Bulletin of the American Physical Society
68th Annual Meeting of the APS Division of Fluid Dynamics
Volume 60, Number 21
Sunday–Tuesday, November 22–24, 2015; Boston, Massachusetts
Session M5: Nonlinear Dynamics: General |
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Chair: Kevin Mitchell, University of California, Merced Room: 104 |
Tuesday, November 24, 2015 8:00AM - 8:13AM |
M5.00001: Topological entropy and symbolic dynamics for three-dimensional fluid mixing Kevin Mitchell, Bryan Maelfeyt, Joshua Arenson Topological entropy provides an important metric of mixing in two-dimensional fluid flows; it has led to a quantification of mixing for various periodic stirring protocols and other chaotic flows. In this context, the topological entropy can be viewed as the exponential growth rate of a material line. In this talk, we explain how one can compute an analogous entropy for topological mixing in three-dimensional flows. This entropy amounts to an exponential growth rate in the size of material sheets. Our approach involves the extraction of symbolic dynamics from the intersections of two-dimensional stable and unstable manifolds of the flow field. We illustrate our theory with a mathematical model of a chaotic ring vortex. [Preview Abstract] |
Tuesday, November 24, 2015 8:13AM - 8:26AM |
M5.00002: Passive scalars chaotic dynamics induced by two vortices in a two-layer geophysical flow with shear and rotation Eugene Ryzhov Vortex motion in shear flows is of great interest from the point of view of nonlinear science, and also as an applied problem to predict the evolution of vortices in nature. Considering applications to the ocean and atmosphere, it is well-known that these media are significantly stratified. The simplest way to take stratification into account is to deal with a two-layer flow. In this case, vortices perturb the interface, and consequently, the perturbed interface transits the vortex influences from one layer to another. Our aim is to investigate the dynamics of two point vortices in an unbounded domain where a shear and rotation are imposed as the leading order influence from some generalized perturbation. The two vortices are arranged within the bottom layer, but an emphasis is on the upper-layer fluid particle motion. Point vortices induce singular velocity fields in the layer they belong to, however, in the other layers of a multi-layer flow, they induce regular velocity fields. The main feature is that singular velocity fields prohibit irregular dynamics in the vicinity of the singular points, but regular velocity fields, provided optimal conditions, permit irregular dynamics to extend almost in every point of the corresponding phase space. [Preview Abstract] |
Tuesday, November 24, 2015 8:26AM - 8:39AM |
M5.00003: Maximal stochastic transport in the Lorenz equations Sahil Agarwal, John Wettlaufer We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh-Benard convection the upper bounds are for heat transport versus Rayleigh number. As might be expected the stochastic upper bounds are larger than the deterministic counterpart of Souza and Doering (2015), but their variation with noise amplitude exhibits surprising behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude. However, in the chaotic regime this monotonicity is lost; at a particular Rayleigh number the bound may increase or decrease with noise amplitude. The origin of this behavior is the coupling between the noise and unstable periodic orbits. This is confirmed by examining the close returns plots of the full solutions to the stochastic equations. Finally, we note that these solutions demonstrate that the effect of noise is equivalent to the effect of chaos. [Preview Abstract] |
Tuesday, November 24, 2015 8:39AM - 8:52AM |
M5.00004: Flow primitives to manipulate the dynamics of inertial particles Senbagaraman Sudarsanam, Phanindra Tallapragada The nonlinear dynamics of inertial particles in many microfluidic settings occurs in flows whose main feature is cell-like structures created due to specific distributions of vorticity. Examples include Dean vortices, Taylor-Couette vortices and streaming vortex cells. To obtain insights into the motion of inertial particles in such complex flows, in possibly confined domains, we develop certain flow primitives generated by point-vortex like structures. We model the the motion of spherical inertial particles by the Maxey-Riley equation. With this governing equation the inertial particles demonstrate sensitive dependence on size and initial conditions in the fluid flow generated by the flow primitives. Size based particle segregation, trapping particles at the centers of vortex cores or on limit cycles is shown to be possible. We demonstrate some of these phenomena using Lagrangian coherent structures (LCS). [Preview Abstract] |
Tuesday, November 24, 2015 8:52AM - 9:05AM |
M5.00005: Neimark-Sacker bifurcation and evidence of chaos in a discrete dynamical model of walkers Aminur Rahman Bouncing droplets on a vibrating fluid bath can exhibit wave-particle behavior, such as being propelled by the waves they generate. These droplets seem to walk across the bath, and thus are dubbed \emph{walkers}. These walkers can exhibit exotic dynamical behavior which give strong indications of chaos, but many of the interesting dynamical properties have yet to be proven. In recent years discrete dynamical models have been derived and studied numerically. We prove the existence of a Neimark-Sacker bifurcation for a variety of eigenmode shapes of the waves from one such model. Then we reproduce numerical experiments and produce new numerical experiments and apply our theorem to the test functions used for that model in addition to new test functions. Further evidence of chaos is shown by numerically studying a global bifurcation. [Preview Abstract] |
Tuesday, November 24, 2015 9:05AM - 9:18AM |
M5.00006: Topology of three-dimensional steady cellular flow in a two-sided lid-driven cavity Francesco Romano, Stefan Albensoeder, Hendrik Kuhlmann The topology of a laminar three-dimensional flow in a rectangular lid-driven cavity is investigated. A two-dimensional flow in the (x,y) plane is driven by two facing walls moving in opposite directions with equal velocities. The cross-sectional aspect ratio in the (x,y)-plane is 1.7. The cavity is assumed to be infinitely extended in the spanwise (z) direction. At a Reynolds number $Re = 212$ the flow becomes three-dimensional via an elliptic instability resulting in a steady cellular flow with spanwise half-period of $\lambda_z/2=1.365$. The nonlinear steady flows at $Re=500$ and $700$ are accurately computed using a Chebyshev spectral collocation method. The flow is analyzed with respect to regular (KAM tori) and chaotic regions. The shape of the KAM tori and associated closed streamlines as well as their dependence on the Reynolds number is discussed. Further considerations will be given to the symmetry, period and minimum distance between the KAM tori and the cavity walls. [Preview Abstract] |
Tuesday, November 24, 2015 9:18AM - 9:31AM |
M5.00007: Using Persistent Homology to Describe Rayleigh-B\'enard Convection Jeffrey Tithof, Balachandra Suri, Mu Xu, Miroslav Kramar, Rachel Levanger, Konstantin Mischaikow, Mark Paul, Michael Schatz Complex spatial patterns that exhibit aperiodic dynamics commonly arise in a wide variety of systems in nature and technology. Describing, understanding, and predicting the behavior of such patterns is an open problem. We explore the use of persistent homology (a branch of algebraic topology) to characterize spatiotemporal dynamics in a canonical fluid mechanics problem, Rayleigh B\'enard convection. Persistent homology provides a powerful mathematical formalism in which the topological characteristics of a pattern (e.g. the midplane temperature field) are encoded in a so-called persistence diagram. By applying a metric to measure the pairwise distances across multiple persistence diagrams, we can quantify the similarities between different states in a time series. Our results show that persistent homology yields new physical insights into the complex dynamics of large spatially extended systems that are driven far-from-equilibrium. [Preview Abstract] |
Tuesday, November 24, 2015 9:31AM - 9:44AM |
M5.00008: Characterizing mixing in time periodic planar flows through the topology of almost cyclic sets Pradeep Rao, Mark Stremler, Shane Ross Almost Invariant Sets (AIS) can be used to identify coherent structures that move as Almost Cyclic Sets (ACS) for time-periodic planar flows. The relative motion of the ACS identified using the second most dominant eigenvector of the reversible matrix obtained from the discretized Perron-Frobenius operator provides a reduced order model for quantifying transport. This has been shown through the application of the Thurston-Nielsen classification theorem to the topology of the motions of the ACS for certain time periodic lid driven cavity Stokes flows. We extend this notion to more general flows with inertial effects. We provide a recipe for identifying the ACS whose dynamics provide a reduced order model for predicting mixing efficiency for such flows. [Preview Abstract] |
Tuesday, November 24, 2015 9:44AM - 9:57AM |
M5.00009: Computing the Evans function via solving a linear boundary value ODE Colin Wahl, Rose Nguyen, Nathaniel Ventura, Blake Barker, Bjorn Sandstede Determining the stability of traveling wave solutions to partial differential equations can oftentimes be computationally intensive but of great importance to understanding the effects of perturbations on the physical systems (chemical reactions, hydrodynamics, etc.) they model. For waves in one spatial dimension, one may linearize around the wave and form an Evans function - an analytic Wronskian-like function which has zeros that correspond in multiplicity to the eigenvalues of the linearized system. If eigenvalues with a positive real part do not exist, the traveling wave will be stable. Two methods exist for calculating the Evans function numerically: the exterior-product method and the method of continuous orthogonalization. The first is numerically expensive, and the second reformulates the originally linear system as a nonlinear system. We develop a new algorithm for computing the Evans function through appropriate linear boundary-value problems. This algorithm is cheaper than the previous methods, and we prove that it preserves analyticity of the Evans function. We also provide error estimates and implement it on some classical one- and two-dimensional systems, one being the Swift-Hohenberg equation in a channel, to show the advantages. [Preview Abstract] |
Tuesday, November 24, 2015 9:57AM - 10:10AM |
M5.00010: Learning Flow Regimes from Snapshot Data Maziar Hemati Fluid flow regimes are often categorized based on the qualitative patterns observed by visual inspection of the flow field. For example, bluff body wakes are traditionally classified based on the number and groupings of vortices shed per cycle (e.g., 2S, 2P, P+S), as seen in snapshots of the vorticity field. Subsequently, the existence and nature of these identified flow regimes can be explained through dynamical analyses of the fluid mechanics. Unfortunately, due to the need for manual inspection, the approach described above can be impractical for studies that seek to learn flow regimes from large volumes of numerical and/or experimental snapshot data. Here, we appeal to established techniques from machine learning and data-driven dynamical systems analysis to automate the task of learning flow regimes from snapshot data. Moreover, by appealing to the dynamical structure of the fluid flow, this approach also offers the potential to reveal flow regimes that may be overlooked by visual inspection alone. Here, we will introduce the methodology and demonstrate its capabilities and limitations in the context of several model flows. [Preview Abstract] |
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