Effect of the Aspect Ratio on the heat transfer enhancement by the
Al2O3-H2O Nanofluid traversing a Heated Shallow Cavity
CHAHRAZED ABDELLAHOUM1, 2, AMINA MATAOUI2
1Faculty of Sciences, M’hamed Bougera Universiy – Boumerdes, UMBB, ALGERIA
2Theoretical and Applied Fluid Mechanics Laboratory, University of Science and Technology Houari
Boumediene — USTHB, ALGERIA
Abstract: - In this work, the effects of the cavity aspect ratio (AR) on the flow of Al2O3-water nanofluid in a
two-dimensional cavity subjected on its lower part to a constant and uniform temperature. The governing
equations are discretized by the finite volume method based one point closure turbulence model. For nanofluid
proprieties, Maxwell-Garnetts model (MG) and Brinkman models are used for the calculation respectively of
the conductivity and viscosity of the nanofluid. The parameters of this study are the shape parameter of the
cavity from 2 to 14, Reynolds number Re between 4.103 and 105 and volume fraction of the nanoparticles
between 0 and =4%. The cavity aspect ratio effect on the flow structure and heat transfer was also examined.
The results confirm that the flow structure and heat transfer are very sensitive to the cavity aspect ratio. The
numerical results highlight the effect of the main parameters on the distribution of Nusselt number and friction
coefficient.
Key-Words: - Forced convection; cavity; heat transfer enhancement; nanofluids
Received: July 15, 2021. Revised: February 17, 2022. Accepted: March 19, 2022. Published: April 21, 2022.
1 Introduction
The open cavity in a channel flow is considered as
an interesting topic for many researchers since it
includes the phenomena of separation and
reattachment in several engineering applications
such as cooling of electronic components, solar
collector, heat exchanger and nuclear reactors. Until
now, the open cavity and the step flow interest many
researchers to study heat transfer required in several
industrial applications. According to the Plentovich
classification, there is two distinct types of cavity
flows, namely, open cavity of (1≤AR≤ 8) and closed
cavity (12≤AR≤14). The cavity of aspect ratio AR =
10 is called transitional cavity flow. Although their
geometrical simplicity, cavity flows are complex;
they consists of complex flow phenomena.
Nevertheless, this flow has been extensively studied,
both experimentally and numerically, and has
received considerable attention ever since the early
work of Krishnamurty [1] and Roshko [2].
However, the flow over a rectangular cavity has
been well described by Roshko [2] through the
results of pressure and mean velocity measurements.
Plentovich [3] investigations included pressure
measurements for two different upstream boundary
layer thicknesses. Its results indicate that as the
boundary layer thickness decreases, positive
pressure levels inside the cavity increase. Zdanski et
al. [4] numerically simulated both laminar and
turbulent flows over shallow rectangular cavities.
They studied the influence of cavity aspect ratio,
turbulence intensity of the incoming flow and
Reynolds number. However the reattachment
phenomenon mainly depends on the aspect ratio of
the cavity, the experiments revealed that this
phenomenon is also sensitive to the characteristics
of the incident flow. A similar phenomenon was
observed by several researchers. Eaton and Johnston
[5] showed that the increase in the Reynolds number
induces a reduction of the reattachment length and
an important increase in the size of the recirculation
zone. Several numerical and experimental works
have been carried out to study the hydrodynamic
and thermal behavior in confined flows. In this
view, Manca et al. [6] studied numerically the effect
of heated wall position on mixed convection in a
channel with an open cavity. They took into account
three heating modes as assisting flow, opposing
flow and heating from below by applying constant
heat flux. In their configuration, there is no inlet
section and it suddenly extends to the cavity. After
that, Manca et al. [7] investigated experimentally
the mixed convection in an open cavity with the
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heated vertical left wall at uniform heat flux.
Atashafrooz et al. [8] studied laminar forced
convection of gas flow over a recess including two
backward and forward facing steps in a horizontal
duct subjected to bleeding condition. The effects of
bleeding coefficient and recess length on the flow
and heat transfer behaviors of the system are
investigated. Because of the efficiency of thermal
properties of nanofluids, they were widely studied in
the last decade for various flow configurations and
different heat transfer processes. Oztop et al. [9]
have performed a numerical study to analyze the
effect of different nanofluids on the distribution of
temperature field in a rectangular cavity. They
confirmed that the increase of Rayleigh number and
the volume fraction of the nanofluids improve
significantly heat transfer. Al-aswadi et al. [10]
investigated numerically the laminar forced
convection flow over confined BFS using different
nanofluids. They confirmed that the recirculation
size and reattachment length increase as the
Reynolds number increases in the laminar regime.
A great number of experimental studies have
focused on heat transfer behavior in a backward
facing step flow. These configurations were
performed under different conditions, which justify
the absence of an accurate database for comparisons
to confirm the best methodology to solve such
problems. Xuan and Li [11] have investigated
experimentally the turbulent convective heat
transfer of a single phase flow of water–Cu
nanofluids in straight heated tubes with a constant
heat flux. Their results showed that nanofluids gave
a substantial enhancement of heat transfer rate
compared to the case of pure water. Kalteh et al.
[12] have studied numerically and experimentally
the convective heat transfer of a laminar flow of a
water–Al2O3 nanofluid inside a rectangular micro
channel heat sink subjected to a constant heat flux.
Their experiments were done on pure water, 0.1%
and 0.2% fraction volume of alumina–water
nanofluid. They investigated the effect of Reynolds
number and nanofluid volume fraction on heat
transfer. Abdellahoum et al. [13] examined several
models of the viscosity of turbulent forced
convection of Al2O3 nanofluid over a heated cavity
in a horizontal duct. They found that this work leads
to further investigations for viscosity of nanofluids
in forced convection in separated flow,
experimentally and numerically.
Abdellahoum et al. [14] investigated the effect of
nanoparticle volume fraction on thermal
conductivity of nanofluid with nanoparticle volume
fraction range between 0% and 4% and presented
new correlations. According to results with
increasing the nanoparticle volume fraction, the
thermal conductivity of nanofluid increases.
This study extends previous work (Abdellahoum et
al. [14]) and aims to analyze numerically the
hydrodynamic and thermal aspect of a turbulent
single-phase flow of a nanofluid composed of solid
particles of alumina (Al2O3) dispersed in a base
fluid (pure water) for different aspect ratios.
The main aim of this study is to investigate
turbulent heat transfer in a channel with cavity
2 Physical Model
Turbulent forced convection of nanofluid flow in a
heated cavity is numerically simulated. The flow is
assumed as Newtonian, steady in average and
incompressible. A schematic of the cavity with
coordinates and boundary conditions are sketched in
Figure 1.
The bottom of the cavity is heated to a constant
temperature (Tw > T0), while the confining wall is
adiabatic. The base fluid (i.e. water) and the
nanoparticles are in thermal equilibrium and no slip
boundary condition occurs between them.
Thermophysical properties of the Al2O3 nanofluid
are assumed to be constant.
Fig. 1. Sketch of the problem geometry and
boundary conditions.
In this study, the applied models for thermophysical
properties of the nanofluid are given in table 1.
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Table 1 Applied models for thermophysical
properties of the nanofluid and hybrid nanofluid
Before starting the discussion of the results of the
simulation in the cavity, we give in the table 2 the
thermophysical characteristics of base fluid and the
nanoparticle used in this work.
3. Governing equation
The governing equations for a homogenous analysis
of forced convection are continuity (eq. 1),
momentum (eq. 2), and Energy (eq. 3) depending on
their nanofluid properties, as follows:
0
i
xi
U
(1)
UU
P
ii
U u u
j i j
nf nf nf
x x x x
j i j j
(2)
nf
nf n f
nf
TT
Uu
ii
x x p x
r
i i i
(3)
In the above equations, the symbols Ui, P and T
correspond to the time averaged flow variables,
while ui and represent the fluctuations of velocity
and temperature. The turbulent shear stress
uu
ij
and
turbulent heat flux
ui
, require modeling. They
may be approximately expressed versus the time
averaged flow variables (velocity or temperature) .
By analogy with molecular transport, for all models
(first or second order models), the Simple Gradient
Diffusion Hypothesis (SGDH) is used. The
following algebraic constitutive law is allows to
deduce the velocity-temperature correlation are
deduced by the following algebraic equations which
based on the Boussinesq assumption.
2()
,,
3
nf nf
u u k U U
t
i j ij i j j i
(4)
nf T
uit
xi
(5)
Although the nanofluids are solid-liquid mixtures,
the approach conventionally used in most studies of
forced convection handles the nanofluid as a single-
phase (homogenous) fluid. In fact, due to the
extreme size and low concentration of the
suspended nanoparticles, the particles are assumed
to move with same velocity as the fluid.
4. Grid arrangement and validation
The two-dimensional Cartesian coordinate system is
used to simulate flow by considering non-uniform
structured grid. Sufficiently fine grids are in the
viscous sub-layer, near each wall where a very high
gradient of variables prevail (Fig. 2). For each cavity
ratio , a grid is generated as shows Figure 2.
x/H
y/H
0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
AR = 6
x/H
y/H
0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
AR = 8
Property
Nanofluid
Density
sfnf 1
Dynamic
viscosity
5.2
1
f
nf
Thermal
conductivity
)(2
)(22
f
k
s
k
f
k
s
k
f
k
s
k
f
k
s
k
f
k
nf
k
Heat capacity
nf
s
p
C
f
p
C
nf
p
C
1
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x/H
y/H
0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
AR =10
x/H
y/H
0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
AR=12
x/H
y/H
0 5 10 15 20 25
0
0.1
0.2
0.3
0.4
0.5
AR =14
Fig. 2. Typical grid of (x, y) plane for each cavity
aspect ratio.
For each case a grid independency test is carried out
by refining and adjusting the grid in the two
directions. For each grid, the heat transfer
characteristics are examined for a wide range of
aspect ratios and Reynolds numbers. In order to
ensure the accuracy as well as the consistency of
numerical results, several non uniform grids have
been tested for each of all considered cases.
a) Local Nusselt number
b) Local Friction coefficient
Fig. 3. Typical grid test : AR = 10, Re = 6.104.
As shows Fig. 3, a good overall agreement was
obtained between grid 3 (36,700 cells) and Grid 4
(37,500 cells). Grid 3 gives satisfactory results on the
number of Nusselt and the coefficient of friction. The
numerical results of the present study are carried out
by the grid 3 since the geometrical parameters remain
unchanged; and to reduce the calculation time.
5. Validation
This study extends the previous work of Abdellahoun
et al. [13] by considering several viscosity models on
water–Al2O3 nanofluids at small volume particle
fraction. The influence of nanoparticle. For validation
the pressure coefficient along the cavity bottom is
compared to experimental data of numerical
predictions of Arous et al. [17] and Esteve et al. [18].
An overall good agreement is obtained as shown in
Fig. 4. One notes that pressure distribution varies
from a concave-up shape to a concave-down shape.
Therefore, Fig. 4 confirms that a cavity of an aspect
ratio of 10 corresponds to an open cavity according to
the classification of Plentovich (Plentovich et al.
[19]).
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0 2 4 6 8 10
-0,1
0,0
0,1
0,2
0,3 Present study
Estève & al. (2000)
Madi Arous et al. (2012)
CP
x/H
Fig. 4. Validation
Pressure coefficient distribution along the
cavity bottom (Re =105, AR = 10).
6. Results and discussion
This part presents the results obtained by the
numerical simulation applied to a turbulent flow of
nanofluid through a shallow cavity. The water–
Al2O3 nanofluid is considered as the working fluid
in this article. The influence of aspect ratios and the
Reynolds number on the flow fields and heat
transfer were examined. The geometry gives rise to
the appearance of complex vortex structures.
Characterized by a main recirculation zone
delimited by the attachment length which
corresponds to the point where the shear layer joins
the lower solid wall of the cavity. Figures 5 shows
the topology of the flow inside the cavity obtained
with k-ω SST turbulence model for a mixture
containing 4% of the nanoparticles (φ = 4%). These
figures are characterized by the presence Contra-
rotating vortices at the level of the walls. It's about
respectively of the main recirculation zone
(Primary) and secondary recirculations. The
turbulent boundary layer takes off at the upstream
edge of the cavity giving birth to a shear layer and
thereby creating a first recirculation zone having
high average velocity gradients, then separates again
just before the downstream wall, Thus creating a
foot whirl. By analyzing the flow through these
structures, we observe the presence of three vortex
structures in the cavities of large aspect ratios (AR =
14, 12 and 10). The corner vortices grow in size as
aspect ratio increases. In the case of the cavity of
aspect ratio equal to 8, the main recirculation bubble
touches the one that is in front of the downstream
step. We also note the presence of a vortex located
at the upstream corner and the total disappearance of
the vortex located on the downstream step. For
AR=6, the main tourbillion merges with the
secondary vortex thus forming a single recirculation
bubble.
Fig. 5. Streamlines contours : effect of cavity aspect
ratio (Re = 105,φ = 4%).
The pressure coefficient makes it possible to classify
the flows of cavities and to examine the overall
characteristics of the flow. Figure 6 illustrate the
evolution of the static pressure coefficient along the
lower wall of the cavities as a function of the
longitudinal distance. It can be noticed that
pressure distribution varies from a concave-up
shape to a concave down shape. In the case of
cavities with large aspect ratios , the pressures
are more important while the cavities of small
aspect ratios are characterized by lower wall
pressures. According to the classification of
Plentovich [20], the flow of the cavity with an aspect
ratio of 14, is that of a closed cavity. Whereas the
cavity of aspect ratio of 6, 8, 10 and 12, the flow is
that of an open cavity.
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Fig. 6. Pressure coeffecient along the cavity bottom: effect
of aspect ratio (Re = 105, φ = 4%).
Figure 6 highlights the influence of aspect ratio on the
pressure coefficient at the bottom of the cavity. The
pressure distribution, In the case of the cavity having an
aspect ratio of 6, 8 and 10, does not show a change in
concavity. It is seen from these results that the pressure
distribution is sensitive to the changes of the aspect
ratios.
In all cases, there is a negative pressure on the step face
followed initially by slight drop in pressure downstreamof
the step, and then by a rather rapid rise of pressure
indicating the reattachment of separated flow. The base
pressure is essentially the same for different cavities and
the pressure rise by reattachement increases slihtly as the
aspect ratio increases
The effect of the volume fraction, Reynolds number and
cavities aspect ratios; are examined through the
distribution of local and averaged Nusselt number.
For the each cavity wall (Bottom and side walls) in the test
section through which the nanofluid flowed, the local
Nusselt number was obtained using the computed values
of the local heat transfer coefficient and is written as:
00
( ) ( )
nf nf
Bottom side xwall
ww
ywall
kk
TT
Nu x and Nu y
T T y T T x
(14)
Figure 7 depicts the effect of the nanoparticle volume
fraction , Reynolds number Re and the aspect ratios AR,
on heat transfer via the values of local Nusselt number
along the cavity bottom. We notice that the curves have a
similar shape. The increase of aspet ratio leads to an
increase in the number of local Nusselt and a displacement
of its maximum value downstream of the cavity. For all
the cases tested, it reaches its maximum value which
corresponds to the gluing area which is the seat of the best
heat exchange hence the Nusselt number is much higher,
Then, it falls to reach a minimum. We also find that all
curves go through a minimum.
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Fig. 7. Effect of the cavity aspect ratios on local
Nusselt number along the cavity bottom.
(a)
(b)
Fig. 8. Variation of local Nusselt number at the
bottom cavity for different aspect ratios:
(a) Versus x/H, (b) Versus x/L.
Thermal exchanges for the flow of the nanofluid
inside the cavity are characterized by the
average Nusselt number which is determined
from the following relation:
The effect of the aspect ratio on the average Nusselt
number is shown in table 3.
It can be seen through this table that the average
Nusselt number undergoes a noticeable increase for
some aspect ratios. The maximum value of Nu is
reached for the ratio AR = 14. In addition, the
evolution of the average Nusselt number for a given
Reynolds number and a volume fraction varies
between 1% and 4%, shows that the smallest heat
transfer values are obtained by the aspect ratio
cavity AR = 6. So, we can conclude that the flow
structure and the Nusselt number (Average and
local) are strongly influenced by the geometrical
characteristics used Abdellahoum et al [21].
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Table 3. Average Nusselt number for différent
values of the aspect ratios
Re = 105
AR
6H
8H
10H
12H
14H
Nu
Avg
Nanofluids
19,66
20,69
24,34
28,40
28,47
Pure water
11,97
12,60
14,82
17,296
17,34
In the light of this study, we established global
correlation quantifying the average Nusselt number
depending on the different characteristic parameters
namely Reynolds number,nanoparticle volume
fraction and the aspect ratio. This correlation is
valid for a range of Reynolds number ranging from
40000 to 100000 and a nanoparticles volume
fraction of 0≤φ≤2%.
834.0
Re.
23.1
.028.0
502.0
.00033767.0
ARNu
A good agreement between this correlation and the
numerical results is observed at the level of the
Figure 9.
This figure compares the evolution of the average
Nusselt number for numerical predictions and with
the proposed correlation for each cavity. We note a
concordance quite satisfactory for all the cases
studied.
Fig.9. Average Nusselt number versus nanoparticle
volume fraction
𝜙
, Reynolds number and aspect
ratio AR.
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7. Conclusion
Based on the finite volume method, we first
described and validated the numerical method by
referring to the previous results. Then, the focus is
on studying the thermal and dynamic fields of
turbulent flow in forced convection. In particular,
we studied the effects of different parameters on the
flow and heat transfer induced by two-dimensional
forced convection within a rectangular cavity heated
from below. Comparisons were made according to
different criteria by using previous results.
In conclusion the results have clearly show that :
1- The pressure rise by reattachement increases
slihtly as the aspect ratio increases
2- maximum heat transfer is achieved at the point
of attachment followed by a decrease towards
the zone of fully developed flow.
3- The local Nusselt number is low in the
recirculation zone and that it is maximal at the
point of recollement, and this maximum
increases with increasing Reynolds number.
4- he increase of aspect ratio enhances heat
transfer, for each nanoparticle volume fraction
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