{'fraction_non_alphanumeric': 0.023378582202111614, 'fraction_numerical': 0.01088127558715794, 'mean_word_length': 4.615849969751966, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Mesoscopic Study of a Square Cavity Ventilated and heated by the bottom

In this paper, we present a numerical study of the laminar mixed convection in a ventilated cavity square; one of the walls is subjected to a constant temperature, while the other walls are considered adiabatic. The thermal lattice Boltzmann method with the double population model was used. A computer code was developed with the D2Q9 model for the velocity field and D2Q5 for the temperature field to determine the whole structure of the flow. The results are presented in the form of thermal fields for the Richardson number Ri = 10. Those show that each time the Richardson number increases, the number of Nusselt increases too what justifies an increase in the heat transfer.


Introduction
The transfer of heat by mixed convection aroused the considerable interest of several researches for technological applications such as: the ventilation of the buildings, the chemical plating of the thin layers, the cooling of the electronic parts, squanderers of heat in the solar collectors and nuclear reactors. Several simple and complex geometrical configurations were examined by report to a theoretical, digital end experimental approach. A large number of numerical studies were interested on fixing of only one entry and exit [1][2][3] with an isothermal wall. Others research considered the wall heated with a heat flow [4][5], where [6] studied various configurations of the entry and exit position in order to detect the best possible position of the entry and exit opening, and to obtain a more effective cooling. Other numerical and experimental [7][8][9] studies treated the effect of the geometry of an obstacle like the source of heat inside the cavity in order to maximize the total conductance. The position of the air entry and the exit has a great effect of the hydrodynamic and thermal structures [10]. One cavity that has several entries [11] improves its ventilation. The lattice Boltzmann method (LBM) being a digital method relatively recent and original which came out at the beginning of the Nineties. It is interested, either with the macroscopic quantities (celerity, pressure and density), but directly the distribution of the various particles constituting a fluid. We speak then about mesoscopic representation. What makes it competing with the other conventional methods such as finite volume, finite elements and the finite differences. It is initially resulting from the lattice gas method [12], from the automata cellular theory [13] and while being based on the physic statistical formalism [14]. It is important to be able to locate its performances compared to those of the classical digital methods used to simulate and reproduce the flows with the heat transfer in the ventilated enclosures. The purpose of the study presented in this work is the analysis of the phenomenon of the heat transfer with mixed flow of the convection in the laminar mode, in a square cavity provided with two openings. The interior walls are supposed to be adiabatic except for the located on the low side, it is considered isothermal. The thermal model of the lattice Boltzmann method with nine celebrities (D 2 Q 9 ) is used to reproduce the dynamic field and that simplified at five celebrities (D 2 Q 5 ) is used for the temperature field. Thus, a thermal analysis will have to be carried out with LBM. That will enable us to determine the performances of this new digital method in this field.

Physical description of the problem
The model chosen is square cavity of coast H filled with two ventilation openings, the first located in the lower left corner of L 1 side and the second located in the upper right side L 2 = L 1 = 20% H. The walls of this cavity are adiabatic except the lower wall. The walls of this cavity are adiabatic except the low wall which is maintained by a source of heat at a constant temperature T h . Air entering through the left opening of the wall with a temperature T 0 and a uniform velocity U 0 as fig.1 shows it. The assumptions used are summarized in the case of a incompressible fluid, Newtonian in two dimensions, laminar, satisfying the assumption with Boussinesq, stationary with a transfer of heat by radiation and a dissipation of heat by effect of viscosity negligible.

Lattice Boltzmann method
In statistical physics, a gas is described by a cloud of particles obeying the Boltzmann equation. This continuous equation as it was proposed by Boltzmann in 1872 can be written as follows: . (1) This equation is the basic equation of the kinetic theory of gases. It describes gas molecules distribution function which is a function of time, space and velocity of the molecule. With the approximation Bhatnagar, Gross and Krook (BGK) [15] The collision operator is written as follows : . (2) The collision frequency is given by: The local equilibrium distribution function is denoted by ; which is the distribution function of Maxwell-Boltzmann. In general the equilibrium distribution function Maxwell Boltzmann [16] is expressed by : . (4) Discretizing equation (10) in space, time and compared with speeds gives the Boltzmann equation system given by the following formula: .
The Local conservation laws: the Density, The velocity and Energy are expressed as follows [17]:

D 2 Q 9 model
For the discretization of speeds, the model of the square lattice nine speeds (Fig. 2) with a unit spacing. Each node of the network is connected to eight by eight other neighbouring links. The particles can only reside on a node and move to their earnest neighbours along these links for a unit time. There are three types of particles on each node with nine different speeds. , and The corresponding weight factors are : The dynamic viscosity is given by the formula:

D 2 Q 5 model :
For the discretization of the temperature, the square lattice pattern of five-speed ( Figure 3) with a spacing unit. Each node of the network is connected to four other neighbours by four links. The particles can only reside on a node and move to their nearest neighbours along these links for a unit time.
Since there is no coupling between the celerity equation and the energy, two different distribution functions have to be solved, for example for speed f and g for energy. The effort provided for the establishment of a stable model for the lattice Boltzmann method thermal (TLBM) has paid off recently. The celerity function equation is the same as that of D 2 Q 9 . For the energy function, the equation is : . (11) The summation of the distribution functions in each D 2 Q 5 lattice node gives us the temperature: .
The distribution equilibrium function is: . (13) e i are the lattice celerities defined by : .
The corresponding weight factors are: .
The equation of the thermal diffusion coefficient is given by .

Numerical code validation
We studied the natural convection in a square cavity closed differentially heated, where two horizontal walls of the top and bottom are considered adiabatic. It emphasizes that the right side wall of the cavity is subjected to a cold temperature and the left at a hot temperature. We compare the results of the value of Nusselt using the two methods with those of the literature. In this study, we perform simulations for moderate Rayleigh numbers 10 3 ≤ Ra ≤ 10 5 and Prandtl number Pr = 0.71. The mesh study based on lattice Boltzmann method with dual population (TLBM) was performed in the laboratory. Resolution mesh was satisfied between 71x71 and 151x151 for different Rayleigh numbers.

Results for the ventilated cavity
The geometrical configuration treated in this study is the ventilated square cavity presented in (fig. 1). The selected fluid inside is the air with (Pr = 0.71). We use the grid of 121*121 for the lattice Boltzmann method in all our calculations. We study the influence of the Richardson number on the total structure of the flow. For this case, the Grashof number is fixed to Gr = 105 and the Richardson number is varied by   Ri ≥ 10, we note a widening of temperature occupying a considerable portion of the cavity, also we observe a horizontal stratification of the layers of an air in which the natural convection is prevalent. The position of the hot wall has an influence on the heat transfer. We notice that the high temperatures are localised in narrow spaces in the vicinity of the hot wall which correspond to the thermal thickness of the boundary layer and which is largely influenced by the Richardson number. Far from the hot wall, the gradients of the temperature are weak.
In figure 5, the variation of the average Nusselt number depending on the Richardson number is presented. We note that the Nusselt number decreases with the increase of the Richardson number what justifies a decrease in the heat transfer.

Conclusion
This work focuses on the numerical study of laminar mixed convection in a ventilated cavity with two openings. The results obtained in this work were used to evaluate the performance and LBM's ability to reproduce the mixed convection in a ventilated cavity. The TLBM model with a double population model used has all the advantages, including good numerical stability and the ability to generally manage the overall heat transfer by convection problems.