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ABSTRACT
The current study deals with the heatline-based analysis of natural convection in porous cavities with the curved top and bottom walls involving the Rayleigh–Bénard heating. The streamline cells are weak, and the wall-to-wall heatlines are observed for all the cases at the low Dam involving two test cases, Prm = 0.015 and 7.2. At the high Dam, the convective force takes the command, and multiple heatline cells are observed for all the concave (except for high wall concavity) and convex cases. The directions of the streamlines (for all Dam) and heatlines (at the high Dam) are exactly opposite for the concave and convex cases. The case 3 (concave) is the efficient case based on the largest heat transfer rate for Prm = 0.015 involving all Dam and for Prm = 7.2 involving the low Dam. At Prm = 7.2 and high Dam, the case 1 (concave or convex) may be the efficient cases compared with the cases involving high wall curvatures.
Nomenclature
| Dam | = | Darcy number |
| g | = | acceleration due to gravity, m s−2 |
| L | = | height of enclosure, m |
| n | = | normal vector in outward direction |
| N | = | total number of nodes |
| Nu | = | local Nusselt number |
| = | average Nusselt number | |
| p | = | pressure, Pa |
| P | = | dimensionless pressure |
| Prm | = | Prandtl number |
| R | = | residual of weak form |
| Ram | = | Rayleigh number |
| S | = | dimensionless distance along the wall |
| Smax | = | length of the curved wall |
| s′ | = | dummy variable |
| T | = | temperature, K |
| Th | = | temperature of hot bottom wall, K |
| Tc | = | temperature of cold top wall, K |
| u | = | x component of velocity |
| U | = | x component of dimensionless velocity |
| v | = | y component of velocity |
| V | = | y component of dimensionless velocity |
| x | = | distance along x coordinate |
| X | = | dimensionless distance along x coordinate |
| y | = | distance along y coordinate |
| Y | = | dimensionless distance along y coordinate |
| α | = | thermal diffusivity, m2 s−1 |
| β | = | volume expansion coefficient, K−1 |
| γ | = | penalty parameter |
| θ | = | dimensionless temperature |
| ν | = | kinematic viscosity, m2 s−1 |
| ρ | = | density, kg m−3 |
| Φ | = | basis functions |
| Π | = | dimensionless heatfunction |
| φ | = | angle made by tangent of curved wall with positive x axis |
| ψ | = | dimensionless streamfunction |
| ξ | = | horizontal coordinate in a unit square |
| η | = | vertical coordinate in a unit square |
| Subscripts | = | |
| b | = | bottom wall |
| k | = | node number |
| l | = | left wall |
| r | = | right wall |
| s | = | surface/wall |
| t | = | top wall |
| Superscripts | = | |
| e | = | element number |
Nomenclature
| Dam | = | Darcy number |
| g | = | acceleration due to gravity, m s−2 |
| L | = | height of enclosure, m |
| n | = | normal vector in outward direction |
| N | = | total number of nodes |
| Nu | = | local Nusselt number |
| = | average Nusselt number | |
| p | = | pressure, Pa |
| P | = | dimensionless pressure |
| Prm | = | Prandtl number |
| R | = | residual of weak form |
| Ram | = | Rayleigh number |
| S | = | dimensionless distance along the wall |
| Smax | = | length of the curved wall |
| s′ | = | dummy variable |
| T | = | temperature, K |
| Th | = | temperature of hot bottom wall, K |
| Tc | = | temperature of cold top wall, K |
| u | = | x component of velocity |
| U | = | x component of dimensionless velocity |
| v | = | y component of velocity |
| V | = | y component of dimensionless velocity |
| x | = | distance along x coordinate |
| X | = | dimensionless distance along x coordinate |
| y | = | distance along y coordinate |
| Y | = | dimensionless distance along y coordinate |
| α | = | thermal diffusivity, m2 s−1 |
| β | = | volume expansion coefficient, K−1 |
| γ | = | penalty parameter |
| θ | = | dimensionless temperature |
| ν | = | kinematic viscosity, m2 s−1 |
| ρ | = | density, kg m−3 |
| Φ | = | basis functions |
| Π | = | dimensionless heatfunction |
| φ | = | angle made by tangent of curved wall with positive x axis |
| ψ | = | dimensionless streamfunction |
| ξ | = | horizontal coordinate in a unit square |
| η | = | vertical coordinate in a unit square |
| Subscripts | = | |
| b | = | bottom wall |
| k | = | node number |
| l | = | left wall |
| r | = | right wall |
| s | = | surface/wall |
| t | = | top wall |
| Superscripts | = | |
| e | = | element number |