Analysis of heatfunction boundary conditions on invariance of heat flow in square enclosures with various thermal boundary conditions

A complete understanding on sensitivity of heatfunction boundary conditions during natural convection in a square cavity is illustrated. The enclosure is subjected to various cases of thermal boundary conditions such as (a) case 1: hot left wall, cold right wall and adiabatic horizontal walls, (b) case 2: hot bottom wall, cold left and right walls and adiabatic top wall and (c) case 3: hot bottom wall and cold left right and top walls. Various possible Dirichlet boundary conditions for heatfunction have been derived for cases 1-3. Galerkin finite element method with penalty parameter is used to solve the nonlinear coupled partial differential equations governing the flow and thermal fields and the finite element method is further used to solve the Poisson equation for streamfunction and heatfunction. Results are obtained to display the streamlines, heatlines and isotherms for various Rayleigh numbers (103 and 105) and Prandtl number (Pr = 0.7). It was concluded that, for all cases of heatfunction boundary conditions, the qualitative trend of heatlines remains identical whereas, the magnitudes of heatline contours varies with reference/datum point. In addition, the local and average Nusselt number plots are also independent of the selection of datum of reference heatfunction for all cases.