Stability analysis of the flow in a cubical cavity heated from below

Authors: D. Puigjaner, C. Sim\'o, F.X. Grau and Francesc Giralt

Abstract

Bifurcations from the conductive state in a laterally insulated cubical
cavity heated from below are examined. The perturbation equations for
velocity and temperature are solved using a Galerkin method with a set
of trial functions whose completeness within the cavity is proved. The
set of trial functions  satisfies boundary conditions and continuity
so that pressure is eliminated and the stability problem reduces to an
eigenvalue problem. Five different bifurcations from the conductive
state, two at $Ra=3389$ and one at $Ra=5902$, $7458$ and $8610$, are
identified together with the corresponding convective structures for
Rayleigh numbers below $10^4$. The stability of the five structures,
one $x$--roll, a diagonal roll ($x$--roll $\pm$ $y$--roll), a four rolls
structure ($x$--roll $+$ $y$--roll with both rolls changing sense of
rotation at the two central orthogonal vertical planes), a $2x$--rolls
$+$ $2y$--rolls and a $2x$--rolls $-$ $2y$--rolls, is examined up to
$Ra=3\times 10^4$. There is good agreement between the critical Rayleigh
numbers and the velocity and temperature fields predicted by stability
analysis and by previous numerical calculations. The effect of changing
the aspect ratios on the critical Rayleigh numbers is also presented.

A non-linear stability analysis of the five convective structures
identified shows that the single $x$--roll and the four rolls structures
are stable, the former from the first birfucation from the conductive
state and the latter beyond $Ra\approx 8900$. The diagonal single-roll
structure is slightly unstable near the first bifurcation. Small numerical
diffusion or experimental imperfections in previous numerical and
experimental studies could cause the stabilization of this diagonal-roll.
Both structures combining  $2x$--rolls and $2y$--rolls are unstable. The
non-linear analysis also shows that the $2x$--roll $-$ $2y$--roll
structure could bifurcate for $Ra>3\times 10^4$. Current results are in
agreement with previous numerical and experimental studies reported in
the literature.