Computational Homology and Fluid Dynamics Workshop

Georgia Tech, Atlanta, GA

March 1-3, 2007

Abstracts

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Eberhard Bodenschatz, Crystalline chaos and resonant structures in spatially forced Rayleigh-Benard convection

We report experiments in spatially forced Rayleigh-Benard convection of a fluid with Prandtl number unity. Spatially periodic forcing was introduced through a micro-machining of the bottom plate. Forcing was investigated for two forcing wavelengths 6/5 and 2 times the critical wavenumber. This way it was possible to study the harmonic and subharmonic response. The observed close to onset behavior of the system will be presented and compared with theoretical predictions. Then the talk will focus on the case of 6/5 forcing, which for sufficiently high temperature differences lies outside of the stability balloon. The new state of crystalline chaos consisting of solitary kink and antikinks will be presented and it will be shown that the spatio-temporal chaotic state can be described by an interaction of localized pattern elements. The physics leading to the these kink and antikink states will be presented. The talk will close with a description of the phase diagram and other localized states observed in the pattern. This work is supported through DMR of the National Science Foundation.

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Predrag Cvitanovic Turbulence: a walk on the wild side

We test the "recurrent coherrent states" description of turbulence on a Kuramoto-Sivashinsky model, deploying a new methods that yield large number of numerical unstable spatiotemporally periodic solutions.

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Karen Daniels What can we learn about granular materials from the geometry and topology of force chain networks?

Within a sheared or compressed granular material, the internal stresses take the form of "force chains," which are roughly co-linear chains of particles which transmit larger-than-average stress. A popular technique in the study of granular materials is to use photoelastic particles and polarizing filters to render these chains visible: particles with large forces appear brighter than those with small forces. This network of connections among the particles is observed to highly heterogeneous, and the magnitude of the stress varies widely over short distances. In analyzing digital images of force chains, it is an open question how we can characterize the geometry and topology of the network itself as a means of understanding the properties and state of the granular material. I will show images from recent experiments on stick-slip motion and sound transmission performed in photoelastic granular materials. In the case of the former, we observe that a granular material under shear fails in patches with a wide distribution of sizes, from a few grains up to system-spanning events. During sound transmission, the wave front induces reversible changes in the force chain network which we visualize using a high-speed camera. In each case, we seek to understand what properties of the network topology and geometry change during the events and how can we quantitatively characterize them.

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John Gibson, Invariant solutions and the dynamics of plane Couette flow

Recent advances in the computation of exact solutions of Navier-Stokes suggest that low-Reynolds turbulence can be described as a chaotic walk between its least unstable invariant solutions. We develop a precise dynamical-systems representation in R^N for plane Couette flow and discuss the computation of equilibria, periodic orbits, and unstable manifolds. State-space portraits reveal dynamical structure and lead to the discovery of new invariant solutions.

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Brian Hunt, State estimation for Rayleigh-Benard convection from shadowgraph data

Using methodology presented in Edward Ott's talk, we estimate the state of a cylindrical Rayleigh-Benard convection cell, in the parameter regime of spiral defect chaos, using a time series of shadowgraph images that indicate the vertically averaged temperature yield. We use a computational model for convection in cylindrical geometry (L. Tuckerman, J. Comp. Phys. 80, 403-441, 1989) to track the approximate state of the system over time, and at each observation time our method adjusts both the approximate temperature and velocity fields according to the information provided by the shadowgraph. We compare our method with a cruder method that simply adjusts the approximate temperature field at each observation time to fit the shadowgraph, without regard to the dynamical correlations between state variables. We also show how to use our method to estimate system parameters.

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Genta Kawahara, Turbulence structures, their control and unstable periodic orbits

Recently found unstable time-periodic solutions to the incompressible Navier--Stokes equation are reviewed to discuss their relevance to plane Couette turbulence and isotropic turbulence. It is shown that the periodic motion embedded in the Couette turbulence exhibits a regeneration cycle of near-wall coherent structures, which consists of formation and breakdown of streamwise vortices and low-velocity streaks. The Kolmogorov universal-range energy spectrum is observed for the periodic motion embedded in high-symmetric turbulence at the Taylor-microscale Reynolds number Re_lambda=67. Spatio-temporal structures of the periodic solution in high-symmetric flow are investigated to characterize the dynamics of coherent structures which appear in the energy cascade process. A laminarization strategy inspired by investigation of the phase-space structure in the vicinity of the unstable periodic orbit is presented for the Couette turbulence.

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Edward Ott, Determining the state of a large spatiotemporally chaotic system

For the purposes of scientific investigation, forecasting, or control it is neccessary to somehow determine a good estimate of the current state of an evolving dynamical system. The problem of state estimation from limited noisy measurements and knowledge of a system dynamical model has a nice classical solution, the Kalman Filter technique. Unfortunately, this classical solution can become impossible to impliment for systems with large dimensionality because it would require computer resources that are many orders of magnitude beyond what is [or will likely become] available. Recently, in the context of weather forecasting, we have addressed this problem and constructed an efficient and accurate solution, applicable to a large class of spatiotemporally chaotic situations. This talk will discuss our technique, called a local ensemble Kalman Filter. Tests of the technique and applications to weather forecasting will be presented. Application of the technique to data from a Rayleigh-Benard laboratory experiment will be given in the subsequent talk by Brian Hunt.

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Mark Paul, Quantifying spatiotemporal chaos in Rayleigh-Benard convection: New insights from numerics

Spatiotemporal chaos is studied using large-scale numerical simulations of Rayleigh-Benard convection in cylindrical domains with experimentally realistic boundary conditions. The Lyapunov exponents and fractal dimension are calculated over a range of system sizes as given by aspect ratios of the cylindrical convection domain between 5 and 15. It is found that the chaos is extensive over this range of system size as illustrated by a linear dependence of the fractal dimension with the square of the aspect ratio. The chaos is extensive even though the convection pattern is found to transition from boundary to bulk dominated dynamics as the system size is increased. An analysis of the Lyapunov vectors is used to yield quantitative information describing the location of the largest growing perturbations which are then correlated with defect structures in the fluid flow field.

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Hermann Riecke, Geometric diagnostics of complex patterns: Spiral defect chaos in convection

Motivated by the observation of spiral patterns in a wide range of physical, chemical, and biological systems we present an approach that aims at characterizing quantitatively spiral-like elements in complex stripe-like patterns. The approach provides the location of the spiral tip and the size of the spiral arms in terms of their arclength and their winding number. In addition, it yields as topological information the number of pattern components (Betti number of order 1), as well as their size and certain aspects of their shape. We apply the method to spiral defect chaos in thermally driven Rayleigh-Benard convection and find that the winding number of the spirals, but not their arclength, is non-monotonic in the heating. The distribution function for the number of spirals is significantly narrower than a Poisson distribution. The distribution function for the winding number decays approximately exponentially. For small Prandtl numbers the analysis reveals a large number of small compact pattern components. Including non-Boussinesq effects, we find that they not only break the up-down symmetry but also strongly increase the number of small, compact convection cells.

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Michael Schatz, Homological analysis of spatiotemporal patterns

We describe the application of computational homology to characterize complex, dynamical patterns. We show that homological measures can detect the onset of spatiotemporal chaos in reaction-diffusion models. We also discuss how homology can be used to reveal symmetry-breaking in data from Rayleigh-Benard convection experiments and simulations.

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Fabian Waleffe, Homotopy of exact coherent states in shear flows

The theoretical study of turbulence has typically focused on the idealized problem of homogeneous, isotropic turbulence. In such studies, turbulence is viewed as a `random' flow and the goal is to characterize the scaling of `structure functions'. But when a fluid flows, there is usually a wall somewhere and observations show that `turbulence' is largely controlled by coherent structures arising in the near-wall regions. This is the case in turbulent convection and in shear flows for instance. Flow in a pipe or a channel are the canonical examples of shear flows. For about 100 years, shear flows were known to be observed in one of 2 states: ultra simple `laminar' flows and ultra complicated `turbulent' flows, with no clear connections between the two. In recent years, a class of 3D steady state, traveling wave and even periodic solutions of the Navier-Stokes have been discovered in all canonical shear flows. These new states are closely connected to one another and are all unstable, yet a single such unstable coherent state may capture the statistics of turbulent flows remarkably well, at low Reynolds number. I will describe the basic characteristics of these states, how they were found and how they are related to one another and to observations.

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