Stability analysis of natural convection in a cavity; walls with uniform heat or mass flux

A linear stability analysis is made of a family of natural convection flows in an arbitrarily inclined rectangular enclosure. The flow is driven by prescribed heat or mass fluxes along two opposing walls. The analysis allows for perturbations in arbitrary directions; however, the purely longitudinal or transverse modes are numerically found to be the most unstable. For the numerical treatment, a finite difference method with automatically calculated differencing molecules, variable order of accuracy, and accurate boundary treatment is developed. In cases with boundary layers, a special scaling is applied. For base solutions with natural (bottom heavy) stratification, critical conditions are solved for as a function of the Rayleigh number, Ra, and the angle of inclination to the bottom-heated case, a, for different Prandtl numbers (Pr), with complete results for Pr = 0.025,0.1,0.7,7,1000, and Pr → ∞. The uniform flux case is found to be much more stable than that of Hart (1971) with fixed wall temperatures, a fact which is attributed to the much larger stratification which occurs in the base solution. As could be expected, instabilities tend to be favoured by a decrease in Pr, an increase in Ra, and a decrease in a; however, exceptions to all these rules could be found. Cases in which the wavenumber is zero, or approaches zero in different ways, are studied analytically. Integral conditions, derived from the unresolved end regions, are applied in the analysis. The results show that all the base solutions with unnatural (top heavy) stratification are unstable to large-wavelength stationary rolls whose axes are parallel with the base flow. Real-valued perturbations are constructed and visualized for some of the modes considered.