Environmental Engineering Reference
In-Depth Information
=
∂v
z
∂
∂
T
v
r
0
at r
∗
=
z
∗
≤
r
∗
=
r
∗
=
0
,
0
≤
L
,
(7)
∂
∂
T
v
n
=
v
t
0
at r
∗
=
z
∗
/ʻ),
z
∗
≤
=
n
∗
=
R
+
a
cos
(
2
ˀ
0
≤
L
,
(8)
∂
where
∗
indicates dimensional variables,
r
and
z
are the radial and axial coordinates,
v
r
and
v
z
are the radial and axial components of velocity, respectively,
ˁ
is the density,
P
is the modified pressure accounting for the hydrostatic effect,
ʽ
is viscosity,
g
is
ʲ
ʱ
the acceleration of gravity,
is the thermal expansion coefficient,
is the thermal
diffusivity,
T
m
=
(
T
1
+
T
2
)/
v
n
and
v
t
ar
e
the norma
l a
n
d
2 is the mean temperature,
tangen
ti
al ve
lo
city components to the wavy sidewall, i.e.
v
n
=
v
·
n
and
v
t
=
v
·
t
,
where
n
and
t
are normal and tangential unit vectors to the wavy wall.
A scale analysis suggests the following dimensionless variables:
z
∗
/
r
∗
/
R
2
v
z
/ʽ
v
r
/ʽ
;
z
=
L
;
r
=
R
;
v
z
=
L
;
v
r
=
R
(9)
T
∗
−
ʵ
=
R
/
L
;
ʛ
=
ʻ/
L
;
ʸ
=
(
T
m
)/(
T
H
−
T
m
)
;
where
is the dimensionless
temperature, and
T
H
is the higher temperature between
T
1
and
T
2
.
The explicit dependence of momentum equations on pressure is dropped and the
dimensionless stream function
ʵ
is the aspect ratio,
ʛ
is the dimensionless wavelength,
ʸ
)
=
ˈ
∗
/μ
ˈ(
r
,
z
L
is introduced as
∂ˈ
∂
r
=
r
v
z
,
(10)
∂ˈ
∂
z
=−
r
v
r
.
(11)
The resulting motion equation is
4
3
2
4
3
A
∂
ˈ
+ B
∂
ˈ
+ C
∂
ˈ
+ D
∂ˈ
∂
∂
ˈ
z
2
+ F
∂
ˈ
r
+ E
(12)
r
4
r
3
r
2
r
2
z
2
∂
∂
∂
∂
∂
∂
r
∂
3
2
2
3
4
+ G
∂
ˈ
z
+ H
∂
ˈ
ˈ
z
2
+ J
∂
ˈ
+ K
∂
ˈ
= L
∂ʸ
∂
z
+ I
r
,
r
2
z
3
z
4
∂
∂
∂
r
∂
∂
∂
∂
and the energy equation is as follows
2
2
N
∂ʸ
∂
z
+ O
∂ʸ
r
+ P
∂
ʸ
+ Q
∂
ʸ
=
0
,
(13)
∂
∂
r
2
∂
z
2
where coefficients
are given in Table
1
of the Appendix. The Rayleigh and
Prandtl numbers are defined as
Ra
A
-
Q
L
3
=
g
ʲ(
T
H
−
T
m
)
/ʽʱ
and
Pr
=
ʽ/ʱ
, respec-
tively.
An analytical coordinate transformation was used to obtain a coordinate frame for
computation in which the irregular domain fits into a square. The new dependent and
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