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In this work, we studied the thermoconvective instabilities in a pentagonal cavity containing a Newtonian fluid. The cavity provided with a side opening is uniformly heated from above by a constant heat flux. The natural ventilation phenomenon in the classic habitat of the hot climate is thus numerically analyzed with unsteady natural convection equations formulated with vorticity and stream-function variables. The finite volume predictions of two-dimensional laminar natural convection at high Rayleigh number are presented. Results show that the incoming fresh air and the hot air discharge begin with the late start of the convection. The phenomenon intensifies with time and the birth of instabilities improves the homogenisation of temperatures which imply the elimination of very cold and very hot areas. However, the competition between the incoming fresh air and the hot air expansion leads to a perpetual displacement of the thermal front. The cross-sections at the opening of the incoming fresh and outgoing hot air are time-varying and the penetration depth of the fresh air is highlighted by the large convective cells originated from the aperture. The non monotonic variation of the Nusselt number reflects not only the multicell nature of the flow but also expresses the heat lost by the active walls due to the fresh air.

In the absence of wind, the temperature gradient between the walls of an open cavity and the external environment causes a natural ventilation phenomenon which is the air movement through the openings. In doing so, by the phenomenon of convection, the apertures provide the thermal comfort and/or chemical decontamination of the enclosures. Interests of researchers are mainly focused on the natural ventilation control parameters in partially or fully open cavities.

The square or rectangular shape with a side opening is widely used to model the type of flow that occurs in household refrigerators and ovens under open-door conditions. Thus, works of Mokhtarzadeh-Dehghan et al. [^{6}. Experimental works performed by Wu and Ching [^{8}. Gan [

The recently numerical study of Kpode et al. [

On the basis of the literature review, it appears that no wok was reported on natural convection in ventilated pentagonal enclosure heated mainly by the top which is nevertheless the most encountered configuration in very hot climate. Indeed, it is the habitat model (residential, commercial building, warehouse, etc.) heated by the roof during a long period of sunshine. Transfers dominated over a long period of the day by conduction, cause a strong thermal stratification, and therefore the stagnant air becomes more and hotter. The birth of the thermoconvective instabilities would have positive effects by improving mixtures and mass transfers. Thus, very hot and very cold areas will be removed. Hence, by providing the pentagonal cavity essentially heated from above with a large side opening, this study consists in using the more general fluid numerical methods such as CFD for the insightful exploration of the thermal and dynamic fields of the open classical habitat of the hot climate for energy efficiency. The state of the system over time for a large Rayleigh number will be analyzed and the impact of the opening on the transfers will be highlighted.

A schematic diagram of the system under consideration is shown in

∂ t θ + ∇ → ⋅ ( V → ⋅ θ − ∇ → θ ) = 0 (1)

∂ t ω + ∇ → ⋅ ( V → ⋅ ω − Pr ⋅ ∇ → ω ) = Pr ⋅ Ra ⋅ ∂ x θ (2)

∇ → ⋅ ( − ∇ → ψ ) = ω (3)

where ψ and ω are such that:

u = ∂ y ψ , v = − ∂ x ψ and ω = ∂ x v − ∂ y u , u and v are respectively horizontal and vertical dimensionless coordinates of the velocity V → in x and y cartesian coordinates system. The reference parameters used to make the problem dimensionless are L , L 2 / α and ( q L ) / λ , which, respectively, represent the length, time and temperature gradient and the dimensionless temperature is such that: θ = λ ( T − T 0 ) / ( q L ) . Thus, it appears in Equation (2) the dimensionless parameters Prandtl number, Pr and Rayleigh number, Ra defined respectively as follows:

Pr = ν / α , Ra = g β L 4 q / ( ν α λ ) ,

where ν = μ / ρ , α = λ / ( ρ c p ) , and β = ( − 1 / ρ 0 ) ⋅ ( ∂ ρ / ∂ T ) P

The above equations are complemented by the following initial and boundary conditions:

when t = 0

u ( x , y , 0 ) = v ( x , y , 0 ) = ψ ( x , y , 0 ) = ω ( x , y , 0 ) = θ ( x , y , 0 ) = 0 ,

when t > 0

hydrodynamic conditions on the walls

ψ = ( n → ⋅ ∇ → ) ψ = 0 and ( n → ⋅ ∇ → ) [ ( n → ⋅ ∇ → ) ψ ] = − ω ,

thermal conditions on the inclined walls

( n → ⋅ ∇ → ) θ = − 1 ,

thermal conditions on the right side wall

( n → ⋅ ∇ → ) θ = 0 ,

thermal conditions on the bottom wall

θ ( x , 0 , t ) = 0 ,

where n is the external normal vector to each wall.

Conditions at the opening x = 0 and 0 ≤ y ≤ A ; where A is the dimensionless aperture (aspect ratio):

In the case of the transfers by natural convection in the open cavities, the major difficulty lies in the treatment of the magnitudes at the openings, which are unknown. To deal with these difficulties, Kettleborough [

∂ x v = ∂ y u = ∂ x ψ = 0 ,

ω = 0 ,

θ = 0 if u ≥ 0 and ∂ x θ = 0 if u < 0 .

The heat energy transmitted by the inclined active walls is characterized by the Nusselt number. The base or the ambient air temperature ( θ = 0 ) is used. Thus, the local and the mean Nusselt numbers of an active wall are:

N u = 1 θ P ,

N u ¯ = 1 S ∫ S N u ⋅ d s _{ }

where θ P is the instantaneous local temperature of the wall and S the length of an active wall. The instantneous volume flow rate [_{ }

V ˙ = 1 2 ∫ 0 A | u ( 0 , y ) | d y

An implicit Euler scheme is used for time integration and the finite volume method [^{−5} over all the grid points for each variable is considered as the convergence criterion. Thus, the computation code based on these above numerical schemes is firstly used to perform the tests with the isosceles right triangle under the work conditions of Flack [^{6} show that the transfers are mainly in the area adjacent to the active inclined walls as reported by Flack [^{4} with aperture size AR = 0.5 and for aperture position AP = 0.5 when the heated wall is inclined φ = 90˚ from the horizontal is used as reported [

Spaces and time steps are chosen to satisfy the necessary conditions of Courant Friedrichs-Lewy (CFL) [^{−}^{5} is then selected (see

The upper walls are inclined at an angle of 45˚ relative to the horizontal plane and the aspect ratio A = h/L = 0.5. The results are relative to the air whose physical properties match a Prandtl number of 0.7 and the Rayleigh number; Ra = 1 × 10^{8}. The streamlines and the isotherms are respectively incremented by Δ ψ = ( ψ max − ψ min ) / n and Δ θ = ( θ max − 0.02 ) / n with n, the number of increments.

The isotherms depicted in

The increasing deformations with a sudden depression to the median line of the isotherms, show that the heat gradually affects the whole cavity. The transfer by convection thus develops and induces the inlet and outlet air phenomenon. The symmetry of the flow is then broken. However, for t < 225, the symmetry is preserved (see

From t = 225, the diffusion depth of the incoming fresh air concomitantly increases with the appearance and development of a second counter-rotating secondary cell (dashed line) (

(Figures 5(e)-(g)). This shows the renewal of the air of almost the whole rectangular part at t = 650 (

Figures 6-8 display the temperature variations and velocity profiles over time at the opening. In time, the portion of the opening where the temperature is equal to the external ambient temperature, decreases and reaches the limit space point (x = 0, y = 0.35) that is greater than the height of the half-opening (

The curves in

which are substantially equal at the first moments, and their strong decrease corresponds to the pseudo-conduction mode and to the existence of the two symmetric main cells in the attic space (see

boundary layer gets thinner and the mean Nusselt numbers grow to a maximum and then drop smoothly as the thickness of the boundary layer increases.

A numerical investigation of two-dimensional unsteady laminar natural convection in a partially open pentagonal cavity was conducted with sunny boundary conditions and for a high Rayleigh number; 1 × 10^{8}. Under these conditions, the fluid flow in the cavity is unstable and enables understanding transfers by natural convection in classic semi-open habitat with a gable roof in a hot climate. When convection increases with the heat expansion at high time, phenomena of the incoming fresh air and hot air discharge begin. But, at the same time, the

fresh incoming air opposes to the heat expansion. This perpetual competition causes the displacement of the thermal front. The volume flow rate which increases with the transfers, is a good parameter for assessing the indoor air change rate. The time when its maximum value is reached would be the right time for effective decontamination or efficient insecticide treatment. The unsteady natural convection allows predicting the adequate diameter of the fresh air intake opening or the polluted air discharge opening because the cross-section of fluids (fresh and hot fluids) at the opening is time-varying. The largest cross-section is reached before the quasi-steady state. The average Nusselt numbers show that the overall variations of the inclined walls temperature over time are at the origin of the self-organisation of the system due to the appearance and disappearance of the hot fluid cores.

The authors declare no conflicts of interest regarding the publication of this paper.

Kpode, K., Nougbléga, Y., Mbow, C. and Banna, M. (2019) Numerical Analysis of the Transient Process of Flow and Heat Transfer at High Rayleigh Number in a Partially Open Habitat Heated from Above. Open Journal of Fluid Dynamics, 9, 253-267. https://doi.org/10.4236/ojfd.2019.93017

Nu: mean Nusselt number

V → : dimensionless velocity vector

A: aspect ratio, = h/L

C_{P}: specific isobaric heat capacity (J∙kg^{−1}∙K^{−1})

Gr: Grash of number, = g β L 4 q / ( ν 2 λ )

h: side heigth (m)

h_{O}: height of the attic space (m)

L: base (m)

Nu: local Nusselt number

Pr: Prandtl number, = ν / α

q: wall heat flux (W/m^{2})

Ra: Rayleigh number, Ra = Gr∙Pr

S: length of an incline wall (m)

t: dimensionless time

T_{O}: temperature of the base wall (K)

u, v: horizontal and vertical dimensionless velocity coordinates

x, y: horizontal and vertical dimensionless coordinates

α : thermal diffusivity (m2∙s−1)

β : coefficient of thermal expansion (K−1)

γ : angle (rad)

λ : thermal conductivity (W∙K^{−1}∙m^{−1})

μ : dynamic viscosity (kg, m^{−1}∙s^{−1})

ν : kinematic viscosity (m2∙s−1)

ω : dimensionless vorticity

ψ : dimensionless stream function

ψ min , ψ max : minimum and maximum value of ψ

ρ : density (kg/m3)

θ : dimensionless temperature, = λ ( T − T O ) / ( q L )