Analytical solution for natural convection of a heat-generating fluid in a vertical rectangular cavity with two pairs of heat source/sink

Abstract

An analytical solution of the steady natural convective flow in a vertical rectangular cavity induced by a uniform volumetric heat generation is established in the case where two source/sink pairs are separately mounted on its vertical sidewalls. The developed analytical solution is based on the perturbation theory and the Fourier series expansion analysis. Primary efforts focus on the significant effect of the source/sink pairs arrangement on the fluid flow and heat transfer characteristics in the cavity. Therefore, three different configurations are considered, and the established theoretical formulas are validated using numerical simulations based on the finite volume method. Comparisons are made for various cavity aspect ratios A ranging from 0.2 to 2, heat source/sink amplitude r from 0 to 0.5, and the transition parameter α from 0 to 0.3. The Rayleigh number, chosen as the perturbation quantity, is fixed at Ra=103. Results clearly show a good agreement regarding streamlines, isotherms, velocity, and temperature profiles. It is observed that for low α values, the local heat transfer along the cavity active walls undergoes an oscillating behavior due to the truncation errors in the mathematical process of integration.