A model of confined thermal convection driven by non-uniform heating from below

Abstract

A model is presented of the two-dimensional boundary-layer and interior flow in a rectangular box resulting from the application of a quadratic temperature variation on its lower surface. The other walls are insulating. It is shown that a similarity form exists for the narrow, thermocline-like layer near the lower surface, and that this can satisfy all known consistency conditions with the interior, together with either laminar or turbulent side-wall regions. The interior temperature and Nusselt number are shown to be insensitive to Prandtl number, and to be primarily functions of the horizontal Rayleigh number Ra L . Specifically, the interior temperature, relative to the coldest applied value, is 60% of the total applied temperature range. The Nusselt number is predicted to vary as $0.26 Ra^{\frac{1}{5}}_L$ for a box with unit aspect ratio. The dynamics of the side-wall region, and the details of the imposed temperature variation appear to be unimportant in determining the overall buoyancy exchange. The solutions are compared with numerical and observational results, and generally good agreement is found.