Hexagonal convection cells under conditions of vertical symmetry

Nucleation, growth, and coarsening of crystalline domains in order-order transitions between lamellar and hexagonal phases

Using the numerical solution of the time-dependent Ginzburg-Landau equation, we study the entire process of transformation between the lamellar and the hexagonal phases from the early-stage nucleation and growth to the late-stage coarsening regime. The metastable crystalline structure that nucleates first is identified in terms of the mean-field theory under the single-wave-number approximation. This has been borne out by the numerically efficient preparation of single-crystal structure developed via the noise-induced self-organization. We also present results for the scaling of the late-time domain growth, which is quantified by two measures: the structure factor and the orientational correlation function. In particular, the growth exponent is shown to be robust and indifferent to conservation of the order parameter.

Hexagons and triangles in the Rayleigh–Bénard problem: quintic-order equations on a hexagonal lattice

On a weakly nonlinear basis, we revisit the pattern formation problem in the Boussinesq convection, for which nonlinear terms of the quadratic order are known to vanish from amplitude equations. It is thus necessary to proceed to the quintic-order approximation in order for the amplitude equations to be generic. By deriving the quintic amplitude equations from the governing PDEs, we examined the bifurcation of steady solutions under rigid–free, rigid–rigid and free–free boundary conditions. Right above the criticality, all the axial solutions are obtained including up- and down-hexagons under the asymmetric boundary conditions and hexagons and regular triangles under the symmetric conditions. Hexagons and regular triangles are unstable whereas rolls are stable as has already been predicted by the cubic-order amplitude equations. Irrespective of the boundary conditions, quintic-order terms stabilize hexagons except near the criticality; rolls and hexagons thus coexist stably in an open region. This suggests that amplitude equations of higher order are possible to predict re-entrant hexagons .

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.

Cellular flow patterns and their evolutionary scenarios in three-dimensional Rayleigh-Bénard convection

The evolution of three-dimensional, cellular convective flows in a plane horizontal layer of a Boussinesq fluid heated from below is studied numerically. Slow motion in the form of a spatially periodic pattern of hexagonal cells is introduced initially. In a further development, the flow can undergo a sequence of transitions between various cell types. The features of the flow evolution agree with the idea of the flow seeking an optimal scale. In particular, two-vortex polygonal cells may form at some evolution stages, with an annular planform of the upflow region and downflows localized in both central and peripheral regions of the cells. If short-wave hexagons are stable, they exhibit a specific, stellate fine structure.

Long-wavelength instabilities of three-dimensional patterns

Long-wavelength instabilities of steady patterns, spatially periodic in three dimensions, are studied. All potentially stable patterns with the symmetries of the simple-, face-centered- and body-centered-cubic lattices are considered. The results generalize the well-known Eckhaus, zigzag, and skew-varicose instabilities to three-dimensional patterns and are applied to two-species reaction-diffusion equations modeling the Turing instability.

Asymmetric squares as standing waves in rayleigh-Benard convection

Possibility of thermal convection in the form of asymmetric squares in a thin layer of Boussinesq fluids of large lateral extension confined between stress-free and conducting flat boundaries is investigated numerically using a seven mode Lorenz-like model. For fluids with moderate and high Prandtl numbers (4<sigma<20) and in a narrow window of high Rayleigh numbers (12<R/R(c)<15), which depend on Prandtl number, the stationary rolls become unstable and asymmetric squares appear as standing waves at the onset of secondary instability. Asymmetric squares, two-dimensional rolls, and again asymmetric squares with their corners shifted by half a wavelength form a stable limit cycle. The oscillatory bifurcation is supercritical.