Civil Engineering Reference
In-Depth Information
y
b
c
a
x
d
c
b
(a)
S
2
S
1
S
3
S
4
(b)
Figure 8.10
(a) Velocity of fully developed flow. (b) Boundary conditions.
■
Solution
For fully developed flow, the velocity profiles at
a-b
and
c-d
are parabolic, as shown in
Figure 8.10a. Denoting the maximum velocities at these sections as
U
ab
and
U
cd
, we have
u
(
x
a
,
y
)
=
U
ab
1
−
y
2
y
b
v
(
x
a
,
y
)
=
0
u
(
x
c
,
y
)
=
U
cd
1
−
y
2
y
b
v
(
x
c
,
y
)
=
0
The volume flow rate is obtained by integrating the velocity profiles as
y
b
y
d
Q
=
2
u
(
x
a
,
y
)d
y
=
2
u
(
x
c
,
y
)d
y
0
0
Substituting the velocity expressions and integrating yields
3
Q
4
y
d
and thus the velocity components at all element nodes on
a-b
and
c-d
are known.
3
Q
4
y
b
U
ab
=
U
cd
=
