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 pr escribed 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 calcula ted differencing molecules, variable order of accuracy, and accurate bounda ry treatment is developed. In cases with boundary layers, a special scaling is applied. For base solutions with natural (bottom heavy) stratification, critical con ditions are solved for as a function of the Rayleigh number, Ra, and the an gle of inclination to the bottom-heated case, alpha, for different Prandtl numbers (Pr), with complete results for Pr = 0.025, 0.1, 0.7, 7, 1000, and Pr --> infinity. The uniform flux case is found to be much more stable than that of Hart (1971) with fixed wall temperatures, a fact which is attribut ed 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, a n increase in Pa, and a decrease in alpha; 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 unresolve d end regions, are applied in the analysis. The results show that all the b ase solutions with unnatural (top heavy) stratification are unstable to lar ge-wavelength stationary rolls whose axes are parallel with the base flow. Real-valued perturbations are constructed and visualized for some of the mo des considered.