Carles Sim\'o

Title: From steady solutions to chaotic flows in a
       Rayleigh--B\'enard problem at moderate Rayleigh numbers

Talk given at the DDAYS in Benic\`assim, Castell\'o, on October 24,2012

Abstract: 

The dynamics of a Rayleigh--B\'{e}nard convection problem in a cubical
cavity at moderate values of the Rayleigh number ($Ra\leq 10^5$) 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 $N\approx 11\,000$ 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 5000units wide). An attracting two-torus, also
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 ($9\times 10^{4}\le Ra\le 10^{5}$) 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.

This is a joint work with Dolors Puigjaner, Joan Herrero and Francesc
Giralt.