Civil Engineering Reference
In-Depth Information
Table 8.2
Velocity Components at Selected Nodes in Example 8.2
Node
u
v
4
40.423
0.480
19
41.019
0.527
20
42.309
0.594
21
43.339
0.516
5
43.676
0.002
or
isotherms
). A direct comparison between this finite element solution and that described
for the stream function approach is not possible, since the element meshes are different.
However, we can assess accuracy of the velocity potential solution by examination of the
results in terms of the boundary conditions. For example, along the upper horizontal
boundary, the
y
-velocity component must be zero, from which it follows that lines of con-
stant
must be perpendicular to the boundary. Visually, this condition appears to be rea-
sonably well-satisfied in Figure 8.8. An examination of the actual data presents a slightly
different picture. Table 8.2 lists the computed velocity components at each node along the
upper surface. Clearly, the values of the
y
-velocity component
v
are not zero, so addi-
tional solutions using refined element meshes are in order.
Observing that the stream function and velocity potential methods are
amenable to solving the same types of problems, the question arises as to which
should be selected in a given instance. In each approach, the stiffness matrix is
the same, whereas the nodal forces differ in formulation but require the same
basic information. Hence, there is no significant difference in the two proce-
dures. However, if one uses the stream function approach, the flow is readily
visualized, since velocity is tangent to streamlines. It can also be shown [2] that
the difference in value of two adjacent streamlines is equal to the flow rate (per
unit depth) between those streamlines.
8.4.1 Flow around Multiple Bodies
For an ideal (inviscid, incompressible) flow around multiple bodies, the stream
function approach is rather straightforward to apply, especially in finite element
analysis, if the appropriate boundary conditions can be determined. To begin
the illustration, let us reconsider flow around a cylinder as in Example 8.1. Ob-
serving that Equation 8.11 governing the stream function is linear, the principle
of superposition is applicable; that is, the sum of any two solutions to the equa-
tion is also a solution. In particular, we consider the stream function to be given
by
2
(
x
,
y
)
(8.42)
where
a
is a constant to be determined. The boundary conditions at the horizon-
tal surfaces (
S
1
)
are satisfied by
1
, while the boundary conditions on the surface
(
x
,
y
)
=
1
(
x
,
y
)
+
a
