Lattice Boltzmann Simulation of Mixed Convection Around a Heated
Elliptic Block Cylinder within a Lid-driven Square Cavity
ABDELHAK DAIZ1,*, AHMED BAHLAOUI1, ISMAIL ARROUB1, SOUFIANE BELHOUIDEG1,
ABDELGHANI RAJI2, MOHAMMED HASNAOUI3
1Research Laboratory in Physics and Sciences for Engineers (LRPSI),
Polydisciplinary Faculty,
Béni-Mellal,
MOROCCO
2Energy and Materials Engineering Laboratory (LGEM),
Faculty of Sciences and Technics,
Béni-Mellal,
MOROCCO
3Laboratory of Fluid Mechanics and Energetics (LMFE),
Faculty of Sciences Semlalia, Marrakech,
MOROCCO
*Corresponding Author
Abstract: - A numerical analysis of mixed convection flows and heat transfer in a square enclosure having a
sliding wall containing an elliptical block heated by isothermal temperature has been carried out. The enclosure
is full of air and cooling from its sides by a cold temperature, whereas the remaining walls of the enclosure are
considered thermally insulated. The mixed convection impact is attained by the heating elliptic block and
moving upper wall. The investigation of fluid’s hydrodynamic and thermal behavior was examined by using
Lattice Boltzmann Method (LBM) at different locations and orientations of the interior elliptical block for
Richardson number, Ri, varying from 0.01 to 100 while the Rayleigh number, Ra, is fixed at 104. The findings
indicate that the temperature pattern and flow structure are very responsive to the position of the elliptical block
and Richardson number. Also, it is found that the heat exchange is very important for the block placed
vertically close to the left wall or horizontally close to the bottom wall. More precisely, for Ri = 0.01, by
moving the vertical block from the center towards the vicinity of the left/ (the right) surface, the heat transfer
rate increases from 5.44 to 11.06/(8.36) with an increase of 103.30%/(53.67%). On the other hand, it is noted
that the horizontal elliptic block favors heat evacuation in comparison with the vertical one. This
study’s real-world impact lies in the potential to improve our understanding and, consequently, design
more efficient cooling systems for electronic devices.
Key-Words: - Numerical analysis, Mixed convection, Lid-driven cavity, Elliptic block, Lattice Boltzmann
method, Block locations, Block orientations, Heat evacuation.
Received: April 15, 2023. Revised: September 21, 2023. Accepted: November 17, 2023. Published: December 31, 2023.
1 Introduction
Mixed convection heat transfer has always attracted
a lot of attention due to its wide applications in
various fields such as cooling of electronic
equipment [1], [2], solar energy collectors [3],
chemical processing equipment [4], [5], and heat
exchangers [6], [7]. The combined free and forced
convection problems are mostly linked to the
ventilated and lid-driven enclosures [8], [9], [10],
[11], [12].
Over the years, many authors have devoted their
attention to studying the characteristics of
convection flow and heat transfer with blocks.
Several studies have considered free convection in
cavities containing blocks with different boundary
conditions. In this context, several studies carried
out free convection inside a cold square enclosure
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Abdelhak Daiz, Ahmed Bahlaoui,
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with a variety of shaped inner blocks [13], [14].
Their findings demonstrated that the maximum heat
exchange is observed at the ellipse’s orientation 30°
and Ra = 106. They discovered that when aspect
ratio and Ra were taken into range consideration,
the heat exchange rises by augmenting aspect ratio
(AR) and Ra. Furthermore, with the exception of
AR = 0.1, it is observed that the inclination of the
rectangular obstruction often increased the heat
exchange rate. It has been demonstrated that the
mean Nusselt number goes up as the Ra increases,
and its lowest value moves further away from the
upper wall of the inner cylinder. In other studies,
[15], [16], [17] have analyzed natural convection
with blocks in a cold vertical-sided square cavity. It
is observed that for low Ra, the inclination angle
had no appreciable influence on the typical Nusselt
number. The size of the central obstruction has a
significant impact on the mean Nusselt number, and
it reduces considerably as the diameter of the
interior block grows. Also, the increase in block size
has enhanced heat exchange caused by increasing
the lower block surface. [18], [19] studied the
influence of various sizes and positions of the
heating cylinder block in a square enclosure. The
obtained outcomes demonstrate that the mean
Nusselt number, Nu, diminishes with the growth of
the block size. Also, they conclude that an increase
in the Nu for all positions of the block and the lack
of the effect of magnetic field (Ha = 0), whereas
augmenting the Hartmann number lowered the mean
Nusselt number.
Several studies dealing with forced convection
have considered channels with blocks inside, [20],
[21] have numerically explored the forced
convection in a rectangular channel containing
heated square blocks. They obtained that because of
the asymmetric fluid flow, the local Nusselt number
evaluated on the upper and bottom walls is not
identical to the heat exchange that a unique
obstruction suffers. The mean Nusselt number rises
by increasing Re. Beyond critical Re, the Nusselt
number calculated on the lower wall is larger than
that calculated on the upper wall. The results reveal
that adding a flat plate with a height of h = 2D in
front of the cylinder block increases the heat
exchange through heated cylinders to enter the fluid.
The altering effect of fin size on heat exchange and
cooling liquid flow behavior in the channel was
numerically investigated by [22]. The author
observed that heat sinks with a fin size of 0.8 mm
have significant heat transfer, which is greater
compared to 0.9 mm and 1.0 mm fin sizes (Totally
enclosed thermal dissipator). [23] experimentally
examined the forced convection in a horizontal duct
containing a set of aluminum blocks. At this point,
the block’s temperature drops by increasing the
Reynolds number while the mean Nusselt number
increases. For small Reynolds numbers, when the
convective coefficient of heat transfer decreases, the
block’s temperature rises. Recently, [24] performed
an analysis of forced convection characteristics in a
horizontal duct with triple hot obstructions by
applying Lattice Boltzmann Method (LBM-MRT).
It is observed that raising the thermal conductivity
of the solid fluid improves the heat exchange. In
addition, as the spacing between obstructions
increases, the heat exchange for second and third
obstructions increases. Also, the use of many jets in
the analyzed array improves heat transfer.
Combined natural and forced convection is
frequently observed in very high-output appliances
when forced convection or free convection is
insufficient to remove all necessary heat. At this
stage, great attention has been dedicated to the
possibility of increasing the heat exchange by mixed
convection in rectangular cavities by the presence of
various block shapes. In this regard, [25] explored a
computational investigation of the laminar flow
mixed convection in a vented enclosure containing a
hot square block situated in the enclosure center.
The findings demonstrate that as the Re and Ri rise,
so does the mean Nusselt number around the heating
interior block. They also found that the influence of
the positions and size (aspect ratio) of the interior
block has an important role in the flow and
temperature patterns. Further studies on the
insertion of blocks inside of the cavity [26], [27].
They concluded that the heat exchange rate
calculated on the heated surface of the cavity rises
for low values of Ri and reduces for greater values
of Ri and by increasing cylinder diameter in both
situations of vertical wall or lower wall heated.
Moreover, the rate of heat transmission increases as
block diameter and heat generation rise, whereas it
decreases when the thermal conductivity ratio rises.
The upsurge of the block diameter, thermal
conductivity ratio, and heat generating
characteristics cause the fluid’s average temperature
to rise. For such configurations, [28] numerically
examined mixed convection of laminar flow in a lid-
driven square enclosure having a square hot obstacle
inside. The authors noticed that the most optimal
heat exchange achieved for the blocking is located
in the top left and lower right-hand angles of the
enclosure. [29] numerically studied the mixed
convection heat transfer in a lid-driven square cavity
containing a circular block inside. Their derived
results demonstrate that the existence of the circular
block entails an augmentation of the mean Nusselt
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number compared with the absence of a circular
block. Also, the mean Nusselt number grows by
growing the cylinder block diameter for different Ri.
The same work has been done by, [30]. It is noticed
that the current lines and the isotherms within the
enclosure are highly dependent on the locations of
the circular block and the relative of the convection
dominates regime (natural, forced, or mixed). [31],
studied the impact of thermally insulated obstruction
in an enclosure with moving vertical walls and
differential heating on mixed convection. The
findings show that the thermal exchange inside the
cavity improves up to a specific block size.
Recently, [32], explored the effect of the triangle
obstruction position with uniform thermal flux on
the vertical median of the square enclosure with a
moving wall. The outcomes of this investigation
indicate a great heat exchange rate in the case of the
block located at the cavity center. Also, the Nusselt
number declines with increasing Ri.
According to the aforementioned review of
literature, a computational examination of combined
free and forced convection from an obstacle in the
shape of an elliptical block enclosed in a square
cavity has never been done. The principal goal of
the current study is to investigate the influence of
position and orientation of heated elliptical
obstruction on mixed convection in a square moving
wall enclosure. The side surfaces are maintained at
cold temperatures whereas the remaining surfaces of
the enclosure are adiabatic. The upper wall is
moving at a uniform speed. The flow-controlling
parameters employed in this study are the
Richardson number (Ri), position, and orientation of
the elliptical cylinder.
2 Problem Formulation
2.1 Problem Statement
The schematic representation of the studied
configuration is illustrated in Figure 1. It consists of
a lid-driven square enclosure, containing a heated
elliptical block by a hot temperature. The size of the
elliptical block is 0.2 and 0.1 on semi-major axis
and semi-minor, respectively. The horizontal walls
of the enclosure are kept thermally insulated while
the side walls are subjected to cold temperatures.
The air, as the working fluid in this work, is
considered Newtonian, incompressible, and
circulating in a laminar regime. All air
thermophysical characteristics are supposed to be
made unchanged, apart from the density in the
floatability expression. The latter (density) is
evaluated using the Boussinesq model.
Fig. 1: Diagram of the studied configuration
2.2 Governing Equations
Based on the preceding hypothesis, the continuity,
momentum, and energy equations, obtained by the
application the conservation of mass, momentum,
and energy, in a two-dimensional Cartesian
coordinate system in dimensionless forms are as
listed below:
(1)
(2)
(3)
(4)
Where
,
,
and
are the Reynolds,
Richardson, Grashof and the Prandtl numbers,
respectively, and is the temperature
difference, [33].
The foregoing equations are rendered
dimensionless by employing the appropriate set of
the next dimensionless variables, [34]:
The limit conditions considered in form of non-
dimensional variables are as follows:
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2.3 Numerical Method
The Lattice Boltzmann Model (LBM) is a robust
method, that has been widely applied to analyze
fluid dynamics in both simple and complex
geometries. The reason for the increase in use of this
technique is due to the high adaptability in
approaching various types of models and the great
ease it offers in the implementation and processing
of boundary conditions, [35]. This method considers
the fluid as a set of particles that move in a uniform
network from one node to neighboring nodes during
the time interval. In this method, the dynamic and
thermal fields are solved by employing two different
distribution functions and . The discretization of
the fluid field into homogeneous sized rectangular
cells. A unit cell contains a certain number of
functions of distribution that indicate the
possibilities for a particle situated in the central
node to move in one of the directions determined by
the discretization model chosen. For two-
dimensional problems, the D2Q9 scheme, illustrated
in Figure 2 is the most widespread, and it is the one
we have retained in our work for the dynamic and
thermal distribution functions. The BGK
(Bhatnagar-Gross-Krook) approximation is a
simplification used in lattice Boltzmann methods
(LBM). Named after its inventors, Bhatnagar,
Gross, and Krook, the BGK approximation
simplifies the collision operator in the Boltzmann
equation, making the computations more tractable,
[36].
Fig. 2: Schematic diagram of D2Q9 model of
lattices discretization.
The governing equations for two distribution
functions using the Bhatnagar-Gross-Krook (BGK)
approximation can be written as follows, [37]:
(5)
(6)
In the foregoing equations, τf and τg are the
relaxation times for energy and momentum
equations, respectively. Also, ck is the lattice speed
in direction k, which is defined below:
(7)
Where c is the lattice speed expressed in
equation below:
(8)
The Δt', Δx and Δy parameters represent the
time step, space step in x and y direction,
respectively.
In equation (5), Fk, is the external force, its
expression is given by, [38]:
(9)
With β is the coefficient of the thermal
expansion, Tm = (TH + TC)/2 is non-dimensional
reference temperature and
is the weighting
parameter given by equation (10):
(10)
and are the local equilibrium functions
of the dynamic and thermal fields. They are given in
the following equations:
(11)
(12)
cs is the speed of sound in the lattice, which
equals to
in the D2Q9 model.
The lattice kinetic viscosity and lattice thermal
diffusivity are related to the Relaxation times of
momentum and energy by the following expressions
2
1
5
847
3
6
0
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and ,
respectively.
The value of Mach number for the present
investigation was Ma = 0.1, which is below the
threshold of 0.3 in order not to break the fluid’s
incompressibility assumption [9]. Lattice viscosity
can be determined as a relation of Prandtl number,
Mach number and Rayleigh number by using
equation (13):
(13)
With M is the number of knots in the y axis.
All distribution functions in the LBM are
defined by collision and streaming steps, although
establishing boundary conditions is required before
computing the distribution functions. The boundary
conditions in the LBM are considerably changed
since all not defined steering functions at every limit
are necessary to be determined.
The classic rebound technique is used on fixed
walls to calculate the unidentified distribution
functions, which stream from outside to inside the
fluid domain. These unidentified functions are set to
their inverse, [39].
(14)
The longitudinal component of speed on the
upper mobile boundary, fluid density, and particle
distribution functions are all calculated in equation
(15) as follows:
(15)
Now, let’s talk about the boundary conditions of
temperature.
For thermally insulated upper and lower walls:
(16)
For the cold left wall:
(17)
For the cold right wall:
(18)
The method proposed in, [14], [40], has been
utilized for the treatment of the velocity and
temperature of the ellipse’s curved boundaries.
The macroscopic variables that are the density,
ρ, the component velocity, u, v, and the temperature,
T, where calculated by the below expressions:
(19)
(20)
(21)
(22)
2.3 Heat Transfer
The local and mean Nusselt numbers characterizing
the heat transfer rate on vertical cold walls are given
by the expressions below:
(23)
(24)
3 Independent Grid and Numerical
Validation
Preliminary tests were also carried out to check the
sensibility of the results to the mesh. The results
presented in Table 1 in terms of average Nusselt
number for horizontal heated elliptic cylinder block
at (0.5;0.5) and for Ra = 104 show that a grid of
141×141 (used in the simulation part) is sufficient to
describe the flow characteristics because it provides
a good balance between the computation time and
the required accuracy. More precisely, the
refinement of the used grid to 161×161 implies
maximum variations less than 0.2 % in terms of
.
The calculations were made using the Windows
operating system with Intel core i5 of frequency
2.60 GHz and 4 GB of RAM. By analyzing the
impact of the grid on CPU time, we found that the
refining grid from 141x141 to 161x161 involves a
maximum increase of about 48% in computational
effort.
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Table 1. Grid’s effect on average Nusselt number
for Ra = 104 and various Richardson numbers
Grids
101×101
121×121
141×141
161×161
Ri = 0.1
6.38
6.43
6.45
6.46
Ri = 5
4.90
4.94
4.96
4.97
We limit ourselves here to presenting
quantitative and qualitative comparisons between
our results and those found by, [29], in the case of
mixed convection in a square enclosure with a
moving wall whose side surfaces are thermally
insulated, the lower surface is kept at a hot
temperature and the upper one is kept at a cold
temperature and moves to right by uniform speed.
An inner cold circular block placed in the center is
used for different Richardson numbers Ri by
varying Ra at fixed Re = 100 and Pr = 0.7. Figure 3
demonstrates that the streamlines and isotherms
generated by, [29], are properly replicated by our
numerical code. The comparative quantitative
results, reported in Table 2 in the form of average
Nusselt numbers assessed on the hot surface,
demonstrate a high concordance with maximal
differences within 0.55 %.
Table 2. Comparative results of the mean Nusselt
number assessed on the hot surface for Re = 100,
r0/L = 0.2 and Pr = 0.7
Ri
Present
study
Ref. [29]
Max
deviations
Ri = 0.01
Ri = 1
Ri = 5
Ri = 10
2.904
3.497
4.689
5.045
2.92
3.50
4.96
5.04
0.55%
0.086%
0.02%
0.09%
a) Present study
(Ψmax=0.003;Ψmin= -0.079)
b) Ref. [29]
(Ψmax=0.002 ;Ψmin= -0.078)
Fig. 3: Comparison in terms of the streamlines and
isotherms for r0/L = 0.2, Re = 100 and Ri = 1: a) our
results and b) Ref. [29].
4 Results and Discussion
The present work is done to examine the influence
of the heated elliptical block on the thermal and
dynamic structures in a square enclosure with
moving top wall and cooled by the side walls. The
calculations were performed for air (Pr = 0.71) as
the operating fluid and a fixed value of Rayleigh
number (Ra = 104). The Richardson number varies
in the range 0.01 and 100 through the variations of
Reynolds number 11.86 ≤ Re ≤ 1186.78, as well as
the position and orientation of the elliptical block
are the control parameters in this work. The typical
outcomes are illustrated in the form of current lines,
isotherms and average Nusselt numbers and mean
temperature.
Figure 4 presents the streamlines obtained for
the heated elliptic cylinder block placed horizontally
at X = 0.5 for different locations and various values
of Richardson number. This later measures the
importance of free convection caused by buoyancy
forces and forced convection caused by lid-driven
wall. For a dominating forced convection
mechanism (Ri = 0.1), the streamlines in Figure 4a
show that for Y = 0.25 the flow field is represented
by a large cell that occupies the whole cavity and
brings together the elliptical cylinder block.
Consequently, the flow is mainly monocellular and
rotates in a clockwise direction imposed by the
kinematic condition given on the upper wall.
Changing the position of the obstruction up to Y =
0.5 and 0.75, the flow structure remains unicellular
but its intensity decreases. This decrease is justified
by the fact that the cylinder block located close to
the moving wall delays the flow. Also, the cell
center is moved progressively closer to the right
wall. For Ri = 1 (Figure 4b), the fluid flow is
intensified by this increase in Ri for all positions of
the cylinder block. In addition, in the lower left
corner of the enclosure, a small trigonometric cell of
very low intensity appears. The appearance of this
cell is due to the increase in heat exchange by free
convection in this area following the noticeable
effect of buoyancy forces. Moreover, by climbing
the obstacle at the top, this small cell changes
relatively in size. The increase of Ri up to 10
(Figure 4c) results in an augmentation of the size
and intensity of the small positive cell to the
expense of the big one. Also, as the body moves
upwards, the small cell undergoes slight changes in
intensity.
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Fig. 4: Effect of horizontal cylinder block locations (X = 0.5) on the streamlines for various Richardson numbers
Ri
For a large Richardson number value Ri = 100 the
free convection regime dominates (Figure 4d), this
increase in Ri indicates that the flow structure
becomes bicellular characterized by two cells of the
same shape and opposite direction and are driven
by the buoyancy force for both cases of the
elliptical cylinder placed close the top and lower
walls of the enclosure, for the elliptical obstruction
placed in the center of the enclosure, the dynamic
structure is composed of four counter-rotating cells
around the heated cylinder. It can also be seen that
the vertical centerline of the cavity is an axis of
symmetry of the flow structure. This symmetry is
due to the symmetrical thermal limit conditions
imposed and the importance of the flow velocity
generated by the buoyancy forces in front of the
velocity caused by the displacement of the upper
wall.
The corresponding isotherms are illustrated in
Figure 5 for the same parameters mentioned in
Figure 4. It is observed from Figure 5a plotted for
Ri = 0.1 that the isotherms are more condensed
around the heated cylindrical block testifying in a
good heat transfer between the heated block and the
surrounding fluid. Also, the isotherms are parallel
and constricted along the cold left vertical wall and
at the top portion of the right one. This finding
proves that the heat released by the block is
evacuated to the outside through the vertical walls
in these places. It is underlined that the movement
of the block upwards involves homogenization of
the temperature within the cavity. By increasing
Richardson number progressively to 1 and 10, the
temperature distribution remains qualitatively
unchanged (Figure 5b and Figure 5c). Figure 5d
shows that by raising the Richardson number from
Y = 0.25
Y = 0.5
Y = 0.75
a) Ri = 0.1
max = 0.096;
min = 0
max = 0.075;
min = 0
max = 0.064;
min = 0
b) Ri = 1
max = 0.107;
min = -0.005
max = 0.086;
min = -0.001
max = 0.068;
min = -0.002
c) Ri = 10
max = 0.156;
min = -0.054
max = 0.108;
min = -0.021
max = 0.076;
min = -0.022
d) Ri = 100
max = 0.352;
min = -0.260
max = 0.213;
min = -0.122
max = 0.162;
min = -0.110
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10 to 100 (dominant natural convection), the
isotherms become practically parallel to cold
vertical walls and symmetrical about the vertical
centerline. This increase in Ri shows that the
increase in thickness of the boundary layer leads to
a reduction in the temperature gradient near the
vertical walls, i.e. a decrease in heat exchange
between the cold walls and the fluid.
Fig. 5: Effect of horizontal cylinder block locations (X = 0.5) on the isotherms for several Richardson numbers
Ri
Figure 6 presents the streamlines obtained for
the hot elliptical block placed vertically at Y = 0.5
for three locations and various Richardson
numbers. For Ri = 0.1 (Figure 6a) when the
cylinder block is positioned near to the left wall,
the streamlines show that the flow structure
consists of a big cell, turning clockwise, containing
the block and occupying almost the whole cavity.
Another small positive cell forms in the lower right
corner. When the block is localized in the center of
the cavity, the flow is perfectly monocellular. It is
indicated that the flow between the inner block and
the rigid walls is similar to a flow in the channel.
By changing the location of the block from the
center to the right (X = 0.75), the dynamic structure
is composed of the same large cell with the
presence of a relatively trigonometric cell situated
in the bottom left corner. These secondary cells,
involved in the buoyancy forces, favor the natural
convection heat transfer. In addition, we note that
the large cell center shifts to the right wall
following the displacement of the block.
Y = 0.25
Y = 0.5
Y = 0.75
a) Ri = 0.1
b) Ri = 1
c) Ri = 10
d) Ri = 100
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Fig. 6: Effect of vertical cylinder block locations at (Y = 0.5) on the streamlines for various Richardson numbers
Ri
From the examination of the Figure 6b, Figure
6c and Figure 6d corresponding to Ri = 1, Ri = 10
and Ri = 100 respectively, we can see that the flow
structure is intensified and a second convective cell
has occurred close to the left vertical wall. The
development in volume and intensity of this latter
cell at the expense of the big cell is favored by X
and Ri. More precisely, for X = 0.75 and Ri = 10,
the second cell occupies almost the left half of the
enclosure and consequently, the flow structure
becomes bicellular. Finally, this change in the
dynamic aspect of the flow causes a drastic change
in the heat exchange rate and also determines the
most dominant convection regime.
Figure 7 displays the isotherms corresponding to
the streamlines presented in Figure 6. Generally,
changing horizontally the position of the block
from left to right of the cavity (increase of X)
modifies the temperature distribution throughout
the cavity and leads to its homogenization. On the
contrary, the augmentation in Ri, i.e. decreases the
speed of the upper wall and favors the natural heat
exchange near the cold isothermal walls since the
isotherms are condensed and layered near these
latter walls.
Variations of the average Nusselt numbers
,
calculated on the cold walls according to the
Richardson number Ri are illustrated in Figure 8
and Figure 9 for the case of the elliptical block
placed horizontally (Figure 8) and vertically
(Figure 9) and three locations for each orientation.
It can be noted that, for all considered orientations
and locations,
undergoes a drastic decrease,
limited in the interval 0.01 ≤ Ri ≤ 1, followed by
asymptotic behavior in the remaining range of Ri.
Such a result indicates that as the heat exchange by
X = 0.25
X = 0.5
X = 0.75
a) Ri = 0.1
max = 0.086;
min = 0
max = 0.074;
min = 0
max = 0.058;
min = -0.002
b) Ri = 1
max = 0.098;
min = 0
max = 0.084;
min = -0.001
max = 0.059;
min = -0.027
c) Ri = 10
max = 0.172;
min = -0.004
max = 0.110;
min = -0.035
max = 0.062;
min = -0.12
d) Ri = 100
max = 0.464;
min = -0.031
max = 0.24;
min = -0.172
max = 0.096;
min = -0.418
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Abdelhak Daiz, Ahmed Bahlaoui,
Ismail Arroub, Soufiane Belhouideg,
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natural convection is increasing, to the detriment of
the forced one, as the mixed heat transfer rate
decreases. For each orientation, the effect of block
location on
is well noted. Indeed, heat transfer
is very important by placing the block vertically
close to the left wall (X = 0.25) or horizontally near
the lower wall (Y = 0.25). In addition, from the
analysis of the curves X = 0.5 and Y = 0.5 we can
deduce that the horizontal orientation is better for
heat transfer compared to the vertical case.
Fig. 7: Effect of vertical cylinder block locations (Y = 0.5) on the isotherms for several Richardson numbers Ri
Fig. 8: Variations of the average Nusselt number vs. Ri for various cylinder locations for a horizontal elliptical
block at X = 0.5
X = 0.25
X = 0.5
X = 0.75
a) Ri = 0.1
b) Ri = 1
c) Ri = 10
d) Ri = 100
0.01 0.1 1 10 100
3
4
5
6
7
8
9
10
11
12
Nu
Ri
Y=0.25
Y=0.5
Y=0.75
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DOI: 10.37394/232012.2023.18.13
Abdelhak Daiz, Ahmed Bahlaoui,
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Fig. 9: Variations of the average Nusselt number
vs. Ri for various locations of the cylinder for
vertical elliptic block at Y = 0.5.
The calculation of the fluid’s mean temperature
inside the cavity is crucial for practical purposes.
For this reason, the fluid’s average temperature
contained within the cavity is presented in Figure
10 and Figure 11 for horizontal and vertical elliptic
blocks, respectively, and for different positions of
the elliptic block. Figure 10 demonstrates that the
fluid’s average temperature contained within the
cavity is characterized by a significant decrease for
both positions Y = 0.25 and Y = 0.75 and for low
values of the Richardson number Ri ≤ 0.1 (forced
convection dominates). By increasing Ri, the mean
temperature remains almost constant for the case
where the elliptical block is located close to the
upper wall Y = 0.75, on the other hand, for the case
where the block is positioned close to the lower
surface Y = 0.25, the average temperature is
characterized by a minimum at Ri = 0.15. The
evolution of the fluid’s average temperature
contained within the cavity is indicated by a
continuous monotonic increase according to the
Richardson number when the elliptical cylinder
block is placed at the cavity center Y = 0.5. As a
result, it can be seen from this figure that the
position of the cylinder near the top wall
(Y = 0.75) leads to a better overall cooling of the
enclosure since the values of the are lower
compared to the other positions for Ri ≤ 0.1. For
the case where the elliptical block is vertically
oriented, the variation of the mean temperature of
the fluid versus Ri and for various block
locations is shown in Figure 11. It can be seen that
the mean temperature follows almost the same
trend as that for the horizontal block and all
positions, with a clear decrease and the minimum
observed becomes large for X = 0.25 compared to
Y = 0.25 in the horizontal case. As a conclusion of
this figure, the cavity is cooler for the cases where
the elliptical cylinder is positioned vertically close
to the left cold wall and in the cavity center for Ri =
0.15 and Ri = 0.01, respectively.
Fig. 10: Variation of mean fluid temperature vs.
Richardson number for various positions of
horizontal elliptic block at X = 0.5.
Fig. 11: Variation of mean fluid temperature vs.
Richardson number for various positions of vertical
elliptic block at Y = 0.5.
5 Conclusions
The problem of combined forced and free
convection inside a square enclosure with a moving
wall, containing a hot elliptical cylinder full of air
is investigated numerically using the Lattice
Boltzmann method. The enclosure is cooled from
its sides by a cold temperature, whereas the
remaining walls of the enclosure are considered
thermally insulated. The mixed convection impact
is attained by the heating elliptic block and moving
upper wall. The combined effects of orientation and
location of the block and Richardson number on
heat transfer and flow behavior are examined. From
the results presented, several interesting
conclusions can be derived.
The temperature distribution and flow structure
are very responsive to geometrical parameters
0.01 0.1 1 10 100
4
5
6
7
8
9
10
11
12
Nu
Ri
X=0.25
X=0.5
X=0.75
0.01 0.1 1 10 100
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
Ri
Y=0.25
Y=0.5
Y=0.75
T
0.01 0.1 1 10 100
0.32
0.34
0.36
0.38
0.40
0.42
0.44
Ri
X=0.25
X=0.5
X=0.75
T
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.13
Abdelhak Daiz, Ahmed Bahlaoui,
Ismail Arroub, Soufiane Belhouideg,
Abdelghani Raji, Mohammed Hasnaoui
E-ISSN: 2224-3461
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Volume 18, 2023
such as the location of the elliptical cylinder
(block) and flow parameters, especially Ri.
It is found that the heat exchange rate can be
enhanced by moving horizontally /(vertically)
elliptical cylinder (block) to the left wall,
(0.25;0.5) / (the lower wall, (0.5;0.25)).
The heat transfer is very important for a
horizontal elliptical cylinder (block) position in
comparison with a vertical one for the different
locations.
The geometric configuration where the
elliptical cylinder (block) is placed horizontally
near to the top wall (0.5;0.75) leads to an
improved cooling of the enclosure for Ri ≥ 1.
For Ri = 0.01, by moving the vertical block
from the center towards the vicinity of the left/
(the right) wall, the heat transfer rate increases
from 5.44 to 11.06/(8.36) with an increase of
103.30%/(53.67%).
This research offers a thorough insight into the
thermal and dynamic behavior of the system and
showcases the Lattice Boltzmann Method’s
efficacy in studying heat transfer within a cavity
featuring an elliptical block. Looking ahead, future
inquiries should encompass the exploration of non-
uniform dynamic and thermal boundary conditions
on the cavity’s walls. Diversifying the geometries
of internal obstacles will contribute valuable
insights into their effects on fluid dynamics and
heat transfer.
Nomenclature
c Lattice speed, c = 1
Discrete vector velocity
Sound speed (m/s)
D2Q9 Lattice arrangement
Dynamical distribution function
Equilibrium dynamical function
Direction of the imposed body force
Gravitational acceleration (m/s2)
Thermal distribution function
Equilibrium thermal function
Grashof number
Length of the cavity (m)
LBM Lattice Boltzmann Method
Local Nusselt number
Average Nusselt number
Dimensional pressure (Pa)
Dimensionless pressure
Prandtl number
Reynolds number
Richardson number
Dimensionless time
Dimensional time (s)
Dimensionless reference temperature
Dimensional temperature (K)
Dimensionless temperature
Dimensional velocity components (m/s)
Dimensionless velocity components
Velocity lid-driven (m/s)
Dimensional coordinates
Dimensionless coordinates
Greek symbols
Thermal diffusivity (m2/s)
Thermal expansion coefficient (1/K)
Time step (s)
Dimensional temperature difference (K)
Density (Kg/m3)
Hydrodynamic relaxation time (s)
Thermal relaxation time (s)
Kinematic viscosity (m2/s)
Dimensionless stream function
Weighting factor
Subscripts
C Cold temperature
H Hot temperature
max Maximum
min Minimum
Superscripts
Dimensional variable
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Abdelhak Daiz, Ahmed Bahlaoui,
Ismail Arroub, Soufiane Belhouideg,
Abdelghani Raji, Mohammed Hasnaoui
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Ismail Arroub, Soufiane Belhouideg,
Abdelghani Raji, Mohammed Hasnaoui
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Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
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Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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