Role of the importance of ‘Forchheimer term’ for visualization of natural convection in porous enclosures of various shapes

Abstract

The present study deals with the importance of the quadratic (Forchheimer) drag force for the flow through porous media during natural convection within various geometrical shapes (square, rhombus, concave and convex). The enclosures consist of the uniformly heated bottom wall, cold side walls and adiabatic top wall. The numerical simulations are performed via the Galerkin finite element method for various Darcy numbers (10-5⩽Dam⩽1), Prandtl numbers (Prm=0.015,0.7 and 1000) at a high Rayleigh number (Ram=106). Two different flow models are considered based on the inclusion of the quadratic drag term (the Forchheimer term); Case 1: the Darcy–Brinkman model and Case 2: the Darcy–Brinkman–Forchheimer model. At the low and moderate Prm (Prm=0.015 and 0.7), the effect of the Forchheimer term on the distributions of streamlines (ψ), heatlines (Π) and isotherms (θ) is significantly large for all Dam at Ram=106 in all the cavities. At the high Prm(Prm=1000), both the magnitudes and qualitative trends of the heat and flow fields are unaffected by the Forchheimer term for all Dam in all the cavities. The significance of the quadratic drag term on the heat flow visualization is addressed in detail via the heatline approach. The variation of the local Nusselt number(Nub) with the distance is also illustrated in detail for the Cases 1 and 2 for all the cavities. The overall heat transfer rate (Nub‾) and percentage error (E^) in Nub‾ for the Cases 1 and 2 are also analyzed. At Prm=0.015, E^ is largest, that decreases with Prm for all the cavities and E^ tends to 0% at Prm=1000. The effect of the Forchheimer term for various shapes are also illustrated via E^ vs Dam. It is found that, E^ is largest for the rhombic cavity at Prm=0.015 and that is largest for the square and convex cavities at Prm=0.7 with Dam>10-4.