interactions between the flame front and the vortices. We measure the subsequent wrinkling of the flame front by computing its fractal dimension and also measure mixing behind the flame front by computing the finite-time Lyapunov exponents. [Preview Abstract] |
Monday, November 22, 2010 4:53PM - 5:06PM |
LF.00007: Characterizing changes in topological entropy via break up of almost-invariant sets Piyush Grover, Mark Stremler, Shane Ross, Pankaj Kumar In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We build upon our earlier work that showed the first application of the TNCT to braiding of almost-invariant sets (AIS). AIS in a fluid flow are regions with high local residence time that can act as stirrers or 'ghost rods'. In the present work, we discuss the break up of the AIS as a parameter value is changed, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes. Hence we make the case that a topological analysis based on spatio-temporal braiding of almost-invariant sets can be used for analyzing chaos in fluid flows. [Preview Abstract] |
Monday, November 22, 2010 5:06PM - 5:19PM |
LF.00008: On the relationship between stretching and homogenization in chaotic Stokes flows Mohsen Gheisarieha, Mark Stremler It is well known that chaotic particle trajectories can be generated in laminar flows by deterministic, time-periodic velocity fields. The exponential stretching of material lines in these flows can be quantified using the `topological entropy'. This measure of chaos is useful because in some circumstances it can be predicted mathematically using very limited information about the flow. We consider the relationship between this stretching and the mixing produced in these flows, which we evaluate by considering homogenization of a passive scalar. We study two different time-dependent, two-dimensional Stokes flow systems as examples: a double-lid-driven cavity flow and a 3-rod stirring system in a cylindrical domain. We will discuss the correspondence between topological entropy and decay in the variance of scalar concentration for varying parameters in these flows. [Preview Abstract] |
Monday, November 22, 2010 5:19PM - 5:32PM |
LF.00009: Eigenmode analysis of scalar transport in distributive mixing Patrick Anderson, M.K. Singh, Michel Speetjes In this study we explore the spectral properties of the distribution matrices of the mapping method and its relation to the distributive mixing of passive scalars. The spectral decomposition of these matrices constitutes a discrete approximations to the eigenmodes of the continuous advection operator in periodic flows. The asymptotic state of a fully-chaotic mixing flow is dominated by the eigenmode corresponding with the eigenvalue closest to the unit circle. This eigenvalue determines the decay rate; its eigenvector determines the asymptotic mixing pattern. The closer this eigenvalue value is to the origin, the faster is the homogenization by the chaotic mixing. Its magnitude can be used as a quantitative mixing measure for comparison of different mixing protocols. Eigenvalues on the unit circle are qualitative indicators of inefficient mixing; the properties of its eigenvectors enable isolation of the non-mixing zones. Results are demonstrated of two different prototypical mixing flows: the time-periodic sine flow and the spatially-periodic partitioned-pipe mixer. [Preview Abstract] |
Monday, November 22, 2010 5:32PM - 5:45PM |
LF.00010: Diffusion of adiabatic invariants and mixing in Stokes flows Alimu Abudu, Dmitri Vainchtein We discuss a quantitative long-term theory of mixing due to scatterings on resonances in 3-D near-integrable flows. As a model problem we use the flow in the annulus between two coaxial elliptic counter-rotating cylinders. We illustrate that the resonance phenomena cause the jumps of adiabatic invariants and mixing. We show that the resulting mixing can be described in terms of a single 1-D diffusion-type for the probability distribution function. Parameters of the diffusion equation are defined by the averaged statistics of a single passage through resonance. [Preview Abstract] |
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