On laminar free convection in inclined rectangular enclosures

Abstract

A class of problems of natural convection in tilted boxes is studied by analytical and numerical methods. The convection is assumed to be driven by uniform fluxes of heat (or mass) at two opposing walls, the remaining walls being perfect insulators. Disregarding end-region effects, an exact analytical solution is derived for the state which occurs after initial transients have decayed. This state is steady except for a spatially uniform, linear growth in the temperature (or the species concentration) which occurs whenever the fluxes are not equal. It is characterized by a uni-directional flow, a linear stratification and wall-to-wall temperature profiles which, except for the difference in absolute values due to the stratification, are the same at each crosssection. The mathematical problem is in essence nonlinear and multiple solutions are found in some parameter regions. The Bénard limit of horizontal orientation and heating from below is found to give a first bifurcation for which the steady states both before and after the bifurcation are obtained analytically. For a tilted Bénard-type problem, a steady state with top-heavy stratification is found to exist and compete with a more natural solution. The analytical solution is verified using numerical simulations and a known approximate solution for a vertical enclosure at high Rayleigh numbers. The presented solution admits arbitrary Rayleigh numbers, inclination angles and heat fluxes. Some restrictions on its validity are discussed in the paper.