Civil Engineering Reference
In-Depth Information
Equation 8.13. The vector product of two nonzero vectors is zero only if the vec-
tors are parallel. Therefore, at any point on a streamline, the fluid velocity vector
is tangent to the streamline.
8.3.1 Finite Element Formulation
Development of finite element characteristics for fluid flow based on the stream
function is straightforward, since (1) the stream function
(
x
,
y
)
is a
scalar
function from which the velocity vector components are derived by differen-
tiation and (2) the governing equation is essentially the same as that for two-
dimensional heat conduction. To understand the significance of the latter point,
reexamine Equation 7.23 and set
=
T
,
k
x
=
k
y
=
1,
Q
=
0
, and
h
=
0
.
The
result is the Laplace equation governing the stream function.
The stream function over the domain of interest is discretized into finite
elements having
M
nodes:
M
(
x
,
y
)
=
N
i
(
x
,
y
)
i
=
[
N
]
{}
(8.14)
i
=
1
Using the Galerkin method, the element residual equations are
N
i
(
x
,
y
)
∂
d
x
d
y
2
2
+
∂
=
0
i
=
1,
M
(8.15)
x
2
y
2
∂
∂
A
(
e
)
or
[
N
]
T
∂
d
x
d
y
2
2
+
∂
=
0
(8.16)
x
2
y
2
∂
∂
A
(
e
)
Application of the Green-Gauss theorem gives
[
N
]
T
∂
[
N
]
T
∂
∂
∂
∂
∂
∂
∂
[
N
]
T
n
x
d
S
−
d
x
d
y
+
n
y
d
S
x
x
x
y
S
(
e
)
A
(
e
)
S
(
e
)
[
N
]
T
∂
∂
∂
∂
−
d
x
d
y
=
0
(8.17)
y
y
A
(
e
)
where
S
represents the element boundary and
(
n
x
,
n
y
)
are the components of the
outward unit vector normal to the boundary. Using Equations 8.10 and 8.14
results in
∂
d
x
d
y
[
N
]
T
∂
[
N
]
T
∂
∂
[
N
]
∂
+
∂
∂
[
N
]
∂
[
N
]
T
(
un
y
−
{} =
vn
x
)d
S
x
x
y
y
A
(
e
)
S
(
e
)
(8.18)
and this equation is of the form
k
(
e
)
{} =
f
(
e
)
(8.19)
