Environmental Engineering Reference
In-Depth Information
independent variables were analytically introduced to transform the mathematical
representation of the conservation principles before the equations were discretized.
The suitable transformation states
ʾ
=
z
,
(14)
r
cos
2
ˀ
ʛ
,
ʷ
=
(15)
1
+
ʓ
where
R
is the dimensionless wave amplitude. Once the coordinate trans-
formation was applied, Eqs.
12
and
13
become
ʓ
=
a
/
2
3
4
2
3
4
3
b
∂
ˈ
∂ʾ
c
∂
ˈ
∂ʾ
d
∂
ˈ
∂ʾ
e
∂
ˈ
∂ʾ∂ʷ
+
f
∂
ˈ
g
∂
ˈ
∂ʾ
h
∂
ˈ
∂ʾ∂ʷ
+
+
+
∂ʷ
+
+
2
2
4
2
3
2
∂ʾ
4
4
2
3
4
i
∂
ˈ
∂
ˈ
k
∂ˈ
∂ʷ
l
∂
ˈ
∂ʷ
m
∂
ˈ
∂ʷ
n
∂
ˈ
∂ʷ
+
∂ʾ
+
j
2
+
+
+
+
=
0
,
(16)
3
2
2
3
4
∂ʷ
∂ʷ
∂ʾ
2
2
2
o
∂ʸ
p
∂
ʸ
∂ʾ
q
∂
ʸ
∂ʾ∂ʷ
+
s
∂ʸ
t
∂
ʸ
∂ʷ
∂ʾ
+
2
+
∂ʷ
+
=
0
,
(17)
2
subject to the following boundary conditions
ˈ
=
∂ˈ
∂ʾ
=
0
,ʸ
=
ʸ
2
at
ʾ
=
0
,
0
≤
ʷ
≤
1
;
(18)
ˈ
=
∂ˈ
∂ʾ
=
0
,ʸ
=
ʸ
1
at
ʾ
=
1
,
0
≤
ʷ
≤
1
;
(19)
ˈ
=
∂ˈ
∂ʷ
=
∂ʸ
∂ʷ
=
0
at
ʷ
=
0
,
0
≤
ʾ
≤
1
;
(20)
A
∂ˈ
∂ʷ
B
∂ˈ
∂ʾ
∂ʸ
∂
ˈ
=
+
=
n
=
0
at
ʷ
=
1
,
0
≤
ʾ
≤
1
;
(21)
again, coefficients
b
-
t
are given in Table
2
of the Appendix, while the coefficients
A and B are given in Table
3
. Equations
16
and
17
were discretized using finite
differences. The resulting square domain was meshed with 110
110 nodes. A
computational code was programmed using the Fortran 95 language to solve the
algebraic equations applying the
LU
inversion matrix algorithm. Since the motion
equation is a fourth order non-linear differential equation an iterative scheme was
employed. The motion equation was found to need around ten iterations to generate a
total residual of order 10
−
9
, whichwas the selected criterion to consider the numerical
solution convergence. The stream function solution was then applied to the energy
equation to compute the temperature distribution. Five iterations were done between
the energy and motion equations to generate a grand total residual of order 10
−
9
.
The total heat transfer through the horizontal walls is
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