Finite element based heatline approach to study mixed convection in a porous square cavity with various wall thermal boundary conditions

Abstract

A penalty finite element method based simulation is performed to analyze the influence of various walls thermal boundary conditions on mixed convection lid driven flows in a square cavity filled with porous medium. The relevant parameters in the present study are Darcy number ( Da = 10 −5 − 10 −3), Grashof number ( Gr = 10 3 − 10 5), Prandtl number ( Pr = 0.7–7.2), and Reynolds number ( Re = 1–10 2). Heatline approach of visualizing heat flow is implemented to gain a complete understanding of complex heat flow patterns. Patterns of heatlines and streamlines are qualitatively similar near the core for convection dominant flow for Da = 10 −3. Symmetric distribution in heatlines, similar to streamlines is observed irrespective of Da at higher Gr in natural convection dominant regime corresponding to smaller values of Re. A single circulation cell in heatlines, similar to streamlines is observed at Da = 10 −3 for forced convection dominance and heatlines are found to emanate from a large portion on the bottom wall illustrating enhanced heat flow for Re = 100. Multiple circulation cells in heatlines are observed at higher Da and Gr for Pr = 0.7 and 7.2. The heat transfer rates along the walls are illustrated by the local Nusselt number distribution based on gradients of heatfunctions. Wavy distribution in heat transfer rates is observed with Da ⩾ 10 −4 for non-uniformly heated walls primarily in natural convection dominant regime. In general, exponential variation of average Nusselt numbers with Grashof number is found except the cases where the side walls are linearly heated. Overall, heatlines are found to be a powerful tool to analyze heat transport within the cavity and also a suitable guideline on explaining the Nusselt number variations.