Spiral defect chaos in a model of Rayleigh-Bénard convection

Dynamical Crystallites of Active Chiral Particles

One of the intrinsic characteristics of far-from-equilibrium systems is the nonrelaxational nature of the system dynamics, which leads to novel properties that cannot be understood and described by conventional pathways based on thermodynamic potentials. Of particular interest are the formation and evolution of ordered patterns composed of active particles that exhibit collective behavior. Here we examine such a type of nonpotential active system, focusing on effects of coupling and competition between chiral particle self-propulsion and self-spinning. It leads to the transition between three bulk dynamical regimes dominated by collective translative motion, spinning-induced structural arrest, and dynamical frustration. In addition, a persistently dynamical state of self-rotating crystallites is identified as a result of a localized-delocalized transition induced by the crystal-melt interface. The mechanism for the breaking of localized bulk states can also be utilized to achieve self-shearing or self-flow of active crystalline layers.

Erratum: Exploring spiral defect chaos in generalized Swift-Hohenberg models with mean flow [Phys. Rev. E 84, 046215 (2011)]

This corrects the article DOI: 10.1103/PhysRevE.84.046215.

Exploring spiral defect chaos in generalized Swift-Hohenberg models with mean flow

We explore the phenomenon of spiral defect chaos in two types of generalized Swift-Hohenberg model equations that include the effects of long-range drift velocity or mean flow. We use spatially extended domains and integrate the equations for very long times to study the pattern dynamics as the magnitude of the mean flow is varied. The magnitude of the mean flow is adjusted via a real and continuous parameter that accounts for the fluid boundary conditions on the horizontal surfaces in a convecting layer. For weak values of the mean flow, we find that the patterns exhibit a slow coarsening to a state dominated by large and very slowly moving target defects. For strong enough mean flow, we identify the existence of spatiotemporal chaos, which is indicated by a positive leading-order Lyapunov exponent. We compare the spatial features of the mean flow field with that of Rayleigh-Bénard convection and quantify their differences in the neighborhood of spiral defects.

Skew-varicose instability in two-dimensional generalized Swift-Hohenberg equations

We apply analytical and numerical methods to study the linear stability of stripe patterns in two generalizations of the two-dimensional Swift-Hohenberg equation that include coupling to a mean flow. A projection operator is included in our models to allow exact stripe solutions. In the generalized models, stripes become unstable to the skew-varicose, oscillatory skew-varicose, and cross-roll instabilities, in addition to the usual Eckhaus and zigzag instabilities. We analytically derive stability boundaries for the skew-varicose instability in various cases, including several asymptotic limits. We also use numerical techniques to determine eigenvalues and hence stability boundaries of other instabilities. We extend our analysis to both stress-free and no-slip boundary conditions and we note a crossover from the behavior characteristic of no-slip to that of stress-free boundaries as the coupling to the mean flow increases or as the Prandtl number decreases. Close to the critical value of the bifurcation parameter, the skew-varicose instability has the same curvature as the Eckhaus instability provided the coupling to the mean flow is greater than a critical value. The region of stable stripes is completely eliminated by the cross-roll instability for large coupling to the mean flow.

Numerical simulation of asymptotic states of the damped Kuramoto-Sivashinsky equation

The damped Kuramoto-Sivashinsky equation has emerged as a fundamental tool for the understanding of the onset and evolution of secondary instabilities in a wide range of physical phenomena. Most existing studies about this equation deal with its asymptotic states on one-dimensional settings or on periodic square domains. We utilize a large-scale numerical simulation to investigate the asymptotic states of the damped Kuramoto-Sivashinsky equation on annular two-dimensional geometries and three-dimensional domains. To this end, we propose an accurate, efficient, and robust algorithm based on a recently introduced numerical methodology, namely, isogeometric analysis. We compared our two-dimensional results with several experiments of directed percolation on square and annular geometries, and found qualitative agreement.

Grain boundary dynamics in stripe phases of nonpotential systems

We describe numerical solutions of two nonpotential models of pattern formation in nonequilibrium systems to address the motion and decay of grain boundaries separating domains of stripe configurations of different orientations. We first address wave-number selection because of the boundary, and possible decay modes when the periodicity of the stripe phases is different from the selected wave number for a stationary boundary. We discuss several decay modes including long wavelength undulations of the moving boundary, as well as the formation of localized defects and their subsequent motion. We find three different regimes as a function of the distance to the stripe phase threshold and initial wave number, and then correlate these findings with domain morphology during domain coarsening in a large aspect ratio configuration.

Hexagons and spiral defect chaos in non-Boussinesq convection at low Prandtl numbers

We study the stability and dynamics of non-Boussinesq convection in pure gases (CO2 and SF6) with Prandtl numbers near Pr approximately 1 and in a H2-Xe mixture with Pr=0.17. Focusing on the strongly nonlinear regime we employ Galerkin stability analyses and direct numerical simulations of the Navier-Stokes equations. For Pr approximately 1 and intermediate non-Boussinesq effects we find reentrance of stable hexagons as the Rayleigh number is increased. For stronger non-Boussinesq effects the usual, transverse side-band instability is superseded by a longitudinal side-band instability. Moreover, the hexagons do not exhibit any amplitude instability to rolls. Seemingly, this result contradicts the experimentally observed transition from hexagons to rolls. We resolve this discrepancy by including the effect of the lateral walls. Non-Boussinesq effects modify the spiral defect chaos observed for larger Rayleigh numbers. For convection in SF6 we find that non-Boussinesq effects strongly increase the number of small, compact convection cells and with it enhance the cellular character of the patterns. In H2-Xe, closer to threshold, we find instead an enhanced tendency toward roll-like structures. In both cases the number of spirals and of targetlike components is reduced. We quantify these effects using recently developed diagnostics of the geometric properties of the patterns.

Hydrodynamic coarsening in striped pattern formation with a conservation law

We observed in numerical simulations that the interaction of striped-pattern-forming instability and a neutrally stable zero mode induces patterns of domains of upflow hexagons coexisting with domains of downflow hexagons. They appear only when hydrodynamic flow is present.

Measurement of mean flows of Faraday waves

We measure the velocities of the mean flows that are driven by curved rolls in a pattern formation system. Curved rolls in Faraday waves are generated in experimental cells consisting of channels with varying widths. The mean flow magnitudes are found to scale linearly with roll curvatures and squares of wave amplitudes, agreeing with the prediction from the analysis of phase dynamics expansion. The effects of the mean flows on reducing roll curvatures are also seen.

Mean flow and spiral defect chaos in Rayleigh-Bénard 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. First, we show that, in the absence of the mean flow, spiral defect chaos collapses to a stationary pattern comprising textures of stripes with angular bends. The quenched patterns are characterized by mean wave numbers that approach those uniquely selected by focus-type singularities, which, in the absence of the mean flow, lie at the zigzag instability boundary. The quenched patterns also have larger correlation lengths and are comprised of rolls with less curvature. Secondly, we describe how the mean flow can contribute to the commonly observed phenomenon of rolls terminating perpendicularly into lateral walls. We show that, in the absence of the mean flow, rolls begin to terminate into lateral walls at an oblique angle. This obliqueness increases with the Rayleigh number.

Harmonic modulation instability and spatiotemporal chaos

It is shown from the conserved Zakharov equations that many solitary patterns are formed from the modulational instability of unstable harmonic modes that are excited by a perturbative wave number. Pattern selection in our case is discussed. It is found that the evolution of solitary patterns may appear in three states: spatiotemporal coherence, chaos in time but the partial coherence in space, and spatiotemporal chaos. The spatially partial coherent state is essentially due to ion-acoustic wave emission, while spatiotemporal chaos characterized by its incoherent patterns in both space and time is caused by collision and fusion among patterns in stochastic motion. So energy carried by patterns in the system is redistributed.

Spiral-defect chaos: Swift-Hohenberg model versus Boussinesq equations

Spiral-defect chaos (SDC) in Rayleigh-Bénard convection is a well-established spatio-temporal complex pattern, which competes with stationary rolls near the onset of convection. The characteristic properties of SDC are accurately described on the basis of the standard three-dimensional Boussinesq equations. As a much simpler and attractive two-dimensional model for SDC generalized Swift-Hohenberg (SH) equations have been extensively used in the literature from the early beginning. Here, we show that the description of SDC by SH models has to be considered with care, especially regarding its long-time dynamics. For parameters used in previous SH simulations, SDC occurs only as a transient in contrast to the experiments and the rigorous solutions of the Boussinesq equations. The small-scale structure of the vorticity field at the spiral cores, which might be crucial for persistent SDC, is presumably not perfectly captured in the SH model.

Pattern formation near onset of a convecting fluid in an annulus

Numerical simulations of the time-dependent Swift-Hohenberg equation are used to test the predictions of Cross [Phys. Rev. A 25, 1065 (1982)] that Rayleigh-Bénard convection in the form of straight rolls or of an array of dislocations may be observed in an annular domain, depending on the values of inner radius r(1), outer radius r(2), reduced Rayleigh number epsilon, and initial states. As r(1) is decreased for a fixed r(2) and for different choices of epsilon and of symmetric and random initial state, we find that there are indeed ranges of these parameters for which the predictions of Cross are qualitatively correct. However, when the radius difference r(2)-r(1) becomes larger than a few roll diameters, a new pattern is observed consisting of stripe domains separated by radially oriented grain boundaries. The relative stabilities of the various patterns are compared by evaluating their Lypunov functional densities.

Spiral patterns in oscillated granular layers

Cell-filling spiral patterns are observed in a vertically oscillated layer of granular material when the oscillation amplitude is suddenly increased from below the onset of pattern formation into the region where stripe patterns appear for quasistatic increases in amplitude. These spirals are transients and decay to stripe patterns with defects. A transient spiral defect chaos state is also observed. We describe the behavior of the spirals, and the way in which they form and decay. Our results are compared with those for similar spiral patterns in Rayleigh-Bénard convection in fluids.

Phase-field modeling of eutectic growth

A phase-field model of eutectic growth is proposed in terms of a free energy F, which is a functional of a liquid-solid order parameter psi, and a conserved concentration field c. The model is shown to recover the important features of a eutectic phase diagram and to reduce to the standard sharp-interface formulation of nonequilibrium growth. It is successfully applied to the study of directional solidification when the solid phase is a single or two phase state. The crystallization of a eutectic compound under isothermal conditions is also considered. For that process, the transformed volume fraction and psi-field structure factor, both measured during numerical simulations, closely match theoretical predictions. Three possible growth mechanisms are also identified: diffusion-limited growth, lamellar growth, and spinodal decomposition.

Complex spatiotemporal convection patterns

This paper reviews recent efforts to describe complex patterns in isotropic fluids (Rayleigh-Benard convection) as well as in anisotropic liquid crystals (electro-hydrodynamic convection) when driven away from equilibrium. A numerical scheme for solving the full hydrodynamic equations is presented that allows surprisingly well for a detailed comparison with experiments. The approach can also be useful for a systematic construction of models (order parameter equations). (c) 1996 American Institute of Physics.

Excitation of Spirals and Chiral Symmetry Breaking in Rayleigh-Bénard Convection

Spiral-defect populations in low-Prandtl number Rayleigh-Bénard convection with slow rotation about a vertical axis were measured in carbon dioxide at high pressure. The results indicate that spirals act like "thermally excited" defects and that the winding direction of a spiral is analogous to a magnetic spin. Rotation about a vertical axis, the spiral analog of the magnetic field, breaks the zero-rotation chiral symmetry between clockwise and counterclockwise spiral defects. Many properties of spiral-defect statistics are well described by an effective statistical-mechanical model.

Order and disorder in fluid motion

The development of complex states of fluid motion is illustrated by reviewing a series of experiments, emphasizing film flows, surface waves, and thermal convection. In one dimension, cellular patterns bifurcate to states of spatiotemporal chaos. In two dimensions, even ordered patterns can be surprisingly intricate when quasiperiodic patterns are included. Spatiotemporal chaos is best characterized statistically, and methods for doing so are evolving. Transport and mixing phenomena can also lead to spatial complexity, but the degree depends on the significance of molecular or thermal diffusion.