ABSTRACT Two-dimensional laminar natural convection in an incline

Manish Kulkarni

Sushanta K Mitra

U N Gaitonde

CiteSeerX

Proceedings of HT2007
2007 ASME-JSME Thermal Engineering Summer Heat Transfer Conference
July 8-12, 2007, Vancouver, British Columbia, CANADA
HT2007-32677
STUDY OF NATURAL CONVECTION IN INCLINED SQUARE ENCLOSURES WITH
UNIFORM HEAT GENERATION
Manish Kulkarni
Department of Mechanical Engineering
Indian Institute of Technology Bombay
Mumbai, Maharastra, 400076
India
Sushanta K. Mitra∗
Department of Mechanical Engineering
Indian Institute of Technology Bombay
Mumbai, Maharastra, 400076
India
Email: skmitra@me.iitb.ac.in
U. N. Gaitonde
Department of Mechanical Engineering
Indian Institute of Technology Bombay
Mumbai, Maharastra, 400076
India
ABSTRACT
Two-dimensional laminar natural convection in an inclined
square enclosure with uniform internal heat generation is stud-
ied here. The steady-state solutions are obtained for inclination
angles of 450,300 and 150 and at Rayleigh number of 1.5×105.
For these cases, the two counter-rotating rolls of fluid are present
in the cavity. Streamlines, isotherms and heat transfer for these
results are compared with the existing experimental results and
are found to be in reasonably good agreement. It is found that
the location of maximum non-dimensional temperature in the in-
clined cavity is higher than that for pure conduction case. The
maximum non-dimensional temperature in the cavity decreases
as the Rayleigh number increases. For Ra > 5× 104, the max-
imum non-dimensional temperature in inclined cavity is almost
independent of the inclination angle. It is also observed that the
local Nusselt number at the top wall is greater than the pure con-
duction solution, whereas that for bottom wall it is lower than
the Nusselt number for pure conduction. The effect of Rayleigh
number and inclination angle on the local Nusselt number and
modified local Nusselt number are also studied. For horizontal
cavity, at Rayleigh number greater than or equal to 5×104, peri-
odic solutions are obtained. In this case, two unstable secondary
rolls are present near the center of top wall, in addition to the
primary rolls. The secondary rolls are dissipated and recreated
during one period of oscillation.
∗Address all correspondence to this author.
NOMENCLATURE
C Heat capacity
L Dimension of square cavity
Ra Rayleigh Number
t Time
u Velocity component in x-direction
v Velocity component in y-direction
µ Viscosity of fluid
ρ0 Density of fluid
γ Inclination angle
INTRODUCTION
Buoyancy driven flows in differentially heated cavities have
been studied extensively in many experimental and computa-
tional works. Natural convection in closed cavities with internal
heat generation is important in various fields like-geophysics, as-
tronomy, and post accident heat removal from nuclear reactors
and storage of nuclear waste. Natural convection markedly en-
hances the rate of heat transfer as compared to a purely conduc-
tive mechanism. Buoyancy driven flows in differentially heated
cavities have been studied extensively in many experimental and
computational works. The problem of enclosures with internal
heat generation is treated by relatively few investigators.
Two-dimensional steady state laminar natural convection
problem in differentially heated rectangular cavity is solved nu-
merically by de Vahl Davis [1] with vorticity-stream function
1 Copyright c© 2007 by ASME
Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/28/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
approach. Markatos and Preicleous [2] have studied laminar and
turbulent natural convection in a differentially heated square cav-
ity. The range of Rayleigh numbers considered in this study is
from 103 to 106. Two-dimensional steady state governing equa-
tions in primitive variables are solved using finite volume ap-
proach. Upwind differencing is used for convective terms.
The benchmark numerical solution for the natural convec-
tion of air in a square cavity with differentially heated side
walls and adiabatic top and bottom walls is provided by de Vahl
Davis [3]. The benchmark results are also provided for four dif-
ferent Rayleigh numbers 103, 104, 105 and 106 [4]. Streamlines,
isotherms and Nusselt number data are provided.
The problem of natural convection in enclosures with uni-
form internal heat generation has been studied both experimen-
tally and numerically. Lee and Goldstein [5] have investigated
this problem experimentally. They have used an enclosure of
square cross section 3.81cm × 3.81cm and 25.4cm long. All
the four walls forming the square cross section of the cavity are
isothermal. The test fluid used is distilled water with NaCl added
to increase the electrical conductivity of the water. The salinity
of water is less than 0.01 molar, so that the thermo-physical prop-
erties of the fluid are close to that of pure water. The results are
presented for four inclination angles 00, 150, 300 and 450. For
each inclination angle, four Rayleigh numbers 104, 5×104, 105
and 1.5×105 are considered.
Therefore, there is a definite need to understand the flow
in enclosures with internal heat generation. Existing literature
suggests that limited study is conducted to understand the natural
convection in inclined cavity. Hence, in this work attention is
focused in the numerical study of natural convection processes
in inclined square cavity for a range of Rayleigh number.
PHYSICAL PROBLEM
Figure 1 shows a square cavity with uniform internal heat
generation. The cavity is sufficiently long in z direction for the
motion to be assumed two dimensional. The cavity is inclined
to the horizontal at an angle γ. It is further assumed that motion
is laminar. All the four walls of the cavity are assumed to be
isothermal. The heat generation causes density changes within
the fluid, which leads to buoyancy-driven flow.
Governing Equations
The Boussinesq approximated unsteady equations for the
given problem are,
Continuity equation:
∂u
∂x
+
∂v
∂y
= 0 (1)
x-momentum equation:
g
L
X
Y
q’’’
L
γ
Figure 1: Schematic of cavity with internal generation
ρ0(
∂u
∂t
+
∂uu
∂x
+
∂vu
∂y
)=−
∂p
∂x
+µ(
∂2u
∂x2
+
∂2u
∂y2
)−ρ0[1−β(T −T0)]gsinγ
(2)
y-momentum equation:
ρ0(
∂v
∂t
+
∂uv
∂x
+
∂vv
∂y
)=−
∂p
∂y
+µ(
∂2v
∂x2
+
∂2v
∂y2
)−ρ0[1−β(T −T0)]gcosγ
(3)
Energy equation:
ρ0C(
∂T
∂t
+
∂uT
∂x
+
∂vT
∂y
) = k(
∂2T
∂x2
+
∂2T
∂y2
)+ q
′′′
(4)
At initial time t = 0, no slip condition holds at all four walls
and temperature throughout the cavity is T0 = 0, which can be
expressed mathematically as
u = v = 0,T = T0 = 0 at t = 0 (5)
At times t > 0, no slip condition holds at walls and temper-
ature is specified at wall, which can be written as,
u = v = 0,T = 0 f or x = 0 and x = L (6)
u = v = 0,T = 0 f or y = 0 and y = L (7)
Solution Procedure
The governing equations in non-dimensional form for the
square cavity with uniform internal heat generation are solved
using finite volume method. SIMPLE algorithm and a staggered
2 Copyright c© 2007 by ASME
Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/28/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
grid is used for calculation. Upwind differencing scheme is used
for discretization convective terms in the governing equations.
Implicit scheme is used for discretization with respect to time.
The grid used is a uniform grid with 50 points in both x and y di-
rections. Discretized equations are solved using ADI method [6].
The parameters of the numerical solutions are chosen in such
a way that a comparison with experiments by Lee and Gold-
stein [5]. Solutions are obtained for four different Rayleigh num-
bers - 104,5×104,105 and 1.5×105 and four different inclination
angles - 00,150,300 and 450. Lee and Goldstein [5] did the exper-
iments with distilled water with small amount of NaCl added to
raise electrical conductivity of water. The thermo-physical prop-
erties of the fluid are very close to that of pure water. For com-
parison of results with these experiments, the Prandtl number is
taken as 7.
It is to be noted that Rayleigh number is defined [7] in
terms of the maximum temperature difference in a layer for one-
dimensional pure conduction mode and the corresponding char-
acteristic length as,
Ra =
gβ
(
q
′′′
L2
8k
)(
L
2
)3
αν
(8)
Results and Discussions
The unsteady governing equations are solved subject to ini-
tial and boundary conditions of the problem. For a given incli-
nation angle, the steady state solutions are approached after an
initial transient. The streamlines and isotherms, for inclination
angles 150,300 and 450, obtained by present computation are now
presented below.
It is observed in Fig. 2 that there are two strong counter-
rotating rolls in the inclined cavity. The hot interior fluid starts
to move up nearly parallel to the gravity vector, which divides
the whole cross-section approximately in two halves. These two
hot flows arrive at almost same position near the top wall. The
flows then divide and move downward separately along the cold
side walls. At inclination angle of γ = 450 the flow and temper-
ature contours are symmetric about the vertical diagonal of the
enclosure.
It is found from the isotherms that the position of the maxi-
mum non-dimensional temperature is higher than that in the pure
conduction situation. This is because of the influence of natural
convection. As the Rayleigh number increases, the position of
the maximum non-dimensional temperature moves toward upper
top corner. This indicates greater influence of natural convection
at higher Rayleigh number, as shown in Fig. 3. With the increase
in Rayleigh number, the wall boundary layers become more and
more pronounced. Hence the streamline and isotherms near the
walls become closely spaced. Figure 5 shows the plot of max-
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
−0
.1
02
−0
.3
04
−0
.5
07
−0
.7
09
0.
10
1
0.
30
3
0.
70
8
0.
50
6
XY
Figure 2: Streamline plot for γ = 450 and Ra = 104
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
−0
.2
78
−0
.8
31
−1
.3
9
−1
.9
4
1.
94
0.
27
6
0.
83
1.
38
XY
Figure 3: Streamlines for γ = 450 and Ra = 1.5×105
imum non-dimensional temperature in the cavity against non-
dimensional time for various Rayleigh numbers. The maximum
dimensionless temperature in the cavity reduces as the Rayleigh
number increases. This can be attributed to the stronger convec-
tive motion in the fluid at higher Rayleigh number which causes
the higher heat transfer coefficient at the cold walls of the cavity.
The local Nusselt number can be defined based on the
maximum temperature difference in pure conduction in one-
3 Copyright c© 2007 by ASME
Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/28/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
1
0.
20
5
0.
17
9
0.
15
4
0.
12
8
0.
10
3
0.
07
69
0.
05
13
0.
02
56
XY
Figure 4: Isotherms for γ = 450 and Ra = 1.5×105
0 0.5 1 1.5
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Non−dimensional time
M
ax
im
um
 n
on
−
di
m
en
si
on
al
 te
m
pe
ra
tu
re
Ra=104
Ra=5*104
Ra=105
Ra=1.5*105
Figure 5: Maximum non-dimensional temperature in the cavity
for γ = 450
dimensional layer containing uniform internal heat source and
the corresponding characteristic length L2 [5]. In terms of dimen-
sionless variables, the local Nusselt number is expressed as,
Nu = −0.5
∂θ
∂N wall
(9)
where, N is the dimensionless wall coordinates and the average
Nusselt number is given by,
Nu =
Z 1
0
NudN (10)
There are four types of local Nusselt numbers: NuL, NuR, NuB
and NuT corresponding to left, right, bottom and top walls of the
cavity, respectively. Figure 6 shows the local Nusselt distribu-
tion on four walls at Ra = 105 and γ = 150. Along the top wall,
the local Nusselt number NuT is greater than that by conduction
only. On the other hand, local Nusselt number along the bottom
wall NuB is lower than that by conduction only. The values of
local Nusselt number along the left and right walls are smaller in
the lower portion and greater in the upper portion of the enclo-
sure than those for conduction only. A comparison of the local
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
X or Y
N
u
Left
Right
Top
Bottom
Conduction only
Figure 6: Local Nusselt number distributions on each wall for
γ = 150 and Ra = 105
heat transfer rate with convection, qconv, to that which would oc-
cur with two-dimensional pure conduction, q2D,cond , is used for
investigating the influence of the natural convection. To describe
the ratio of these two heat transfer rates, a modified local Nusselt
number is defined as [5],
Nu+ =
qconv
q2D,cond
(11)
Hence, the modified local Nusselt number can be expressed as,
Nu+ =
π2Nu
16∑∞n=0
sin[(2n+1)πx/L]. tanh[(2n+1)π/2]
(2n+1)2
(12)
It is to be noted that the local Nusselt number Nu is related to the
absolute magnitude of the heat transfer rate, whereas the modi-
fied local Nusselt number Nu+ is related to the heat transfer rel-
ative to that of a two-dimensional conduction case.
As can be seen from Fig. 7, NuL+ decreases as the inclina-
tion angle γ increases. Figure 8 shows that NuR+ increases in the
4 Copyright c© 2007 by ASME
Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/28/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Y
N
u L
+
150
300
450
Figure 7: NuL+ distribution for different values of γ at Ra = 105
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Y
N
u R
+
150
300
450
Figure 8: NuR+ distributions for different values of γ at Ra = 105
lower portion but remains almost constant in the upper portion of
the enclosure as the inclination angle increases.
It is interesting to note that, in case of horizontal cavity,
the steady-state is not approached for Rayleigh number equal
to or greater than 5 × 104. Figures 9 to 12 show streamlines
during twelve time instants during one period of oscillation for
Ra = 105. At τ = 0.64 a pair of secondary counter-rotating rolls
is situated near center of the top wall of the cavity in addition
to main rolls. Then at τ = 0.675, left secondary roll joins with
right main roll and the right secondary roll is destroyed gradu-
ally (τ = 0.675,τ = 0.71,τ = 0.74). At τ = 0.835 two new sec-
ondary rolls appear and gradually enlarge. From τ = 0.87, the
whole process repeats itself being reflected at the vertical sym-
metry line of the cavity until same flow pattern as at τ = 0.64
is reached again. At τ = 0.87, the right secondary roll gets at-
tached to the left main roll and left secondary roll gets destroyed
gradually(τ = 0.900,τ = 0.925,τ = 0.950). At τ = 1.045 new
secondary rolls appear near the top wall of cavity.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−1.13
−0.673
−0.22
1.14
0.686
0.233
1.14
0.686
−0.673
−0.22
X
Y
Figure 9: Streamline plot for γ = 00,Ra = 105, and τ = 0.640
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−1.25
−0.76
−0.273
1.19
1.19
0.699
0.213
−0.273
X
Y
Figure 10: Streamline plot for γ = 00,Ra = 105, and τ = 0.675
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−1.39
−0.99
−0.594
−0.197
1.39
0.993
0.597
0.2
X
Y
Figure 11: Streamline plot for γ = 00,Ra = 105, and τ = 0.800
Conclusions
Two-dimensional laminar natural convection in an inclined
square enclosure containing uniform internal heat generation is
5 Copyright c© 2007 by ASME
Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/28/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−1.37
−0.999
−0.628
−0.258
1.22
0.854
0.483
0.113
0.483
−1.37
−0.628
−0.999
0.113
X
Y
Figure 12: Streamline plot for γ = 00,Ra = 105, and τ = 0.870
studied here. Two-dimensional unsteady governing equations
are solved by implicit scheme and ADI method. SIMPLE al-
gorithm with staggered grid is used for calculation. For incli-
nation angles of 450, 300 and 150 and Rayleigh numbers up to
1.5×105, steady-state solutions are approached after initial tran-
sient period. For these cases two counter-rotating rolls of the
fluid are present in the cavity. In case of horizontal cavity, for
Rayleigh number greater than or equal to 5 ×104, periodic solu-
tions are obtained. For such cases, two unstable secondary rolls
are present near the center of top wall, in addition to main rolls.
The secondary rolls are dissipated and recreated during one pe-
riod of oscillation.
REFERENCES
[1] De Vahl Davis G. Laminar natural convection in an enclosed
rectangular cavity. Int. J. Heat Mass Transfer, 11:1675–
1693, 1968.
[2] Markatos N. C. and Pericleous K. A. Laminar and turbulent
natural convection in an enclosed cavity. Int. J. Heat Mass
Transfer, 27(5):755–772, 1984.
[3] De Vahl Davis G. and Jones I. P. Natural convection in a
square cavity: A comparison exercise. Int. J.For Numerical
Methods in Fluids, 3:227–248, 1983.
[4] De Vahl Davis G. Natural convection of air in a square cav-
ity: A bench mark solution. Int. J.For Numerical Methods in
Fluids, 3:249–264, 1983.
[5] Lee J. H. and Goldstein R. J. An experimental study on
natural convection heat transfer in an inclined square enclo-
sure containing internal energy sources. J. Heat Transfer,
110:345–349, 1988.
[6] Anderson D. A., Pletcher R. H., and Tannehill J. C. Com-
putational Fluid Mechanics and Heat Transfer. Taylor and
Francis Publishers, 1997.
[7] Kulacki F. A. and Goldstein R. J. Thermal convection
in a horizontal fluid layer with uniform volumetric energy
source. J. Fluid Mech., 55(2):271–287, 1972.
6 Copyright c© 2007 by ASME
Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/28/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use


Study of natural convection in inclined square enclosure with uniform heat generation.

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=E55C7E90414EC9EFC7D55E95C7FE58BC?doi=10.1.1.1069.4093&rep=rep1&type=pdf

Study of natural convection in inclined square enclosure with uniform heat generation.

Abstract

Similar works

Full text

Available Versions

CiteSeerX