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A two-dimensional numerical study of natural convection in a tunnel whose plane is sinusoidal with an opening on the horizontal ceiling is presented. In this work, we study the thermoconvective instabilities of air in a tunnel closed at both ends and heated by the floor. The study was carried out for different cases of geometry by varying the thickness of the sinusoid and the height of the tunnel. In order to create a transverse movement of the air, we heated the floor to the temperature T
* _{f}* and kept the horizontal ceiling at the temperature T

*such that T*

_{g}*< T*

_{g}*. This work has a lot of scope of application ranging from geothermal flow to the civil engineering like ventilation in the case of tunnels and thermal comfort of buildings. The Navier Stokes equations that model this problem have been solved using the numerical method with simplifying assumptions such as the Boussinesq hypothesis. And finally an interpretation of the results of the simulations has been made taking into account the variation of the form factors such as the amplitude of the sinusoid for Rayleigh number 2 x 10*

_{f}^{4}≤

*R*

_{a}≤ 7 x 10

^{6 }and Prendtl number P

_{r}= 0.71.

Modeling heat transfer in cavities like tunnels, by natural convection, has been the subject of many researches ranging from geothermal modeling to bioengineering or electronics. The effect of geometry on the thermodynamic behavior of these cavities has aroused much interest among researchers [_{g} presents an opening and we progressively vary the amplitude of the sinusoidal floor brought to temperature T_{f}. Since the appearance of the first tunnels a lot of investigation going from the ventilation to the smoke extraction came into being in order to make more practical and to maintain these structures [

different values of the form factors like the height of the tunnel

amplitude of the sinusoid δ presented (

To establish the natural convection we are studying, we have applied the formalism vorticity-stream function to the adimensional equations of Navier stocks and energy that govern the flow of fluid in our tunnel whose form factors are shown in the table below.

With vorticity

Λ | |||
---|---|---|---|

δ = 2a |

Peclet number (dimensionless)

Reynolds number (dimensionless)

Our problem, which is described by the equations above, does not admit analytical solutions except in very simplified cases. Therefore obtaining exact analytical solution of the equations being almost impossible; To solve this problem we used the finite volume method and the governing equations with appropriate boundary conditions are solved under steady, incompressible conditions with the Boussinesq approximation using the comlputational fluide dynamic(CFD) [

In the logic of validating our work, we have approached the results obtained (isothrems and streamelines) with the results obtained by P. K. Das [^{4} and P_{r} = 0.71. In the following, to better visualize the effects of form factors on the isothrems and streamelines we will display the results from the input of the sinusoid to the right wall.

We performed the simulation for different Rayleigh numbers ranging from 2 × 10^{4} to 7 × 10^{6} (^{4} and (Λ = 1/8) we find a symmetry of the two convolutional convective cells in the middle of the tunnel which is deformed with the increase of the amplitude of the sinusoid. This is due to the fact that as the amplitude of the sinusoid is increased, the heated fluid tends to cool rapidly before reaching the ceiling, thus causing the enlargement of one of the cells to the detriment the other. In addition to the low Rayleigh number cooling the heated fluid is favored by the opening in the middle of the ceiling with an increase in the amplitude of the current function due to a sharp increase in the velocity amplitude at the roof opening. This phenomenon becomes more and more important for the Rayleigh number R_{a} = 10^{5} and (Λ = 1/4) with the total disappearance of the cell at the entrance of the sinusoid to form a single cell in the middle of the tunnel at δ = 3/2. For Rayleigh = 7 × 10^{6} and (Λ = 3/8) already we find that from the first value of δ = 1/2 we have the presence of a single convective cell occupying the entire length of the sinusoid. This cell disappears progressively for δ = 1, giving rise to superimposed daughter cells perpendicular to the length of the tunnel. For Λ = 3/8 and δ = 3/2 a recirculation zone caused by a rise in temperature is formed in the middle of the tunnel and extends a little far from the outlet of the sinusoid. Thus, we can observe the appearance of areas of turbulence on the level of the sunisoid that increases by increasing the amplitude of the sinusoid.

Local Nusselt number distribution at the wavy bottom wall is shown in

periodic symmetry is due to the fact that the amplitude of the sinusoid is weak so we are closer to a plane plate with undisturbed convective cells. We find that for any value of δ the decrease of the number of convective cells as a function of the increase in Λ leads to a decrease in the number of ripples in the Nusselt number. And as the amplitude of the sinusoid increases, we notice a deformation of the periodic form in the middle of the tunnel caused by the appearance of turbulence zones in the middle of the sinusoid. Λ = 3/8 coresponding to R_{a} = 7 × 10^{6}

the height of the tunnel is so sufisant that the exchange between the fluid and the floor is disturbed and from where the diformity on the wave profile of Nusselt number at the level of the sinusoid. So we can say that as the amplitude of the sinusoid is superior or equal to 1 for Λ = 3/8 instability develops inside the sinusoid creating the birth of many daughter cells.

Based on the numerical results carried out in this research programme, the numerical results we were allowed to pull a better understanding of the influence of shape factors on the thermodynamic transfer of the area in a confined space such as the case of the tunnel studied in this problem. The effects of the wavy surface, aspect ratio, and the variation in the number of Rayleigh on the local Nusselt number have examined using numerical simulations with Ansys Fluent [

This work is the brainchild of the research group of Fluid Mechanics and Applications of Cheikh Anta Diop University of Dakar in collaboration with Dr. Ing. Bode Florin in Department of Mechanical Engineering at the Technical University of Cluj-Napoca, Romania.

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

Drame, O., Mbow, C. and Bode, F. (2019) Numerical Simulation of the Natural Convection in a Tunnel Whose Floor Has a Variable Sinusoid with a Roof Opening. Open Journal of Fluid Dynamics, 9, 326-333. https://doi.org/10.4236/ojfd.2019.94021