Evolution of two-dimensional periodic Rayleigh convection cells of arbitrary wave-numbers

Abstract

By introducing small controlled perturbations prior to the onset of motion, two-dimensional convection cells of arbitrary width-depth ratios are produced in a horizontal fluid layer heated from below. The conditions employed correspond to the finite-amplitude Rayleigh stability problem for a constant viscosity, large Prandtl number, Boussinesq fluid with rigid, conducting boundaries. It was found that two-dimensional cells with width-depth ratios close to unity are stable at all Rayleigh numbers investigated (Rc [Lt ] R [Lt ] 2·5Rc). Cells whose widthdepth ratios are moderately too large or too small tend to undergo size adjustments toward a preferred value of about 1·1. If the width-depth ratios are much too large or too small, they tend to develop three-dimensional instabilities in the form of cell boundary distortions and transverse secondary cells, respectively. Eventually the flow settles into a new family of rolls with a more preferred widthdepth ratio. It is suggested that these observations may have implications on nonlinear interchange instability problems and geophysical flows.