3

Emirates Journal for Engineering Research, 17 (1), 17-30 (2012)
(Regular Paper)
17
TURBULENT NATURAL CONVECTION INSIDE AN INCLINED
SQUARE ENCLOSURE WITH BAFFLES
Khudheyer S. Mushatet
College of Engineering, Thiqar University
Nassiriya, Iraq
(Received January 2011 and Accepted November 2011)
‫دا‬ ‫اﻟﺤﺮ‬ ‫اﻷﺿﻄﺮاﺑﻲ‬ ‫ﻟﻠﺤﻤﻞ‬ ‫ﻋﺪدﻳﺔ‬ ‫دراﺳﺔ‬ ‫أﺟﺮﻳﺖ‬‫ﻟﻠﺤﺮارة‬ ‫ﻣﻮﺻﻠﺔ‬ ‫اﻋﺎﻗﺔ‬ ‫ﺻﻔﻴﺤﺘﺎ‬ ‫ﻋﻠﻰ‬ ‫ﻳﺤﺘﻮي‬ ‫ﻣﺎﺋﻞ‬ ‫ﻣﺮﺑﻊ‬ ‫ﺣﻴﺰ‬ ‫ﺧﻞ‬.
‫اﻟﻤﺤﺪد‬ ‫اﻟﺤﺠﻢ‬ ‫ﺗﻘﻨﻴﺔ‬ ‫وﺑﺄﻋﺘﻤﺎد‬ ‫اﻟﻀﻤﻨﻴﺔ‬ ‫اﻟﺨﻮارزﻣﻴﺔ‬ ‫ﺑﺎﺳﺘﺨﺪام‬ ‫ﺣﻠﻬﺎ‬ ‫ﺗﻢ‬ ‫اﻟﻄﺎﻗﺔ‬ ‫وﻣﻌﺎدﻟﺔ‬ ‫وﺳﺘﻮآﺲ‬ ‫ﻧﺎﻓﻴﺮ‬ ‫ﻣﻌﺎدﻻت‬.‫ﺗﻤﺖ‬
‫ﻣﻮدﻳﻞ‬ ‫ﺑﺄﺳﺘﺨﺪام‬ ‫اﻷﺿﻄﺮاب‬ ‫ﺗﺄﺛﻴﺮات‬ ‫ﻧﻤﺬﺟﺔ‬k-ε.‫زاوﻳﺔ‬ ‫ﻣﺜﻞ‬ ‫ﻋﻮاﻣﻞ‬ ‫ﻋﺪة‬ ‫ﺗﺄﺛﻴﺮ‬ ‫ﺗﺤﺖ‬ ‫اﻟﺤﺎﻟﻴﺔ‬ ‫اﻟﻤﺴﺄﻟﺔ‬ ‫درﺳﺖ‬
‫ا‬‫راﻳﻠﺔ‬ ‫وﻋﺪد‬ ‫اﻟﺼﻔﻴﺤﺘﻴﻦ‬ ‫ﺑﻴﻦ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ،‫اﻟﺼﻔﻴﺤﺘﻴﻦ‬ ‫ﺑﻴﻦ‬ ‫اﻟﻨﺴﺒﻲ‬ ‫اﻟﻄﻮل‬ ،‫ﻟﻤﻴﻼن‬.‫وﺿﻌﺖ‬ ‫ﺑﻴﻨﻤﺎ‬ ‫ﺟﺰﺋﻴﺎ‬ ‫اﻟﺤﻴﺰ‬ ‫ﺳﺨﻦ‬
‫اﻟﺴﻔﻠﻲ‬ ‫اﻟﺠﺪار‬ ‫ﻋﻠﻰ‬ ‫اﻷﻋﺎﻗﺔ‬ ‫ﺻﻔﻴﺤﺘﺎ‬.‫ﻣﻊ‬ ‫ازداد‬ ‫ﻗﺪ‬ ‫اﻟﺤﺮارة‬ ‫اﻧﺘﻘﺎل‬ ‫ﻣﻌﺪل‬ ‫أن‬ ‫اﻟﺪراﺳﺔ‬ ‫هﺬﻩ‬ ‫ﻣﻦ‬ ‫اﻟﻤﺴﺘﺤﺼﻠﺔ‬ ‫اﻟﻨﺘﺎﺋﺞ‬ ‫ﺑﻴﻨﺖ‬
‫راﻳﻠﺔ‬ ‫وﻋﺪد‬ ‫اﻟﻤﻴﻼن‬ ‫زاوﻳﺔ‬ ‫زﻳﺎدة‬.‫أ‬ ‫اﻟﻨﺘﺎﺋﺞ‬ ‫أوﺿﺤﺖ‬ ‫آﺬﻟﻚ‬‫ﻣﻠﺤﻮﻇﺔ‬ ‫ﺑﺼﻮرة‬ ‫ﺗﺄﺛﺮ‬ ‫ﻗﺪ‬ ‫وﺷﻜﻠﻬﺎ‬ ‫اﻟﻨﺎﺗﺠﺔ‬ ‫اﻟﺪواﻣﺎت‬ ‫ﻋﺪد‬ ‫ن‬
‫اﻟﺼﻔﻴﺤﺘﻴﻦ‬ ‫ﺑﻴﻦ‬ ‫اﻟﻨﺴﺒﻲ‬ ‫واﻟﻄﻮل‬ ‫اﻟﻤﻴﻼن‬ ‫زاوﻳﺔ‬ ‫ﺗﻐﻴﺮ‬ ‫ﻋﻨﺪ‬.
The turbulent natural convection inside an inclined air filled square enclosure having two
conducting solid baffles has been numerically investigated. Fully elliptic Navier-Stokes and
energy equations based on a finite volume method were solved by using a SIMPLE algorithm.
The effect of turbulence was modeled using k-ε turbulence model. The problem was investigated
for different parameters such as the angle of inclination, relative baffles height, the distance
between baffles and Rayleigh number. The enclosure was differentially heated and the conducting
baffles were placed on the bottom wall. The obtained results show that the rate of heat transfer is
increased as the angle of inclination increases for the studied Rayleigh numbers. Also the results
show that the number of resulting vortices and their elongation were significantly affected with
angle of inclination and the relative baffles height.
Keywords: turbulent natural convection, baffles, square enclosure.
1. INTRODUCTION
The turbulent natural convection inside an enclosed
enclosure is considered one of the important topics
that attracted the researchers for many years. The
importance arises from its diverse implications such
as solar collectors, cooling of electronic devices,
energy transfer in building, thermal insulation and
geothermal applications. Inspite of the current
noticeable advances in experimental techniques and
numerical methods, the understanding of the complex
flow and thermal fields needs more attention. When
reviewing the related previous studies, there is no
study was reported on turbulent natural convection
inside an inclined square baffled enclosure. So this
work is aim to enhance the academic research in this
field. An experimental bench mark study on turbulent
natural convection in an air-filled square cavity was
performed by Ampofo and Karayiannis[1]. The cavity
was differentially heated giving a Rayleigh number of
1.58 ×109. The local velocities and temperature were
measured at different locations in the cavity. Also the
local and average Nusselt numbers, the wall shear
stress as well as the turbulent kinetic energy were
presented. The study was considered as an
experimental benchmark data. Markatos and
Pericleous[2] presented a computational method to
obtain the solutions of the buoyancy-driven laminar
and turbulent flow and heat transfer in a square cavity
with differentially heated side walls. A series of
Rayleigh numbers ranging from 103 to 1016 was
studied. Donor-cell differencing is used and mush
refinements studies have been performed for all
considered Rayleigh numbers. The results were
presented in tabular and graphical form and
correlations of the Nusselt and Rayleigh numbers.
Acharya and Jetli[3] presented a numerical study on
laminar natural convection heat transfer inside a
partially divided square box. The results showed that
the thermal stratification between the divider and cold
wall played a significant role. Also they verified that
the effect of the divider position on the overall heat
transfer coefficient was small. A numerical study was
accomplished by Khalifa and Sahib[4] to investigate
the natural convection in a rectangular enclosure
fitted with adiabatic position. The position of the
partition was at the middle of the enclosure. The
enclosure was differentially heated. Water was used a
working fluid to obtain a Rayleigh number range of
1011 to 7 ×1011. Correlations for the tested

18 Emirates Journal for Engineering Research, Vol. 17, No.1, 2012
configuration was reported and the percentage
reduction in heat transfer for each case was compared
to that of a single room. Barozzi and Corticelli[5]
investigated the two dimensional numerical
simulation of the two vertical plates with uniform
heat generation. In their study a rectangular heating
block with constant wall temperature was placed in
the center of the enclosure. The study was done for
Gr ranging from 4×104 to 108. The laminar and
turbulent natural convection in an enclosure with
partial partitions was studied by Bilgen[6]. The
considered values of Ra were ranged form 104 to
1011 . Different values of partitions location ratio,
aspect ratio and height ratio were tested. The results
were presented as streamlines and isotherm contours
for different values of geometrical conditions and Ra.
SaiD et al. [7] investigated the turbulent natural
convection in a partition enclosure. The study
presented Numerical solutions for the buoyancy
driven flows in an inclined two dimensional
rectangular enclosure. One of the inclined walls was
heated and the other was cold, The low Reynolds
number k-ε model was used to model the turbulence.
The flow field and average Nusselt number was
investigated for different angles of inclination and
Rayleigh numbers. The Nusselt number was
increased as Rayleigh number increases. The laminar
and turbulent natural convection in an inclined
enclosure was investigated by Kuper et al.[8]. The k-ε
model was used to model the turbulence. They
concluded that the angle of inclination showed a
significant effect on the Nusselt number. Sey et al. [9]
studied the transient laminar mixed convection in a
two dimensional enclosure partitioned by a
conducting baffle. The interaction between the
external forced air stream and the buoyancy driven
flow was exhibited by streamlines and isotherms. The
effect of divider on heat transfer mechanism was
found to be significant. Fu et al. [10] investigated the
transient laminar natural convection in an enclosure
partitioned by an adiabatic baffle. The obtained
results showed that stream function strength was
significantly dependent on the location of the baffle
and Rayleigh number. Zekeriya and Ozen [11]
presented a numerical study for laminar natural
convection in a tilted rectangular enclosures with a
vertically situated hot plate. The plate was very thin
and isothermal on both lateral ends and act as a heat
source. Yucel and Ozdem[12] numerically
investigated the natural convectioin heat transfer in a
partially divided enclosure. They demonstrated that
the average rate of heat transfer was decreased with
the decrease of number and height of partitions. The
laminar natural convection in a tilted cavity
containing two baffles attached to its isothermal walls
was numerically studied by Ghassemi et al. [13]. The
governing equations was used by using a finite
volume code. They showed that the Nusselt number
decreases as baffles length increases. Shi et al [14]
studied the laminar natural convection inside a
differentially heated square cavity with a fin on the
hot wall. The obtained results indicated that the flow
field is enhanced for high Rayleigh numbers
regardless of fin length and position. Khalifa and
Abdulla [15] investigated numerically the turbulent
natural convection in a partitioned rectangular
enclosure. The study was performed for Rayleigh
number range up to 1.5×108 . The correlations and
the effect of the enclosure inclination angle besides to
the number of partitions on the flow and thermal
fields were reported. Tansnim and Collins[16] studied
the effect of the presence a baffle (placed on the hot
wall of the square cavity) on the flow and thermal
field. They indicated that the baffle has a significant
effect on the Nusselt Number variation. Viktor et al
[17] studied numerically the fluid motion and laminar
free convection heat transfer in a square enclosure
with staggered dividing partitions. They showed that
the surface average heat transfer coefficient was
decreased with a rise of partition height and Nusselt
number increased significantly.
In this work, the turbulent natural convection heat
transfer inside an inclined square enclosure
containing two conducting solid baffles has been
numerically studied. The working fluid was air and Pr
=1. As Fig.1 shows, the vertical walls are fixed at
different isothermal temperatures while the horizontal
walls are insulated. The enclosure angle of inclination
was ranged as
0
1800 ≤≤ α
o
. The studied Rayleigh
numbers was ranged as
148
1010 ≤≤ Ra ,the distance
between baffles 5.02.0 ≤≤ c while the relative
baffles height was 6.02.0 ≤≤ h . To the
knowledge of the researcher, there is no study
documented on this situation(in the turbulent natural
convection) up to date. So, the present work aim to
show how the angle of the inclination can affect the
flow and thermal field characteristics of the inclined
baffled enclosure in the turbulent natural convection.

Turbulent Natural Convection Inside an Inclined Square Enclosure with Baffles
Emirates Journal for Engineering Research, Vol. 17, No. 1, 2012 19
2. MATHEMATICAL MODEL
The turbulent flow and heat transfer of the working
fluid (air) are described by the time averaged Navier-
Stokes and energy equations. The thermo physical
properties of air are assumed to be constant except
the density in the gravity force which is assumed to
be linearly dependent on the temperature introduced.
So, the considered equations are described as
follows.
0=
∂
∂
+
∂
∂
y
v
x
u
(1)
( ) αβρμ
μμρρ
sin0
2
TTg
x
v
eff
y
y
u
eff
yx
u
eff
xx
p
y
u
v
x
u
u
−+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
−=
∂
∂
+
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
(2)
( ) αβρμ
μμρρ
cos0
2
TTg
y
u
eff
x
y
v
eff
yx
v
eff
xy
p
y
v
v
x
v
u
−+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
−=
∂
∂
+
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
(3)
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
Γ
∂
∂
+
∂
∂
Γ
∂
∂
=
∂
∂
+
∂
∂
y
T
eff
yx
T
eff
xy
T
v
x
T
u ρρ (4)
teff μμμ +=
,
( ) 2/0 hC TTT += , 0/1 T=β
(5)
PrPr
,
t
Teff
μμ
+=Γ
(6)
The effect of turbulence was modeled using k-ε
model as follows[18].
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
Γ
∂
∂
+⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
Γ
∂
∂
=
∂
∂
+
∂
∂
ρε
ρρ
G
y
k
y
x
k
xy
k
v
x
k
u
keff
keff
,
,
(7)
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
Γ
∂
∂
+⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
Γ
∂
∂
=
∂
∂
+
∂
∂
yyxxy
v
x
u effeff
εεε
ρ
ε
ρ εε ,,
k
CG
k
C
2
21
εε
εε ++ (8)
The shear production term was defined as:
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
+
∂
∂
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
+⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
222
22
x
v
y
u
y
v
x
u
G tμ (9)
The turbulent viscosity is defined as:
ε
ρ μ
2
μ
k
ct =
(10)
The constants in the turbulence model are (σk ;σЄ
;C1є ;C2є ; Cµ) = (1.0 , 1.3 , 1.44 , 1.92 , 0.09 )
respectively.
To get the distribution of the stream function (ψ), the
following Poisson equation is solved with the
boundary condition ψ = 0 at the solid walls.
Figure 1. Schematic diagram of the problem
α

x
v
y
u
yx ∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
2
2
2
2
ψψ
(11)
The temperature distribution through the conducting
solid baffles was obtained by solving the two
dimensional steady state heat conduction equation.
02
2
2
2
=
∂
∂
+
∂
∂
y
T
x
T
(12)
2.1. Boundary Conditions
To complete the solution of the mathematical
model, the following boundary conditions are
considered
At the walls: u = v = 0. and wall function laws were
used for the near wall grid points[19].
For vertical walls: at x = 0, T = Th, at x =L , T = Tc
For the horizontal walls,
0=
∂
∂
y
T
.
The local dimensionless Nusselt number along the
left vertical hot wall is obtained as follows:
Ch TT
H
x
T
X
Nu
−∂
∂
=
∂
∂
=
θ
(13)
sf
s
f nk
k
n
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=⎟
⎠
⎞
⎜
⎝
⎛
∂
∂ θθ
(14)
On the baffles boundaries:
where fk
and sk are the thermal conductivities of
the fluid and solid respectively, n is a unit normal
vector.
The Nusselt number is a function of Rayleigh
and grid points. The number of grid points for
108
1010 ≤≤ Ra is 41×41 and for
1410
1010 ≤≤ Ra is 84×82 . It can be mentioned
here that the increase in Ra needs more grid points
and computational time to obtain a converged
solutions. The large part of grid points for all the
studied Rayleigh numbers was concentrated near the
walls in all directions.
3. NUMERICAL ANALYSIS
The governing equations of the flow and thermal
fields for the working fluid(air) inside an inclined
baffled square enclosure were discretized by using a
finite volume approach. The resulting discretized
equations represent an algebraic equations. The
solutions of these equations was done by using a
SIMPLE algorithm along with a simi-implicit line by
line Gause elimination scheme. Non-uniform grids in
all directions were used and these grids were finer
near the solid walls where steep gradients of the
dependent variables are important. In general, the
computational grids are staggered for vector
variables and not staggered for the scalar one. Grid
dependency is controlled by monitoring the
considered variable values and residuals of the
iteration process. The residual sum for each of the
variables is computed and stored at the end of each
iteration. The criteria
Max
( ) ( ) 51
10,, −−
≤− jiji kk
φφ
was the
convergence history required for all dependent
variables. Because of the inherent coupling and non
linearity in the governing equations, underlaxation
factors were used. The factors used for velocity
components, energy and turbulence quantities are
0.5, 0.8, 0.7 respectively. A computer program is
developed to get the results using the pressure
velocity coupling (SIMPLE algorithm)[19]. The
validation of the present code was performed through
the comparison with published results[2] for
Pr=0.71as shown in table1. Noting that the number
of grid points and refinements are taken as found in
the mentioned reference. From the comparison, it is
concluded that a good agreement has been obtained
Table 1. Comparison between the present and published results.
Ra Nuav (present results) Nuav (published results)
106 8.748 8.754
108 32.1 32.04
1010 156.85 156.8
1012 8 40.8 840.1
1014 3627 3624
1016 11229.9 11226

d.
o
09=α
Figure 2 Stream function distribution for different angles of inclination, h=0.2,
W=0.12, c=0.2 and Ra=1E10
a.
o
0=α
b.
o
03=α c.
o
06=α
e.
o
012=α f.
o
015=α
g.
o
018=α
a.
o
0=α
b.
o
03=α
c.
o
06=α
d.
o
09=α
e.
o
012=α
f.
o
015=α
g.
o
018=α
Figure 3 Isotherm contours distribution for different angles of inclination, h=0.2,
W=0.12, c=0.2 and Ra=1E10

a.
o
0=α
b.
o
03=α c.
o
06=α
d.
o
09=α
e.
o
012=α
f.
o
015=α
g.
o
018=α
W=0.12, c=0.2 and Ra=1E10
a.
o
0=α
b.
o
03=α
d.
o
09=α
e.
o
012=α
f.
o
015=α
g.
o
018=α
W=0.12, c=0.2 and Ra=1E10
c.
o
06=α

c.
o
06=α
a.
o
0=α
b.
o
03=α
d.
o
09=α
e.
o
012=α
f.
o
015=α
g.
o
018=α
W=0.12, c=0.2 and Ra=1E10
a.
o
0=α
b.
o
03=α
c.
o
06=α
d.
o
09=α
e.
o
012=α f.
o
015=α
g.
o
018=α
W=0.12, c=0.2 and Ra=1E10

a. Ra=1E8
b. Ra=1E10
c. Ra=1E12
Figure 9 Effect of Ra on isotherm contours distribution for h=0.4,
o
06=α
W=0.12 and c=0.2
a. Ra=10E8 b. Ra=10E10 c. Ra=10E12
Figure 8 Effect of Ra on stream function distribution for h=0.4,
o
0=α
W=0.12 and c=0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Y
0
5
10
15
20
25
30
35
40
45
Nu
Ra=1E8
Ra=1E10
Ra=1E12
Ra=1E13
Ra=1E14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Y
0
5
10
15
20
25
30
Nu
Ra=1E8
Ra=1E10
Ra=1E12
Ra=1E13
Ra=1E14
a.
o
0=α b.
o
06=α
Figure 10 Effect of Rayleigh number on local Nusselt number distribution for
W=0.12, c=0.2 and h=0.4

0 0.25 0.5 0.75 1
X
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
V
angle=0
angle=30
angle=60
a. b.
0 0.25 0.5 0.75 1
Y
-0.6
-0.4
-0.2
0
0.2
0.4
U
angle=0
angle=30
angle=60
Figure 14 Horizontal and vertical velocity velocities(m/s), a.(x=L/2), b.(y=H/2) , for
W=0.12, c=0.2, Ra=1E10 and h=0.4
Figure 13 Effect of relative baffles height on average heat transfer distribution for
W=0.12, c=0.2,
a. stream,
o
03=α , c=0.4
b. stream,
o
06=α , c=0.4
a. Ra=1E8 b. Ra=1E10 c. Ra=1E12
0 30 60 90 120 150 180
Y
0
5
10
15
20
25
30
35
40
Nuav
h=0.2
h=0.4
h=0.6
0 50 100 150
Y
10
15
20
25
30
35
40
Nuav
h=0.2
h=0.4
h=0.6
0 30 60 90 120 150 180
Y
10
15
20
25
30
35
40
Nuav
h=0.2
h=0.4
h=0.6
angle angle angle

c. isotherm,
o
03=α , c=0.4 d. isotherm,
o
06=α , c=0.4
Figure15 Effect of dimensionless distance between baffles on stream function and isotherm distribution for W=0.12,
Ra=1E10 and h=0.4
0 0.25 0.5 0.75 1
Y
25
30
35
40
45
50
Nu
c=0.2
c=0.4
c=0.5
Figure 16 Effect of dimensionless distance between baffles on local Nusselt number distribution for W=0.12,
Ra=1E10 and h=0.4
a.
o
0=α a.
o
06=α
0 0.25 0.5 0.75 1
Y
-0.6
-0.4
-0.2
0
0.2
0.4
U
c=0.2
c=0.4
c=0.5
0 0.25 0.5 0.75 1
X
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
V
c=0.2
c=0.4
c=0.5
a. b.
Figure 17. Horizontal and vertical velocities(m/s), a.(x=L/2), b.(y=H/2) , for
W=0.12, Ra=1E10,
o
0=α and h=0.4
a.
o
0=α
0 0.25 0.5 0.75 1
Y
0
5
10
15
20
25
30
35
40
45
Nu
c=0.2
c=0.4
c=0.5

Figure 18 Horizontal and vertical velocities(m/s), at x=L/2 and y=H/2 respectively for different angles of inclination,
w=0.12, c=0.4, and h=0.4
0 0.25 0.5 0.75 1
Y
-0.2
0
0.2
0.4
0.6
0.8
U
angle=0
angle=30
angle=60
0 0.25 0.5 0.75 1
X
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
V angle=0
angle=30
angle=60
c. Ra=1E10 d. Ra=1E10
0 0.25 0.5 0.75 1
Y
-0.05
0
0.05
0.1
0.15
U
angle=0
angle=30
angle=60
0 0.25 0.5 0.75 1
X
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
V
angle=0
angle=30
angle=60
a. Ra=1E8 b. Ra=1E8
Figure 20. Comparison between the present and published results for isotherm contours at Ra=10E12,
o
0=α
a. present results b. published results[2]

4. RESULTS AND DISCUSSION
The computed results of the turbulent natural
convection inside an inclined baffled square
enclosure has been reported. The stream function,
streamlines, velocities and local and average Nusselt
number were investigated for different parameters
such as Rayleigh number, relative height of baffles
and the distance between baffles.
Fig.2 presents the distribution of stream function for
different angles of inclination and h=0.2. It can be
seen that the angle of inclination has a significant
effect on the distribution of stream function. When
α =
o
0 , there is four rotating vortices, two above the
first baffle and the other lie at the right of the second
baffle. When α =
o
03 as shown in (b), the two
vortices above the first baffle are elongated along
with appearing two small vortices. The two vortices
at the left of the second baffles are elongated to one
vorticity occupying larger zone. At α =
o
06 , all the
mentioned vortices are elongated to two vortices, one
at the left and the other at the right of the second
baffle. This dramatic change in stream function
distribution is continuing with increasing the angle of
inclination as in (d) where the size and location of
the two vortices were changed besides to appear a
small vorticity between the two baffles. In (e), the
first vorticity is elongated to occupy the most region
of the enclosure while the second vorticity is shrunk
to a very small vorticity at the left of the second
baffle. This trend is extended at
o
150=α . However
the elongation of the first vorticity is large and
penetrates behind the second baffle. At
o
180=α ,
the two vortices tries to form again but in the new
shapes. The contours of the isotherm lines are
depicted in Fig.3 for different angles of inclination
and h=0.2. For,
o
0=α , it is clear that the heat is
transferred from the hot wall to the cold wall through
the working fluid(air). Also the conduction occurs
through the two solid baffles. At the enclosure
region(no baffles), the isotherm lines are inclined
and that confirm the occurrence of the convection
process. The presence of baffles distorts the
distribution of these lines and the convection occurs
between the baffles and the fluid is faster than the
conduction within the baffles. The rate of heat
transfer is decreased near the first baffle while it is
increased above the second baffles and then
decreases. This is confirmed through Fig.11.a. It is
evident that the relative baffles height has a
noticeable effect on the distribution of stream
function and isotherm lines. At Fig.4, the number of
vortices, the size and the location are changed
compared with Fig.2(h=0.2). Also the Figure shows,
the strength of vortices increases as angle of
inclination increases. This is expected to enhance the
rate of heat transfer as shown in Fig.12.b. It can be
seen that the two vortices above the first baffle are
elongated to one vorticity while one of the two other
vortices behind the second baffle is elongated. The
cause behind this is when the relative height of the
baffles increases, the first two elongated vortices are
pushed upward and that prevent the right-hand
vortices behind the second baffle from penetration
towards the left. When
o
30=α , the heat transfer
from the hot wall to the cold one seems to be slower
and that confirmed at Fig.11.a. Also this effect on the
isotherm lines distribution is extended to
o
60=α .
When
o
90=α , the rate of heat transfer from the hot
wall to the cold wall is improved. When
o
120=α ,
the heat transfer is faster and that confirmed through
Fig.11.a. This trend is increased at (f) and (g). The
effect of relative baffles height on stream function
and isotherm lines distribution for different angles of
inclination is shown at Fig.4. and Fig.5. At Fig.4,
when h=0.4, it is evident that the rate of heat transfer
is increased especially in the region above the baffles
and then decreased. This effect is enhanced with
increasing the angle of inclination. This is confirmed
through Fig.12.b. this trend is extended to Figs.6-7.
In the Fig.6, the number of vortices, location and the
size are significantly effected with the relative
baffles height. The strength of vortices is larger and
that leads to enhance the rate of heat transfer from
the hot wall to the cold one and through the solid
baffles as shown in Fig.7. and Fig.12.b. The effect of
Ra on stream function distribution for
o
0=α and
o
60=α is depicted in Figs.8-9. At
o
0=α , It is
evident that the strength of vortices are increase as
Ra increases. The number of vortices, location and
size are significantly changed with increasing Ra.
The increase in Ra increase the buoyancy forces and
hence increase the convection currents as shown in
Fig.10. At
o
60=α , dramatic changes are occurred
on the size, location and distribution of the resulting
vortices. Fig.10 demonstrates the effect of Ra on
variation of Nusselt number along the hot wall. It is
clear that the Nusselt Number increase as Ra
increases. However at
o
60=α , the maximum
values of Nu is decreased significantly. Also it is
observed that at
o
60=α , the two values of Nu

decreases as Ra>1012. The effect of Ra on Nusselt
number variation for different angles of inclination
and for different relative baffles height is depicted in
Fig.11. For h=0.2, it can be seen that the Nusselt
number decrease as
0
600 ≤≤ α
o
and increase as
0
18006 ≤≤ α
o
. This trend is extended to h=0.4.
However there is a difference in the magnitude and
location of the maximum and minimum values of
Nusselt number due to the increase of the baffles
height. For h=6, the maximum and minimum values
of Nu are shifted to new locations due to the increase
of relative baffles height. Also it is observed that the
there un symmetric increase or decrease in Nu with
the increase of angle of inclination. Fig.12 shows the
variation of the local Nusselt number for different
relative baffles height. For
o
60=α (h=0.2), it can be
seen that the Nu decreases for 25.00 ≤≤ Y and
then increases for 55.025.0 ≤≤ Y after that
decrease. When h=0.4, the Nu decreases along the
hot wall. However, it starts from the higher value
compared with h=0.2. This trend is shown at Fig.6.
From (a), it can be included that the maximum
values of Nusselt number lie at the left corner and
the minimum at the upper corner. Also it can be
included that the maximum values of Nusselt number
increase as h increases. When α increases as shown
in (b), the values of Nusselt number increase as h
increase. However, the minimum and maximum
values of Nusselt number is decreased significantly
compared with (a). Also, it can be observed that the
location of minimum values of Nusselt number do
not occur at the upper corner of the enclosure. The
effect of the angle of inclination on the average
Nusselt number for different relative baffles height
and Ra is seen at Fig.13. It can be observed that the
average value of Nusselt number is increased as Ra
increases. For Ra =108, the minimum value of
Nusselt number occurs at
o
90=α for h=0.2 and
h=0.4 and at
o
120=α for h=0.6 while the
maximum values occur at
o
180=α for h=0.4 and
h=0.6 and at
o
120=α for h=0.2 . for Ra>108, the
minimum values of Nu occur at
o
60=α for h=0.2
and h=0.4 and at
o
120=α for h=0.6 while the
maximum values at
o
180=α . The horizontal and
normal velocities at the center of the enclosure are
depicted in Fig.14 for h=0.4 and for different angles
of inclination. For (a), it is evident that the values of
horizontal velocity is increased with the increase of
angle of inclination. Also, the values of normal
velocity increase as angle of inclination increases.
The effect of the distance between baffles on stream
function and isotherm lines is shown at Fig.15. It can
be seen that when the distance between the baffles
increases, the size, location and the trend of
elongated vortices are noticeably changed. This trend
is changed according to the increase of angle of
inclination as in (b). Also the increase in the distance
between the baffles affected the isotherm lines and
the rate of heat transfer as shown in the next Figure.
The effect of distance between the baffles on Nusselt
number variation is depicted in Fig.16. In (a), it is
clear that the Nusselt number is increased with the
increase of the distance (c) between the baffles.
However this increase is continued until c=0.4, after
that the Nusselt number decreases. For (b), the same
trend is seen when α increases. However the values
of Nu is less than that of
o
0=α . The variation of
horizontal and vertical velocities at the enclosure
center for different distances between the baffles. It
can be seen that the velocities are increased as angle
of inclination increases. The horizontal and axial
velocity are increased as Ra increases for all the
considered angles of inclination as shown in Fig.18.
The increase of velocities is due to increase the
buoyancy forces as a result of the increase of Ra. The
validation of the present code is also examined
against the published results and acceptable
agreement is obtained as shown in Fig.19.
5. CONCLUSIONS
The turbulent flow and thermal fields inside an
inclined square baffled enclosure has been
numerically investigated. The k-ε model is used to
simulate the turbulence for high Rayleigh numbers.
It is worth to mention here the most important
concluding remarks .
The rate of heat transfer increases as Ra increases.
However
o
0=α indicated the maximum increase.
The rat of heat transfer decreases as angle of
inclination increases. However this decrease is
limited for 25.00 ≤≤ Y and 9.06.0 ≤≤ Y .
The rate of heat transfer increases as the relative
baffles height increases. This increase is larger at
α >
o
0 .
The rate of heat transfer increases as the distance
between the baffle(c) increases. However when
c>0.4, this trend is reflected. The distance(c) equal to
0.4 indicated the optimum rate of heat transfer.
The horizontal and vertical velocities are increased
as angle of inclination increases for all the studied
Rayleigh numbers.
The number of resulting vortices, their shapes and
elongation is greatly effected with the angle of

inclination, relative baffles height and the distance
between baffles.
REFERENCES
1. Ampofo, F. Karayiannis, T.G.,” Experimental
bench data for turbulent natural convection in an
air filled square cavity” International J. Heat and
Mass Transfer, vol.46, pp.3551-3572,2003.
2. Marakatos, N.C., Pericleous, k. A.,” Laminar and
turbulent natural convection in an enclosed
cavity”, Int. J. Heat Mass Transfer, vol.27,
pp.755-772,1984.
3. Acharya S., Jetli, R., “Heat transfer due to
buoyancy in a partially divided square box”,
vol.33, pp.640-647, 1990.
4. Khalifa, A.J., N., Sahib, W.K., “Turbulent
buoyancy driven convection in partially divided
enclosure”, J., Energy Convergion and
Management, vol.43, pp.2115-2121,2002.
5. Barozzi G. S., Corticelli, M. A.,” Natural
convection in cavities containing internal source
“,J. Heat and Mass Transfer ,vol.36, pp. 473-
480,2000.
6. Bilgen E., “Natural convection in enclosure”,
Renewable Energy, vol. 26, pp. 257-270,
2002.
7. Said, S.A.M., Habib, M.A., Khan, A.R.,
“Turbulent Natural Convection in Partitioned
enclosure”, Computers and Fluids, vol.26,
pp.547-563,1997.
8. Kuyper, R.A, Van Der Meer, C.J., Henkes, R.A.,
Hoogendoorn, C.J.” Numerical study of laminar
and turbulent natural convection in an inclined
square cavity”, Int.J. Heat Mass Transfer, vol.36,
pp.2899-2911,1993.
9. Sey. Ping How, Tsan-Hui Hsu,” Transient Mixed
Convection in a partially divided enclosure”,
Computer Math. Applic, vol.36, No.8, 1988.
10. Fu, W.S., Perng, J.C., Shieh, W.J.,” Transient
Laminar natural Convection in an Enclosure
Partitioned by an adiabatic baffles”, Numerical
Heat Transfer, Part A, pp.335-350, 1989.
11. Zekeriya Atlas, Ozen Kurtul, “Natural
convection in tilted rectangular enclosures with a
vertically situated hot plate”, Applied Thermal
Engineering, vol. 27, pp.1832-1840, 2007.
12. Yucel N, Ozdem, AH.,” Natural convection in
partially divided square enclosure”, J. Heat Mass
Transfer, vol.40, pp. 167-175, 2003.
13. Ghassemi, M., Pirmohammadi, M., Sheikhzadeh
“, A numerical study of natural convection in a
tilted cavity with two baffles attached to its
isothermal walls”, Transactions on Fluid
Mechanics, vol.2, pp. 61-68 ,2007.
14. Shi M, X, Khodadadi, J. M., “Laminar natural
convection heat transfer in differentially heated
square cavity due to a thin fin on the hot wall”,
ASME J. Heat Transfer , vol.125., pp. 624-634,
2003
15. Tansnim S. T., Collins M. R., “ Numerical
analysis of heat transfer in a square cavity with
baffle on the hot wall”, Int. J. Heat Transfer,
vol.31, pp.639-650,2003.
16. Kalifa A.J. N., Abdullah S. E.,” Buoyancy driven
convection in undivided and partially divided
enclosures”, Energy Conversion and
Management,vol.40, pp. 717-727, 1999.
17. Viktor I. Terekhov, Alexander V. Chichindaev,
Ali L. Ekaid, “ Buoyancy heat transfer in
staggered dividing square enclosure”, Thermal
Science, vol.15, pp.409-422, 2011.
18. Jones, W.P., Lunder, B.E., “The prediction of
laminarization with a two equation model of
turbulence”, J. Heat and Mass transfer, 1972.
19. Versteege H., Meer W., An introduction of
computational fluid dynamics, Hemisophere
publishing corporation, United State of America,
1995.
NOMENCLATURE
a thermal diffusivity, m2/s
c dimensionless distance between baffles(e/H)
G generation term by shear, Kg/m.sec3
h relative height of the baffles (L1/L2)
H height of the enclosure, m
k turbulent kinetic energy, m2/s2
Nu local Nusselt number
Nuav average Nusselt number
P pressure, N/m2
Pr Prandtl number
Ra Rayleigh number
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
ν
β
a
TTHg ch(3
TC cold wall temperature, Ċ
Th hot wall temperature, Ċ
x, y Cartesian coordinates, m
X dimensionless Cartesian coordinate
⎟
⎠
⎞
⎜
⎝
⎛
H
x
Greek symbols
є turbulance dissipation rate, m2/s3
µ dynamic viscosity, N.s/m2
µt turbulent viscosity, N.s/m2
α angle of inclination, deg.
θ dimensionless temperature
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
ch
c
TT
TT

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