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

AUTHORS:
D. Puigjaner (1), C. Sim\'{o} (2), F.X. Grau (3) and Francesc Giralt (4)

(1) dpuigja@etse.urv.es 
Dept. Enginyeria Inform\`{a}tica i Matem\`{a}tiques, ETSE, 
Universitat Rovira i Virgili, 
Tarragona, Catalunya, Spain.

(2) carles@maia.ub.es 
Dept. Matem\`{a}tica Aplicada i An\`{a}lisi, Fac. Matem\`{a}tiques, 
Universitat de Barcelona, 
Barcelona, Catalunya, Spain.

(3) Dept. Enginyeria Mec\`{a}nica, ETSEQ, 
Universitat Rovira i Virgili, 
Tarragona, Catalunya, Spain.

(4) fgiralt@etseq.urv.es
Dept. Enginyeria Qu\'{\i}mica, ETSEQ, 
Universitat Rovira i Virgili, 
Tarragona, Catalunya, Spain.

ABSTRACT:
A numerical study of bifurcations and stability of stationary convective
flows in cubical and rectangular enclosures heated from below was carried
out using a Galerkin spectral method with a complete, divergent-free set of
trial functions satisfying all boundary conditions. A path-continuation
method was applied to determine the stationary solution of the non--linear
governing equations as a function of Rayleigh number within the range 
Ra_c < Ra < 70 000 . The eigenvalue problem associated with
stability analysis of the non-linear stationary solutions along the
bifurcated branches was solved using the QR algorithm. Four different
bifurcations from the conductive state were identified for Ra<10^{4} . At
the first transition (Ra_c < 3389) a x--roll and a diagonal--roll
were formed. While the former is stable, the latter is slightly unstable
with instability increasing with Ra. The second bifurcation yielded an
unstable four--roll structure that stabilized at Ra = 8900 . The third
and fourth bifurcations resulted in highly unstable structures. The effect
of changing aspect ratios on the bifurcation Ra for rectangular cavities
was also studied. There is reasonable agreement between the critical
Rayleigh numbers, the type of structures developed, and the velocity and
temperature fields predicted by the current stability analysis and previous
numerical and experimental results reported in the literature. The
convergence of the method is consistent with the number of modes used and
the results clarify inconsistences of previous DNS and experimental results
published in the literature.