Nonlinear pattern formation near the onset of Rayleigh-Bénard convection
- 1 July 1984
- journal article
- research article
- Published by American Physical Society (APS) in Physical Review A
- Vol. 30 (1), 398-428
- https://doi.org/10.1103/physreva.30.398
Abstract
A two-dimensional relaxational model equation is studied numerically to investigate the role of lateral boundaries and nonlinear terms in pattern formation. The model reduces in perturbation theory to the same amplitude equation as the one derived from the three-dimensional Boussinesq equations for thermal convection. State-of-the-art numerical methods are described that solve the initial-boundary-value problem efficiently and accurately in large rectangular cells and for long times, for both rigid and periodic boundary conditions. The results of simulations for different aspect ratios, Rayleigh numbers, and initial conditions are discussed in detail. The interaction of defects, the effect of lateral boundaries on the growth and saturation of linear instabilities, and the origin of the long-time scales needed to reach a stationary state are studied. Wave-number selection is investigated using spatial Fourier analysis, and evidence is presented that the band of stable wave numbers is not uniformly occupied as a pattern evolves from random initial conditions of all length scales. These results are in good agreement with many of the observed experimental features of pattern formation in small- and large-aspect-ratio cells, and show some new features that have not yet been seen.Keywords
This publication has 65 references indexed in Scilit:
- Ingredients of a theory of convective textures close to onsetPhysical Review A, 1982
- Roads to turbulence in dissipative dynamical systemsReviews of Modern Physics, 1981
- Symmetry breaking far from equilibriumPhysical Review A, 1981
- Non-linear properties of thermal convectionReports on Progress in Physics, 1978
- Convective instability: A physicist's approachReviews of Modern Physics, 1977
- Hydrodynamic fluctuations at the convective instabilityPhysical Review A, 1977
- INSTABILITIES, OSCILLATIONS, AND CHAOSLe Journal de Physique Colloques, 1976
- Hydrodynamic fluctuations near the convection instabilityPhysical Review A, 1974
- Convection in boxes: experimentsJournal of Fluid Mechanics, 1972
- Evolution of two-dimensional periodic Rayleigh convection cells of arbitrary wave-numbersJournal of Fluid Mechanics, 1968