Civil Engineering Reference
In-Depth Information
■
Solution
First let us note the analogies
u
=−
∂
=−
k
x
∂
T
∂
x
∂
x
⇒
q
x
v
=−
∂
=−
k
y
∂
T
∂
y
∂
y
⇒
q
y
so that, if
k
x
=
k
y
=
1
, then the velocity potential is directly analogous to temperature and
the velocity components are analogous to the respective flux terms. Hence, the boundary
conditions, in terms of thermal variables become
q
x
=
Uq
y
=
0
on
a-b
q
x
=
q
y
=
0
on
b-c
and
a-e-d
T
=
constant
=
0
on
c-d
(the value is arbitrary)
Figure 8.8 shows a coarse mesh finite element solution that plots the lines of constant
velocity potential
(in the thermal solution, these lines are lines of constant temperature,
4
19
20
21
5
41
47
22
37
33
23
46
35
40
32
24
27
39
30
25
36
44
26
1
31
34
42
45
7
8
28
9
29
43
10
3
15
14
13
12
11
2
Figure 8.8
Lines of constant velocity potential for the finite
element solution of Example 8.2.
