Natural Convection Coupled with Radiation Heat Transfer in Slanted Square and Shallow Enclosures Containing an Isotropic Scattering Medium

Abstract

A finite-volume method (FVM) is used to address combined heat transfer, natural convection, and volumetric radiation with an isotropic scattering medium, in a tilted shallow enclosure. Numerical simulations are performed in the in-house fluid flow software GTEA. All the results obtained by the present FVM agree very well with the numerical solutions in the references. The effects of various influencing parameters such as the Planck number (0.0001 ≤ Pl ≤ 10), the scattering albedo (0 ≤ ω ≤ 1), the inclination angle (−60° ≤ α ≤ 90°), and aspect ratio (1 ≤ AR ≤ 5) on flow and heat transfer have been numerically studied. For a constant Pl number, flow is slightly intensified at the midplane as the Ra number increases from 106 to 5 × 106. As the scattering albedo increases, the effect of radiation heat transfer decreases on both slanted and horizontal enclosures. In positive tilt angles, the influence of α on heat transfer is quite remarkable. The highest Nurad appears at α = 30° (ω = 1)and 0° (ω = 0, 0.5), whereas Nurad is maximum at α = − 15° (ω = 1) and −45° (ω = 0, 0.5). At α = −15°, the maximum and minimum values of Nurad are presented for ω = 0, AR = 1 and ω = 1, AR = 5.

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Numerical Heat Transfer, Part A: Applications
An International Journal of Computation and Methodology
ISSN: 1040-7782 (Print) 1521-0634 (Online) Journal homepage: http://www.tandfonline.com/loi/unht20
Natural Convection Coupled with Radiation Heat
Transfer in Slanted Square and Shallow Enclosures
Containing an Isotropic Scattering Medium
Lirong Fu, Wenping Zhang, Pingjian Ming, Meng Zhang & Daming Ni
To cite this article: Lirong Fu, Wenping Zhang, Pingjian Ming, Meng Zhang & Daming Ni (2015)
Natural Convection Coupled with Radiation Heat Transfer in Slanted Square and Shallow
Enclosures Containing an Isotropic Scattering Medium, Numerical Heat Transfer, Part A:
Applications, 68:12, 1369-1393
To link to this article: http://dx.doi.org/10.1080/10407782.2015.1052314
Published online: 12 Jul 2015.
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NATURAL CONVECTION COUPLED WITH RADIATION
HEAT TRANSFER IN SLANTED SQUARE AND
SHALLOW ENCLOSURES CONTAINING AN
ISOTROPIC SCATTERING MEDIUM
Lirong Fu
1
, Wenping Zhang
1
, Pingjian Ming
1
, Meng Zhang
2
,
and Daming Ni
1,3
1
College of Power and Energy Engineering, Harbin Engineering University,
Harbin, People’s Republic of China
2
School of Energy Science and Engineering, Harbin Institute of Technology,
Harbin, People’s Republic of China
3
Aeronautical Science & Technology Research Institute of COMAC, Beijing,
People’s Republic of China
A finite-volume method (FVM) is used to address combined heat transfer, natural
convection, and volumetric radiation with an isotropic scattering medium, in a tilted shallow
enclosure. Numerical simulations are performed in the in-house fluid flow software GTEA.
All the results obtained by the present FVM agree very well with the numerical solutions in
the references. The effects of various influencing parameters such as the Planck number
(0.0001 Pl 10), the scattering albedo (0 x1), the inclination angle (60
a90), and aspect ratio (1 AR 5) on flow and heat transfer have been numerically
studied. For a constant Pl number, flow is slightly intensified at the midplane as the Ra
number increases from 10
6
to 5 10
6
. As the scattering albedo increases, the effect of
radiation heat transfer decreases on both slanted and horizontal enclosures. In positive tilt
angles, the influence of aon heat transfer is quite remarkable. The highest Nu
rad
appears at
a¼30(x¼1)and 0(x¼0, 0.5), whereas Nu
rad
is maximum at a¼15(x¼1) and
45(x¼0, 0.5). At a¼15, the maximum and minimum values of Nu
rad
are presented
for x¼0, AR ¼1 and x¼1, AR ¼5.
1. INTRODUCTION
Combined natural convection and radiation heat transfer play an important
role in a number of engineering applications, such as electronic cooling devices, solar
energy collectors, spacecrafts, aero-engine combustors, thermal insulation, etc. The
radiation problem is commonly treated in two ways: as surface radiation with non-
participating media and the other by including a radiatively participating medium
between the surfaces. The first case can be solved by algebraic equations taking into
Received 12 February 2015; accepted 17 April 2015.
Address correspondence to Pingjian Ming, No. 145, Nantong Street, Nangang District, Harbin
City, Heilongjiang Province 15001, China. E-mail: pingjianming@hrbeu.edu.cn.
Color versions of one or more of the figures in the article can be found online at
www.tandfonline.com/unht.
Numerical Heat Transfer, Part A, 68: 1369–1393, 2015
Copyright #Taylor & Francis Group, LLC
ISSN: 1040-7782 print=1521-0634 online
DOI: 10.1080/10407782.2015.1052314
1369
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account view factors. For the later one, a solution of radiative transfer equation
(RTE) is required [1].
The problem of coupled heat transfer involving surface radiation and natural
convection has been widely studied in numerous works. Bouali et al. [2] numerically
studied the effects of surface radiation and inclination angle on heat transfer and flow
structures in an inclined rectangular enclosure. It was shown that the isotherm struc-
tures and the streamline values were considerably affected by the inclination angle in
the cavity. Alvarado et al. [3] conducted a study on the effect of inclination angle on
the thermal performance of tilted slender cavities, which heated from a side by con-
sidering the interaction between the decoupled and coupled surface radiation and
natural convection for Ra numbers of 10
4
–10
6
, aspect ratios of 8, 12, and 16, and incli-
nation angles varying from 15to 35. Diaz and Winston [4] investigated the effect of
surface radiation on natural convection for parabolic geometry. Ayachi et al. [5]
NOMENCLATURE
Aarea of the cell-surface
AR cavity’s aspect ratio (¼W=L)
c
p
specific heat
Dmn
idirectional weights
fradiation flux on the cell-surface
gacceleration vector of gravity
Iradiation intensity
I
b
blackbody radiation intensity (¼r
T
4
=p)
i, j, k unit vector in x, y, z
kthermal conductivity
Llength of the cavity
nnormal outward unit vector
N
h
total number of polar angles
N
/
total number of azimuthal angles
Nu
conv
average convective Nusselt number
Nu
rad
average radiative Nusselt number
Nu
total
average total Nusselt number
Ppressure
Pl Planck number (¼(k=L)=(4rT3
0))
Pr Prandtl number (¼mc
p
=k)
q
r
radiative heat flux
Ra Rayleigh number (¼gbq
2
(T
h
T
c
)L
3
c
p
=mk)
rposition vector (¼ixþjyþkz)
sdirection vector (¼isinhcos/þjsinh
sin/þkcosh)
S
r
the radiative source term
T
0
reference temperature
T
h
,T
c
hot and cold wall temperature
Ttemperature
Uvelocity vector
u, v velocity components
Vvolume of the control volume
Wwidth of the cavity
ainclination angle
bcoefficient of thermal expansion
b
0
extinction coefficient (¼jþr
s
)
ctemperature ratio (¼T
c
=T
h
)
eemissivity of the radiative wall
hpolar angle
jabsorption coefficient
mdynamic viscosity
qdensity
q
0
reference density
rStefan–Boltzmann constant (¼5.67 
10
8
)
r
s
scattering coefficient
soptical thickness (¼(jþr
s
)L)
/azimuthal angle
xscattering albedo (¼r
s
=b
0
)
X
mn
discrete control angle
H
0
reference temperature ratio (¼T
0
=
(T
h
T
c
))
Subscripts
br, tr radiative Nusselt number on the
bottom and top walls
in, out radiation flux direction on the cell
face
i, j index of the cell-surface
Iupwinding nodal point of P
Pcenter of present cell
wwall
Superscripts
m, n index of radiation directions on polar
angle hand azimuthal angle /
m,mþstarting and ending values of m
n,nþstarting and ending values of n
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numerically analyzed the coupling between natural convection and surface radiation
in a square cavity with its vertical walls subjected to different heating models. Velu-
samy et al. [6] performed a numerical study of the interaction effects of surface radi-
ation with turbulent natural convection of a transparent medium in rectangular
enclosures, covering a wide range of Ra numbers from 10
9
to 10
12
and aspect ratios
of 1–200. Mezrhab et al. [7] studied the problem of surface radiation and natural con-
vection in heated greenhouses. In their work, it was found that the overall heat trans-
fer within the greenhouse increases with the rise in the value of Rayleigh number.
Recently, Martyushev et al. [8–9] conducted a numerical analysis of combined natural
convection and surface thermal radiation with a heat source. The primary focus was
on the influences of five types of factors such as Ra number, an emissivity of internal
surfaces, a thermal conductivity ratio, the ratio of solid wall thickness, and the dimen-
sionless time on the velocity and temperature fields. Other related works on such
combined heat transfer have been carried out by S. K. Jena and S. K. Mahapatra
[10], A. Muftuoglu and E. Bilgen [11], and C. Balaji and H. Herwig [12].
The interaction between volumetric radiation and natural convection has been
investigated by several researchers. Colomer et al. [13] analyzed the natural convec-
tion with radiation for transparent and participating media in a three-dimensional
differentially heated cavity. The effects of Rayleigh and Planck numbers, as well
as the optical thickness, are studied. Yu
¨cel et al. [14] used the discrete ordinates
method to study combined natural convection and radiation from a scattering
medium for a vertical (h¼90) and an inclined (h¼60) differential heated cavity
of equal sides for Ra ¼510
6
. It was shown that the influence of the inclination
angle on the total wall heat transfer is more pronounced for a non-radiating fluid
than for a radiating fluid. Chiu et al. [15] investigated the radiation effect on the
characteristics of mixed convection fluid flow and heat transfer in inclined ducts.
Moufekkir et al. [16] studied the numerical solution for natural convection and volu-
metric radiation in an isotropic scattering medium within a heated square cavity
using a Hybrid Thermal Lattice Boltzmann Method (HTLBM), and found that
the average Nusselt number is considerably augmented by the volumetric radiation
effects and the maximum of heat transfer is carried out for low optical thickness.
Kumar et al. [1] investigated numerically combined radiation and natural convection
in slanted cavities of angles 45and 60using the finite-volume method (FVM). The
radiative optical length of the medium and the wall emissivity show a strong effect
on heat transfer, but the scattering albedo has hardly any influence on the overall
heat transfer. Moufekkir et al. [17] studied coupled double-diffusive convection
and volumetric radiation in a tilted and differentially heated square enclosure filled
with a gray fluid participating in absorption, emission, and non-scattering. The
results show that the isotherms and the iso-concentrations inclined in the cavity;
although the flow is more stabilized in the presence of cooperating flows, multicellu-
lar flow is favored in the opposite flows. Moufekkir et al. [18] studied natural con-
vection in an asymmetrically heated square cavity including non-scattering
medium, using HTLBM. Moreover, the effects of parameters such as the Rayleigh
number, the optical thickness, and the inclination angle were numerically analyzed.
Based on the above-mentioned literature reviews and the authors’ knowledge,
the effects of tilted angle and aspect ratio on the interaction between natural
convection and thermal radiation considering an isotropic scattering medium have
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rarely been studied. Hence, the present work is focused on such an investigation of
combined radiation and natural convection in slanted shallow enclosures. The influ-
ences of the Planck number, the scattering albedo, the inclination angle, and the
aspect ratio on isotherms, streamlines, nondimensional temperature, and velocity
in the midplane and the average Nusselt number are also discussed in detail. This
paper is briefly summarized as follows. The next section presents the general assump-
tions and the governing equations. The FVM method for the RTE and boundary
conditions are reported in the third section. Grid independency test and code vali-
dation are presented in Section 4. Then the numerical results and discussion concern-
ing natural convection and radiation interaction with=without scattering medium,
and parametrical study cases are given in Section 5. Finally, conclusions of the
numerical study are provided in Section 6.
2. MATHEMATICAL MODEL
The physical model of the considered problem is displayed in Figure 1with
boundary conditions. It concerns a two-dimensional enclosure filled with a homo-
geneous, gray, absorbing, emitting, and isotropic scattering medium. The length
and width of the cavity are Land W, respectively. The two horizontal walls are ther-
mally insulated, whereas the left and right vertical walls are maintained isothermal
with T
c
and T
h
. The working fluid is assumed as Newtonian and incompressible.
2.1. Fluid Flow Equations
With the above-mentioned assumptions, the governing conservation equations
of mass, momentum, and energy are obtained as follows [19]:
rU¼0ð1Þ
rðqUUÞ¼rpþrmrUþðrUÞT
hi
þqbð2Þ
Figure 1. Physical model and boundary conditions.
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r qUcpT

¼r krTðÞþSrð3Þ
where the Boussinesq approximation
qb¼q0bðTT0Þgð4Þ
was used in the buoyancy terms to allow for the change in density with temperature.
Here qis the density, Uis the velocity vector, pis the pressure, mis the dynamic
viscosity, c
p
is the specific heat, kis the thermal conductivity, Tis the temperature, S
r
is the radiative source term and its expression is presented in section 2.3, q
0
is the
reference density at reference temperature T
0
,bis the thermal expansion coefficient,
and gis the acceleration vector of gravity.
For Eqs. (1)–(3), the initial conditions are u¼v¼0, p¼0, and T¼T
0
. For the
velocity field the no-slip boundary condition is assumed.
2.2. Radiative Transfer Equation
Considering the absorbing, emitting, and scattering gray media, the RTE in the
Cartesian coordinate system is written as
[20]
dIðr;sÞ
ds ¼b0Iðr;sÞþjIbðrÞþrs
4pZX0¼4p
Iðr;s0ÞUðs0;sÞdX0ð5Þ
where I(r,s) is the radiation intensity, which is a function of the spatial position rand
direction s.I
b
is the blackbody radiation intensity. U(s, s0) is the scattering phase
function for radiation from the incoming direction s0to the scattered direction s,
where s0is the unit vector of the solid angle direction X0.U(s, s0) is unit when the
medium is isotropic scattering. b
0
¼jþr
s
is the extinction coefficient. jand r
s
are the absorption and scattering coefficients, respectively. x¼r
s
=jis the scattering
albedo.
The boundary condition on the diffusely emitting and reflecting wall is given as
Iðrw;sÞ¼ewIbðrwÞþ1ew
pZs0nw<0
Iðrw;s0ÞdX0ð6Þ
where e
w
is the wall emissivity and n
w
is the normal outward unit vector.
2.3. Coupling Natural Convection and Radiation
The radiation source term S
r
appearing in Eq. (3) is given by [19]
Sr¼rqr¼j4rT4Z
4p
IsðÞdX
0
@1
A¼j4rT4X
N/
n¼1X
Nh
m¼1
ImnXmn
!
ð7Þ
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The radiative heat flux at the wall can be calculated from the following [20]:
qr¼ZX¼4p
Irw;sðÞsnw
ðÞdX¼X
N/
n¼1X
Nh
m¼1
Imn
wDmn
i;in þDmn
i;out

ð8Þ
For the adiabatic boundary condition, the wall temperature of the problem is
unknown. This is part of the solution provided the heat balances between the
radiation and convection at the wall:
krTnwqr¼0ð9Þ
2.4. Heat Transfer
The overall heat transfer involves the contribution of the convective and radi-
ative Nusselt numbers. The average Nusselt number is computed as follows [21]:
Nuc¼k
qcond ZL
0
qT
qxdy ð10Þ
Nur¼1
qcond ZL
0
qrdy ð11Þ
qcond ¼kðThTcÞ=Lð12Þ
Nutotal ¼Nuconv þNurad ð13Þ
3. NUMERICAL METHODS
The mass, Navier–Stokes, and temperature equations and the RTE are itera-
tively solved using the FVM by compiling a FORTRAN computation code. The
pressure–velocity coupling in the mass and momentum equations is solved by the
SIMPLE algorithm [22]. The discretization processes for Eqs. (1)–(3) have been pre-
sented in [23]; therefore, we provide only the discretization details for Eq. (5) in the
section.
3.1. Finite-Volume Method
To obtain the discretization equation, Eq. (5) is integrated over a control vol-
ume, V, and a control angle, X
mn
. The following finite-volume formulation can be
written as
ZXmn ZV
dI
dsdVd X¼ZXmn ZV
b0IþjIbþrs
4pZXi¼4p
IUðs0;sÞdX0

dVdXð14Þ
The volume integral is transferred to the surface integral based on the Gauss
theorem; the left-hand side of Eq. (14) can be written in the following form:
ZVZXmn
dI
dsdVd X¼ZXmn IA
IðsnÞdAdX¼X
i¼1;NZXmn
Imnðsmn niÞAidXð15Þ
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Here we suppose that the radiation intensity is uniform in the cell faces, so the
following formulation holds:
X
i¼1;NZXmn
Imnðsmn niÞAidX¼X
i¼1;N
Imn
iAiDmn
ið16Þ
By assuming that the magnitude of the intensity is constant but its direction
varies within the control volume and control angle given, the right-hand side of
Eq. (14) can be derived:
ZXmn ZV
b0IþjIbþ
rs
4pRXi¼4pIðr;s0ÞUðs0sÞdX0

dVdX¼b0IþjIbþrs
4pX
l
IlUðXmn;XlÞXl
!
VXmn
ð17Þ
Substitution of Eqs. (16)and(17) into Eq. (14) yields
X
i¼1;N
Imn
iAiDmn
i¼b0Imn
PþjIb;Pþrs
4pX
l
Il
PUðXmn;XlÞXl
!
VXmn ð18Þ
where the expressions for Dmn
i,s
mn
,n
i
, and X
mn
have been given by [19].
To solve Eq. (5) numerically, it is obvious that not only spatial discretization
but also the angular integration over the solid angle needs to be considered. There-
fore, both spatial domain and angular domain must be discretized. The spatial
domain is discretized into a serial of triangles or quadrilaterals with nonoverlapping
small cells, and all the solved variables and properties are stored at the cell center.
The solid angle, 4psteradians, is divided into N
h
N
/
direction, where the polar
angle hchanges from 0 to pand /is an azimuthal angle ranging from 0 to 2p.
The schematic of hybrid grids and solid angle are shown in Figures 1and 2of
ref [19], respectively. On the other hand, an upwind scheme in which a downstream
facial intensity is set equal to the upstream nodal value is adopted. For this scheme, a
typical relation between facial and nodal intensities is determined as follows:
Imn
iDmn
i¼Imn
PDmn
i;out þImn
IDmn
i;in ð19Þ
where I
P
and I
I
represent nodal intensities corresponding to facial intensities I
i
.
The surface weight factor fis defined for a given direction. In fact, it is similar
to the flux in CFD; the flux into the cell is negative and the flux out of the cell is
positive:
fi;in ¼AiDmn
i;in
fi;out ¼AiDmn
i;out
where the expressions for Dmn
i;out and Dmn
i;in can be found in [20].
Moreover, we can find that the sum of the flux is zero for geometry
conservation
X
i
fi;out þX
i
fi;in ¼0
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Based on the result reported in [19], the left-hand side of Eq. (18) can be written
as
X
i¼1;N
Imn
iAiDmn
i¼X
i
fi;in Imn
IImn
P
 ð20Þ
By substituting Eq. (20) into Eq. (18), we can obtain the following algebraic
equation:
amn
pImn
p¼X
I¼1;N
amn
IImn
Iþbmn
pð21Þ
Here, coefficients and source terms are
amn
p¼ðjþrsÞVXmn þX
i¼1;N
DAiDmn
i;out ð22Þ
Figure 2. Computation grids for square: (a)4040; (b)5050; (c)6060; and (d)7070.
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amn
I¼DAiDmn
i;in ð23Þ
bmn
p¼ðjþrsÞSmn
P

VXmn ð24Þ
Smn
P¼ð1x0ÞIb;Pþx0
4pX
l
Il
PUðXmn;XlÞXlð25Þ
The coefficient matrix is an asymmetric positive definite one. We adopt a
conjugate gradient square method with under-relaxation.
3.2. Boundary Conditions
The boundary condition can be discretized as follows:
Imn
w¼ewIbw þ1ew
pX
sjnw<0
Imjnj
wDmjnj
w;in
;sjnw>0ð26Þ
where
Dmn
w;in ¼ZXmn ðsnwÞdX;snw<0ð27Þ
4. GRID SIZE SENSITIVITY TEST AND CODE VALIDATION
To study the effect of grid size on the numerical results, a grid size sensitivity
test is carried out for Ra ¼510
6
,Pr ¼0.71, Pl ¼0.02, e
1,2,3,4
¼1, s¼1, x¼0,
AR ¼1, a¼0. The reference temperature ratio is taken to be H
0
¼1.5, and the ratio
cis chosen to be equal to 0.5. As can be seen in Figure 2, four nonuniform grid
patterns from 40 40 to 70 70 are considered. Thin grids are used in the vicinity
of the boundary layers. The corresponding values of radiative and total Nusselt
numbers are listed in Table 1, for all the grids. The numerical solutions become
independent of grid size at 60 60, because the maximum absolute difference in
the Nusselt numbers compared with the 50 50 grid is 0.197%. Hence, the grid size
of 60 60 is selected for all cases herein considered.
Furthermore, the algorithm for natural convection and volumetric radiation in
a square cavity filled with an isotropic scattering medium is validated for different
Rayleigh numbers (10
4
Ra 10
5
) at two values of scattering albedo (x¼0.5
and 1). The conditions for the results shown include the wall emissivity of e
1, 2
¼1,
Table 1. Mesh independence study at Ra ¼510
6
,Pl ¼0.02, e
1,2,3,4
¼1, s¼1, x¼0, AR ¼1, and a¼0
Mesh size 40 40 50 50 60 60 70 70
Nu
rad
31.248 30.963 30.902 30.905
%Error 0.912 0.197 0.009
Nu
total
39.44 38.8 38.751 38.704
%Error 1.623 0.126 0.121
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Table 2. Comparison of the present results with the results reported in ref [16]. for e
1,2
¼1, e
3,4
¼0, and
s¼1 on the hot wall
Nu
rad
Nu
total
Ref [16]. Present %Error Ref [16]. Present %Error
Ra ¼10
4
x¼0.5 25.145 24.956 0.752 27.574 26.826 2.713
x¼1 25.191 25.145 0.183 27.652 27.385 0.966
Ra ¼10
5
x¼0.5 26.042 26.476 1.667 28.849 29.132 0.981
x¼1 25.191 25.162 0.115 27.864 29.691 6.557
Figure 3. Isotherms and streamlines for Ra ¼510
6
,Pl ¼0.02, e
1,2,3,4
¼1, AR ¼1, a¼0,s¼1, and
x¼0, 0.5 (a), (c), (e), (g): Ref. [14], (b), (d), (f), (h): present work): (a)–(d)x¼0; (e)–(h)x¼0.5.
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e
3, 4
¼0, and the extinction coefficient of s¼1. As shown in Table 2, the maximum
absolute differences in total average Nusselt numbers and average radiative Nusselt
number on the hot wall compared to those in Moufekkir [16] are 6.557% and
1.667%, respectively.
The comparisons of isotherms and streamlines reported by Yu
¨cel et al. [8] are
presented in Figures 3–4. As in ref [8]. we consider two different scattering albedos
(x¼0 and 0.5) and two values of a¼0and 30. There also, we obtained a good
agreement.
The above-mentioned studies confirm that the method developed in this article
can be successfully applied to analyze the heat transfer by natural convection and
volumetric radiation by taking into account scattering in an enclosure.
5. RESULTS AND DISCUSSION
In this investigation, parameters such as the Prandtl number, temperature
ratio, reference temperature ratio, and optical thickness are, respectively, fixed to
Pr ¼0.71, c¼0.5, H
0
¼1.5, and s¼1. The relevant parameters, such as the Planck
number, scattering albedo, tilt angle, and cavity aspect ratio, are studied.
Figure 4. Isotherms and streamlines for Ra ¼510
6
,Pl ¼0.02, e
1,2,3,4
¼1, AR ¼1, a¼30,s¼1, and
x¼0: (a), (c) present work; (b), (d) Ref [14].
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5.1. Effect of the Planck Number
In this section, the objective is investigating the effect of the Planck number
on velocity and temperature for two values of Ra number, where the following
parameters are fixed, namely AR ¼1, a¼0, and x¼0. On the other hand, the
vertical walls are assumed to be black (e
1,2
¼1), whereas the horizontal walls are
adiabatic and purely reflecting (e
3,4
¼0).
The impact of various Pl numbers on the isotherm is depicted in Figure 5for
two Ra (left-hand side: Ra ¼10
6
, right-hand side: 5 10
6
). As the Planck number
increases from 0.02 to 1 (see Figures 5a, c), the thermal stratification is more homo-
geneous at the core of the cavity because of radiation effect. However, the structure
of the isotherm is approximately the same for Pl ¼1 and 10. A similar finding was
reported in a recent study by Moufekkir et al. [16]. On the other hand, the Rayleigh
number varies from 10
6
to 5 10
6
. The horizontal temperature gradients reach a
narrow layer in the vicinity of the cold and hot walls, which implies the influence
of radiation decreases toward the walls.
In Figure 6, the nondimensional temperature along the midplane at y=L¼0.5
is displayed. For a low value of Pl namely 0.001, the temperature distribution
is almost linear in the middle of the cavity. However, it can be observed that the
temperature gradient is steeper adjacent to the active walls for Pl ¼0.02, 0.1, 1,
and 10. In the meanwhile, the temperature profiles seem to be horizontal lines at
the center. These results are also in good agreement with those in the literature
[16]. For Pl ¼1, 10, the results are completely identical. A comparison of
Figures 6a, b shows that the Rayleigh number slightly affects the temperature profile
at y=L¼0.5.
The effect of Planck number on the horizontal and vertical velocities at the
vertical and horizontal midplanes is given in Figures 7and 8, respectively. The
velocity gradually increases with the reduction of Pl number. Furthermore, we
find that the velocity increases when the Ra number increases. In fact, the driven
buoyancy force becomes strong due to the increasing Ra number, and thus larger
velocities are obtained.
5.2. Effect of Scattering Albedo
Now, our focus is on the discussion of the effect of different scattering albedos
on the flow and heat transfer. For this study, two configurations are considered for a
fixed Ra number, Pl number and optical thickness (Ra ¼10
6
,Pl ¼0.02, s¼1): the
first one relates to a cavity whose vertical walls are black (e
1,2
¼1, e
3,4
¼0), and
the second is an enclosure with black walls (e
1,2,3,4
¼1).
Figure 9illustrates the effects of the scattering albedo xon the streamlines
and isotherms, where xranges from 0 to 1. It is clear that both the emissivity of hori-
zontal walls and the scattering albedo have an impact on the structures of isotherms
and streamlines. For e
1,2,3,4
¼1, the isotherm in the vicinity of the insulated wall
slants against e
1,2
¼1, e
3,4
¼0, which is due to the radiative energy exchanged
between the radiative surfaces. For x¼0, 0.2, regardless of the emissivity of horizon-
tal walls, the structure of the streamline and the isotherm is likely similar. However,
the scattering albedo increases from 0.2 to 1, and the thermal stratification is more
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Figure 5. Planck number effect on isotherms at e
1,2
¼1, e
3,4
¼0, x¼0, AR ¼1, and a¼0(left-hand side
at Ra ¼10
6
; right-hand side at Ra ¼510
6
): (a)Pl ¼0.02; (b)Pl ¼0.1; (c)Pl ¼1; and (d)Pl ¼10.
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Figure 8. Vertical velocity component at y=L¼0.5 for Pl ¼0.001, 0.02, 0.1, 1, and 10, e
1,2
¼1, e
3,4
¼0,
x¼0, AR ¼1, a¼0:(a)Ra ¼10
6
;(b)Ra ¼510
6
.
Figure 7. Horizontal velocity component at x=W¼0.5 for Pl ¼0.001, 0.02, 0.1, 1, 10, e
1,2
¼1, e
3,4
¼0,
x¼0, AR ¼1, a¼0:(a)Ra ¼10
6
;(b)Ra ¼510
6
.
Figure 6. Temperature profiles at y=L¼0.5 midplane for Pl ¼0.001, 0.02, 0.1, 1, 10, e
1,2
¼1, e
3,4
¼0,
x¼0, AR ¼1, a¼0:(a)Ra ¼10
6
;(b)Ra ¼510
6
.
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Figure 9. Albedo effect on isotherms at Ra ¼10
6
,Pl ¼0.02, AR ¼1, and a¼0for x¼0–1 and two values
of the horizontal emissivity (e
3,4
¼0, 1).
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uniform at the core region. In the meantime, the center of the vortex moves toward
the lower part of the cold wall. This is attributed to the fact that as the scattering
albedo increases, the effect of radiative heating decreases. On the other hand, the
influence on flow and temperature field due to the emissivity of horizontal walls
becomes strong by setting xin the range 0.5–1, and the temperature distribution
is more uniform for e
1,2
¼1, e
3,4
¼0. For a pure scattering medium (x¼1),
when e
1,2
¼1, e
3,4
¼0, we observe that the unicellular structure of flow exhibited
by the streamlines disappears and the temperature and flow fields become
centro-symmetric. Moreover, the isotherm and the streamline are somewhat similar
to those in the pure natural convection case as shown in Figure 3of ref [14]. There-
fore, the effect of radiation decreases with the scattering albedo increasing.
Figure 10 shows the dimensionless temperature profiles in the horizontal
midplane, i.e., y=L¼0.5, for five values of scattering albedo and for e
1,2
¼1, e
3,4
¼0,
and e
1,2,3,4
¼1. It can be observed that the difference between these profiles for
x¼0, 0.2 is very little. Furthermore, for a higher value of x, the core of the cavity is
more cold and a strong temperature gradient is noted near the hot boundary. A com-
parison of Figures 10a, b shows that the emissivity of the horizontal walls has a vanish-
ingly small effect on the temperature profile for x¼0, 0.2. However, when xincreases
from 0.5 to 1, the temperature of the core part is enhanced compared to e
1,2
¼1, e
3,4
¼0.
The effects of the scattering albedo on the U- and V-velocity profiles at the
midplanes x=W¼0.5 and y=L¼0.5, respectively, are displayed in Figures 11 and
12. Results show that for e
1,2,3,4
¼1, both the horizontal and vertical velocities are
slightly enhanced with respect to the one corresponding to e
1,2
¼1, e
3,4
¼0, whereas
the effect of scattering albedo xdecreases with an increase in the value of x.
5.3. Effect of Tilt Angle
In the following section, the results are analyzed with particular interest on the
influence of inclination angle aon the flow fields and heat transfer. We set the Ra
number of 5 10
6
, wall emissivity of e
1,2,3,4
¼1, for four cases of s¼0 and s¼1,
Figure 10. Temperature profiles at midplane y=L¼0.5 with Pl ¼0.02, Ra ¼10
6
,AR ¼1, a¼0for x¼0,
0.2, 0.5, 0.8, and 1: (a)e
1,2
¼1, e
3,4
¼0; (b)e
1,2,3,4
¼1.
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x¼0, 0.5, 1. The ratio AR is fixed at 1. The inclination angle is varied from 60to
90by an angle step equal to 30. The results are summarized in Figures 13–16.
The variations of the isotherm for inclination angle and scattering albedo are
depicted in Figures 13a, b for negative angles and in Figures 13d, f for positive
angles. At s¼0, irrespective of how the angle achanges, the isotherms in the central
part of the cavity are nearly perpendicular to the gravitation. For negative angles,
the temperature gradient adjacent to the isothermal walls is relatively larger than
in the cases of the positive angles. In Figures 13a, b, the structure of the isotherms
is almost similar for x¼0 and 0.5; moreover, the scattering alters significantly the
thermal distributions in the range of xfrom 0.5 to 1. However, for positive tilt
angles, thermal stratification exists and the isotherms are horizontal in the core area.
In fact, this phenomenon means less heat transfer. It is seen that the isotherm struc-
tures in the enclosure are slightly affected by the scattering albedo seen in
Figures 13d, f.
Figure 12. Vertical velocity component at y=L¼0.5 with Pl ¼0.02, Ra ¼10
6
,AR ¼1, a¼0for x¼0, 0.2,
0.5, 0.8, and 1: (a)e
1,2
¼1, e
3,4
¼0; (b)e
1,2,3,4
¼1.
Figure 11. Horizontal velocity component at x=W¼0.5 with Pl ¼0.02, Ra ¼10
6
,AR ¼1, and a¼0for
x¼0, 0.2, 0.5, 0.8, and 1: (a)e
1,2
¼1, e
3,4
¼0; (b)e
1,2,3,4
¼1.
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Figure 13. Angle inclination effect on isotherms for s¼0 and s¼1, x¼0, 0.5, and 1: (1) s¼0; (2) s¼1,
x¼0; (3) s¼1, x¼0.5; and (4) s¼1, x¼1.
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Figure 14. Temperature profiles at y=L¼0.5: (a)s¼0; (b)s¼1, x¼0; (c)s¼1, x¼0.5; and (d)s¼1,
x¼1.
Figure 15. Convective, radiative, and total Nusselt number on the hot wall for the scattering albedos of
x¼0, 0.5, and 1.
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For a transparent fluid (s¼0), Figure 14ashows that the dimensionless tem-
perature profile is symmetric for all tilt angles. The high gradient near the thermally
active wall is produced at a¼0.Ata¼60, the temperature profile is somewhat lin-
ear except near the boundary layer. As for a¼90, the temperature profile is com-
pletely linear, signifying that conduction is the primary mode of heat transfer. For
the three cases (x¼0, 0.5, and 1), when inclination angles aare greater than 0,
the trend of temperature profiles is quite similar. Meanwhile, the temperature gradi-
ent is nearly the same next to the cold boundary and the temperature gradient
increases with decreasing anear the hot wall. Furthermore, the temperature gradient
slightly increases in the middle of the cavity for a pure scattering medium. From
Figure 14a(s¼0) and b (s¼1, x¼0), it is clear that the core region of the enclosure
is more heated for the latter case (s¼1, x¼0). The effects of the parameter are almost
similar to those in Figures 14a, b of ref [18]. for Ra ¼510
5
. Comparing the
non-scattering case (x¼0) and an absorbing-scattering fluid (x¼0.5), it seems that
the temperature profiles do not have any noticeable differences between the two cases.
On the other hand, the temperature profiles obtained at a¼0,30,and60are
almost horizontal at the core region and there are little discrepancies among these
results. By comparison cases with x¼0.5 and 1 (Figures 14c, d), however, the effect
of scattering albedo on the temperature profiles is dramatic, except for a¼90, where
the effect is almost negligible. At a¼0, its effect has been presented for the case of
Ra ¼10
6
(Figure 10b). Notice, at a¼30, the temperature gradients in the center
part of the cavity are negative, presented in Figures 14a, d. As for a¼60, it is evi-
dent that the temperature is largely lower in the center part of the cavity for the latter
case. Furthermore, for large angles of 30and 60, the temperature gradient slightly
increases in the middle of the cavity for a pure scattering medium.
Variation of the mean convective, Nu
conv
, the radiative, Nu
rad
, and the total
Nusselt numbers on the hot wall with respect to the tilt angle and the scattering
albedo is discussed in Figure 15. It is found that the inclination angle and scattering
Figure 16. Variation of the mean radiative Nusselt number on the adiabatic walls with tilt angle.
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albedo considerably change the heat transfer in the cavity. We can see that Nu
total
gradually increases when the scattering albedo decreases corresponding to the same
value of a. However, for Nu
rad
, it increases as the scattering albedo reduces for 60
a60and then it slightly increases with increasing scattering albedo at abetween
60and 90. For x¼0 and 0.5, Nu
conv
has little change with avarying from 60to
45. When xincreases to 1, the convective Nusselt number increases for the incli-
nation angle ranging from 60to 30. However, for 45a90, the lower value
of convective Nusselt number is observed on increasing the scattering albedo from 0
to 1. For three values of x,Nu
conv
and Nu
total
slightly change for a negative incli-
nation angle, and then there is a significant reduction when the tilt angle exceeds
0. In cases of x¼0 and 0.5, the effect of tilt angle on Nu
rad
is nearly similar to
Nu
conv
and Nu
total
for a negative tilt angle. However, for x¼1, Nu
rad
is slightly aug-
mented at 60a30, and then decreases for the tilt angle range from 30to 90.
Variation of the mean radiative Nusselt number on the adiabatic walls with tilt
angle range from 60to 75and scattering albedo of x¼0, 0.5, and 1 is plotted in
Figure 16. Regardless of scattering albedo, the maximum of the mean radiative Nus-
selt number on the bottom wall occurs at a¼30. For x¼0 and 0.5, the maximum
of the mean radiative Nusselt number on the top wall occurs at a¼30. However,
for x¼1, the highest value of heat transfer on the wall is observed for a¼45.
The mean radiative Nusselt number largely increases with increasing scattering
albedo on the bottom wall. However, for the top wall, when the tilt angles are less
than 0,Nu
rad
is almost equal to the cases of x¼0 and 0.5 but it is higher than
the case of x¼1. The effect of scattering albedo on the mean radiative Nusselt num-
ber along the top wall is similar to that on the bottom wall, when the tilt angle varies
from 15to 75. For x¼0, Nu
rad
along the top wall and the bottom wall is largely
enhanced by an increase of tilt angles from 60to 15and 15to 30, and
decreases as the tilt angles vary from 15to 15and 30to 75. For the case of
x¼0.5, Nu
rad
on the two adiabatic walls increases with the tilt angles ranging from
Figure 17. Variation of mean radiative Nusselt number on the hot wall for aspect ratios AR ¼1, 2, 3, 4,
and 5, and scattering albedo at a¼15.
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60to 30, and then decreases for 30a75. On the other hand, for x¼1, the
effect of tilt angles on Nu
rad
along the bottom wall is similar to that for x¼0.5.
Meanwhile, the variation of Nu
rad
along the top wall is quite similar to the value
along the bottom wall, except that it is slightly high at a¼30compared to that
at a¼45.
5.4. Effect of Aspect Ratio
The mean radiative Nusselt number for enclosures tilted by 15with five
aspect ratios, AR ¼1, 2, 3, 4, and 5, and five scattering albedos, x¼0, 0.2, 0.5,
0.8, and 1, are depicted in Figure 17. As it can be observed, the average Nu number
Figure 18. Isotherms for x¼0, 0.2, 0.5, 0.8, and 1 at AR ¼5 tilted 15.
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gradually steadily decreases with increasing scattering albedo for a fixed aspect ratio.
Meanwhile, for all values of scattering albedo, the radiative Nu number is consider-
ably reduced by an increase of aspect ratio from 1 to 5. As an expected result, at
a¼15, the highest and lowest values of heat transfer are obtained for x¼0,
AR ¼1andx¼1, AR ¼5, respectively. The influence of scattering albedo on the iso-
therm is presented in Figure 18, for the case of AR ¼5 tilted by 15. The scattering
albedo has a slight effect on the isotherm for xincreasing from 0 to 0.8; however,
its effect is quite obvious when xincreases to 1. The value of temperature in the
middle region is lower in the case of pure scattering medium because more energy
is scattered.
6. CONCLUSIONS
This paper presented a numerical investigation of the interaction between the
heat transfer by natural convection and volumetric radiation with a gray and iso-
tropic scattering medium. The Planck number, scattering albedo of the medium, tilt
angle, and the aspect ratio of the enclosure have been systematically varied as main
parameters. The present investigation concludes as follows:
.The temperature in the inner region and velocity decrease on increasing the Pl
number from 0.02 to 1, and they remain nearly the same for a Pl value up to
10. For a fixed Pl number, the horizontal temperature gradient exists in a thinner
boundary layer and the velocities are intensified as the Ra number increases from
10
6
to 5 10
6
owing to stronger driven buoyancy force.
.Regardless of the scattering albedo, when e
3,4
vary from 0 to 1, the isotherm lines
adjacent to the adiabatic walls become slanted and the flow is slightly accelerated
at the midplane. The emissivities of horizontal walls and scattering albedo signifi-
cantly affect the isotherm and streamline except for the cases of x¼0 and 0.2. As
the scattering albedo increases, the effect of radiation heat transfer decreases in a
horizontal square cavity.
.For negative avalues, there is a higher temperature gradient next to the iso-
thermal walls and the scattering albedo considerably changes the structure of
the isotherm when xvaries from 0.5 to 1 against the positive angle. However,
the differences of temperature profile between x¼0 and x¼0.5 are nearly negli-
gible. Obviously, at a¼30, the negative temperature gradients in the center
part of the cavity appear for the cases of s¼0 and x¼1.
.Concerning the average total Nu number on the hot wall, it gradually increases as
xdecreases with respect to the same value of a. The influence of scattering albedo
on Nu
rad
is quite similar to that on Nu
total
for 60a60; however, the trend
of influence is opposite at abetween 60and 90. The effect of aon the three Nu
numbers is very noticeable only for its positive values.
.For the three values of x, irrespective of how avaries, radiation heat transfer is
always the principal mode in slant square cavity including an isotropic scattering
and Nu
br
is much higher compared to Nu
tr
.
.For the enclosure tilted by 15, the augmentation in aspect ratio and scattering
albedo decreases the value of Nu
rad
along the hot wall. As a result, the mean radi-
ative Nu number approaches the maximum for the case of x¼0, AR ¼1.
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FUNDING
This work was supported by National Natural Science Foundation of China
(51206031, 51479038).
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University, Harbin, China, 2008.
NATURAL CONVECTION COUPLED WITH RADIATION HEAT TRANSFER 1393
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