ASCHT 2015
The Asian Symposium on Computational Heat Transfer and Fluid Flow
June 21 – 24 2015
Busan, Korea
EFFECTS OF THERMAL BOUNDARY CONDITIONS ON NATURAL
CONVECTION IN A SQUARE ENCLOSURE WITH AN INNER
CIRCULAR CYLINDER
Minsung Kima and Man Yeong Hab*
a
b
Department, School or etc., Address, City, Postal, Country
School of Mechanical Engineering, Pusan National University, San 30, Jangjeon-dong, Geumjeong-gu, Busan 609735, Korea
E-mail: kms514v@pusan.ac.kr (1st author E-mail address)
ABSTRACT
Two-dimensional numerical simulations are carried out for natural convection in an enclosure with an hot inner
cylinder located at the center for four different Rayleigh numbers of , , and . The immersed boundary method
(IBM) was used to handle the virtual surface of the inner circular cylinder with a no-slip boundary condition. The
Prandtl number Pr was taken to be 0.7 corresponding to that of air. This study focuses on the effect of the
temperature variation of bottom wall of the enclosure on thermal and flow structures of natural convection. The
results indicate negligible changes in thermal and flow structures based on variations in the size of the local heating
zone on the bottom
Keyword : Natural convection, variation temperature of bottom wall, vortex structure, heat transfer.
NOMENCLATURE
aZY
[-]
3
CGTP
[m K/W]
T
x
y
z
[K]
[m]
[m]
[m]
Special characters
α
[-]
A
[m]
b
ε
γ
[m]
[-]
[-]
Z
[m]
Subscripts
C
eff
ext
int
M
max
y-z view aspect ratio of rectangular region between two
adjacent cooling layers
Coefficient dependent on geometric, thermal and material
property values
Temperature
Cartesian axis direction
Cartesian axis direction
Cartesian axis direction
Volume fraction ratio
Half centre-to-centre offset distance between
neighbouring cooling inserts in the x direction
Half y directional dimension of rectangular cooling insert
Relative E% value in terms of maximum E%
Ratio between thermal conductivities of cooling insert
and the heat-generating medium
Half z directional dimension of rectangular cooling insert
Cooling layer
Volumetric effective expression
External: Towards external heat sink
Internal: Between cooling and heat-generating layers
Heat generating medium
Maximum
INTRODUCTION
Natural convection phenomena are encountered in many
practical applications such as the energy conservation in
buildings, cooling of electronic equipment, cooling of nuclear
reactor systems, solar engineering, and environmental and
geothermal fluid dynamics. These application areas include the
cooling of electronic devices, double-pane windows, heating
and cooling of building, refrigerators, room ventilating, heat
exchangers, solar collectors and so on. Due to a wide range of
applications, fundamental studies on the natural convection in
an enclosure, and Rayleigh-Bénard convection in a horizontal
layer of the fluid confined between two parallel plates, have
been performed by many investigators over the last few
decades. Gelfgat [1] provided a complete numerical solution of
a formulated benchmark problem devoted to the parametric
study of Rayleigh-Bénard instability in rectangular twodimensional (2-D) and three-dimensional (3-D) boxes. The
results of the parametric calculations were presented in [1] as
characteristic curves showing the dependence of the critical
Rayleigh number on the aspect ratio of the cavity. Quertatani et
al. [2] numerically performed a study on a classical RayleighBénard convection problem, and reported the characteristics of
Table 1 Grid dependency test results for the surface-averaged
Nusselt number around the inner cylinder Nucyl when
NUMERICAL METHODOLOGY
A schematic of the system is shown in Figure 1. The system
consists of a square enclosure with sides of length L , within
which a circular cylinder of radius R (= 0.2 L ) is located at the
center. As shown in Figure 1, the top and side walls of the
enclosure are kept at a constant low temperature Tc , whereas
the bottom wall of the enclosure is kept at a constant
temperature Tb whose value changes as a parameter in the
present computation. The cylinder surface is kept at a constant
high temperature Th .
Ra  106 and b  0.0 .
Grid number
Nucyl
202  202
252  252
302  302
352  352
402  402
4.9921
4.9902
4.9978
4.9998
5.0009
Difference (%)
0.18
0.21
0.06
0.02
-
RESULTS AND DISCUSSION
Figure 3 shows the distribution of isotherms and streamlines
in the enclosure for the case of b  0.0 for different Rayleigh
Figure 1 Computational domain and coordinate system
along with boundary conditions
The
governing
equations
describing
unsteady
incompressible viscous fluid flow and thermal fields are the
continuity, momentum, and energy conservation equations in
their non-dimensional forms, which are defined as
ui
q  0
xi
(1)
ui
u
 2ui
P
uj i  
 Pr
 RaPri 2  fi (2)
t
x j
xi
x j x j


 2
uj

h
t
x j x j x j
(3)
where the dimensionless variables in equations (1)-(3) are
defined as follows:
t
T *  Tc*
xi*
t *
P* L2
ui* L
x

,
,
,
,


P

u

i
i
L
L2
 2

Th*  Tc*
(4)
In equation (4), the superscript (*) denotes the dimensional
variables;  and  represent the density and the thermal
diffusivity of the fluid, respectively; P ,  , and t represent the
dimensionless pressure, the dimensionless temperature, and the
dimensionless time, respectively; xi represents the
dimensionless Cartesian coordinates; and ui represents the
numbers of Ra  103 , 104 , 105 and 106 . The flow and
thermal fields eventually reached the steady state for all
simulation cases at four different Rayleigh numbers considered.
In terms of general fluid motion occurring due to natural
convection as shown in Figure 3, the heated lighter fluid is
lifted along the hot surface of the inner cylinder due to
buoyancy as a driving force. As the fluid flow approaches the
cold top wall of the enclosure, it becomes gradually colder and
denser. The fluid cools further as it moves along the cold top
wall in the lateral direction. Finally, a denser fluid cooled
moves downward along the cold side walls of the enclosure.
Thus, the main circulation of the convection flow is formed in
the enclosure.
CONCLUSIONS
Two-dimensional numerical simulations were performed for
natural convection in a square enclosure with an hot inner
cylinder for four different Rayleigh numbers of 103 , 104 , 105
and 106 , by using the immersed boundary method to provide
an in-depth analysis of various phenomena associated with
natural convection such as the formation of the vortex structure
and the corresponding heat transfer based on the various
temperature conditions of bottom wall.
When Ra  103 and Ra  104 , variations in the value of
the bottom temperature have little effect on thermal and flow
structures, although there are small variations in the convection
velocity in the enclosure. Consequently, there is a little
difference in the overall heat transfer capacity in terms of the
Nusselt number between the top wall and the cylinder surface
based on variations  b . A primary vortex pair showing a
mirror symmetric pattern at Ra  103 and Ra  104 . For the
cases of Ra  103 , the temperature variation of the bottom.
ACKNOWLEDGEMENT
This research was supported by ABCDE of National
Research Foundation of Korea(NRF).
REFERENCES
[1] Van Wyk J.D., Strydom J.T., Zhao L., and Chen R., Review of the
development of high density integrated technology for
electromagnetic power passives, Proceedings of the 2nd
International Conference on Integrated Power Systems (CIPS),
2002, pp. 25-34
[2] Strydom, J.T., and Van Wyk, J.D., Electromagnetic design
optimisation of planar integreted passive modules, Proceedings of
the 33rd I.E.E.E. Power Electronics Specialist Conference (PESC),
Australia, Vol. 2, June 2002, pp. 573-578
[3] Strydom, J.T., Van Wyk, J.D., and Ferreira, J.A., Some units of
integrated L-C-T modules at 1MHz, I.E.E.E Transactions on
Industry Applications, Vol. 37, No. 3, May/June 2001, pp. 820-828
[4] Dirker, J., Liu, W., Van Wyk, J.D., Meyer, J.P., and Malan, A.G.,
Embedded solid state heat extraction in integrated power electronic
modules, Accepted for publication in the I.E.E.E. Transactions on
Power Electronics, vol. 20, No. 3, May 2005.
[5] Almogbel M., and Bejan A., Conduction trees with spacing at tips,
International Journal of Heat and Mass Transfer, Vol. 42, 1999, pp.
3739-3756
[6] Bejan A., and Almogbel M., Constructal T-shaped fins,
International Journal of Heat and Mass Transfer, Vol. 43, 2000, pp.
2101-2115
[7] Almogbel M., and Bejan A., Cylindrical trees of pin fins,
International Journal of Heat and Mass Transfer, Vol. 43, 2000, pp.
4285-4297