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This current study focuses on the simulation of natural convection in square cavity filled with a porous medium considered homogenous, isotropic and saturated by a Newtonian fluid obeying the law of Darcy and the hypothesis of Boussinesq. The lower horizontal wall of the enclosure is subjected to a temperature varying sinusoidally with the space while the upper horizontal wall is maintained adiabatic. The vertical walls are kept cold isotherm. In order to generalize the results, all governing equations are put into dimensionless form, discretized by the Finite Difference Method and solved by the relaxed Gauss Seidel (SUR) Algorithm. A code has been proposed in Fortran 95, in order to solve numerically the equations of the problem. The study parameters are the Rayleigh-Darcy number (Ra) and the amplitude (A
_{r}) of the hot wall temperature. The effects of the Rayleigh-Darcy number and amplitude on the dynamic and thermal field, the horizontal velocity distribution and the mean horizontal temperature distribution (y = 0.5) were presented and discussed. It emerges from this study that the increases of the amplitude and Rayleigh-Darcy number intensify the flow and the global transfer of heat in our physical domain.

Nowadays convection transfer in a porous medium saturated by a fluid is a paramount importance in the field of engineering and Physics. Its applications are encountered in several fields such as geothermal energy, oil recovery, solar energy storage systems, transport of pollutants into a soil and storage chemicals materials to name but a few.

The problem of convection in fluid saturated porous media has been widely studied in the past decades for its applications in thermal, electronics, and engineering [

However, few studies on natural convection in porous media with variable heating have been conducted throughout the world. Schaladow et al. (1989) [

The main objective of this work is to continue to enrich this kind of problem by conducting a numerical study of natural convection in a porous square cavity saturated by a Newtonian fluid whose lower horizontal wall of the enclosure is totally subjected to Temperature varying sinusoidally with the space whereas the upper horizontal wall is maintained adiabatic (

kept cold isotherm. This study is very important because this kind of temperature variation occurs in the applications when a cylindrical heater is placed on a flat wall. The influence of the amplitude on the one hand on the thermal and dynamic fields and on the temperature and velocity distributions of the mean horizontal plane (y = 0.5) on the other hand has been investigated for a best control of convection in our cavity.

The system studied here is a porous square cavity saturated with a Newtonian fluid. The lower horizontal wall of the enclosure is subjected to a temperature varying sinusoidally with the space while the upper horizontal wall is maintained adiabatic. The vertical walls are kept cold isotherm. The simplifying assumptions adopted are as follows:

1) The flow is laminar and two-dimensional and follows the law of Darcy;

2) The porous medium saturated by the fluid is assumed to be homogeneous and in thermal equilibrium.

Δ v = 0 (1)

v = − K μ f ( ∇ p − ρ f g e z ) (2)

( ρ c ) * ∂ T ∂ t + ( ρ c ) f ( v ⋅ ∇ ) T = ∇ ( λ * ∇ T ) (3)

( ρ c ) * = φ ( ρ c ) f + ( 1 − φ ) ( ρ c ) s (4)

Under the boundary conditions are as follows:

u ( 0 , y ) = 0 , v ( 0 , y ) = 0 , T ( 0 , y ) = 0 , u ( H , y ) = 0 , v ( H , y ) = 0 , T ( H , y ) = 0 (5)

u ( x , 0 ) = 0 , v ( x , 0 ) = 0 , T ( x , 0 ) = 0 , u ( x , H ) = 0 , v ( x , H ) = 0 , (6)

T H ( x , 0 ) = T H + A r ( T H − T C ) cos ( π x H ) , ∂ T ( x , H ) / ∂ x = 0

Equations (1)-(3) can be written in terms of stream function (5)-(6) by positing:

u = ∂ ψ ∂ y , v = − ∂ ψ ∂ x (7)

x * = x H , y * = y H , t * = t H 2 ( ρ c ) * λ * ,

t 0 = H 2 ( ρ c ) * λ * , T * = T − T 0 T w − T 0 , v * = V λ * H ( ρ c ) f , P * = P + ρ 0 g z λ * μ f K ( ρ c ) f , ψ * = ψ λ * ( ρ c ) f (8)

∂ 2 ψ * ∂ x * 2 + ∂ 2 ψ * ∂ y * 2 = − R a ( ∂ T * ∂ x * ) (9)

∂ T * ∂ t * + ∂ ψ * ∂ y * ∂ T * ∂ x * − ∂ ψ * ∂ x * ∂ T * ∂ y * = ∇ 2 T * (10)

where the Darcy-Rayleigh number is defined by:

R a = g β K ( T H − T C ) α ν (11)

with: α = λ * ( ρ c ) f (12)

The Boundary Conditions (5)-(6) become:

y * = 0 , 0 ≤ x * ≤ 1 : T * ( 0 , x * ) = 1 + A r cos ( π x * ) , ψ * = 0 (13a)

y * = 1 , 0 ≤ x * ≤ 1 : ∂ T * ∂ y * = 0 , ψ * = 0 (13b)

x * = 1 , 0 ≤ y * ≤ 1 : T * = 0 , ψ * = 0 (13c)

x * = 0 , 0 ≤ y * ≤ 1 : T * = 0 , ψ * = 0 (13d)

An important physical quantity in heat transfer is the Nusselt number, which is defined by the flux transferred in convection on the flux transferred in conduction state.

The local number of Nusselt at a bottom wall heated by a variable temperature is defined as follows:

N u = 1 Δ T * ( − ∂ T * ∂ y * ) y * = 0 (14)

The average Nusselt number is given by the following formula:

N u ¯ = ∫ 0 1 N u d x * (15)

The differential Equations (9) and (10) governing the physical situation are translated into Algebraic equations using the finite difference scheme centered and accurate on the second order. The system of dimensionless algebraic equations with boundary conditions associated (13a)-(13d) is iteratively solved by the relaxed Gauss seidel algorithm. A mesh size of 120 × 121 was selected. We have developed a numerical code with FORTRAN 95.

The same convergence criterion is imposed in terms of relative error for temperature and stream function. The calculation stopped when the follows inequalities were satisfied:

Max [ ∑ | ψ i , j n + 1 − ψ i , j n | ∑ | ψ i , j n + 1 | ] ≤ 10 − 5 (16)

Max [ ∑ | T i , j ∗ n + 1 − T i , j ∗ n | ∑ | T i , j ∗ n + 1 | ] ≤ 10 − 5 (17)

^{−4}. The results provided by the present complete code solving the problem of natural convection in porous media have been confronted with the results obtained from the work of Manole and Lage (1999), Bejan (1979), and Baytas and pop (2002), for a stationary state of convection in a porous cavity saturated by a Newtonian fluid and whose vertical walls are active (The left vertical wall is heated by a constant temperature while the vertical wall on the right is cooled with a constant temperature) while the horizontal ones are insulated.