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 H35: Chaos, Fractals, and Dynamical Systems II: Analysis, Prediction, and Control |
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Chair: George Haller, Eidgenossische Technische Hochschule Zurich Room: 406 |
Monday, November 25, 2013 10:30AM - 10:43AM |
H35.00001: Novel sampling strategies for dynamic mode decomposition Jonathan H. Tu, Dirk M. Luchtenburg, Clarence W. Rowley, Steven L. Brunton, J. Nathan Kutz Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. We present a theoretical framework that extends the understanding of DMD to nonsequential, potentially rank-deficient, time series and strengthens the connections between DMD and Koopman operator theory. This is in contrast to existing DMD theory, which deals primarily with sequential time series for which the snapshot (measurement) size is much larger than the number of snapshots (measurements). We demonstrate the benefits of applying DMD to nonsequential time series using two examples. First, we sample the simulated flow past a two-dimensional cylinder nonuniformly in time. The result is a more efficient DMD computation, with little effect on the accuracy of the dominant DMD modes and eigenvalues. Next, we combine particle image velocimetry data from multiple runs of a bluff-body wake experiment in a single DMD computation. This greatly mitigates the effects of noise and more clearly isolates the dominant modes. [Preview Abstract] |
Monday, November 25, 2013 10:43AM - 10:56AM |
H35.00002: Effects of small noise on the DMD/Koopman spectrum Shervin Bagheri Koopman modes and Dynamic Mode Decomposition (DMD) have quickly become popular tools for extracting coherent structures associated with different frequencies from both (nonlinear) numerical and experimental flows. It is often expected (see Rowley et al, JFM, vol 641, 2009) that all the eigenvalues have zero growth rate (e.g. that they are located on the unit circle). However, in practice parabolic shapes and branches are observed in nearly all DMD spectra, and it is often the case that the tails of the parabolas are sensitive to the quality of the data set. In this talk, we provide a theoretical explanation for this parabolic form of the spectrum, and show that it arises due to the presence of noise. We show analytically that the presence of noise induces a damping on the eigenvalues, which increases quadratically with the frequency, and linearly with the non-normality of the linearized (Floquet) system. Thus the location of Koopman eigenvalues in the complex plane varies depending on the amount of noise in the environment, and one cannot expect any variant of the Dynamic Mode Decomposition algorithm to be fully robust to noise. [Preview Abstract] |
Monday, November 25, 2013 10:56AM - 11:09AM |
H35.00003: Attractor Identification from Empirical Data Using Diffusion-Mapped Delay Coordinates Zrinka Greguric Ferencek, Tyrus Berry, Timothy Sauer, John Cressman Nonlinear driven system can exhibit a diverse range of dynamics, from highly ordered to chaotic. These systems are ubiquitous, from atmospheric phenomena to brain function. In many cases, the governing equations for these systems are unknown. Here we present a dimensionality reduction algorithm based on diffusion-mapped delay coordinates that identifies the dimension and volume of the system's underlying attractor from empirical data. We generate data in the form of movies that are governed by the R\"{o}ssler and Lorenz systems, as well as purely noisy and simple period dynamics. We show that this algorithm can be used to identify the dimensionality and volume of these attractors from empirical data. We then go on to apply this algorithm to a small electroconveting liquid crystals that supports multistable states that are characterized by patterns of creation, evolution, and annihilation of defects in the sample. We are able to identify the dimension and volume of their dynamics and use them to discriminate between these states. [Preview Abstract] |
Monday, November 25, 2013 11:09AM - 11:22AM |
H35.00004: Nonlinear analysis of polymer electrolyte fuel cell dynamics with cathode two-phase flow Michael Burkholder, Shawn Litster Water management in polymer electrolyte fuel cells (PEFCs) must be optimized to minimize parasitic costs. Removing water with excessive air flow rates at low-current, low-power conditions can be very parasitic, but these conditions can be unstable from the two-phase slug flow in the cathode air-delivery microchannels that occurs from the intrinsically low air and water flow rates. In this work, we use nonlinear analysis to understand the effect that varying currents and air flow rates have on PEFC dynamics. We estimate the dimension and entropy invariants indicative of dynamical complexity and stability from a reconstructed state space embedded using PEFC voltage data. We show that the estimated invariants can be correlated to the channel two-phase flow regime. We also investigate autocorrelation in the voltage signal by using diffusion analysis to estimate Hurst exponents. Lastly, we propose a reduced order map for application to real time PEFC water management. [Preview Abstract] |
Monday, November 25, 2013 11:22AM - 11:35AM |
H35.00005: Is Chaotic Advection Inherent to Porous Media Flow? Daniel Lester, Guy Metcalfe, Mike Trefry All porous media, including granular and packed media, fractured and open networks, are typified by the inherent topological complexity of the pore-space. This topological complexity admits a large number density of stagnation points under steady Stokes flow, which in turn generates a 3D fluid mechanical analouge of the Bakers map, termed the Baker's flow. We demonstrate that via this mechanism, chaotic advection at the pore-scale is inherent to almost all porous media under reasonable conditions, and such dynamics have significant implications for a range of fluid-borne processes including transport and mixing, chemical reactions and biological activity. [Preview Abstract] |
Monday, November 25, 2013 11:35AM - 11:48AM |
H35.00006: Collaborative tracking and control in time-dependent stochastic dynamical systems Eric Forgoston, Ani Hsieh, Ira Schwartz, Philip Yecko We consider the problem of stochastic prediction and control in a time-dependent stochastic environment, such as the ocean, where escape from an almost invariant region occurs due to random fluctuations. Lagrangian Coherent Structures (LCS) are found using collaborative tracking, and a control policy is formulated that utilizes knowledge of the LCS. The control strategy enables mobile sensors to autonomously maintain a desired distribution in the environment, and is evaluated with experimental data. [Preview Abstract] |
Monday, November 25, 2013 11:48AM - 12:01PM |
H35.00007: Nonlinear dynamic estimation with sparse sensors Steven Brunton, Jonathan Tu, Nathan Kutz We show that dimensionality reduction and compressive sensing strategies can be combined to estimate the state and/or parameters of a complex nonlinear system using only sparse measurements. $L^2$ based dimensionality reduction techniques, such as the proper orthogonal decomposition, are used to construct libraries spanning many dynamic phenomena, and sparse sensing is used to identify and reconstruct the dynamics from the library elements. This technique provides an objective and general framework for characterizing the underlying dynamics, stability, and bifurcations of the complex system. These methods are demonstrated on the complex Ginzburg-Landau equation using sparse, noisy measurements, as well as on the two-dimensional Navier-Stokes equation at low Reynolds number. Various spatiotemporal sampling strategies are investigated, with an emphasis on practical engineering considerations. We demonstrate that using a data-driven basis facilitates accurate nonlinear estimation from far fewer sensors than would typically be required of compressive sensing in a generic transform basis. [Preview Abstract] |
Monday, November 25, 2013 12:01PM - 12:14PM |
H35.00008: Understanding the evolution of complex multiscale systems: Dynamic renormalization, non-equilibrium entropy and stochasticity Marc Pradas, Markus Schmuck, Grigorios Pavliotis, Serafim Kalliadasis We present a novel methodology that enables the study the complex dynamics of dissipative systems. By means of a generic reduced equation which is also computationally efficient we tackle a fundamental problem in science: Many time-dependent problems are generally too complex to be fully resolved and hence some information needs to be neglected. A central question is then how can one systematically and reliably reduce the complexity of such high-dimensional systems without neglecting essential information. Popular examples of this are models for climate prediction, cell biology processes, or economics. We combine elements from nonlinear science, statistical physics, and information theory to develop a new stochastic strategy that rigorously shows how to replace the non-relevant degrees of freedom of an infinite-dimensional system by a finite random process statistically well defined [1]. A dynamic renormalization group approach reveals that the neglected information can be described in terms of an appropriately defined entropy for dissipative non-equilibrium processes which seems to have universal characteristics, thus providing a rational and systematic means for quantifying the evolution of dissipative systems. \\[4pt] [1] Schmuck, Pradas, Kalliadasis, Pavliotis. PRL 110, 244101(2013). [Preview Abstract] |
Monday, November 25, 2013 12:14PM - 12:27PM |
H35.00009: Semi-automatic reduced order models from expert-defined transients Andreas Class, Dennis Prill Boiling water reactors (BWRs) not only show growing power oscillations at high-power low-flow conditions but also amplitude limited oscillations with temporal flow reversal. Methodologies, applicable in the non-linear regime, allow insight into the physical mechanisms behind BWR dynamics. The proposed methodology [1] exploits relevant simulation data computed by an expert choice of transient. Proper orthogonal modes are extracted and serve as Ansatz functions within a spectral approach, yielding a reduced order model (ROM). Required steps to achieve reliable and numerical stable ROMs are discussed, i.e. mean value handling, inner product choice, variational formulation of derivatives and boundary conditions.Two strongly non-linear systems are analyzed: The tubular reactor, including Arrhenius reaction and heat losses, yields sensitive response on transient boundary conditions. A simple natural convection loop is considered due to its dynamical similarities to BWRs. It exhibits bifurcations resulting in limit cycles. The presented POD-ROM methodology reproduces dynamics with a small number of spectral modes and reaches appreciable accuracy. \\[4pt] [1] Prill, D.\& Class, A. Semi-automated POD-ROM non-linear analysis for future BWR stability analysis, Annals of Nuclear Energy, 20 [Preview Abstract] |
Monday, November 25, 2013 12:27PM - 12:40PM |
H35.00010: Cluster-based reduced-order modelling of a mixing layer Eurika Kaiser, Bernd R. Noack, Laurent Cordier, Andreas Spohn, Marc Segond, Markus Abel, Guillaume Daviller, Robert K. Niven We propose a novel cluster-based reduced-order modelling (CROM) strategy of unsteady flows. CROM builds on the pioneering works of Gunzburger's group in cluster analysis (Burkardt et al.\ 2006) and Eckhardt's group in transition matrix models (Schneider et al.\ 2008) and constitutes a potential alternative to POD models. This strategy processes a time-resolving sequence of flow snapshots in three steps. First, the snapshot data is clustered into a small number of representative states in the phase space. The states are sorted by probability and transition considerations. Secondly, the transitions between the states are dynamically modelled via a Markov process. Finally, physical mechanisms are distilled by a refined analysis of the Markov process. The resulting CROM is applied to the Lorenz attractor as illustrating example and velocity fields of the spatially evolving incompressible mixing layer. For these examples, CROM is shown to distill non-trivial quasi-attractors and transitions processes. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, and for the identification of precursors for desirable and undesirable events. [Preview Abstract] |
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