Natural convection resulting from exponentially varying wall heating in a square enclosure

The numerical investigation in this study explores the effects of non-uniform wall heating in a square cavity and its influence on natural convection behavior. A non-uniform heat source is applied to the left vertical wall of the cavity, whereas the right vertical wall is uniformly cooled. The remaining horizontal walls are thermally isolated. The main focus is on the heat transfer and fluid mixing caused by the convection occurring within the cavity. The governing equations are tackled with the help of the Dual Reciprocity Boundary Element Method (DRBEM). In the DRBEM procedure, the fundamental solution of the Laplace equation is used for solving the stream function equation, while for the vorticity transport and temperature equations-initially converted into the modified Helmholtz form-the fundamental solution of the modified Helmholtz equation (MHD) is applied. In order to transform the equations into this form, a relaxation parameter is applied to the corresponding term within the Laplace terms, and a forward difference scheme is employed for the time derivatives. In addition to the benefit of solving smaller-sized systems resulting from the boundary discretization in DRBEM, there is no requirement for an additional time integration scheme for the vorticity transport and energy equations, thus removing any potential stability issues. Calculations were performed for Rayleigh numbers of 10³, 10⁴, 10⁵ and 10⁶ and beta parameters $-2,-1,0,1,2$. Obtained results show that the average Nusselt number was found to increase with increasing Ra and β parameter, indicating enhanced convective heat transfer. Thus, it has been concluded that the heater position is quite effective in heat transfer.