Effect of Wavy Interface on Natural Convection in Square Cavity
Partially Filled with Nanofluid and Porous Medium using Buongiorno
Model
CHERIFA BENYGZER1, MOHAMED BOUZIT1, ABDERRAHEM MOKHEFI2
1LSIM Maritime Science and Engineering Laboratory, Faculty of Mechanical Engineering,
Mohamed Boudiaf University of Science and Technology, El Maouar, Bp 1505, Bir Eldjir 31000,
Oran,
ALGERIA
2L2ME Modeling and Experimentation Laboratory, Faculty of Sciences and Technology,
Bechar University B.P.417, 08000, Bechar,
ALGERIA
Abstract: - Convective heat transfer improvement from wavy surfaces presents a new solution in industrial
engineering for composite materials, including porous medium, and nanofluids to address the wavy irregular
surfaces in heat transfer devices such as a wavy solar collector, energy absorption and filtration, thermal
insulation, and geothermal power plants. This technique enables the performance of engineering applications.
The numerical study is performed to examine the effects of a wavy interface separating two layers in the
enclosure on heat exchange rates. This paper investigates numerically the natural convection flow in a square
cavity partially filled with nanofluid-porous layers separated by a wavy horizontal interface. The left and right
walls of the cavity are maintained at constant hot and cold temperatures, whereas the other walls are adiabatic.
The Buongiorno model is used to describe nanofluid motion, taking into account the brownian and
thermophoresis effects in the cavity. The Galerkin finite element method was applied to solve the differential
governing equations. The dynamic, thermal field and heat transfer have been analyzed for various parameters
such as Rayleigh number (103 ≤ Ra ≤ 106), the amplitude of interface (0 ≤ A ≤ 0.1), and undulation number (0 ≤
n ≤ 9). The results reveal that the flow intensity induced by buoyancy forces is more significant in the nanofluid
layer than in the porous layer, since the heat transfer is enhanced while the flow is not sensitive to variations in
amplitude and number undulation, and accordingly, the decline of average Nusselt and Sherwood numbers is
insignificant. The effects of controlled parameters on the structure of nanofluid flow, heat, and mass transfer
rate are insignificant.
Key-Words: - Wavy interface, nanofluid, heat transfer, Buongiorno model, natural convection, porous medium,
layers, amplitude, undulation number, Galerkin finite element method.
Received: June 14, 2023. Revised: February 26, 2024. Accepted: April 11, 2024. Published: June 4, 2024.
1 Introduction
To improve the thermal conductivity of fluids,
scientific research has multiplied in recent years and
has made it possible to synthesize nano-sized
particles dispersed in the base fluid, which
constitute nanofluids used in several fields such as
biochemistry and the oil industry.
The applications of nanofluids in thermal
systems filled with porous medium and of different
geometries, flow regimes, boundary conditions, and
different types of nanofluids and thermophysical
properties. Many researchers have investigated the
nanofluids used in porous media due to their thermal
characteristics.
The addition of nanoparticles to the pure fluid
enhances considerably the heat transfer in the
enclosure, [1], [2], [3], [4], [5], [6].
Also, the authors of [7], [8], [9], [10], [11], [12],
[13], [14], [15], use hybrid nanofluids, which
significantly improve the dynamic viscosity and
thermal conductivity of the base fluid.
The horizontal and vertical orientation of the
porous layer in the enclosure was reported, [16],
[17], [18], [19].
The natural convection in the wavy cavity-
saturated Cu-water nanofluid has been studied by
some authors, [20], [21], they examined the effect of
the Rayleigh number, the wave amplitude, and the
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undulations number on the heat transfer rate, the
amplitude ratio was found to be significant in the
rise of the heat transfer in a porous cavity with
sinusoidal heating on both sidewalls, [22], therefore
the convective flow is attenuated with growth of
undulation under the effect of thermal dispersion
[23], while the wave number can control the free
convective motion and heat transport within the
wavy cavity, [24]. Thus, the heat transfer rates were
sensitive to the variation of undulation property,
convection intensity, [25].
The choice of different fluids has been the
subject of some authors. The paper [26] investigates
the effect of the sinusoidal interface on the heat and
mass transfer of laminar clear fluid and porous
medium.
The nanofluid is modeled using Buongiorno’s
model, incorporating thermophoresis and Brownian
motion effects, [27]. We cite in the literature a
numerical study of natural convection inside a
porous, wavy cavity filled with a nanofluid using
the Forchheimer-Buongiorno approach, [23].
Laminar natural-convective flow in a square
composite vertically layered enclosure consisting of
porous and nanofluid layers separated by a
sinusoidal corrugated interface has been reported by
[28], they found that the increase in Darcy's number
of porous media with a low value of the non-
uniform porous layer thickness enhances the
convective heat transfer in the cavity. On the other
hand, a decrease in the Darcy parameter brings a big
resistance force for the fluid flow and an increase in
heat transfer by conduction, [29], [30].
The heat transfer increases with the increase of
modified Rayleigh number, volume fraction, and
amplitude in a porous square cavity saturated by a
nanofluid (Al2O3-Water) in the presence of a
corrugated heat source, [31].
In addition, the heat transfer rates through the
nanofluid and solid phases are found to be better for
high values of the undulation amplitude, [32].
The impacts of various effective parameters,
which include nanoparticle volume fraction, Darcy
number, modified conductivity ratio, the number of
undulations, and the amplitude, on the heat transfer
rate in a wavy-walled porous enclosure are being
analyzed, [33].
The study of natural convection in a two-
dimensional enclosure with horizontal wavy walls
layered by a porous medium, saturated by Cu-
Al2O3/water hybrid nanofluid has been performed
in [34].
The increase in porous layer width leads to a
decline in the average Nusselt and Sherwood
numbers in a square cavity filled with power-law
fluid and porous media separated by a wavy
interface, [35]. The studies of [36], [37], [38]
reported the natural convection in a square
enclosure divided by a corrugated porous partition,
either horizontally or vertically, in porous and fluid
regions and a porous cavity filled with a hybrid
nanofluid and non-newtonian layers. Higher
amplitude and the undulation number of the
sinusoidal interface between the nanofluid and
porous medium layer lead to a decrease in the
average Nusselt number of natural convection in a
square cavity filled by a nanofluid (Cu-water)
/porous-medium, [39].
In this paper, a numerical investigation of the
natural convection in a square cavity partially filled
with porous and nanofluid separated by wavy was
performed. The effects of amplitude, undulation
number of the interface on the dynamic and thermal
fields, heat, and mass transfer rate are analyzed for
various values.
In reviewing the literature, we found no
published work for composite porous cavities
saturated with nanofluid flow modeled using the
Buongiorno model.
This work provides an original contribution: to
develop high-performance materials with minimum
power consumption for future industrial
applications.
2 Physical Model
The configuration considered in the present study is
shown in Figure 1. The physical domain consists of
a 2D square cavity heated from the left vertical wall
and cold from the right wall at constant
temperatures Th and Tc, respectively. The horizontal
walls are considered adiabatic. This cavity is filled
with a nanofluid and porous layer separated by a
horizontal, wavy interface. The nanofluid occupies
the lower part, and the porous medium is placed
above the mid-plane of the cavity.
- H is the height of the cavity.
- H/2 is the width of the porous and nanofluid
layers.
- The fluid flow is considered to be laminar
and incompressible, and the porous medium
is assumed to be homogenous and isotropic.
- The constant thermophysical properties of
water and nanofluids at 25oC are given
in Table 1.
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Adiabatic wall
H
Th
H/
2
Porous
Medium
Tc
Nanofluid
Adiabatic wall
Fig. 1: Physical model
Table 1. Thermophysical properties of water
2.1 Mathematic Model
The two-dimensional equations governing the
stationary flow of nanofluid in natural convection
inside a square cavity using the Buongiorno
mathematical model are described as follows:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
The continuity, momentum, and energy
equations can be written in dimensionless form as
follows by introducing dimensionless variables:
The flow is assumed to be stationary and
incompressible.
(8)
(8)
(9)
(10)
(3)
(11)
(4)
(12)
(13)
(5)
New parameters appear in the dimensionless
equations: Prandtl number, Darcy number, Rayleigh
number, buoyancy ratio, Brownian motion,
thermophoresis and Lewis number.
(14)
2
H
K
Da
(15)
(16)
(17)
(18)
(19)
(20)
The physical quantities of heat transfer are
local, average Nusselt and Sherwood numbers
respectively Nu loc, Sh loc, Nu avg, Sh avg are defined
as below:
0
y
v
x
u
tf
u
Ky
u
x
u
x
p
y
u
v
x
u
u
t
uff
2
2
2
2
v
Ky
v
x
v
y
p
y
v
v
x
v
u
t
vff
2
2
2
2
gCCTTC fscfcfc
1
2
2
2
2
y
T
x
T
y
T
v
x
T
u
t
T
2
2
y
T
x
T
D
D
y
C
y
T
x
C
x
T
D
C
T
B
2
2
2
2
2
2
2
2
y
T
x
T
D
D
y
C
x
C
D
y
C
v
x
C
u
t
C
C
T
B
b
B
Bd
TK
D3
0
b
B
Bd
TK
D3
0
f
fch
C
ch
c
f
pH
P
CC
CC
TT
TTvuH
VU
H
yx
YX 2
2
,,,
,
,,
,
,
0
Y
V
X
U
U
DaY
U
X
U
X
P
Y
U
V
X
U
UPr
Pr 2
2
2
2
NrRaV
DaY
V
X
V
Y
P
Y
V
V
X
V
U
Pr
Pr
Pr 2
2
2
2
2
2
2
2
YX
Y
V
X
U
22
YX
Nt
YYXX
Nb
2
2
2
2
2
2
2
2
1
YXLeNb
Nt
YXLeY
V
X
U
k
Cp
Pr
ff
chfc HTTgC
Ra
3
1
chffc
fsch
TTC
CC
Nr
1
f
f
s
Bch
BCp
CpDCC
N
f
ch
f
s
c
T
TTT
Cp
Cp
T
D
N
B
f
D
Le
Physical property
Water
Cp (J/kg K)
4179
ρ (kg /m3)
997.1
k (W/m .K)
0.613
Β (K-1)
21e-5
μ (kg/m s)
8.55e-4
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(21)
(22)
2.2 Boundary Conditions
The proposed problem provides a boundary
condition in the dimensionless form of the cavity
walls:
At X = 0, 0 Y 1 U, V = 0 and
At X = 1, 0 Y 1 U, V = 0 and
At Y = 0, 1 and 0Y 1 U = V = ,
= 0 ,
= 0
The continuity conditions at the sinusoidal
horizontal interface are described as follows:
3 Numerical Method and Validation
The finite element method is applied to discretize
the dimensionless partial differential equations
provided in Eqs. (9) to (13) that are associated with
the boundary conditions based on the Galerkin
scheme. The convergence criterion is 1e-6 for each
variable.
3.1 Grid Independence Test Mesh
A 2D triangular mesh was generated to cover the
computational domain of the considered
configuration with 82268 elements.
A grid independence test series was proposed in
this study to select the optimum grid size with the
following number of elements: 51104, 55786,
62634, 82268, and 111136 to determine the
appropriate study mesh size (Table 2).
Figures 2 (a) and (b) show the meshes of the
studied configuration; the interface line between the
porous and nanofluid layers was refined to capture
the flow behaviors.
(a)
(b)
Fig. 2: Computational mesh of the physical domain
(a), (b) refined mesh of the interface line
Table 2. Independence test of mesh
E Nr
51104
55786
62634
82268
111136
Time
31608 s
30276 s
2686 s
9435 s
44345 s
5.17445
5.17570
5.17647
5.17727
5.17761
0.59358
0.59360
0.59361
0.59362
0.59362
1.02117
0.98350
0.97116
0.95820
0.95366
Nu avg
3.54361
3.54496
3.54575
3.54652
3.54687
Sh avg
3.54363
3.54497
3.54575
3.54652
3.54688
Five numbers of elements are compared to check
the mesh independence at Ra = 105, Da = 10-3, Pr
=5.82, A = 0.05, n = 9, and Le = Nt = Nb = Nr =
0.1. The deviations of Nu avg and Sh avg between the
grids G4 and G5 are very small. The gird G4 seems
to be the most suitable for this study.
3.2 Validation
The results of the present study for average Nusselt
number variation with volume fraction Al2O3
nanoparticles are compared with those of [40],
investigating heat transfer convective in a square
cavity .
The validation plot (Figure 3) was completed
under identical conditions, and both the previous
studies' findings are in good agreement. The error of
the average Nusselt number value is estimated at
2.03% for the Darcy number value 10-3 as shown in
Table 3.
Fig. 3: Comparison of the present result with the
numerical results of [40]
11
,
X
L
X
LX
Sh
X
Nu
dY
X
ShdY
X
Nu avgavg
1
0
1
0
,
, , , , nf
p nf p nf P nf P nf nf p
f
k
U U V V P P n k n
,nf
p nf nf p
f
CC
CC nn
Nu avg
0 0.05 0.1
0
2
4
6
8
Da = 10-3
Da = 10-3
Frame 001 20 Jul 2023
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Table 3. Validation of average Nusselt number
values
Da
Hea
d
Ra
Sheikhzadeh et al (2013)
(2013)
Present study
Error %
10-3
10-5
7.25
7.40
2.03
4 Results and Discussions
The main interest in this study is focused on the
effect of the horizontal wavy interface on the
structure of the flow in the partially filled cavity
with nanofluid and porous medium. The results of
the different numerical simulations are presented by
streamlines, isotherms, and iso-concentration
contours, as well as local, average Nusselt and
Sherwood numbers.
The simulations are performed for the fixed
values of Rayleigh number Ra = 105, Darcy number
Da = 10−3, the dimensionless amplitude of interface
A = 0.05, undulation number n = 9, Lewis number
Le, buoyancy ratio Nr, brownian motion Nb,
thermophoresis parameter Nt and Le = Nt = Nb =
Nr = 0.1.
The results will consecutively present the effect
of various values of the Rayleigh number (103 ≤ Ra
≤ 106), the dimensionless amplitude (0 ≤ A ≤ 0.1),
and the undulations number (0 ≤ n ≤ 9).
4.1 Effect of Rayleigh Number
Figure 4 represents the distribution of streamlines,
isotherms, and iso-concentrations for various values
of the Rayleigh number (103 ≤ Ra ≤ 106).
For a low Rayleigh number (Ra = 103), the main
cell circulation is located on the nanofluid portion of
the cavity and rotates clockwise;the flow is
immobile in the porous portion.
In the porous layer, temperature and
concentration gradients are weak, indicating a
conductive flow [30].
The cell circulation is intensified by increasing
the Rayleigh number from Ra = 104 to 105.
The cell circulation is elongated horizontally
and is still located on the nanofluid layer, It is
noticed that at Ra = 106, the infiltration of nanofluid
is detected in the porous layer « By increasing
the Ra number, remarkable penetration in the
direction of the porous layer is detected, due to the
elongation of the main vortex » [35].
Streamline intensity indicates that a high-
temperature gradient contributes to enhancing heat
transfer and the dominance of convective heat
transfer« Streamlines shown at Ra = 105 and 107
indicate that the fluid circulation strength becomes
higher when the mode of heat transfer is transferred
from conduction to convection with more vortices
and distortion in the flow pattern [36].
The isotherms and iso-concentrations are
parallel to the vertical wall at Ra = 103, denoting the
conductive heat transfer in the porous and nanofluid
layers.
By increasing Rayleigh's number from Ra = 104 to
106, isotherm lines become horizontal and very
dense close to the left hot wall into the nanofluid
layer. In the porous layer, they remain diagonal.
A high-temperature gradient was observed due to
the convective heat exchange. The flow passes from
a conductive to a convective regime by
increasing the Ra number [30].
Iso concentrations are modified to a horizontal
shape by increasing Ra =105 in both layers. At Ra
=106, an important nanoparticle concentration is
observed near the hot left wall, signifying important
convective heat transfer.
The distribution of streamlines, isotherms, and
iso-concentrations confirms that the strength of flow
is more significant in the nanofluid layer than in the
porous layer« The intensity of circulation is always
much stronger in the fluid region than in the porous
region. The magnitude of penetration of flow from
the fluid region to the porous region is substantially
influenced by the Rayleigh number » [37].
High buoyancy forces reinforce nanofluid
circulation in the cavity and consequently,
convective transfer is enhanced.
4.2 Effect of Dimensionless Amplitude
Figure 5 shows the effect of the various values of
dimensionless amplitude (0 ≤ A ≤ 0.1) on the
streamlines, isotherms, and iso-concentrations
contours.
By increasing the amplitude of interface A from
0 to 0.05, the cell circulation intensity does not
change; the flow is located in the nanofluid layer,
while the flow in the porous region is immobile;
indeed, the convection dominates the flow. The
streamline intensity decreases slightly from A =
0.075 to 0.1. The flow is not sensitive to variations
in dimensionless amplitude.
Isotherms are horizontal in the nanofluid layer
due to the dominance of convection mode; in the
porous layer, their shape is diagonal.
Iso-concentrations are completely stratified at the
bottom cavity; we explain this phenomenon by
nanoparticle concentrations due to the
thermophoretic force effect.
The dimensionless amplitude effect is not
significant for nanofluid circulation, temperature, or
concentration of nanoparticles.
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Streamlines
Isotherms
Iso-concentrations
Ra = 103
Ra = 104
Ra = 105
Ra = 106
Fig. 4: Variation of Ra on streamlines, isotherms and iso-concentrations, A = 0.05, n = 0.9, Da = 10-3,
Le = Nr = Nb = Nt = 0.1
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Streamlines
Isotherms
Iso-concentrations
A= 0
A= 0.025
A= 0.05
A= 0.075
A= 0.1
Fig. 5: Variation of A on streamlines, isotherms and iso-concentrations , Ra =105, Da = 10-3, n = 0.9 ,
Da = 10-3, Le = Nr = Nb = Nt = 0.1
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Streamlines
Isotherms
Iso-concentrations
n = 0
n = 3
n = 6
n = 9
Fig. 6: Variation of n on streamlines, isotherms and iso-concentrations Ra =105, Da = 10-3, A = 0.05,
Da = 10-3, Le = Nr = Nb = Nt = 0.1
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4.3 Effect of the Undulation Number
Figure 6 represents the distribution of the
streamlines, the isotherms, and the iso-
concentrations for various values of the undulation
number (0 ≤ n ≤ 9).
It is observed that the intensity of the cell
circulation does not change, whatever the increase
in the undulation number. The heat transfer is
convective.
The isotherm lines are diagonal in the porous
layer, while in the nanofluid layer, the isotherms are
horizontal to the left hot wall and become denser,
indicating the convection mode.
The iso-concentration stratification zone was
detected in the nanofluid layer, which means a high
nanoparticle concentration near the hot left wall due
to the thermophoresis effect.
The undulation number doesn’t influence the
dynamic and thermal fields.
4.4 Compute of local, Average Nusselt and
Sherwood numbers
Figure 7, Figure 8, Figure 9, Figure 10, Figure 11
and Figure 12 represent the evolution of the local,
average Nusselt and Sherwood d numbers along the
left hot wall as a function of Y coordinate for
various values of Rayleigh number (103 ≤ Ra ≤ 106),
dimensionless amplitude (0 ≤ A ≤ 0.1) and
undulation number (0 ≤ n ≤ 9).
4.4.1 Impact of the Rayleigh number
Figure 7 represents the variation of the local Nusselt
Nu loc (a) and Sherwood Sh loc (b) numbers for
various values of the Rayleigh number (103 ≤ Ra ≤
106).
In Figure 7 (a) and (b), it is observed that the
high values of local Nusselt and Sherwood numbers
are obtained at the bottom of the hot left wall (Y =
0) and decrease by moving along the height of the
hot left wall at Y = 1.
A high increase in local Nusselt and Sherwood
numbers from Ra = 105 to 106 was marked « It is
noticeable that the local Nu intensifies significantly
from Ra = 105 to 106» [38]; this increase is less
important from Ra = 103 to104 due to the lower heat
and mass exchange between the porous and
nanofluid layers, where conduction dominates in the
cavity.
Figure 8 shows the variation of average Nusselt
and Sherwood numbers with Rayleigh number from
Ra = 103 to 106. At low Rayleigh number Ra = 103
to 104, an insignificant increase in Nu avg and Sh avg
was noticed due to the conductive regime, while for
higher Rayleigh number from 105 to 106, this
increase is more significant.
Higher values of average Nusselt and Sherwood
numbers are interpreted by the dominance of
convection.
A significant heat and mass transfer occurred due
to increasing buoyancy forces.
Variations in Rayleigh number values indicate
the same heat and mass transfer rate.
4.4.2 Impact of Dimensionless Amplitude
Figure 10 (a) and (b) illustrate the evolution of the
local Nusselt and Sherwood numbers according to
the value of the dimensionless amplitude (0 ≤ A ≤
0.1).
It is clear that the increase in amplitude A does
not influence the local Nusselt and Sherwood
numbers curves in the porous layer, however a
slight decline was observed in the nanofluid layer
when A increased for the two numbers.
Figure 9 shows the evolution of the average Nusselt
and Sherwood numbers as a function of amplitude
A. It is noticed that Nu avg and Sh avg decrease
linearly as the amplitude increases « higher
amplitude and the undulation number of the
sinusoidal interface between the nanofluid and
porous medium layer lead to a decrease in the
average Nusselt number», [39].
The decrease of Nu avg and Sh avg is not
significant, which is reflected in a low rate of heat
and mass transfer.
Heat and mass transfer are not sensitive to the
change in amplitude.
The amplitude of wavy interface variation
provides the same rate of heat and mass transfer.
4.4.3 Impact of the Undulation Number
Figure 11 (a) and (b) demonstrate the variation of
local Nusselt and Sherwood numbers for various
undulation numbers (0 ≤ n ≤ 9).
In Figure 11 (a) and (b), the Nu loc and Sh loc
decrease slightly by increasing the undulation
number n in the nanofluid layer «The average
Nusselt number decreases by increasing the
undulation number» [34], while in the porous layer,
the increase of n does not change the value of the
Nu loc and Sh loc.
Undulation number increase has no impact on
local Nusselt and Sherwood numbers variation.
Figure 12 shows the evolution of the average
Nusselt and Sherwood numbers with undulation
number n.
By increasing the undulation number n from 0
to 3, a slight increase in average Nusselt and
Sherwood numbers was noticed. At n = 3 to 9, the
numbers’ Nu avg and Sh avg decrease with undulation
numbers n.
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The maximum value of Nu avg and Sh avg is
reached at n = 3, while the minimum value is at n =
9 for the two numbers.
The variation of undulation number n does not
affect heat and mass exchange rates.
The heat and mass transfer rates have the same
value by increasing the undulation number of the
wavy horizontal interface.
5 Conclusion
Numerical investigation of bi-dimensional laminar
natural convection in a square cavity partially filled
with nanofluid and porous medium separated by a
wavy interface using the Buongiorno model to
resolve differential governing equations based on
the Galerkin finite element method was studied. The
effects of the Rayleigh number, the amplitude, and
the undulation number of the interface on the
dynamic and thermal fields reveal that the flow
intensity is more significant in the nanofluid layer
than in the porous layer by increasing buoyancy
forces, which contribute to reinforcing nanofluid
circulation in the cavity and enhancing convective
heat transfer.
Temperature and concentration nanoparticle
distributions are not affected by varying amplitude
and undulation number values.
The decline of local, average Nusselt,
and Sherwood numbers is not important by
increasing amplitude and undulation number values,
from which we conclude that the flow is not
sensitive to the variation in amplitude and
undulation number of the wavy interface.
The present configuration allows for reducing
the product development time and costs. And
achieving engineering-accurate performance
predictions of a specific geometry
This study can be extended to 3-D and
incorporate an anisotropic porous medium with
variable porosity, hybrid nanofluid, or two- phase
fluide using the Buongiorno model.
Acknowledgement:
I would like to thank Mr. Bouzit Mohamed for his
support, Mokhefi Abderrahim, and all the elements
of the maritime science and engineering laboratory
USTO.
Nomenclature
Subscripts
c cold
h hot
np nanoparticles
p porous medium
B
thermal expansion coefficient
C
concentration dimensionnel
DB
Brownian diffusion coefficient
DT
thermodiffusion coefficient
CP
specific heat capacity
Da
Darcy number
g
gravitational acceleration
h
heat transfer coefficient
H
height dimensionless
k
thermal conductivity
K
Permeability
L
characteristic length
Le
Lewis number
Nb
Brownian number
Nr
buoyancy ratio
Nt
thermophoresis
parameter
Nu avg
average Nusselt
number
Nu loc
local Nusselt number
p
pressure
P
Dimensionless
pressure
Pr
Prandtl number
Ra
Rayleigh number
Sh avg
average Sherwood
number
Sh loc
local Sherwood
number
T
dimensional temperature
u, v
dimensional velocity components
U, V
dimensionless velocity components
W
heat flux density
x, y
dimensional space coordinates
X, Y
dimensionless space
coordinates
Greek symbol
ν
kinematic viscosity
Ρ
pressure
thermal diffusivity
θ
dimensionless temperature
volume fraction
nanoparticle
φ
dimensionless concentration
stream function
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2024.19.22
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References:
[1] Kasaeian A., Eshghi A.T., Sameti M., Kearns
D.M. A review on the applications of
nanofluids in solar energy systems,
Renewable and Sustainable Energy Reviews,
Vol. 43, 2014, 584-598.
[2] Chamkha A.J., Molana M., Rahnama A.,
Ghadami F. On the nanofluid Application
microchannels, A comprehensive review,
Powder technology. 332(5).2018.
[3] Ramezanizadeh M., Nazari M.A., Ahmadi
M.H., Açikkalp E. Application of nanofluid
in thermosyphons: A review Journal of
Molecular Liquids 272, 2018, 395-402.
[4] Kasaeian A., Daneshazarian R., Mahian O.,
Kolsi L., Chamkha A.J., Wongwises S., Pop
I. Nanofluid flow and heat transfer in porous
media: A review of the latest developments,
International Journal of Heat and Mass
Transfer, 107, 2017, 778-791.
[5] Tahmasebi A., Mahdavi M., Ghalambaz M.
Local thermal non equilibrium conjugate
naturalconvection heat transfer of nanofluids
in a cavity partially filled with porous media
using Buongiorno’s model, Numerical Heat
Transfer, Part A: Applications 73(4), 2018,
254-276.
[6] Mehryan S., Ghalambaz M., Izadi M.
Conjugate natural convection of nanofluids
inside an enclosure filled by three layers of
solid, porous medium and free nanofluid
using Buongiorno’s and local thermal non-
equilibrium models, Journal of Thermal
Analysis and Calorimetry 135(2) 1047-1067.
[7] Suresh S., Venkitaraj K., Selvakumar P.,
Chandrasekar M. Synthesis of Al2O3–
Cu/water hybrid nanofluids using two step
method and its thermo physical properties,
Colloids and Surfaces A: Physicochemical
and Engineering Aspects, 388(1-3) (2011)
4148.
[8] Mehryan S.A., Kashkooli F.M., Ghalambaz
M., Chamkha A.J. Free convection of hybrid
Al2O3-Cu water nanofluid in a differentially
heated porous cavity, Advanced Powder
Technology 28(9),2017, 2295-2305.
[9] Ghalambaz M., Mehryan S., Izadpanahi E.,
Chamkha A.J., Wen D. MHD natural
convection of Cu–Al2O3 water hybrid
nanofluids in a cavity equally divided into
two parts by a vertical flexible partition
membrane, Journal of Thermal Analysis and
Calorimetry,138(2),2019,1723-1743.
[10] Chamkha A.J., Miroshnichenko I.V.,
Sheremet M.A. Numerical analysis of
unsteady conjugate natural convection of
hybrid water-based nanofluid in a
semicircular cavity, Journal of Thermal
Science and Engineering Applications,2017,
9(4): 041004.
[11] Ghalambaz M., Sheremet M.A., Mehryan, S.,
Kashkooli, F.M., Pop I. Local thermal
nonequilibrium analysis of conjugate free
convection within a porous enclosure
occupied with Ag–MgO hybrid nanofluid,
Journal of Thermal Analysis and
Calorimetry 135(2), 2019, 1381-1398.
[12] Selimefendigil F., Öztop H.F. Conjugate
Natural convection in a cavity with different
nanofluids on different sides of the partition,
Journal of Molecular Liquids 216, 2015, 67-
77.
[13] Sahoo R.R., Ghosh P., Sarkar J. Performance
analysis of a louvered fin automotive radiator
using hybrid nanofluid as coolant, Heat
Transfer—Asian Research 46(7),2016, 978-
995.
[14] Gorla R., Siddiqa S., Mansour M., Rashad
A., Salah T. Heat source /sink effects on a
hybrid nanofluid-filled porous cavity,
Journal of Thermophysics and Heat Transfer
31(4), 2017, 847-875.
[15] Izadi M., Mohebbi R., Delouei A.A., Sajjadi
H. Natural convection of a magnetizable
hybrid nanofluid inside a porous enclosure
subjected to two variable magnetic fields,
International Journal of Mechanical
Sciences, 151, 2018, 154-169.
[16] Arpino F., Massarotti N., Mauro A.
Efficient three-dimensional FEM based
algorithm for the solution of convection in
partly porous domains, International journal
of heat and mass transfer 54(21-22),
2011.4495-4506.
[17] Vasseur P., Wang C., Sen M. Thermal
instability and natural convection in a fluid
layer over a porous substrate, Wärme-und
Stoffübertragung, 24(6), 1989, 337-347.
[18] AL-Srayyih B.M., Gao S., Hussain S.H.
Effects of linearly heated left wall on natural
convection within a superposed cavity filled
with composite nanofluid-porous layers,
Advanced Powder Technology, 30(1), 2018,
55-72.
[19] Sathe S., Lin W.Q, Tong T. Natural
convection in enclosures containing an
insulation with a permeable fluid-porous
interface, International journal of heat and
fluid flow 9(4),1988,389-395.
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DOI: 10.37394/232013.2024.19.22
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E-ISSN: 2224-347X
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Volume 19, 2024
[20] Boulahia Z., Wafik A., Sehaqui R. Modeling
of free convection heat transfer utilizing
nanofluid inside a wavy enclosure with a pair
of hot and cold cylinders, Frontiers in Heat
and Mass Transfer (FHMT), 8.14, 2017.
[21] Khanafer K., AL-AzmI B., Marafie A., Pop
I. Non-Darcian effects on natural convection
heat transfer in a wavy porous enclosure,
International Journal of Heat and Mass
transfer ,7–8, 2009, 1887-1896.
[22] Sivasankaran S., Bhuvaneswari M. Natural
Convection in a Porous Cavity with
Sinusoidal Heating on Both Sidewalls.
Numerical Heat Transfer, Part A:
Applications. An International Journal of
Computation and Methodology, 63, 2012,
14-30.
[23] Sheremet M., Cornelia R., Pop I. Free
convection in a porous wavy cavity filled
with a nanofluid using Buongiorno's
mathematical model with thermal dispersion
effect, Applied Mathematics and
Computation, 299, 2017:1-15.
[24] Alsabery A., Tayebim T., Chamkha A.J.,
Hashim I. Natural convection of Al2O3-water
nanofluid in a non-Darcian wavy porous
cavity under the local thermal
non-equilibrium condition, Scientific
Reports,10 (1)2020:1-22.
[25] Alhashash A., Saleh H. Natural convection
induced by undulated surfaces in a porous
enclosure filled with nanoliquid, Advances in
Mechanical Engineering, 11(9), 2019, 1-9.
[26] De Lemos, M.J.S., Silva R.A. Laminar Flow
Around a Sinusoidal Interface Between a
Porous Medium and a Clear Fluid,
IMECE2003-41452, pp. 283-290.
[27] Benygzer C., Bouzit M., Mokhefi A., Khelif
F.Z. Unsteady natural convection in a porous
square cavity saturated by nanofluid using
Buongiorno Model: Variable permeability
effect on hmogeneous porous medium, CFD
letters, 2022, 14(7):42-61.
[28] Kadhim H.T., Jabbar F.A., Rona A. Cu-
Al2O3 hybrid nanofluid natural convection
in an in an inclined enclosure with wavy
walls partially layered by porous medium,
International Journal of Mechanical
Science,186 (2), 2020,105889.
[29] Aly A.M., Raizah Z.A.S. Mixed Convection
in an Inclined Nanofluid Filled-Cavity
Saturated With a Partially Layered Porous
Medium, J. Thermal Sci. Eng. Appl,2019,
11(4):041002.
[30] Mehdaoui R., Elmir M., Mojtabi A. Effect of
the Wavy permeable Interface on Double
Diffusive Natural Convection in a Partially
Porous Cavity, The International Journal of
Multiphysics, Vol. 4 No. 3, 2010, 217-231.
[31] Bouafia I., Mehdaoui R., Kadri S., Elmir M.
Natural Convection in a Porous Cavity Filled
with Nanofluid in the Presence of Isothermal
Corrugated Source, International Journal of
Heat and Technology Vol. 38, No. 2, 2020,
pp. 334-342.
[32] Alsabery A. I., Tayebi T., Abosinnee A.S .,
Raizah Z. A. S., Chamkha A.J., Hashim I.
Impacts of Amplitude and Local Thermal
Non-Equilibrium Design on Natural
Convection within NanoflUid Superposed
Wavy Porous Layers, Nanomaterials 2021,
11, 1277.
[33] Kadhim H.T., Al-Manea A., Al-Shamani
A.N., Yusaf T. Numerical analysis of hybrid
nanofluid natural convection in a wavy
walled porous enclosure: Local thermal non-
equilibrium model, International Journal of
Thermofluids, 15 (2022) 100190.
[34] Kadhim H.T., Al Dulaimi Z.M., Rona A.
Local Thermal Non-equilibrium Analysis of
Cu-Al2O3 Hybrid Nanofluid Natural
Convection in a Partially Layered Porous
Enclosure with Wavy Walls, Journal of
Applied and Computational Mechanics 9(3),
(2023), 712-727.
[35] Kolsi L., Hussain S., Ghachem K., Jamal M.,
Maatki C. Double Diffusive Natural
Convection in a Square Cavity Filled with a
Porous Media and a Power Law Fluid
Separate by a Wavy Interface, Mathematics
10(7), 2022, 1060.
[36] Javed S., Deb N., Saha S. Natural convection
and entropy generation inside a square
chamber divided by a corrugated porous
partition, Results in Engineering 2023,18
(4):101053.
[37] Nishimura T., Kawamura Y., Ozoe H.
Numerical analysis of natural convection
heat transfer in a square enclosure
horizontally or vertically divided into fluid
and porous region, Technol Rep Yamaguchi
Univ, 1991, 4(5):331-344.
[38] Omri M., Jamal M., Hussain S., Kolsi L.,
Maatki C. Conjugate Natural Convection of a
Hybrid Nanofluid in a Cavity Filled with
Porous and Non-Newtonian Layers: The
Impact of the Power Law Index,
Mathematics 2022, 10, 2044.
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[39] Tuan N.M., ALY A.M., LEE S.W. Effect of
a wavy interface on the natural convection of
a nanofluid in a cavity with a partially
layered porous medium using the ISPH
method, Numerical Heat transfer
Application, 72(1), 2017, 68-88.
[40] Sheikhzadeh G.A., Nazari S. Numerical
Study of Natural Convection in a Square
Cavity Filled with a Porous Medium
Saturated with Nanofluid, Trans. Phenom.
Nano Micro Scales, 1(2), 2013, 138-146.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
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.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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APPENDIX
(a)
(b)
Fig. 7: Local Nusselt and Sherwood numbers as a function of Ra
Fig. 8: Average Nusselt and Sherwood numbers as a
function of Ra
Fig. 9: Average Nusselt and Sherwood numbers as a
function of A
(a)
(b)
Fig. 10: Local Nusselt and Sherwood numbers as a function of A
Y
Nu loc
0 0.2 0.4 0.6 0.8 1
0
5
10
15
Ra = 103
Ra = 104
Ra = 105
Ra = 106
Frame 001 24 May 2022
Y
Sh loc
0 0.2 0.4 0.6 0.8 1
0
5
10
15
20
Ra = 103
Ra = 104
Ra = 105
Ra = 106
Frame 001 24 May 2022
Ra
Nu avg, Sh avg
0 500000 1E+06
1
2
3
4
5
6
7
8
Nu avg
Sh avg
Frame 001 28 May 2023
A
Nu avg, Sh avg
0 0.02 0.04 0.06 0.08 0.1
3.53
3.54
3.55
3.56
Nu avg
Sh avg
Frame 001 29 May 2023
Y
Nu loc
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8A = 0
A = 0.025
A = 0.05
A = 0.075
A = 0.1
Frame 001 15 May 2022
Y
Sh loc
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8A = 0
A = 0.025
A = 0.05
A = 0.075
A = 0.1
Frame 001 15 May 2022
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245
Volume 19, 2024
(a)
(b)
Fig. 11: Local Nusselt and Sherwood numbers as a function of n
Fig. 12: Average Nusselt and Sherwood numbers as a function of n
Y
Nu loc
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8n = 0
n = 3
n = 6
n = 9
Frame 001 15 May 2022
Y
Sh loc
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8n = 0
n = 3
n = 6
n = 9
Frame 001 15 May 2022
n
Nu avg, Sh avg
0 2 4 6 8 10
3.54
3.56
3.58
3.6
Nu avg
Sh avg
Frame 001 29 May 2023
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