Civil Engineering Reference
In-Depth Information
Note that if, for example, a three-node triangular element is used, the velocity
components as defined in Equation 8.32 have constant values everywhere in the
element and are discontinuous across element boundaries. Therefore, a large
number of small elements are required to obtain solution accuracy. Application
of the stream function to a numerical example is delayed until we discuss an
alternate approach, the velocity potential function, in the next section.
8.4 THE VELOCITY POTENTIAL FUNCTION
IN TWO-DIMENSIONAL FLOW
Another approach to solving two-dimensional incompressible, inviscid flow
problems is embodied in the velocity potential function. In this method, we
hypothesize the existence of a
potential function
(
x
,
y
)
such that
=−
∂
∂
u
(
x
,
y
)
x
(8.33)
=−
∂
∂
v
(
x
,
y
)
y
and we note that the velocity components defined by Equation 8.33 automati-
cally satisfy the irrotationality condition. Substitution of the velocity definitions
into the continuity equation for two-dimensional flow yields
2
2
∂
u
∂
v
∂
+
∂
x
+
y
=
=
0
(8.34)
x
2
y
2
∂
∂
∂
∂
and, again, we obtain Laplace's equation as the governing equation for 2-D flow
described by a potential function.
We examine the potential formulation in terms of the previous example of a
converging flow between two parallel plates. Referring again to Figure 8.4a, we
now observe that, along the lines on which the potential function is constant, we
can write
∂
∂
+
∂
∂
d
=
d
x
d
y
=−
(
u
d
x
+
v
d
y
)
=
0
(8.35)
x
y
Observing that the quantity
u
d
x
+
v
d
y
is the magnitude of the scalar product of
the velocity vector and the tangent to the line of constant potential, we conclude
that the velocity vector at any point on a line of constant potential is
perpendic-
ular
to the line. Hence, the streamlines and lines of constant velocity potential
(
equipotential lines
) form an orthogonal “net” (known as the
flow net
) as de-
picted in Figure 8.5.
The finite element formulation of an incompressible, inviscid, irrotational
flow in terms of velocity potential is quite similar to that of the stream function
approach, since the governing equation is Laplace's equation in both cases. By
