Bulletin of the American Physical Society
66th Annual Meeting of the APS Division of Fluid Dynamics
Volume 58, Number 18
Sunday–Tuesday, November 24–26, 2013; Pittsburgh, Pennsylvania
Session L35: Chaos, Fractals and Dynamical Systems III: Miscellaneous Topology and Model Characterization |
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Chair: Steve Brunton, University of Washington Room: 406 |
Monday, November 25, 2013 3:35PM - 3:48PM |
L35.00001: Describing Chaotic Dynamics in Experimental Rayleigh-B\'enard Convection Using Persistent Homology Theory Jeffrey Tithof, Balachandra Suri, Miroslav Kramar, Vidit Nanda, Mu Xu, Mark Paul, Konstantin Mischaikow, Michael Schatz We employ a new technique for describing the dynamics of spatiotemporal chaos in Rayleigh-B\'enard convection. We collect shadowgraph images of multiple time series of weakly chaotic flows, each starting from similar initial conditions which we impose using a laser. We then encode the topological characteristics of each frame into a so-called persistence diagram, measure the distance across all diagrams, and study the dynamical behavior. Results are compared to similar analyses of simulation data. This new methodology provides unique insight into the time evolution of this dynamical system and the chaotic evolution across separate runs, in both experiment and simulation. [Preview Abstract] |
Monday, November 25, 2013 3:48PM - 4:01PM |
L35.00002: Analyzing the dynamics of pattern formation in the space of persistence diagrams Miroslav Kramar, Konstantin Mischaikow, Michael Schatz, Jeffrey Tithof, Mark Paul, Mu Xu Persistence diagrams are a relatively new topological tool for describing and quantifying complicated patterns in a simple but meaningful way. We will demonstrate this technique on patterns appearing inÂ Rayleigh-Benard convection. This procedure allows us to transform experimental or numerical data from experiment or simulation into a point cloud in the space of persistence diagrams. There are a variety of metrics that can be imposed on the space of persistence diagrams. By choosing different metrics one can interrogate the pattern locally or globally, which provides deeperÂ insight into the dynamics of the process of pattern formation. Because the quantification is being done in the space of persistence diagrams this technique allows us to compare directly numerical simulations with experimental data. [Preview Abstract] |
Monday, November 25, 2013 4:01PM - 4:14PM |
L35.00003: Burning Invariant Manifold Theory and the Bipartite Digraph Representation of Generalized Dynamical System Formed by One-way Barriers John Li, John Mahoney, Kevin Mitchell The recently developed \emph{Burning Invariant Manifold} (BIM) theory took a dynamical system approach to understand front propagation in \emph{Advection-Reaction-Diffusion} systems and successfully predicted both the short-term and asymptotic front behavior by finding the unstable BIMs which act as barriers to front propagation. Unlike separatrices in traditional dynamical system being two-way barriers, the BIMs are one-way barriers. This asymmetry gives rise to a much richer dynamical behavior than traditional dynamical systems. Through numerical simulations, we found that the stable BIMs are the basin boundaries. Based on the properties of BIM theory, we further derived a theory to investigate a dynamical system consists of one-way barriers and the cooperative behavior of these barriers. This theory reveals the global structure of both stable and unstable BIMs by first using a systematic algorithm to convert the flow to a bipartite digraph and then extracting information of the steady states of fronts and corresponding basins of attraction from the digraph. [Preview Abstract] |
Monday, November 25, 2013 4:14PM - 4:27PM |
L35.00004: Experimental studies of mixing barriers and reaction fronts in a steady, three-dimensional flow Harrison Mills, Tom Solomon We present experiments studying chaotic mixing and front propagation in a steady, three-dimensional (3D) flow composed of nested vortices. Passive mixing is characterized by tracking almost-neutral, fluorescent tracer particles in the flow. A fluorescent dye is also used, and the spreading of this dye is monitored with a scanning laser system and a camera that images a stack of cross-sectional images. Using both methods, we find evidence of both ordered and chaotic regions of mixing in the flow. We also present preliminary results of studies of behavior of the Ruthenium-catalyzed, excitable Belousov-Zhabotinsky chemical reaction in this flow. Propagating fronts of this reaction are characterized in 3D by the same laser-scanning system. The goal of these experiments is to determine barriers to front propagation and to compare these reaction barriers to the barriers observed for passive mixing in the same flow. Ultimately, a generalization of the {\em burning invariant manifold} theory\footnote{J. Mahoney, D. Bargteil, M. Kingsbury, K. Mitchell and T. Solomon, Europhys. Lett. {\bf 98}, 44005 (2012).} to 3D will be used to explain these barriers. [Preview Abstract] |
Monday, November 25, 2013 4:27PM - 4:40PM |
L35.00005: Pinning of reaction fronts by burning invariant manifolds Peter Megson, Tom Solomon We present experiments that study the behavior of the excitable Belousov-Zhabotinsky chemical reaction in a translating, regular array of vortices. In a reference frame moving with the translating vortices, the flow is equivalent to a stationary vortex array with an imposed uniform wind. Under a wide range of wind speeds, reaction fronts pin to the vortex flow, neither propagating forward against the wind nor being blown back. We explain this pinning behavior with the use of a recent theory\footnote{J. Mahoney, D. Bargteil, M. Kingsbury, K. Mitchell and T. Solomon, Europhys. Lett. {\bf 98}, 44005 (2012).} of {\em burning invariant manifolds} (BIMs) that act as one-way barriers against any propagating reaction front. When the reaction fronts are pinned, several BIMs combine to form an extended barrier that determines the shape of the pinned fronts. The location of the BIMs are calculated numerically with an analytical approximation of the velocity field and are compared with experimental images of the pinned fronts. We also study transient behavior that helps elucidate the one-way nature of the BIMs. [Preview Abstract] |
Monday, November 25, 2013 4:40PM - 4:53PM |
L35.00006: Deterministic Aperiodic Sickle Cell Blood Flows Louis Atsaves, Wesley Harris In this paper sickle cell blood flow in the capillaries is modeled as a hydrodynamical system. The hydrodynamical system consists of the axisymmetric unsteady, incompressible Navier-Stokes equations and a set of constitutive equations for oxygen transport. Blood cell deformation is not considered in this paper. The hydrodynamical system is reduced to a system of non-linear partial differential equations that are then transformed into a system of three autonomous non-linear ordinary differential equations and a set of algebraic equations. We examine the hydrodynamical system to discern stable/unstable, periodic/nonperiodic, reversible/irreversible properties of the system. The properties of the solutions are driven in large part by the coefficients of the governing system of equations. These coefficients depend on the physiological properties of the sickle cell blood. The chaotic nature of the onset of crisis in sickle cell patients is identified. [Preview Abstract] |
Monday, November 25, 2013 4:53PM - 5:06PM |
L35.00007: Bifurcation analysis of an oscillating cylinder wake Matthew Chu Cheong, Jonathan Tu, Clarence Rowley The flow past a transversely oscillating cylinder gives rise to distinct vortex patterns in the wake, with the particular pattern depending on Reynolds number, Strouhal number, and reduced frequency. In this work, we perform a numerical bifurcation analysis of the transitions between 2S, P+S, and disordered wakes at Reynolds numbers 100. Due to the high dimensionality of fluid flow simulations, standard tools such as AUTO are not applicable. Instead, we turn to Krylov-subspace-based algorithms. The coherent wake patterns (2S, P+S) are stable periodic orbits whose common period is that of the forced oscillation. To identify bifurcations, we perform stability analyses of the Poincare map by stroboscopically sampling the flow. We find the following bifurcations as the reduced frequency is held constant and the Strouhal number is increased : (1) the transition from a 2S wake to a P+S wake is a supercritical pitchfork bifurcation, (2) the transition from a P+S wake back to a 2S wake is another supercritical pitchfork bifurcation, and (3) the transition from a 2S wake to a disorganized wake is a torus bifurcation. Consistent with these bifurcations, we confirm the existence of unstable 2S wakes at Strouhal numbers where P+S and disordered wakes are observed. [Preview Abstract] |
Monday, November 25, 2013 5:06PM - 5:19PM |
L35.00008: Probing the dynamics of Rayleigh-B\'{e}nard convection using numerical simulations for the conditions of experiment Mu Xu, Jeffrey Tithof, Miro Kramar, Balachandra Suri, Vidit Nanda, Michael Schatz, Konstantin Mischaikow, Mark Paul We present results from large-scale parallel numerical simulations of Rayleigh-B\'{e}nard convection for the precise conditions of experiment. We are interested in cylindrical convection domains of moderate aspect ratio with a Prandtl number of order 1. We compute the leading order Lyapunov vector and Lyapunov exponent and use these to quantify the dynamics. We explore time periodic dynamics and also the breakdown of patterns with prescribed initial conditions toward weakly chaotic dynamics. We directly compare our results from numerical simulation with experimental measurements where possible. The numerics yield physical insights into the spatiotemporal dynamics of convection that we can use to connect with ideas from topology such as persistence diagrams. [Preview Abstract] |
Monday, November 25, 2013 5:19PM - 5:32PM |
L35.00009: Chaotic flow and the finite-time Lyapunov exponent: Competitive autocatalytic reactions in advection-reaction-diffusion systems Richard M. Lueptow, Conor P. Schlick, Paul B. Umbanhowar, Julio M. Ottino We investigate chaotic advection and diffusion in competitive autocatalytic reactions. To study this subject, we use a computationally efficient method for solving advection-reaction-diffusion equations for periodic flows using a mapping method with operator splitting. In competitive autocatalytic reactions, there are two species, B and C, which both react autocatalytically with species A (A$+$B$\to $2B and A$+$C$\to $2C). If there is initially a small amount of spatially localized B and C and a large amount of A, all three species will be advected by the velocity field, diffuse, and react until A is completely consumed and only B and C remain. We find that the small scale interactions associated with the chaotic velocity field, specifically the local finite-time Lyapunov exponents (FTLEs), can accurately predict the final average concentrations of B and C after the reaction is complete. The species, B or C, that starts in the region with the larger FTLE has, with high probability, the larger average concentration at the end of the reaction. If species B and C start in regions having similar FTLEs, their average concentrations at the end of the reaction will also be similar. Funded by NSF Grant CMMI-1000469. [Preview Abstract] |
Monday, November 25, 2013 5:32PM - 5:45PM |
L35.00010: On a novel approach to anomalous transport in turbulent fluid and plasma Dhurjati Prasad Datta New nonclassical self similar intermediate asymptotics considered recently [1,2] in the context of linear differential equations are shown to have interesting applications in offering a novel explanation of the origin of anomalous transport phenomena in turbulent flows in fluids and plasma devices. The intermediate asymptotics, in the late time or in the inviscid limit, conspire to produce smooth multifractal measures on a turbulent fluid medium leading naturally to generation of stretched Gaussian distributions for passive scalar tracer concentration from the turbulent, integral order, advection-diffusion equation. Such heavy tailed stretched Gaussian distributions can explain the observed anomalous scaling of the average and mean square displacements of tracer particles in a turbulent medium.We also point out that the present novel mechanism for generation of multifractal measure can actually be interpreted as a new class of instabilities leading to turbulence. \\[4pt] [1] D. P. Datta, On a novel signature of late time asymptotics: a new route to nonlinearity, (2013), Communicated.\\[0pt] [2] D. P. Datta, Novel Late Time Asymptotics: Applications to Anomalous Transport in Turbulent Flows, REDS,(2013), accepted. [Preview Abstract] |
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