Bulletin of the American Physical Society
62nd Annual Meeting of the APS Division of Fluid Dynamics
Volume 54, Number 19
Sunday–Tuesday, November 22–24, 2009; Minneapolis, Minnesota
Session MD: Chaos and Fractals II |
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Chair: Tom Solomon, Bucknell University Room: 101D |
Tuesday, November 24, 2009 8:00AM - 8:13AM |
MD.00001: Sensitivity of LCS identification to flow field resolution and random errors Ali B. Olcay, Tait S. Pottebaum, Paul S. Krueger Ridges in the finite-time Lyapunov exponent (FTLE) fields obtained from fluid flows provide separatrices that are transport barriers. These ridges, called Lagrangian coherent structures (LCSs), can be used to identify vortex boundaries and study mixing processes in complex, time-dependent flows. The accuracy of LCS identification is inherently dependent on the accuracy of the underlying velocity field used to compute the FTLE field. To quantify the effect of velocity field errors, this study considers the two canonical flows: a steady 2D vortex pair and an axisymmetric vortex ring generated by a starting jet. The velocity field for the vortex pair was determined analytically while the vortex ring flow was obtained from CFD for jet \textit{Re} = 1000. Velocity field errors were introduced by smoothing and sub-sampling the data (mimicking DPIV interrogation window size effects) and adding random noise. The results show that the LCS location can be shifted as much as 0.2$D$ when sub-sampling the velocity field from a resolution of 0.01$D$ to 0.125$D$ ($D$ is the vortex diameter) and the effect of noise on the mean LCS location is small compared to the resolution effect. [Preview Abstract] |
Tuesday, November 24, 2009 8:13AM - 8:26AM |
MD.00002: Characterization of Mixing Using Experimentally Derived Velocity Fields and Derived Lagrangian Coherent Structures Douglas Bohl, Naratip Santitissadeekorn, Erik Bollt In this work a flat rectangular plate is rotated along its long axis and parallel to the z-axis of a circular cylinder. The blade position is varied with respect to the cylinder wall to allow investigation of the effect of the no slip boundary on the flow structure and mixing field. The cylinder is filled with viscous Newtonian fluids and driven at low Reynolds numbers (8-100). Particle Image Velocimetry is used to measure the velocity in the plane perpendicular to the rotation of the plate (i.e. in the r-$\theta $ plane of the cylinder). The experimental velocity field is used to 1. Determine numerically the motion of 100,000 simulated zero mass particle tracers for up to 25 cycles of the blade and 2. Calculate the Lagrangian Coherent Structures (LCS) of the flow field. Mixing rates and length scales are found using the particle tracers. Results show that the fluid is segregated into distinct regions with limited interaction by the ridges in the short term LCS field. The results also show that if the Lagrangian Coherent Structures are calculated for long times the resulting LCS field is similar to that of the particle distribution field for the same time number of rotations. [Preview Abstract] |
Tuesday, November 24, 2009 8:26AM - 8:39AM |
MD.00003: Identifying Exact Coherent Structures in 2D Turbulence: Experiments and Simulations Michael Schatz, Jon Paprocki, Christopher Lesesne, James Andrews Recent theoretical advances suggest ways to find unstable exact Navier Stokes solutions that capture many features of coherent structures, which have long been observed in turbulent flow. At present, it remains unknown whether these solutions, termed Exact Coherent States, can describe observations of turbulent flow in laboratory experiments. We describe our experimental and numerical investigations, which search for unstable solutions in quasi-2D flows driven by electromagnetic forces. In the experiments, time series of velocity fields are obtained from images of the visualized flow. In the simulations, long time series of velocity fields are calculated for flows with forcing similar to that in the experiments. The velocity field data from both experiments and simulations are used to construct recurrence plots that provide evidence for the existence of unstable periodic orbits. [Preview Abstract] |
Tuesday, November 24, 2009 8:39AM - 8:52AM |
MD.00004: Characterizing the Chaotic Degrees of Freedom of High-Dimensional Fluid Convection Alireza Karimi, Mark Paul The variation of the Lyapunov exponent spectra and fractal dimension with system parameters can yield fundamental insights into the nature of spatiotemporal chaos. We explore this numerically for two systems: the Lorenz 96 model and Rayleigh-Benard convection. The Lorenz 96 model is a phenomenological model that captures important features of atmosphere dynamics. We compute the fractal dimension as a function of system size and external forcing for very long times and over many initial conditions. When varying system size we find extensive chaos with significant deviations from extensivity for small changes in system size and also the power-law growth of the dimension with increasing forcing. We use large-scale parallel numerical simulations to study chaotic Rayleigh-Benard convection for experimentally accessible conditions. We compute the variation of the fractal dimension with system size, Prandtl number, and Rayleigh number. Using statistical properties of the Lyapunov exponents and Lyapunov vectors we connect these features with the dynamics of the flow field pattern. [Preview Abstract] |
Tuesday, November 24, 2009 8:52AM - 9:05AM |
MD.00005: Dynamically Evolving Topology in Spatiotemporal Chaos Nicholas T. Ouellette, Douglas H. Kelley Recent advances in Lagrangian measurements have allowed the robust experimental location of the hyperbolic and elliptic stagnation points in two-dimensional incompressible flow. Here, we extend these techniques to study the dynamics of the stable and unstable manifolds of the hyperbolic points as our experimental quasi-2D electromagnetically driven flow evolves in time. We compare results from nearly stationary flows, where the hyperbolic points move only slightly, and spatiotemporally chaotic flows, where hyperbolic and elliptic points can be created or annihilated in pairs. This work is supported by the National Science Foundation. [Preview Abstract] |
Tuesday, November 24, 2009 9:05AM - 9:18AM |
MD.00006: Uncertainties in Lagrangian mixing Wenbo Tang, Alex Mahalov Lagrangian Coherent Structures have been discovered to be the building blocks of chaotic mixing in turbulent flows and mathematical tools have been developed to extract the invariant manifolds that highlight Lagrangian mixing. These mathematical tools are based on deterministic velocity fields and it is unclear how random processes can modify chaotic mixing. In this talk we discuss archetypical geophysical flow examples embedded in an environment of Gaussian white noise. We examine how the deterministic nonlinear background flow fields alter the Gaussian statistics and the consequences of stochastic processes on Lagrangian mixing. [Preview Abstract] |
Tuesday, November 24, 2009 9:18AM - 9:31AM |
MD.00007: A Chaotic Periodically Reoriented Irrotational Flow: Experiments, Theory, and Applications to Geophysical Transport Guy Metcalfe$^1$, Daniel Lester, Murray Rudman, Klaus Regenauer-Lieb, Mike Trefry, Alison Ord$^3$, Bruce Hobbs$^3$, Pandurang Kulkarni, Park Kwan Fung, Zhurui Xu, Jeff Morris A source-sink pair in a Hele-Shaw cell generates an irrotational dipole flow. In a disk with 360 wells around its periphery and a rotatable manifold controlling which well pairs are open, we have created a periodically reoriented dipole flow which is an open chaotic dynamical system with properties controlled by the reorientation angle and duration of flow. Despite being open the flow can have island regions where fluid that starts in the disk remains there forever. Theory and experiments determine the island existence boundary in control parameter space, the variation in island size, and bifurcations. We also briefly describe possible applications to problems in geophysical transport. [Preview Abstract] |
Tuesday, November 24, 2009 9:31AM - 9:44AM |
MD.00008: The Dynamics of Ratcheting States in Cellular Flames Michael Gorman Cellular flames form ordered states of two concentric rings of brighter, hotter cells, separated by darker, cooler cusps and folds. In ratcheting states one or both rings of cells of cells rotate slowly ($\sim $1 deg/sec), speeding up and slowing down in a manner characteristic, which depends on the numbers of cells in the inner and outer rings and the degree of coupling between the two rings. We present measurements of the velocities of 4 such states and video clips of the motions of 20 other ratcheting states. The characteristics of these states have not yet been explained. The nature of ratcheting motion has not yet been described in the context of bifurcations with symmetry. [Preview Abstract] |
Tuesday, November 24, 2009 9:44AM - 9:57AM |
MD.00009: Cryptography with Chaos and Shadowing Nejib Smaoui, Ali Kanso A novel approach to encrypt a message using chaos and shadowing is presented. The approach is based on two steps: First, a numerical chaotic orbit of the logistic map is used in the shadowing algorithm of Smaoui \& Kostelich [Intern. J. Computer. Math. (1998) 70] to show that there exists a finite number of true orbits that shadow the numerical orbit. Then, Baptista's algorithm [Phys. Lett. A (1998) 240] is used on the finite number of maps to encrypt each character of the message. It is shown that the use of the shadowing method in the encryption process enhances the security level. [Preview Abstract] |
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