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ABSTRACT
This paper presents some novel problems associated with the steady natural convection flow in an inclined square cavity filled with a saturated porous medium. The proposed method is a high-accurate spectral method based on the Fourier–Galerkin technique. The numerical results have demonstrated the advantage for the following reasons. (a) The high-accurate method deals with inclined geometries successfully. (b) The streamlines, isotherms, and the average Nusselt numbers are affected significantly by the inclination of the cavity for high values of Rayleigh number. (c) In contrast with the finite element method a highly accurate and efficient solution with less computational effort is obtained.
Nomenclature
| A | = | stream function coefficients |
| B | = | temperature coefficients |
| g | = | acceleration due to gravity |
| k | = | permeability |
| L | = | cavity square side |
| Nc | = | total number of Fourier coefficients |
| Ne | = | total number of mesh elements |
| Nm | = | stream function truncation order in X |
| Nn | = | stream function truncation order in Y |
| Nr | = | temperature truncation order in X |
| Ns | = | temperature truncation order in Y |
| Nu | = | local Nusselt number |
| P | = | pressure |
| Ra | = | local Rayleigh number |
| RF | = | flow equation residual |
| RH | = | heat transfer equation residual |
| t | = | time |
| T | = | temperature |
| ϕ | = | the inclination angle |
| u | = | velocity component in the x direction |
| U | = | dimensionless velocity component in the x direction (= uL/α) |
| v | = | velocity component in the Y direction |
| V | = | dimensionless velocity component in the Y direction (= vL/α) |
| Umax | = | dimensionless maximum horizontal at the mid-plane X = 0.5 |
| Vmax | = | dimensionless maximum vertical velocity at the mid-plane Y = 0.5 |
| X | = | dimensionless horizontal coordinate (= x/L) |
| x,y | = | vertical and horizontal coordinate |
| Y | = | dimensionless vertical coordinate (= y/L) |
| α | = | effective thermal diffusivity |
| β | = | coefficient of thermal expansion of fluid |
| δ | = | Kronecker delta function |
| ΔT | = | temperature difference |
| ε | = | ratio of composite material to convective fluid heat capacities |
| ρ | = | fluid density |
| μ | = | fluid viscosity |
| υ | = | temperature change of variable |
| θ | = | dimensionless temperature |
| ϖ | = | dimensionless stream function |
| φ | = | stream function |
| χ | = | trial function for heat transfer equation |
| ϑ | = | trial function for the flow equation |
| γ | = | matrix coefficient |
| ξ | = | matrix coefficient |
| λ | = | matrix coefficient |
| τ | = | matrix coefficient |
| Γ | = | matrix coefficient |
| Λ | = | matrix coefficient |
| ϒ | = | matrix coefficient |
| Φ | = | vector coefficient |
| Subscripts | = | |
| c | = | cold fluid |
| h | = | hot fluid |
Nomenclature
| A | = | stream function coefficients |
| B | = | temperature coefficients |
| g | = | acceleration due to gravity |
| k | = | permeability |
| L | = | cavity square side |
| Nc | = | total number of Fourier coefficients |
| Ne | = | total number of mesh elements |
| Nm | = | stream function truncation order in X |
| Nn | = | stream function truncation order in Y |
| Nr | = | temperature truncation order in X |
| Ns | = | temperature truncation order in Y |
| Nu | = | local Nusselt number |
| P | = | pressure |
| Ra | = | local Rayleigh number |
| RF | = | flow equation residual |
| RH | = | heat transfer equation residual |
| t | = | time |
| T | = | temperature |
| ϕ | = | the inclination angle |
| u | = | velocity component in the x direction |
| U | = | dimensionless velocity component in the x direction (= uL/α) |
| v | = | velocity component in the Y direction |
| V | = | dimensionless velocity component in the Y direction (= vL/α) |
| Umax | = | dimensionless maximum horizontal at the mid-plane X = 0.5 |
| Vmax | = | dimensionless maximum vertical velocity at the mid-plane Y = 0.5 |
| X | = | dimensionless horizontal coordinate (= x/L) |
| x,y | = | vertical and horizontal coordinate |
| Y | = | dimensionless vertical coordinate (= y/L) |
| α | = | effective thermal diffusivity |
| β | = | coefficient of thermal expansion of fluid |
| δ | = | Kronecker delta function |
| ΔT | = | temperature difference |
| ε | = | ratio of composite material to convective fluid heat capacities |
| ρ | = | fluid density |
| μ | = | fluid viscosity |
| υ | = | temperature change of variable |
| θ | = | dimensionless temperature |
| ϖ | = | dimensionless stream function |
| φ | = | stream function |
| χ | = | trial function for heat transfer equation |
| ϑ | = | trial function for the flow equation |
| γ | = | matrix coefficient |
| ξ | = | matrix coefficient |
| λ | = | matrix coefficient |
| τ | = | matrix coefficient |
| Γ | = | matrix coefficient |
| Λ | = | matrix coefficient |
| ϒ | = | matrix coefficient |
| Φ | = | vector coefficient |
| Subscripts | = | |
| c | = | cold fluid |
| h | = | hot fluid |