Lattice Boltzmann Simulation of the Three-dimensional Natural

Convection with a Regularly Heated Cavity Slab

El Bachir Lahmer

, Jaouad Benhamou

, Youssef Admi

, Mohammed Amine Moussaoui

Mohammed Jami

and Ahmed Mezrhab

Mechanics & Energetics Laboratory, Faculty of Sciences, Mohammed First University, 60000 Oujda, Morocco.

moussaoui.amine@gmail.com, jamimed7@gmail.com, amezrhab@yahoo.fr.

Keywords: Lattice Boltzmann Method, Heated Cavity, Cubical cavity, Nusselt Number Calculation, Heat Transfer,

LBM-MRT.

Abstract: In this paper, the Lattice Boltzmann method (LBM) associated with the Multi-Relaxation Times (MRT) is

performed to investigate the three-dimensional (3D) natural convection of the air in a cubical cavity for

various values of the Rayleigh number ranged between 10

<Ra<10

. The main 3D-LBM-MRT code was

compared and validated with the experimental and numerical work for different cases. The main objective of

this work is to analyze the heat transfer between the two cold vertical walls and the hot cavity base by using

the D3Q7 model for thermal field, and the evolution of the velocity field employing the D3Q19 model.

Furthermore, the effects of the conjugate heat transfer also take into consideration to investigate the heat

transfer rate by using different values of the thermal conductivity.

1 INTRODUCTION

The natural convection in a cubical cavity within a

heat generation source is commonly used to

investigate the fluid flow and heat transfer

enhancement for the various fields. This prototype

system is considered one of the main subjects of many

researchers because of its privileges in many

industrial and engineering applications related to the

thermal insulation of buildings, cooling the electronic

chips, etc. (Admi, 2020; Li, 2019). The study of

natural convection heat transfer using a three-

dimensional for complex configuration presents a

great challenge to understand the mechanism and the

behavior of the flow of fluid and heat transfer from

numerical simulations related to real physical

problems. Time and cost for experimental studies are

some of the main problems which oriented most

researchers to find alternative solutions to reduce the

time and lack of the experimental equipment. The

numerical simulation enables to solve concrete

problems masterly and efficiently, where it offers

valuable solutions by providing a clear view of

complex systems.

https://orcid.org/0000-0001-5435-1429

https://orcid.org/0000-0003-0920-0618

In this context, the lattice Boltzmann method

(LBM), as a numerical simulation, allows to

describing the fluid behavior and the heat transfer

using different models. Moussaoui et al. (Moussaoui,

2019) analyzed the effect of the Rayleigh number of

two-dimensional heated obstacle inside a square

enclosure by using the double MRT-LBM, they

conclude that the increase of obstacle dimension

enhances the heat transfer by increasing the surface

exchange. The recirculation zone loses its symmetric

when the position of the heated obstacle varied, and

the isotherm field also disturbs because of this

variation in the local heating. Chavez-Modena et al.

(Chávez-Modena, 2020) realized the under-resolved

turbulent flow simulation using the D3Q19 model

with multi-relaxation time with the central moment.

They interpret that the model used to obtain a

satisfactory result compared with the results BGK and

MRT-CM model for the under-resolved systems.

Wang et al. (Wang, 2017) performed a numerical

investigation of a three-dimensional heated cubical

cavity for a high Rayleigh number. This study focuses

on the evolution of the heat transfer and fluid

behavior inside the differential heated cavity, where

Lahmer, E., Benhamou, J., Admi, Y., Moussaoui, M., Jami, M. and Mezrhab, A.

Lattice Boltzmann Simulation of the Three-dimensional Natural Convection with a Regularly Heated Cavity Slab.

DOI: 10.5220/0010730500003101

In Proceedings of the 2nd International Conference on Big Data, Modelling and Machine Learning (BML 2021), pages 175-180

ISBN: 978-989-758-559-3

175

the obtained numerical results conclude that the

thermal and velocity contour formed close to the

adiabatic walls, and the thickness of the contour

layers narrowness near these walls for high values of

Rayleigh number. Liu et al. (Liu, 2019) also

investigate 3D convection heat transfer in porous

media at the REV scale. They found that the model

employed for studying the porous channel is suitable

to investigate the heat transfer and the fluid flow.

The present work was performed to analyze the

heated cavity slab with different values of thermal

conductivity which is related to the nature and

thickness of the heating source.

2 NUMERICAL STATEMENT

DETAILS

2.1 Configuration System

The configuration system studied is a cubical cavity

consist of two vertical cold walls and a heated floor

as shown in Figure 1. The cavity base undergoes a

dimensionless temperature T

=1 according to

different values of the thermal conductivity (Ks).

While the cold surfaces kept fixed cold dimensionless

temperatures Tc=0. The front, rear, and top cavity

surfaces are insulated. The thickness of the cavity

base characterized by a different thermal conductivity

(ks) compared to the thermal conductivity of the

airflow (kf). The continuity of the temperature and

heat flux is taken into account for the interface

solid/fluid (See Figure 1).

Figure 1: 3D Geometry system of a cubical cavity.

2.2 Discrete Numerical Model

In this study, The Natural convection of the air in a

cubical cavity was performed using the three-

dimensional lattice Boltzmann method including the

D3Q19 and D3Q7 model to define the velocity and

the thermal field (Liu, 2019; Wang, 2017),

respectively as illustrate in Figure 2.

Figure 2: a) Discrete Velocities for D3Q19 model b)

discrete temperature for D3Q7 model.

The above figure describes the nineteen discrete

speeds for the D3Q19 model

⟨

𝑒



𝑖0,….,18

⟩

such

as the speed coordinate was proposed as below:

𝑒



𝑐

0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0

0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1

0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1



(1)

The quantity c considers the lattice speed related

to the speed of sound where:

𝑐



3𝑐





(2)

Furthermore, the seven discrete speed was

proposed for this investigation for the D3Q7 scheme

where the temperature direction is correspondent to

seven speed direction of the D3Q19 scheme (Liu,

2019).

2.3 Numerical Approach

Recently, the lattice Boltzmann method associated

with multi-relaxation times (LBM-MRT) consider

one of the powerful numerical estimation to describe

the physics phenomena, the precision, and the

simplicity of the implementation in the code allows to

define of most physics and engineering problem for

various complex geometries. This approach makes it

possible to found satisfactory results with a reduced

computation time and for weak grids compared with

the other numerical approaches.

The lattice Boltzmann equation (LBE) defines

two main parts: the propagation and the collision

process of the fluid popularity, such that the study of

complete fluid is done in such a way that the fluid

discretized into a finite set of particles, namely the

distribution function or the probability of finding a

group of particles in a position x

with velocity e

time t (Lahmer, 2019; Admi, 2020; Li, 2019;

Moussaoui, 2017).

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2.3.1 Lattice Boltzmann Equation for the

Flow Field

The LBM-MRT equation related to the D3Q19 model

for the velocity field is expressed as follows:

𝒇



𝒙𝒆𝒊 ∆𝒕,𝒕



𝒇



𝒙,𝒕



𝑴

𝟏

𝑺



𝒎



𝒙,𝒕



𝒎

𝒆𝒒



𝒙,𝒕





𝑭

(3)

Where M and S denote the 19×19 transformation

matrix and the relaxation matrix, respectively, which

is related to the moment m and the equilibrium

moment m

. The quantity F is the body force related

to the gravity aspect.

The numerical resolution of this equation makes it

possible to describe the fluid flow behavior from the

macroscopic quantities (Lahmer, 2019).

2.3.2 Lattice Boltzmann Equation for the

Thermal Field

The thermal LBE-MRT equation is adopted to

simulate the temperature field using the D3Q7 model

as shown in the following expression:

𝒈



𝒙𝒆𝒊 ∆𝒕,𝒕



𝒈



𝒙,𝒕



𝑵

𝟏

𝑸



𝒏



𝒙,𝒕



𝒏

𝒆𝒒



𝒙,𝒕





(4)

The quantities N and Q define the 7×7

transformation matrix and the relaxation matrix,

respectively. During the numerical computation, the

data obtained from the thermal Boltzmann equation

allow modelling the temperature field at each point of

the computation domain. This process was performed

by the transformation of obtained data to macroscopic

quantities which is the temperature.

All the parameters used for this study have been

presented in detail in the reference (Liu, 2019; Wang,

2017).

2.4 Boundary Conditions

2.4.1 Velocity Boundary Condition

The boundary conditions used in the numerical

calculation procedure correspond to the Boltzmann

equation considered as conditions easy to implement

in the calculation code.

Regarding the fluid flow, the “bounce-back”

scheme was employed to describe the cavity walls as

shown in the expression below:

𝑓





𝑥,𝑦,𝑧



𝑓





𝑥,𝑦,𝑧



𝑖0,…,18 (5)

Where i denotes the direction of the discrete

velocity, and 



defines the opposite speed direction.

The positions x, y, and z describe the position of each

cubical cavity walls.

2.4.2 Thermal Boundary Condition

For the thermal problem, the boundary conditions

adopted to define the cold and hot walls which are

expressed by the equation related to the macroscopic

quantity:

𝑇

∑

𝑔







(6)

From the equation above, the boundary conditions

can be expressed for the hot bottom wall and the cold

vertical walls by the following expressions:

⎩

⎪

⎨

⎪

⎧

𝑧0

→ 𝑔





𝑥,𝑦



𝑇





∑

𝑔









𝑦0

→ 𝑔





𝑥,𝑧



𝑇





∑

𝑔









𝑔





𝑥,𝑧



𝑇





∑

𝑔









(7)

Concerning the adiabatic walls, Neumann

Boundary Condition was executed by the following

expressions:



𝑧𝐻

→ 𝑔





𝑥,𝑦



𝑔





𝑥,𝑦



𝑥0

→ 𝑔





𝑦,𝑧



𝑔





𝑦,𝑧



𝑥𝐻

→ 𝑔





𝑦,𝑧



𝑔





𝑦,𝑧



(8)

3 VALIDATION AND

NUMERICAL RESULTS

To cognize the consistency and accuracy of our

simulation code, we executed Intensive studies in

order to know that the current code used is more

suitable for studying the fluid flow and convective

heat transfer in various geometries.

3.1 Validation Code for Double

MRT-LBM

In this context, the developed code is compared to the

literature results (Corvaro, 2008). The first

comparison was made with the experimental results

of the isotherm and velocity contour obtained by

Corvaro and Paroncini. The results showed a good

agreement with the reference (Corvaro, 2008) for

Rayleigh number equal Ra=2.02×10

and δ=0.5. (See

Figures 3 and 4). The interest of studying the Ra

number is to characterize the heat transfer within a

fluid during its flow.

Lattice Boltzmann Simulation of the Three-dimensional Natural Convection with a Regularly Heated Cavity Slab

177

Figure 3: Comparison between the 3D and 2D projection of

LBM-MRT simulation result, and the experimental

measurement using the double-exposure interferogram

(Corvaro, 2008) For Ra=2.02×10

, δ=0.5.

Figure 4: Comparison study between the 3D projection of

LBM-MRT simulation results, and the numerical

simulation of the streamline For Ra=2×10

, δ=0.5.

To reinforce the reliability of our simulation code,

the Nusselt number is carried out for different values

of the Rayleigh number where δ=0.5. Note that the

role of the Nusselt number is to characterize the

nature of thermal transfer involved between the hot

cavity base and the two vertical cold walls as shown

in the expression below:

𝑁𝑢𝐿











(9)

Where Lc denotes the characteristic length

correspond to the heated part of the cubical cavity.

The experimental and numerical results obtained by

the literature (Corvaro, 2008) are converged towards

the results obtained by the numerical simulation

carried out by the double MRT-LBM as illustrate in

Table 1.

Table 1: Comparison of the average Nusselt number

between the experimental (Corvaro,2008) and numerical

results for δ=0.5.

Ra Nu (exp) Nu (num)

|∆|[%]

1.71E+05 6.3 6.306 0.095

1.98E+05 6.45 6.488 0.59

2.32E+05 6.65 6.697 0.70

2.50E+05 6.81 6.8 0.15

Furthermore, the maximum deviation of each

average Nusselt number results under 1%. Therefore,

the code used in the current study able to make

numerical simulations that correspond to real physical

problems of the fluid flow and thermal transfer.

3.2 Results and Discussion

Isotherm contours, velocity fields, and average

Nusselt number for different values of the thermal

conductivity ratio (Kr=K

/Kf) and Rayleigh number

are investigated in this section. The objective

proposed in this study is the analysis of the heat

transfer evolution by the air cooling of the electronic

equipment inside a cubical box.

Figure 5: 2D projection for isotherms, velocity field and the

3D isosurfaces temperature for Ra=3.10

, Kr=10.

Figures 5 and 6 present the evolution of the fluid

flow and heat transfer by natural convection for two

different values of Rayleigh number with fixed

thermal conductivity ratio. The results show that the

energy transport increase promptly upwards of the

cavity such that the fluid flows symmetrically relative

to the cavity mid-length. Two recirculation zone

appear clockwise and anticlockwise. Furthermore,

the isotherms field evolves symmetrically where the

heat transfer enhances in the left and right of the cold

cavity walls. This explains that the fluid flow and

convective heat transfer behavior refer to the thermal

exchange between the cold walls and the heated slab,

where the magnitude of heat transfer more enhanced

at the two corners of the cavity slab.

Figure 6: 2D projection of isotherm and velocity pattern,

and 3D isosurface of the thermal field for Ra=5.10

, Kr=10.

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The evaluation of the thermal transfer and fluid

flow was also investigated for two various values of

thermal conductivity ratio (Kr=10 and Kr=6600) with

fixed Ra. As shown in Figures 6 and 7, the slab

became almost isothermal as the thermal conductivity

ratio increases. In addition, the energy transport of the

heated air near the slab propagates rapidly upward of

the cavity. The presence of the cold walls in the

vicinity of the heated base provokes an efficient

cooling compared with the center of the heated slab.

Figure 7: 2D projection of isotherm and velocity pattern,

and 3D isosurface of thermal field for Ra=5.10

, Kr=6600.

To understand substantially the mechanisms of

heat transfer involved in this work, the local Nusselt

number evolution was realized for several values of

Ra and Kr for the heated cavity base (see Figure 8).

The thermal exchange increases as Ra increases from

3.10

to 5.10

. While the heat exchange reduced from

5.10

to 6.10

. This diminishes of the Nu refers to the

fact that the fluid receives more heat through

convective heat transfer, and the buoyancy forces

increase, and in turn, thermal boundary layer

thickness near the hot wall decreases. In order to

enhance the heat transfer in this case, for a high

thermal conductivity ratio, the heat exchange

improves for Ra=5×10

, Kr=6600.

Figure 8: Local Nusselt number evolution.

4 CONCLUSION

In this work, the examination of the three-

dimensional natural convection heat transfer with a

regularly heats cubical cavity slab is illustrated. The

results obtained using the double MRT-LBM method

had a good agreement with the experimental

investigation which gives credibility to our

simulation code. To achieve the perfect cooling of

electronic components whatever its nature in a

cubical enclosure, one can reconsider that the effects

of introducing a large Rayleigh number provoke a

decrease in heat exchange if it exceeds the critical

point which is equal to 6.10

. On the other hand, the

increase in thermal conductivity considers a strong

point to reinforce the heat exchange for the case

where the high Rayleigh number.

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