Civil Engineering Reference
In-Depth Information
Table 8.1
Selected Nodal Stream Function and Velocity Values for Solution of Example 8.1
Node
FE
Exact
V
FE
V
Exact
1
0
0
75.184
80
2
0
0
1.963
0
8
0
0
38.735
38.4
16
123.63
122.17
40.533
40.510
20
142.48
137.40
44.903
42.914
21
100.03
99.37
47.109
45.215
22
67.10
64.67
51.535
49.121
23
40.55
39.36
57.836
55.499
24
18.98
18.28
68.142
65.425
45
67.88
65.89
41.706
40.799
46
103.87
100.74
42.359
41.018
This solution is actually for a cylinder in a uniform stream of indefinite extent in both the
x
and
y
directions (hence, the use of the oxymoron, approximately exact) but is sufficient
for comparison purposes. Table 8.1 lists values of
obtained by the finite element solu-
tion and the preceding analytical solution at several selected nodes in the model. The
computed magnitude of the fluid velocity at those points is also given. The nominal errors
in the finite element solution versus the analytical solution are about 4 percent for the
value of the stream function and 6 percent for the velocity magnitude. While not shown
here, a refined element mesh consisting of 218 elements was used in a second solution
and the errors decreased to less than 1 percent for both the stream function value and the
velocity magnitude.
Earlier in the chapter, the analogy between the heat conduction problem and
the stream function formulation is mentioned. It may be of interest to the reader
to note that the stream function solution presented in Example 8.1 is generated
using a commercial software package and a two-dimensional heat transfer
element. The particular software does not contain a fluid element of the type
required for the problem. However, by setting the thermal conductivities to unity
and specifying zero internal heat generation, the problem, mathematically, is the
same. That is, nodal temperatures become nodal values of the stream function.
Similarly, spatial derivatives of temperature (flux values) become velocity com-
ponents if the appropriate sign changes are taken into account. The mathematical
similarity of the two problems is further illustrated by the finite element solution
of the previous example using the velocity potential function.
EXAMPLE 8.2
Obtain a finite element solution for the problem of Example 8.1 via the velocity potential
approach, using, specifically, the heat conduction formulation modified as required.
