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ABSTRACT
A new benchmark with a high accurate solution is proposed for the verification of numerical codes dealing with double-diffusive convection in a heterogeneous porous medium. The new benchmark is inspired by the popular problem of square porous cavity by assuming a stratified porous medium. A high accurate steady state solution is developed using the Fourier–Galerkin method. To this aim, the unknowns are expanded in double infinite Fourier series. The accuracy of the developed solution is assessed in terms of the truncation orders of the Fourier series. Comparison against finite element solutions highlights the worthiness of the proposed benchmark for numerical code validation.
Nomenclature
| A | = | stream function coefficient |
| B | = | temperature coefficient |
| C | = | dimensionless concentration |
| c | = | specific heat at constant pressure [J kg−1 K−1] |
| = | concentration imposed at the left wall | |
| = | concentration imposed at the right wall | |
| D | = | mass diffusivity [m2 s−1] |
| E | = | concentration coefficient |
| g | = | intensity of gravity [m s−2] |
| GrS | = | solutal Grashof number based on H and the fluid properties |
| GrT | = | thermal Grashof number based on H and the fluid properties |
| H | = | size of the square enclosure [m] |
| K | = | permeability of the porous medium [m2] |
| k | = | thermal conductivity [W m−1 K−1] |
| K0 | = | permeability at the origin |
| = | average permeability | |
| Le | = | lewis number (= α/D) |
| N | = | buoyancy ratio (=GrS/GrT) |
| Nc | = | total number of Fourier coefficients |
| Nx | = | truncation order in the x direction |
| Ny | = | truncation order in the y direction |
| = | average Nusselt number | |
| p | = | dimensionless total pressure |
| Pr | = | Prandtl number (= ν/α) |
| Rk | = | ratio of thermal conductivity (=km/kf) |
| Ra | = | local Rayleigh number |
| Ra0 | = | Rayleigh number at y = 0 |
| = | average Rayleigh number | |
| RF | = | residual for the flow equation |
| RH | = | residual for the heat transfer equation |
| RM | = | residual for the mass transfer equation |
| = | average Sherwood number | |
| T | = | dimensionless temperature |
| t | = | time [s] |
| = | temperature imposed at the left wall [K] | |
| = | temperature imposed at the right wall [K] | |
| u | = | dimensionless horizontal velocity |
| umax | = | maximum dimensionless horizontal velocity |
| v | = | dimensionless vertical velocity |
| vmax | = | maximum dimensionless vertical velocity |
| x | = | dimensionless horizontal coordinate |
| y | = | dimensionless vertical coordinate |
| α | = | thermal diffusivity of the fluid [m2 s−1] |
| βC | = | coefficient of expansion with mass |
| βT | = | coefficient of expansion with temperature |
| ϵ | = | porosity of the porous medium |
| δ | = | Kronecher delta function |
| Γ | = | matrix coefficient [Eq. (23)] |
| λ | = | |
| ν | = | kinematic viscosity of the fluid [m2 s−1] |
| Ω | = | matrix coefficient [Eq. (23)] |
| ω | = | |
| ϕ | = | concentration change of variable |
| Π | = | |
| Ψ | = | Galerkin trial function for flow equation |
| ψ | = | dimensionless stream function |
| ρ | = | density [kg m−3] |
| ρ0 | = | reference density of fluid [kg m−3] |
| σ | = | ratio of specific heat (= (ρc)m/(ρc)f) |
| τ | = | |
| Θ | = | Galerkin trial function for heat and mass equations |
| θ | = | temperature change of variable |
| εd | = | |
| φ | = | |
| ζ | = | rate of change of the permeability in the y direction |
| Subscripts and Superscripts | = | |
| f | = | fluid |
| m | = | fluid-saturated porous medium |
| s | = | solid matrix of the porous medium |
| * | = | dimensionless parameter |
Nomenclature
| A | = | stream function coefficient |
| B | = | temperature coefficient |
| C | = | dimensionless concentration |
| c | = | specific heat at constant pressure [J kg−1 K−1] |
| = | concentration imposed at the left wall | |
| = | concentration imposed at the right wall | |
| D | = | mass diffusivity [m2 s−1] |
| E | = | concentration coefficient |
| g | = | intensity of gravity [m s−2] |
| GrS | = | solutal Grashof number based on H and the fluid properties |
| GrT | = | thermal Grashof number based on H and the fluid properties |
| H | = | size of the square enclosure [m] |
| K | = | permeability of the porous medium [m2] |
| k | = | thermal conductivity [W m−1 K−1] |
| K0 | = | permeability at the origin |
| = | average permeability | |
| Le | = | lewis number (= α/D) |
| N | = | buoyancy ratio (=GrS/GrT) |
| Nc | = | total number of Fourier coefficients |
| Nx | = | truncation order in the x direction |
| Ny | = | truncation order in the y direction |
| = | average Nusselt number | |
| p | = | dimensionless total pressure |
| Pr | = | Prandtl number (= ν/α) |
| Rk | = | ratio of thermal conductivity (=km/kf) |
| Ra | = | local Rayleigh number |
| Ra0 | = | Rayleigh number at y = 0 |
| = | average Rayleigh number | |
| RF | = | residual for the flow equation |
| RH | = | residual for the heat transfer equation |
| RM | = | residual for the mass transfer equation |
| = | average Sherwood number | |
| T | = | dimensionless temperature |
| t | = | time [s] |
| = | temperature imposed at the left wall [K] | |
| = | temperature imposed at the right wall [K] | |
| u | = | dimensionless horizontal velocity |
| umax | = | maximum dimensionless horizontal velocity |
| v | = | dimensionless vertical velocity |
| vmax | = | maximum dimensionless vertical velocity |
| x | = | dimensionless horizontal coordinate |
| y | = | dimensionless vertical coordinate |
| α | = | thermal diffusivity of the fluid [m2 s−1] |
| βC | = | coefficient of expansion with mass |
| βT | = | coefficient of expansion with temperature |
| ϵ | = | porosity of the porous medium |
| δ | = | Kronecher delta function |
| Γ | = | matrix coefficient [Eq. (23)] |
| λ | = | |
| ν | = | kinematic viscosity of the fluid [m2 s−1] |
| Ω | = | matrix coefficient [Eq. (23)] |
| ω | = | |
| ϕ | = | concentration change of variable |
| Π | = | |
| Ψ | = | Galerkin trial function for flow equation |
| ψ | = | dimensionless stream function |
| ρ | = | density [kg m−3] |
| ρ0 | = | reference density of fluid [kg m−3] |
| σ | = | ratio of specific heat (= (ρc)m/(ρc)f) |
| τ | = | |
| Θ | = | Galerkin trial function for heat and mass equations |
| θ | = | temperature change of variable |
| εd | = | |
| φ | = | |
| ζ | = | rate of change of the permeability in the y direction |
| Subscripts and Superscripts | = | |
| f | = | fluid |
| m | = | fluid-saturated porous medium |
| s | = | solid matrix of the porous medium |
| * | = | dimensionless parameter |