Civil Engineering Reference
In-Depth Information
■
Velocity Potential
Given the assumptions and geometry, we need consider only one-fourth of the flow field,
as in Figure 8.6b, because of symmetry. The boundary conditions first are stated for
the velocity potential formulation. Along
x
=
0
(
a-b
), we have
u
=
U
=
constant and
v
=
0
. So,
u
(0,
y
)
=
U
=−
∂
∂
x
v
(0,
y
)
=
0
=−
∂
∂
y
and the unit (outward) normal vector to this surface is
(
n
x
,
n
y
)
1, 0)
. Hence,
for every element having edges (therefore, nodes) on
a-b
, the nodal force vector is
known as
=
(
−
U
f
(
e
)
=−
[
N
]
T
(
un
x
+
[
N
]
T
vn
y
)d
S
=
d
S
S
(
e
)
S
(
e
)
and the integration path is simply
d
S
=
d
y
between element nodes. Note the change in
sign, owing to the orientation of the outward normal vector. Hence, the forces associated
with flow into the region are positive and the forces associated with outflow are negative.
(The sign associated with inflow and outflow forces depend on the choice of signs in
Equation 8.33. If, in Equation 8.33, we choose positive signs, the formulation is essen-
tially the same.)
The symmetry conditions are such that, on surface (edge)
c-d
, the
y
-velocity compo-
nents are zero and
x
=
x
c
,
so we can write
=−
∂
d
(
x
c
,
y
)
d
y
v
∂
y
=−
=
0
This relation can be satisfied if
is independent of the
y
coordinate or
(
x
c
,
y
)
is con-
stant. The first possibility is quite unlikely and requires that we assume the solution form.
Hence, the conclusion is that the velocity potential function must take on a constant value
on
c-d
. Note, most important, this conclusion
does not
imply that the
x
-velocity compo-
nent is zero.
Along
b-c
, the fluid velocity has only an
x
component (impenetrability), so we can
write this boundary condition as
∂
∂
n
=
∂
∂
x
n
x
+
∂
∂
y
n
y
=−
(
un
x
+
vn
y
)
=
0
and since
v
=
0
on this edge, we find that all nodal forces are zero along
b-c
,
but the values of the potential function are unknown.
The same argument holds for
a-e-d.
Using the symmetry conditions along this sur-
face, there is no velocity perpendicular to the surface, and we arrive at the same conclu-
sion: element nodes have zero nodal force values but unknown values of the potential
function.
0
and
n
x
=
