Civil Engineering Reference
In-Depth Information
equation, so that the stream function can be represented by [3]
3
(
x
,
y
)
(8.48)
where
a
and
b
are constants to be determined. Again, we know that each inde-
pendent solution in Equation 8.48 must satisfy Equation 8.11 and, recalling that
the stream function must take on constant value on an impenetrable surface, we
can express the boundary conditions on each solution as
1
=
(
x
,
y
)
=
1
(
x
,
y
)
+
a
2
(
x
,
y
)
+
b
Uy
b
on
S
1
1
=
0
on
S
2
and
S
3
2
=
0
on
S
1
and
S
3
2
=
1
on
S
2
(8.49)
3
=
0
on
S
1
and
S
2
1
on
S
3
To obtain a solution for the flow problem depicted in Figure 8.9, we must
1.
Obtain a solution for
1
satisfying the governing equation and the boundary
conditions stated for
1
.
2.
Obtain a solution for
2
satisfying the governing equation and the boundary
conditions stated for
2
.
3.
Obtain a solution for
3
satisfying the governing equation and the boundary
conditions stated for
3
.
4.
Combine the results at (in this case) two points, where the velocity or
stream function is known in value, to determine the constants
a
and
b
in
Equation 8.48. For this example, any two points on section
a-b
are appro-
priate, as we know the velocity is uniform in that section.
As a practical note, this procedure is
not
generally included in finite element
software packages. One must, in fact, obtain the three solutions and hand calcu-
late the constants
a
and
b
, then adjust the boundary conditions (the constant val-
ues of the stream function) for entry into the next run of the software. In this case,
not only the computed results (stream function values, velocities) but the values
of the computed constants
a
and
b
are considerations for convergence of the
finite element solutions. The procedure described may seem tedious, and it is to
a certain extent, but the alternatives (other than finite element analysis) are much
more cumbersome.
3
=
8.5 INCOMPRESSIBLE VISCOUS FLOW
The idealized inviscid flows analyzed via the stream function or velocity poten-
tial function can reveal valuable information in many cases. Since no fluid is
truly inviscid, the accuracy of these analyses decreases with increasing viscosity
