Session J30: Focus Session: Characterizing Spatio-Temporal Complexity in Fluids and Materials

Topological Analysis of Spatial Temporal Patterns

It is fairly easy to collect large amounts of high dimensional data describing the time dependent spatial structures of materials or fluids either through experimentation or numerical simulation. In this talk I will describe how techniques from computational topology can be used to reduce both the size and dimension of the data sets and still provide useful statistics for parameter identification, model selection, and quantification of the spacio-temporal complexity of the dynamics. These ideas will be presented in the context of experimental working involving spiral defect chaos for Rayleigh- Benard convection and numerical simulations of stochastic and deterministic Cahn-Hilliard equations.   

Coarse-grained velocity gradients in turbulence

In a turbulent flow, energy cascades from large length and time scales, where it is injected into the flow, to small scales, where it is dissipated by the action of molecular viscosity. At small scales, this energy dissipation is characterized by the velocity gradient tensor. At larger scales, however, different dynamics must apply. We therefore present measurements of a velocity gradient tensor coarse-grained over inertial-range scales in an intensely turbulent laboratory water flow. We discuss the potential of these coarse-grained gradients as a probe of the scale-to-scale energy transfer in the turbulent cascade and their relation to Large Eddy Simulation. This work was supported both by the National Science Foundation and by the Max Planck Society.   

Turbulent-Laminar Patterns in Shear Flows

We study computationally turbulent-laminar patterns in very-large-aspect-ratio plane Couette flow. These states consist of large-scale alternations of turbulent and laminar flow oriented obliquely to the steamwise direction. Such flow patterns are now believed to be typical of many transitional shear flows when observed on long length scales. For a fixed pattern orientation of $24^{circ}$, suggested by experiment, the basic scenario observed in computations as the Reynolds number is decreased is the following: From uniform turbulence there is a transition to intermittent patterns at $Re\simeq 420$, then to steady, spatially periodic patterns at $Re\simeq 390$. The wavelength increases as the Reynolds number is decreased until $Re\simeq 310$, where the flow consists of localized turbulence within a laminar background. This scenario can depend on pattern orientation -- at $90^{circ}$ with respect to the flow direction, we observe spatio-temporal intermittency in which turbulent patches that repeatedly disappear abruptly and then re-nucleate gradually. We present an analysis of these flows in terms of mean quantities and discuss the difficulties of determining critical bifurcation parameters for such turbulent-laminar systems.   

Turbulence structures and unstable periodic orbits

Recently found unstable time-periodic solutions to the incompressible Navier-Stokes equation are reviewed to discuss their relevance to near-wall turbulence and isotropic turbulence. It is shown that the periodic motion embedded in plane Couette turbulence exhibits a regeneration cycle of near- wall coherent structures, which consists of formation and breakdown of streamwise vortices and low-velocity streaks. In phase space a turbulent state wanders around the corresponding periodic orbit for most of the time, so that the root-mean-squares of velocity fluctuations of the Couette turbulence agree very well with the temporal averages of those along the periodic orbit. 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.   

Computational Homology in Rayleigh-Benard convection experiments

Computational homology is used to analyze the spiral defect chaos (SDC) state in Rayleigh-Benard convection. Image time series of flows visualized by shadowgraphy are used as input; the homology analysis yields Betti numbers, which counts the number of connected components and holes in the flow patterns. Probability distributions and entropies derived from the Betti number measurements are used for identifying and characterizing different states in the SDC regime.   

State and Parameter Estimation of Spatio-Temporally Chaotic Systems: Application to Rayleigh-Benard Convection

Data assimilation refers to the process of obtaining an estimate of a system's state from a time series of incomplete and noisy measurements along with a model (possibly approximate) for the system's time evolution. Here we demonstrate the applicability of a recently developed data assimilation method, the Local Ensemble Transform Kalman Filter (LETKF), to Rayleigh-Benard convection, a non-linear, high dimensional, spatio-temporally chaotic fluid system. Using this technique we are able to extract the full temperature and velocity fields, including the mean flow, from experimental images of shadowgraphs. In addition, we describe extensions of the algorithm for estimating fluid parameters.   

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-B\'enard 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.   

Pattern selection and control via localized feedback

Many theoretical analyses of feedback control of pattern-forming systems assume that feedback is applied at every spatial location, something that is often difficult to accomplish in experiments. We consider an experimentally more feasible scenario where feedback is applied at a sparse array of discrete spatial locations. We use generalized linear stability analysis to determine how dense the actuator array needs to be to select or maintain control of a given pattern state in the presence of noise. The one-dimensional Swift-Hohenberg equation is used to illustrate our theoretical results and explain earlier experimental observations on the control of the Rayleigh-B\'enard convection.   

Structural analysis of particulate suspensions under simple shear flow

A new simulation platform that takes the interaction between fluid and particle has been developed. We analyzed three-dimensional microstructures of repulsive and weakly aggregating suspensions under simple shear flow. Two-dimensional Fourier Transform of the particle images and pair distribution functions were used for microstructure analysis. Particles are well aligned in repulsive suspension while there is an anisotropic configuration of particle clusters in weakly aggregating suspension. We could observe a vorticity-directional motion even in a simple shear flow for aggregating particle suspension, which was recently reported by scattering techniques. Helical motion towards the vorticity direction appears because the flow field is disturbed by the extra stress of the particles. High local shear rate regime is also observed near the fast helical streamlines. This result will provide a clear outlook for the simple shear flow of particulate suspensions.   

Light Propagation in Quasi-Ordered Media

We evaluate the use of light to probe patterns in optical media whose index of refraction is modulated in the direction of propagation over length scales large relative to the optical wavelength. First, we show how a quadratic index waveguide with periodic axial variations induces a parametric instability in the geometric optics limit. A fundamental scaling allows us to examine a wide range of physical conditions and explore nonlinear behavior such as resonance and chaos. Second, we show that a periodic array of cylinders acts as a waveguide and also shows resonances. We consider the possibility for using these results to probe order-disorder phenomena in systems as widely different as fluid flows and living tissues.   

Oscillons and reciprocal oscillons

Formation of spatially localized oscillations in parametrically driven systems is studied, focusing on the dominant 2:1 resonance tongue. Both damped and self-exciting oscillatory media are considered. The forced complex Ginzburg-Landau equation is used to identify two types of such states, small amplitude oscillons and large amplitude reciprocal oscillons resembling holes in an oscillating background. In addition a variety of front-like states with nonmonotonic profiles is described. A systematic analysis of the origin and stability properties of these states is provided. In many regimes all three states are related to the presence of a Maxwell point between finite amplitude spatially homogeneous phase-locked oscillations and the zero state, leading to a large multiplicity of coexisting stable states of different types.