Analysis of heatlines for natural convection within porous trapezoidal enclosures: Effect of uniform and non-uniform heating of bottom wall

Abstract

In this paper, the numerical investigation of natural convection in a porous trapezoidal enclosures has been performed for uniformly or non-uniformly heated bottom wall. Penalty finite element analysis with bi-quadratic elements is used for solving the Navier–Stokes and energy balance equations. The numerical solutions are studied in terms of streamlines, isotherms, heatlines, local and average Nusselt numbers for a wide range of parameters Da(10 −5–10 −3), Pr(0.015–1000) and Ra( Ra = 10 3–10 6). At low Darcy number ( Da = 10 −5), heat transfer is primarily due to conduction for all φ’s as seen from the heatlines which are normal to the isotherms. As Da increases to 10 −4, convection is initiated and the thermal mixing has been observed at the central regime for all φ’s. Distribution of heatlines illustrate that most of the heat transport for high Darcy number ( Da = 10 −3) occurs from hot bottom wall to the top portion of cold side walls. It has been found that secondary circulations appear at the top corners of the cavity for φ = 45°, 60° and bottom corners of the cavity for φ = 90° with Pr = 0.015, Da = 10 −3 and Ra = 10 6. The physical interpretation of local and average Nusselt numbers are illustrated using heatlines. For uniformly heated bottom wall with cold side walls, Nu b values are maximum near the corners of bottom wall for all φ’s irrespective to Da and Pr. In contrast, for non-uniformly heated bottom wall, the local Nusselt number ( Nu b ) is found to be minimum near the corners of bottom wall and that is also found to be a sinusoidal variation with distance at high Da for all angles ( φ). The Nu s distribution is similar in both cases along the side wall except near the junction of hot and cold wall for all φ’s. Overall, heat transfer analysis for bottom and side walls is presented in terms of average Nusselt numbers ( Nu b ¯ , Nu s ¯ ) . The critical Ra numbers corresponding to the onset of convection are obtained at Da = 10 −3 for Pr(0.015–1000). For Da = 10 −3, average Nusselt numbers ( Nu b ¯ and Nu s ¯ ) increase exponentially beyond the critical Ra. Overall, the heat transfer rate is large for square cavity ( φ = 90°) compared to other angles ( φ) irrespective of heating patterns.