Page 131 - Aerodynamics for Engineering Students

1 14 Aerodynamics for Engineering Students

Since the flow due to a line vortex gives streamlines that are concentric circles, the
equipotentials, shown to be always normal to the streamlines, must be radial lines
emanating from the vortex, and since
qn = 0, q5is a function of 8, and

Therefore
r
d+ =-de
27r
and on integrating
r
@ = -6 + constant
2n
By defining q5 = 0 when 8 = 0:
r
+=-e (3.11)
2n
Compare this with the stream function for a source, i.e.

Also compare the stream function for a vortex with the function for a source. Then
consider two orthogonal sets of curves: one set is the set of radial lines emanating
from a point and the other set is the set of circles centred on the same point. Then, if
the point represents a source, the radial lines are the streamlines and the circles are the
equipotentials. But if the point is regarded as representing a vortex, the roles of
the two sets of curves are interchanged. This is an example of a general rule: consider
the streamlines and equipotentials of a two-dimensional, continuous, irrotational
flow. Then the streamlines and equipotentials correspond respectively to the equi-
potentials and streamlines of another flow, also two-dimensional, continuous and
irrotational.
Since, for one of these flows, the streamlines and equipotentials are orthogonal,
and since its equipotentials are the streamlines of the other flow, it follows that the
streamlines of one flow are orthogonal to the streamlines of the other flow. The same
is therefore true of the velocity vectors at any (and every) point in the two flows. If
this principle is applied to the sourcesink pair of Section 3.3.6, the result is the flow
due to a pair of parallel line vortices of opposite senses. For such a vortex pair,
therefore the streamlines are the circles sketched in Fig. 3.17, while the equipotentials
are the circles sketched in Fig. 3.16.

3.3.3 Uniform flow
Flow of constant velocity parallel to Ox axis from lei? to right
Consider flow streaming past the coordinate axes Ox, Oy at velocity U parallel to Ox
(Fig. 3.9). By definition the stream function $ at a point P(x, y) in the flow is given by
the amount of fluid crossing any line between 0 and P. For convenience the contour