This work presents a simulation of the phenomena of natural convection in an enclosure with a variable heating regime by the lattice Boltzmann method (LBM). We consider a square enclosure of side _{0} for ^{7}. The amplitude and period effect on the results is evaluated and discussed. The amplitude of the temperature at the heart of the enclosure increases with the increase in amplitude. This also increases with the period (_{0}) of the imposed temperature, something that is not observable on the global Nusselt number.

Natural convection is a widely observed phenomenon found in various systems, such as residential heating and cooling, heating circuits, and electronic devices [^{9}. Salat et al. [^{5} ≤ ^{6}) and revealed the influence of heating frequency on heat transfer, streamlines, and isotherms. They observed non-periodic solutions at higher Rayleigh numbers. Wang et al. [

In this particular study, the LBM was employed to replicate the phenomenon of heat transfer through natural convection, incorporating the element of variable heating. The investigation revolves around a 2D enclosure (

The LB framework utilized in this context aligns with the one applied in prior studies [

The lattice Boltzmann equation under the BGK approximation and in the absence of external forces can be expressed as stated in references [

The equilibrium distribution can be expressed mathematically by the following expression [

where

For D2Q9 (

The weights factors

Finally, the BGK approximation for the LB equation discretized and simplified with external forces can be written as:

For momentum

Taking into account the Boussinesq approximation, the external force denoted as

Finally, for the model D2Q9, the density

For the temperature, other distribution functions for the lattice Boltzmann method are defined as:

where

The equilibrium distribution functions for the temperature can be used at first-order [

The temperature

The boundary conditions pertaining to the investigated problem are outlined below:

Dynamic boundary conditions:

For all the walls:

Thermal boundary conditions:

The dimensionless temperature can be defined as:

On the adiabatic walls, we used this condition

Here,

For the imposed temperature on the hot and the cold portion, the bounce-back conditions can be used:

For example: (

On the cold portion:

On the hot portion:

The Nusselt number at the hot part can be calculated by the expression below:

The FORTRAN code implemented in this study was validated and tested through a comparison of the outcomes against those reported by Le Quéré [^{7}. _{max}), the highest y-velocity value along the horizontal midpoint (_{max}), and the mean Nusselt number (

Results obtained | Le Quéré [ |
Deviation | |
---|---|---|---|

_{max} |
144.050 | 148.580 | 3.04% |

0.878 | 0.879 | ||

_{max} |
694.179 | 699.236 | 0.72% |

0.023 | 0.021 | ||

16.364 | 16.523 | 0.96% |

In the following section, the flow structures, temperature fields and heat transfer rates are inspected for divers values in the series of amplitude 0.2 ≤ _{0} and _{0} = 2 × _{0}, by using air (^{7}. Numerically in two dimensions, we have divided the physical domain into N = 210 nodes along the x axis, and M = 210 along the y axis. The stability of the numerical method used, the constraints of divergence code FORTRAN, and computing time are taken into consideration. We need the term

Notice that the model is dimensionless, and all results will be presented in dimensionless forms and discussed in the last periods.

The effect of the variation of the amplitude “_{o} at _{o} _{o}

The examination of the isotherms shows that thermal stratification is generated near to the hot walls and at the top of the cavity when _{0} _{0}

Note that these figures are represented at the final instant of the calculation time 8.10^{5} = 8_{0} = 4_{0}_{0} and _{0} _{0}

The temporal variations of dimensionless temperatures, denoted as

For all values of the amplitude “_{h} < 1 due to the important hot air movement when

In _{0} and _{0}_{0}, in directive to evaluate the period effect of

The time evolution of the dimensionless temperature ^{7} and

This correlation gives an idea of the maximum temperature, or Expected temperature value at the center of the cavity for heating conditions already specified.

The investigation of heat transfer within the enclosures involves computing the Nusselt number (

_{0} and _{0}_{0}. From these obtained results, it can be deduced that the amplitude of the Nusselt number (

Generally, when heating is maintained at a constant temperature, during the steady-state phase, the temporal evolution of the average Nusselt number exhibits a consistent value for low Rayleigh numbers (Ra). However, for higher Ra values, it tends to oscillate around a mean value. This is due to the appearance of the phenomena of transition to turbulence and the calculation method of Nusselt number ^{7} and the heating is non-uniform. We observe at each step that there is a zigzag appearing in the evolution of the global Nusselt number represented in

The global Nusselt number oscillates around an average value with a maximum ^{7} and in the range of the amplitude 0.2 ≤

This correlation is used to estimate the maximum of heat transfer in the enclosure for a well-determined heating rate.

In this article, we have conducted a numerical solution of heat transfer to analyze the effect of sinusoidal heating from below on the thermal field, dynamic field and Nusselt number ^{7}. The Results found in this work show that:

An increase in amplitude “

Increasing the parameter “a” results in an amplification of the forces propelling the cells upwards, leading to the emergence of upward-moving air puffs that rotate around their own axes.

The time evolution of

The variation of the period of _{0}” increases.

The Nusselt number _{0}”.

In the range of amplitudes 0.2 ≤

For temperature:

For Nusselt number:

Amplitude

Discrete Lattice velocities

Dimension of space

External buoyancy force

Single particle distribution function for density and temperature

Gravity vector

Height of the enclosure

Heat portion length

Nusselt number

Prandtl number

Rayleigh number

Temperature

_{0}

Period

Time

Macroscopic velocity

Cartesian coordinates

Thermal diffusivity

Coefficient of thermal expansion

Lattice time step

Lattice space step

Kinematic viscosity

Weights factors

Fluid density

Dimensionless temperature (_{c})/(_{h}_{c})

Relaxation times for momentum and for scalar

Cold

Hot

Lattice link number

Opposite

Scalar

Momentum

Equilibrium

Maximum

The authors wish to express their gratitude to the National Center for Scientific and Technical Research (CNRST) for granting access to computational resources.

The authors received no specific funding for this study.

All authors played a role in shaping the study's concept and design. The tasks of material preparation, data collection, and analysis were carried out by Noureddine Abouricha; Chouaib Ennawaoui and Mustapha El Alami. The initial draft of the manuscript was written by Noureddine Abouricha, and all authors provided feedback on earlier iterations of the document. Subsequently, all authors reviewed and endorsed the final version of the manuscript.

Data will be made available on request.

The authors declare that they have no conflicts of interest to report regarding the present study.

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