From steady solutions to chaotic flows in a Rayleigh–Bénard problem at moderate Rayleigh numbers

Abstract

The dynamics of a Rayleigh–Bénard convection problem in a cubical cavity at moderate values of the Rayleigh number ( Ra ≤ 10 5 ) and a Prandtl number of Pr = 0.71 (with extensions to Pr = 0.75 and 0.80) was investigated. The cubical cavity was heated from below and had perfectly conducting sidewalls and uniform temperature distributions on the two horizontal walls. A system of ordinary differential equations with a dimension of typically N ≈ 11 000 was obtained when the conservation equations were discretized by means of a Galerkin method. Previous knowledge of the bifurcation diagram of steady solutions, reported in the literature, was used to identify the origin of several branches of periodic orbits that were continued with Ra. Half a dozen of such periodic orbits were found to be stable within narrow ranges of Ra (at most, some 5000 units wide). An attracting two-torus, restricted to a very narrow region of Ra, was also identified. It was found that the instabilization of periodic orbits quite often resulted into the development of complex dynamics such as the creation of homoclinic and heteroclinic orbits. Instances of both types of global bifurcations were analyzed in some detail. One particular instance of chaotic dynamics (a strange attractor) was also identified. Chaotic dynamics has been found at Pr = 0.71 in a flow invariant subspace, which can be interpreted as a fixed-point subspace in terms of equivariant theory; this subspace is not attracting. However, some regions of attracting chaotic dynamics for moderate Rayleigh numbers ( 9 × 10 4 ≤ R a ≤ 10 5 ) were found at values of Pr slightly above 0.71. The role of a particular homoclinic solution found at Pr = 0.71 in the generation of these chaotic regions was analyzed.