Mean flow and spiral defect chaos in Rayleigh-Benard convection

We describe a numerical procedure to construct a modified velocity field that does not have any mean flow. Using this procedure, we present two results. Firstly, we show that, in the absence of mean flow, spiral defect chaos collapses to a stationary pattern comprising textures of stripes with angular bends. The quenched patterns are characterized by mean wavenumbers that approach those uniquely selected by focus-type singularities, which, in the absence of mean flow, lie at the zig-zag instability boundary. The quenched patterns also have larger correlation lengths and are comprised of rolls with less curvature. Secondly, we describe how mean flow can contribute to the commonly observed phenomenon of rolls terminating perpendicularly into lateral walls. We show that, in the absence of mean flow, rolls begin to terminate into lateral walls at an oblique angle. This obliqueness increases with Rayleigh number.Comment: 14 pages, 19 figure

Numerical study of pattern formation following a convective instability in non-Boussinesq fluids

We present a numerical study of a model of pattern formation following a convective instability in a non-Boussinesq fluid. It is shown that many of the features observed in convection experiments conducted on $CO_{2}$ gas can be reproduced by using a generalized two-dimensional Swift-Hohenberg equation. The formation of hexagonal patterns, rolls and spirals is studied, as well as the transitions and competition among them. We also study nucleation and growth of hexagonal patterns and find that the front velocity in this two dimensional model is consistent with the prediction of marginal stability theory for one dimensional fronts.Comment: 9 pages, report FSU-SCRI-92-6

Pattern Formation and Dynamics in Rayleigh-B\'{e}nard Convection: Numerical Simulations of Experimentally Realistic Geometries

Rayleigh-B\'{e}nard convection is studied and quantitative comparisons are made, where possible, between theory and experiment by performing numerical simulations of the Boussinesq equations for a variety of experimentally realistic situations. Rectangular and cylindrical geometries of varying aspect ratios for experimental boundary conditions, including fins and spatial ramps in plate separation, are examined with particular attention paid to the role of the mean flow. A small cylindrical convection layer bounded laterally either by a rigid wall, fin, or a ramp is investigated and our results suggest that the mean flow plays an important role in the observed wavenumber. Analytical results are developed quantifying the mean flow sources, generated by amplitude gradients, and its effect on the pattern wavenumber for a large-aspect-ratio cylinder with a ramped boundary. Numerical results are found to agree well with these analytical predictions. We gain further insight into the role of mean flow in pattern dynamics by employing a novel method of quenching the mean flow numerically. Simulations of a spiral defect chaos state where the mean flow is suddenly quenched is found to remove the time dependence, increase the wavenumber and make the pattern more angular in nature.Comment: 9 pages, 10 figure

Microextensive Chaos of a Spatially Extended System

By analyzing chaotic states of the one-dimensional Kuramoto-Sivashinsky equation for system sizes L in the range 79 <= L <= 93, we show that the Lyapunov fractal dimension D scales microextensively, increasing linearly with L even for increments Delta{L} that are small compared to the average cell size of 9 and to various correlation lengths. This suggests that a spatially homogeneous chaotic system does not have to increase its size by some characteristic amount to increase its dynamical complexity, nor is the increase in dimension related to the increase in the number of linearly unstable modes.Comment: 5 pages including 4 figures. Submitted to PR

Efficient Algorithm on a Non-staggered Mesh for Simulating Rayleigh-Benard Convection in a Box

An efficient semi-implicit second-order-accurate finite-difference method is described for studying incompressible Rayleigh-Benard convection in a box, with sidewalls that are periodic, thermally insulated, or thermally conducting. Operator-splitting and a projection method reduce the algorithm at each time step to the solution of four Helmholtz equations and one Poisson equation, and these are are solved by fast direct methods. The method is numerically stable even though all field values are placed on a single non-staggered mesh commensurate with the boundaries. The efficiency and accuracy of the method are characterized for several representative convection problems.Comment: REVTeX, 30 pages, 5 figure

Square Patterns and Quasi-patterns in Weakly Damped Faraday Waves

Pattern formation in parametric surface waves is studied in the limit of weak viscous dissipation. A set of quasi-potential equations (QPEs) is introduced that admits a closed representation in terms of surface variables alone. A multiscale expansion of the QPEs reveals the importance of triad resonant interactions, and the saturating effect of the driving force leading to a gradient amplitude equation. Minimization of the associated Lyapunov function yields standing wave patterns of square symmetry for capillary waves, and hexagonal patterns and a sequence of quasi-patterns for mixed capillary-gravity waves. Numerical integration of the QPEs reveals a quasi-pattern of eight-fold symmetry in the range of parameters predicted by the multiscale expansion.Comment: RevTeX, 11 pages, 8 figure

Logarithmically Slow Expansion of Hot Bubbles in Gases

We report logarithmically slow expansion of hot bubbles in gases in the process of cooling. A model problem first solved, when the temperature has compact support. Then temperature profile decaying exponentially at large distances is considered. The periphery of the bubble is shown to remain essentially static ("glassy") in the process of cooling until it is taken over by a logarithmically slowly expanding "core". An analytical solution to the problem is obtained by matched asymptotic expansion. This problem gives an example of how logarithmic corrections enter dynamic scaling.Comment: 4 pages, 1 figur

Measurement in biological systems from the self-organisation point of view

Measurement in biological systems became a subject of concern as a consequence of numerous reports on limited reproducibility of experimental results. To reveal origins of this inconsistency, we have examined general features of biological systems as dynamical systems far from not only their chemical equilibrium, but, in most cases, also of their Lyapunov stable states. Thus, in biological experiments, we do not observe states, but distinct trajectories followed by the examined organism. If one of the possible sequences is selected, a minute sub-section of the whole problem is obtained, sometimes in a seemingly highly reproducible manner. But the state of the organism is known only if a complete set of possible trajectories is known. And this is often practically impossible. Therefore, we propose a different framework for reporting and analysis of biological experiments, respecting the view of non-linear mathematics. This view should be used to avoid overoptimistic results, which have to be consequently retracted or largely complemented. An increase of specification of experimental procedures is the way for better understanding of the scope of paths, which the biological system may be evolving. And it is hidden in the evolution of experimental protocols.Comment: 13 pages, 5 figure

Pattern fluctuations in transitional plane Couette flow

In wide enough systems, plane Couette flow, the flow established between two parallel plates translating in opposite directions, displays alternatively turbulent and laminar oblique bands in a given range of Reynolds numbers R. We show that in periodic domains that contain a few bands, for given values of R and size, the orientation and the wavelength of this pattern can fluctuate in time. A procedure is defined to detect well-oriented episodes and to determine the statistics of their lifetimes. The latter turn out to be distributed according to exponentially decreasing laws. This statistics is interpreted in terms of an activated process described by a Langevin equation whose deterministic part is a standard Landau model for two interacting complex amplitudes whereas the noise arises from the turbulent background.Comment: 13 pages, 11 figures. Accepted for publication in Journal of statistical physic

On the validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation

We consider the problem of the speed selection mechanism for the one dimensional nonlinear diffusion equation $u_t = u_{xx} + f(u)$. It has been rigorously shown by Aronson and Weinberger that for a wide class of functions $f$, sufficiently localized initial conditions evolve in time into a monotonic front which propagates with speed $c^*$ such that $2 \sqrt{f'(0)} \leq c^* < 2 \sqrt{\sup(f(u)/u)}$. The lower value $c_L = 2 \sqrt{f'(0)}$ is that predicted by the linear marginal stability speed selection mechanism. We derive a new lower bound on the the speed of the selected front, this bound depends on $f$ and thus enables us to assess the extent to which the linear marginal selection mechanism is valid.Comment: 9 pages, REVTE