Civil Engineering Reference
In-Depth Information
so
2
=
y
b
U
ab
. At any point on
a-b
, we have
=
(
2
/
y
b
)
y
=
U
ab
y
, so the
value of the stream function at any finite element node located on
a-b
is known.
Similarly, it can be shown that
=
(
2
/
y
c
)
y
=
U
ab
(
y
b
/
y
c
)
y
along
c-d
, so nodal
values on that line are also known. If these arguments are carefully considered,
we see that the boundary conditions on
at the “corners” of the domain are con-
tinuous and well-defined.
Next we consider the force conditions across sections
a-b
and
c-d
. As noted,
the
y-
velocity components along these sections are zero. In addition, the
y
com-
ponents of the unit vectors normal to these sections are zero as well. Using these
observations in conjunction with Equation 8.21, the nodal forces on any element
nodes located on these sections are zero. The occurrence of zero forces is equiv-
alent to stating that the streamlines are normal to the boundaries.
If we now utilize a mesh of triangular elements (for example), as in Fig-
ure 8.4c, and follow the general assembly procedure, we obtain a set of global
equations of the form
[
K
]
{}={
F
}
(8.23)
The forcing function on the right-hand side is zero at all interior nodes. At the
boundary nodes on sections
a-b
and
c-d
, we observe that the nodal forces are zero
also. At all element nodes situated on the line
y
=
0
, the nodal values of the
stream function are
=
0
, while at all element nodes on the upper plate profile
the values are specified as
=
y
b
U
ab
. The
=
0
conditions are analogous to
the specification of zero displacements in a structural problem. With such con-
ditions, the unknowns are the forces exerted at those nodes. Similarly, the speci-
fication of nonzero value of the stream function
along the upper plate profile
is analogous to a specified displacement. The unknown is the force required to
enforce that displacement.
The situation here is a bit complicated mathematically, as we have both zero
and nonzero specified values of the nodal variable. In the following, we assume
that the system equations have been assembled, and we rearrange the equations
such that the column matrix of nodal values is
{
0
}
{
s
}
{
u
}
(8.24)
{}=
where
{
0
}
represents all nodes along the streamline for which
=
0,
{
s
}
rep-
resents all nodes at which the value of
is specified, and
{
u
}
corresponds to all
nodes for which
is unknown. The corresponding global force matrix is
{
F
0
}
{
F
s
}
{
(8.25)
{
F
}=
0
}
and we note that all nodes at which
is unknown are internal nodes at which the
nodal forces are known to be zero.
