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k
r
∗
∂
R
+
a
T
dr
∗
.
q
w
=−
2
ˀ
(22)
∂
z
∗
0
z
∗
=
0
,
L
If
h
a
v
is the average convection heat transfer coefficient on the horizontal walls,
then the average Nusselt number,
Nu
a
v
=
h
a
v
L
/
k
, is computed as
∂ʸ
∂ʾ
2
1
0
ʷ
Nu
a
v
=−
ʷ.
d
(23)
ʾ
=
0
,
1
4 Results and Discussion
There are two possible cases for the natural convection here studied, the cylindrical
enclosure heated frombelow, and the case where the cavity is heated from the top. For
both cases the wavy wall is assumed to be adiabatic. The dimensionless parameters
were varied in order to evaluate their effect on the thermal convection. The studied
parameters were, for the aspect ratio
ʵ
=
.
,
.
,
.
0
1
0
3
0
5; dimensionless wavelength
ʛ
=
1
/
10
,
1
/
5
,
1
/
3; dimensionless amplitude
ʓ
=
0
.
05
,
0
.
1
,
0
.
3; Rayleigh number
from 10
3
to 10
6
, and constant Prandtl number,
Pr
=
7.
4.1 Enclosure Heated from Below
Figure
2
shows the dimensionless temperature and stream function for Rayleigh
numbers from 10
3
to 10
6
. The dimensionless temperature
0 corresponds to
the average temperature
T
m
. When the Rayleigh number is of order 10
3
, the stream
function shows multiple convection cells near the wavy wall. Moreover, the cavity
presents a stratified temperature distribution which yields low velocity convection
cells. When the Rayleigh number is
Ra
ʸ
=
10
4
, the fluid flow is little intensified
according to the stream function values, which increase around one order of magni-
tude, nevertheless, the velocity is relatively low and the thermal stratification persists.
For a Rayleigh number of 10
5
the flow presents two convection cells, the lower cell
with a clockwise rotation. Such a flow removes thermal stratification, increasing the
temperature gradient near the upper and lower walls. For a Rayleigh number of 10
6
the flow shows two convection cells, the upper one flowing faster than the lower one,
as shown by the streamlines in Fig.
2
d. Moreover, the temperature distribution for
Ra
=
10
6
presents intense temperature gradients near the upper and lower walls.
The dimensionless wavelength of the wavy wall modifies considerably the con-
vection flow and the heat transfer process, particularly for high Rayleigh numbers.
Figure
3
shows that there is an almost stagnant thermally stratified core and multiple
convection cells near the wavy wall when the dimensionless wavelength is small.
On the other hand, there are two convection cells with no thermal stratification when
=
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