Eur. Phys. J. B (2014) 87: 278
https://doi.org/10.1140/epjb/e2014-50468-6

Regular Article

Low-Prandtl-number Rayleigh-Bénard convection with stress-free boundaries

¹ Department of Mathematics, National Institute of Technology, 713209 Durgapur, India
² Department of Physics, Indian Institute of Technology, 721302 Kharagpur, India

^a e-mail: pinaki.math@gmail.com

Received: 10 July 2014
Received in final form: 26 September 2014
Published online: 24 November 2014

Abstract

Results of direct numerical simulations on Rayleigh-Bénard convection in low-Prandtl-number convection with stress-free horizontal boundaries are presented. Simulations are done in a three dimensional rectangular simulation box of dimensions L_x × L_y × 1. Instabilities and the corresponding fluid patterns near onset of convection are investigated by varying the horizontal aspect ratio η = L_y/L_x in a range 1 ≤ η ≤ 4. Fluid patterns are complex and unsteady at the instability onset for η ≥ 2. They consist of wavy rolls, rhombic patterns and oblique wavy rolls. The patterns near onset are time periodic for η < 2. We observe periodic wavy rolls for η = 4 / 3. Homoclinic bifurcations are observed for η = 1 for 0 ≤ Pr ≤ 0.03. We observe spontaneous breaking of a single limit cycle in two and again spontaneous merging of two limit cycles into one in a simulation box with η = 1, as the reduced Rayleigh number r = Ra/Ra_c is raised at a fixed value of Pr. The effect of Prandtl number on the homoclinic bifurcations is also investigated. We also present a low-dimensional model, which captures the instability sequence quite accurately for η = 1.