Onset of oscillatory double-diffusive buoyancy instability in an inclined rectangular cavity

Double-diffusive buoyancy convection in an inclined rectangular closed cavity with imposed temperatures and concentrations along two opposite sidewalls is considered. Attention is restricted to the case where the opposing thermal and solutal buoyancy effects are of equal magnitude (buoyancy ratio Rρ = −1). In this case a quiescent equilibrium solution exists and can remain stable up to a critical thermal Grashof number Grc. For both infinite and finite layers, linear stability analysis shows that, when the cavity inclination α with respect to gravity decreases from 0° to −90°, Grc for the onset of stationary instability increases exponentially while that for the onset of oscillatory instability decreases exponentially. Below a critical αc, the first onset of instability is oscillatory, rather than stationary. For the infinite layer, the influences of α on the critical wave number and frequency of the oscillatory mode are shown and the corresponding flow structure of the eigenfunction consists of counterrotating vortices travelling from one end to the other. For a bounded layer, the neutral stability curves of the first two oscillatory modes, centrosymmetric and anticentrosymmetric, cross each other successively at a series of double Hopf bifurcation points as the aspect ratio increases. These two curves are not smooth, but each contains several abrupt changes, after every one of which a pair of counterrotating vortices is added to the flow field and thus the parity of the mode remains unchanged. The neutral curves showing the influences of Pr and Le are also obtained. The present work expands the work of Bergeon et al. (1999) [8] in which the same physical problem was studied and yet no oscillatory onset of instability was considered.