Pattern formation in spatially ramped Rayleigh-B'enard systems

Abstract

We study pattern formation and selection in Rayleigh-Bénard systems confined between well conducting horizontal boundaries and subjected to a weak horizontal gradient of the Rayleigh number. The study is based on the numerical integration of the Swift-Hohenberg equation and addresses the questions of the preferred orientation of the patterns with respect to the gradient of the Rayleigh number, boundary effects observed at subcritial sidewalls, the characteristics of long-term evolution of the patterns with emphasis on the wavelength selection, and the effect of non-potential modifications of the Swift-Hohenberg equation.It is shown that, contrary to common belief, the rolls do not align with the direction of the horizontal temperature gradient (Dewel, 1989;Malomed, 1993), due to the influence of the walls. Rather, rolls approach a sidewall perpendicularly when the local bifurcation parameter is sufficiently beyond the threshold but tend to be parallel to a subcritical or critical sidewall. Simulations performed with non-potential modifications of the Swift-Hohenberg equation lead in most cases, to asymptotic time-dependent behaviours.

Numerical scheme for solving the nonuniformly forced cubic and quintic Swift-Hohenberg equations strictly respecting the Lyapunov functional

“…Good agreement was achieved. As an additional verification procedure, we also found good agreement between the results with the SH3 forced with a spatial ramp of , and those presented by Pontes et al (2008) [28]. In that work, the authors adopted GDBC and a first order in time scheme.…”

“…Configuration 1 shows a pattern emerging in a uniformly forced system with GDBC, in qualitative agreement with previous results [22,25,26,28]. The case was run in the verification framework our numerical code.…”

“…Configuration 2 reproduces the result of a pattern developed in presence of a ramped control parameter , with a subcritical region [27,29,28]. A structure with smaller amount of defects emerges, thanks to the fact that the weak structure of rolls parallel to the wall appears in the subcritical region.…”

“…In the present endeavor, we extended a numerical scheme proposed by Christov & Pontes [28] to investigate pattern formation modelled by the cubic Swift-Hohenberg equation in two dimensions. The original scheme presents second order representation of all derivatives, strict implementation of the associated Lyapunov functional, rigid boundary conditions (GDBC), and a semi-implicit time discretization of the terms, aiming at constructing negative definite operators that assure the unconditional stability of the scheme.…”

“…Other authors employed nonuniform distributions (ramps and gaussians) for the control parameter and presented very interesting results as well, displaying rich competition between bulk and boundary effects. A ramped control parameter mostly appears in the context of rigid boundaries [27,28], which will be addressed here as Generalized Dirichlet boundary conditions (GDBC), following [27,29,28].…”