The natural convection in inclined side-heated rectangular boxes with adiabatic walls is theoretically and numerically investigated. The study is focused on the characterization of the convection patterns arising at the core of the basic steady unicellular flow and covers the whole range of Prandtl numbers $(0\mathrm{}<~\mathrm{Pr}<~\infty )$ and inclinations (from $\alpha =0\mathrm{}°,$ heated-from-below vertical cavities, to $\alpha =180\mathrm{}°).\mathrm{}$ The onset of the flow instabilities depends on the core Rayleigh number $R\equiv K\mathrm{Ra},$ defined in terms of the local streamwise temperature gradient, $K\Delta T/L.\mathrm{}$ The critical value of R for transversal and longitudinal modes is determined by the linear stability analysis of the basic plane-parallel flow, which also provides the stability diagram in the $\left(\mathrm{Pr}-\alpha \right)$ chart. Anyhow, the effect of confinement can decisively change the stability properties of the core: if the steady unicell reaches the boundary layer regime (BLR) the local temperature gradient vanishes at the core leaving a completely stable core region. A theoretical determination of the frontier of the BLR in the space of parameters (α, R, and cavity size) yields an extra criterion of stability that has been displayed in the stability diagram. As confirmed by numerical calculations, the core-flow instabilities can only develop for $\mathrm{Pr}<O\left(1\right)\mathrm{}$ whereas, at larger Pr the core region remains stable and the instabilities may only develop at the boundary layers. The analysis of the instability mechanisms reveals several couplings between the momentum and temperature fields that are not possible in the horizontal $(\alpha =90\mathrm{}°)$ or vertical limits. For instance, by tilting the cavity with respect to $\alpha =0\mathrm{}°,$ the (Rayleigh-Bérnard) stationary thermal mode is suppressed in cavities whose depth is smaller than a theoretically predicted cutoff wavelength. The inclination also alters the properties of the oscillatory longitudinal instability, extensively investigated in the horizontal Hadley configuration at low Pr (liquid metals). An analytical relationship for its frequency in terms of $\alpha ,$ Ra, and Pr is derived. Throughout the paper, numerical calculations in two- and three-dimensional enclosures illustrate each type of multicellular flow and examples of instability interactions near the codimension-2 lines predicted by the theory.

• Received 19 October 2000

DOI:https://doi.org/10.1103/PhysRevE.64.016303