Patterns of Double-Diffusive Natural Convection With Opposing Buoyancy Forces: Comparative Study in Asymmetric Trapezoidal and Equivalent Rectangular Enclosures

The patterns arising from instabilities of double-diffusive natural convection due to vertical temperature $T$ and solute concentration $c$ gradients in confined enclosures are investigated numerically with the finite-volume method, for mixtures with Lewis numbers Le both $Le<1$ (e.g., air-water vapor) and $Le>1$⁠. The problem originated from the need to gain better understanding of the transport phenomena encountered in greenhouse-type solar stills. Therefore, an asymmetric, composite trapezoidal geometry is here the original geometry of interest, for which no studies of stability phenomena are available in the literature. However, this is first related to the simpler and more familiar rectangular geometry having the same aspect ratio $A$ equal to 0.3165, a value lying in a range for which available results are also limited, particularly for air-based mixtures. The case of opposing buoyancy forces is studied in particular (buoyancy ratios $N<0$⁠), at values $N=−1$⁠, $−0.5$ and $N=−0.1$⁠, for which a wide spectrum of phenomena is present. The thermal Rayleigh number Ra is varied from the onset of convection up to values where transition from steady to unsteady convective flow is encountered. For $Le=0.86$ in the rectangular enclosure, a series of supercritical, pitchfork steady bifurcations (primary and secondary) is obtained, starting at $Ra≈13,250$⁠, with flow fields with three, four, and five cells, whereas in the trapezoidal enclosure the supercritical bifurcation is always with two cells. For higher values of Ra $(Ra≥165,000)$⁠, oscillatory phenomena make their appearance for all branches, with their onset differing between branches. The oscillations exhibit initially a simple periodic pattern, which subsequently evolves into a more complex one, with changes in the structure of the respective flow fields. For $Le=2$ and 5, subcritical branches are also encountered and the onset of convection is in most cases periodic oscillatory (overstability). This behavior manifests itself in the form of standing, traveling and modulated waves (SWs, TWs and MWs, respectively) and with an increase of Ra there is a transition from oscillatory to steady convection, either directly or, most often, through an intermediate range of Ra with aperiodic oscillations. In the trapezoidal enclosure, oscillations at onset of convection appear only for $N=−1$ in the form of traveling waves (TWs), succeeded by aperiodic and then steady convection, while for $N=−0.5$ and $−0.1$⁠, the bifurcations are transcritical, comprising a supercritical branch with two flow cells originating at $Ra=0$ and a subcritical branch with either two or four cells.