Finite element simulation of natural convection within porous trapezoidal enclosures for various inclination angles: Effect of various wall heating

The phenomena of natural convection within a trapezoidal enclosure filled with porous matrix for linearly heated vertical wall(s) with various inclination angles φ has been studied numerically. A penalty finite element analysis with bi-quadratic elements is used for solving the Navier-Stokes and energy balance equations. Wide range of parameters such as Rayleigh number, Ra(103⩽Ra⩽106), Prandtl number, Pr(0.026⩽Pr⩽1000) and Darcy number, Da(10-5⩽Da⩽10-3) have been used. Numerical results are presented in terms of streamlines, isotherms and Nusselt numbers. It has been found that secondary and tertiary circulations appear at the bottom half of the cavity for φ = 30° and φ = 0° with Pr=0.026and0.7,Da=10-3andRa=106 for linearly heated side walls. On the other hand, for linearly heated left wall and cold right wall, multiple circulations occur near the top portion of the cavity. For linearly heated side wall, the local Nusselt number (Nub) shows sinusoidal behavior with distance at high Darcy number for all tilt angles whereas increasing trend in Nus is observed in the upper half of the side wall for all tilt angles. For linearly heated left wall with cold right wall, increasing trend in Nub is observed irrespective to Da and Pr and Nub is even larger for higher Da. Increasing trend is also observed in Nul for all tilt angles for linearly heated left wall. Due to discontinuity in right corner, Nur first decreases and thereafter that increases for Y⩾0.2. The average Nusselt number remains constant up to Ra=106 at low Da for Pr = 0.026 whereas for Pr = 1000 and high Ra, that starts to decrease for bottom wall whereas that starts to increase for side walls due to convection dominant effect at high Da. In general, the average Nusselt number increases with the increase of Da and Ra for higher Da.