We retain this study, generally when we submit vertical walls of a cavity filled with air at a constant temperature difference; the flow generated depends on several parameters, the main ones:

Ÿ Rayleigh number R a H = g β Δ T H 3 α υ .

Ÿ Prandtl number P r = υ α .

Ÿ Vertical aspect ratio A υ = H l .

In this configuration, the horizontal walls are adiabatic and the two vertical others are differentially heated. The flow is then monocellular with the upward fluid along the hot wall and descending following the cold wall. We take air as fluid convection within this square cavity.

The two horizontal surfaces are subjected to a radiation flux and taking into account the values used by Wang et al. [15] as shown in Figure 1. The other faces are vertical to a temperature deviation.

Ÿ The hypothesis:

• The fluid is Newtonian and incompressible.

• The interior walls are considered gray, diffuse, opaque, and have the same emissivity value Ԑ and Lambertien (emission and reflection with an isotropic luminance).

• The flux densities and temperatures elementary surfaces are uniform.

• The cavity is filled with a transparent fluid (air), homogeneous and isotropic whose refractive index is equal to unity.

• The air movement is governed by Navier-Stokes equations under the assumption of Boussinesq.

The equations in Cartesian coordinates are given taking into account the assumptions listed above.

Ÿ Continuity equation:

∂ U ∂ X + ∂ V ∂ Y = 0 (1)

Ÿ Conservation equation for momentum:

Along the x axis: the mass forces are zero, so:

∂ U ∂ t + U ∂ U ∂ X + V ∂ V ∂ Y = υ ( ∂ 2 U ∂ X 2 + ∂ 2 U ∂ Y 2 ) − 1 ρ 0 ∂ P ∂ X (2)

Along the y axis: Due to buoyancy in natural convection;

ρ ˙ = ρ 0 − ρ

d V d t = ( ρ 0 − ρ ) g + μ Δ V − ∂ P ∂ Y (3)

In natural convection, the density of the fluid ρ₀ is variable, if we use the Boussinesq approximation, ρ = ρ 0 ( 1 − β ( T − T 0 ) ) assuming that ρ remains constant in the other terms such as, ρ = ρ₀ then we get:

d V d t + U ∂ U ∂ X + V ∂ V ∂ Y = g β ( T − T 0 ) + υ ( ∂ 2 V ∂ X 2 + ∂ 2 V ∂ Y 2 ) − 1 ρ 0 ∂ P ∂ Y (4)

with β the coefficient of thermal expansion.

Ÿ Energy conservation equation:

For a two-dimensional system (0, x, y): d i v ( g r a d T → ) = ( ∂ 2 T ∂ X 2 + ∂ 2 T ∂ Y 2 ) and a = λ ρ C t p is the thermal diffusivity.

So the equation becomes:

∂ T ∂ t + U ∂ T ∂ X + V ∂ T ∂ Y = a ( ∂ 2 T ∂ X 2 + ∂ 2 T ∂ Y 2 ) (5)

Initial Condition at t = 0 , U = 0 , V = 0 , T ( 0 , X , Y ) = T 0 .

- We assume a non-slip condition to the fluid particles on the walls of the rigid and impermeable pregnant, so that:

{ U ( X = 0 , Y ) = V ( X = 0 , Y ) = 0 U ( X = L , Y ) = t V ( X = L , Y ) = 0 U ( X , Y = 0 ) = V ( X , Y = 0 ) = 0 U ( X , Y = H ) = V ( X , Y = H ) = 0 (6)

- With regard to the condition of stationarity of hot and cold temperatures on vertical walls, the following applies:

{ T = T C     for   X = 0 ,     0 ≤ Y ≤ H T = T f     for   X = L ,     0 ≤ Y ≤ H (7)

- For horizontal adiabatic walls, the following applies: ∂ T ∂ Y / Y = 0 , H = 0 .

Dimensioning equations involves making equations dimensionless by using reduced variables.

Let’s reduce these equations to dimensionless form. To do this, let’s define the characteristic quantities as:

∂ u ∂ τ + u ∂ u ∂ x + v ∂ u ∂ y = 1 ρ 0 P r R a ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) − ∂ P ∂ x ∂ v ∂ τ + u ∂ v ∂ x + v ∂ v ∂ y = θ + 1 ρ 0 P r R a ( ∂ 2 υ ∂ x 2 + ∂ 2 υ ∂ y 2 ) − ∂ P ∂ y (8)

- The energy equation:

∂ T ∂ t + U ∂ T ∂ X + V ∂ T ∂ Y = Δ T R a P r ( ∂ 2 T ∂ X 2 + ∂ 2 T ∂ Y 2 ) (9)

- At the initial instant, τ = 0 ,   θ ( x , y , 0 ) = 0 .

- For the temperature on the boundaries of the physical domain: { θ = 0           for   x = 0   ;   0 ≤ y ≤ H θ = 1           for   x = L   ;   0 ≤ y ≤ H .

- Whereas for adiabatic wall ∂ θ ∂ y / y = 0 , 1 = 0 .

The theoretical analysis undertaken has made it possible to reduce the number of variables, and to demonstrate that the system to be solved is a function of two main quantities: the Prandtl number Pr and the Rayleigh number Ra.

In the presence of radiation and for a semi-transparent medium, the adiabatic condition is expressed by the equilibrium between convective and radiative fluxes. The coupling of natural convection with surface radiation occurs solely through the thermal boundary conditions. For the two horizontal walls of the cavity, we have:

λ ∂ T ∂ y / y = 0 − φ r / y = 0 = 0 (11)

∂ T ∂ y / y = H − φ r / y = H = 0 (12)

λ is the thermal conductivity of air.

Dimensionless governing equations.

To scale the conservation equations, we will use the same reference quantities as those used for pure natural convection. The reference quantities are the same.

Ÿ Dimensioning boundary conditions:

θ = T − T 0 T C − T f , Q r = φ r σ T 0 4 , y = Y H , Δ T = T C − T f , T 0 = T C + T f 2 .

For y = 0 , we have: − λ ∂ T ∂ y y = 0 − φ r = 0 ⇒ λ ∂ T ∂ y = − φ r ,

λ Δ T ∂ θ H ∂ y = φ r σ T 0 4 ⇒ ∂ θ ∂ y = H λ Δ T φ r σ T 0 4 (13)

∂ θ ∂ y = N r Q r So N r = H σ T 0 4 λ Δ T .

The radiative condition:

λ ∂ T ∂ y / y = 0 = N r Q r (14)

λ ∂ T ∂ y / y = 1 = − N r Q r (15)

N r is the number of radiation ( N r = H σ T 0 4 λ Δ T ).

Q r is the dimensionless radiative flux Q r = φ r σ T 0 4 .

The above equations are to solve using COMSOL software as the simulation tool.

Different velocity and temperature curves were found and compared with those of Hong Wang et al. [7] .

We investigate the evolution of temperature and vorticity as a function of time at Ra = 10³ at different positions in the cavity. (x = 0.04; y = 0.04), (x = 0.2; y = 0.2), (x = 0.4; y = 0.4).

Figure 2 above shows different temperature fields as a function of the Rayleigh number in the differentially heated cavity. The two horizontal sides are adiabatic and the other two vertical sides are at T_c = 298.5˚K and T_f = 288.5˚K. A high temperature is seen close to the hot wall and as the temperature increases, it is distributed towards the middle of the cavity, before dropping to a low.

Figure 2 shows the evolution of the temperature as a function of time at different selected points in the cavity. It can be seen that, close to the active wall at a hot temperature, the fluids undergo very significant variations (x = 0.04; y = 0.04) in temperature. At the point (x = 0.4; y = 0.4), these variations become progressively smaller. The dynamic behavior varies with time before reaching an asymptotic limit.

Figure 3 shows the temperature profiles as a function of x for different values of Ra. The temperature distribution is linear for Ra = 10³, and a strong temperature gradient in the vicinity of the isothermal faces is observed as Ra increases.

Figure 4 and Figure 5 show the temperatures near the upper wall and the lower wall respectively. When the fluid first comes into contact with the hot wall, the hot air rises. There is a high level of heat exchange but as the fluid moves along the high adiabatic wall, it cools. Figure 6 shows that the air cooled by the altitude flows back down towards the low adiabatic wall.

Figure 6 shows the isotherms of the temperature for different values of the Rayleigh number. It shows that for a low value of the Rayleigh number, the fluid (air) exhibits vertical thermal stratification due to heat transfer solely by conduction which does not generate any convective flow. From the result shown in this figure, we can see that the energy transfer for 0 < x < 1 is one-dimensional. This can be explained by the fact that the isotherms are perpendicular to the main direction of heat transfer, the ox direction. For Ra > 10³, the isotherms are distorted and contract at the vertical interfaces, mainly due to the relatively high Rayleigh number Ra = 10⁷, and the steep temperature gradient at this interface.

Figure 7 shows the evolution of the velocity at different positions for Ra = 10³. It can be seen that close to the hot wall, the fluid particles have a low velocity (x = 0.04; y = 0.04) but as they move towards the hot wall, the fluids have a high velocity and then this velocity stabilises and takes a horizontal direction.

Figure 8, which shows a velocity field in the cavity for different values of Ra, shows that near the hot wall, the fluid subjected to a high temperature becomes light and therefore moves upwards towards the ceiling, whereas near the cold wall and in the presence of gravity, the fluid close to this wall and therefore heavy, moves downwards towards the low adiabatic wall (floor). This movement continues until the fluid reaches thermal equilibrium.

The temperature profile in the case without radiation Ԑ = 0 and in the case with

radiation Ԑ = 0.2 for Ra = 10⁴, the influence of emissivity on the isotherms by varying the Rayleigh number Ra and Ԑ = 0; 0.1; 0.2; 0.3; 0.8.

Analysis of this curve (Figure 9) shows that when the emissivity of the surfaces is increased, the temperature decreases along the cavity. Figure 9 shows that for Ra = 10⁴, the influence of surface radiation on thermal stratification decreases as a function of emissivity.

In Figure 10, the emissivity of the surfaces does not modify the temperature of the upper and lower walls. This can be explained by the fact that the upper wall loses heat (essentially negative net radiative flux) and the lower wall receives heat (essentially positive net radiative flux). Qualitatively speaking, surface radiation has the effect of inducing a flow structure that more closely resembles that in a cavity with perfectly conducting horizontal walls.

According to the analysis of Figure 11 of the velocity field, the increase in the Rayleigh number leads to an acceleration in the velocity of the flow close to the horizontal and vertical walls and at the centre of the cavity (in blue). By observing the velocity field for Ra = 10⁴ and Ԑ = 0.1; Ԑ = 0.2 respectively (b), (d), or for Ra = 10⁵ and Ԑ = 0.1; Ԑ = 0.6 respectively (c), (e), we can see that the variation in emissivity does not modify the velocity of the flow and therefore does not influence the dynamic behavior of the flow.

Figure 12 shows horizontal velocities as a function of y at different values of Ra (10³, 10⁴, 10⁵) and with emissivity Ԑ = 0.1. The velocity profiles indicate a strong acceleration near the horizontal walls, the negative velocities observed over half the cavity (y < 0) are characteristic of the recirculation zone. It can be seen that particle velocity increases with increasing Rayleigh number.