Page 142 - Aerodynamics for Engineering Students

Potential flow 125

3.3.7 A source set upstream of an equal sink
in a uniform stream

The stream function due to this combination is:

(3.31)

Here the first term represents a source and sink combination set with the source to
the right of the sink. For the source to be upstream of the sink the uniform stream
must be from right to left, i.e. negative. If the source is placed downstream of the sink
an entirely different stream pattern is obtained.
The velocity potential at any point in the flow due to this combination is given by:
m I1
$=-ln-- Ursine (3.32)
27r r2
or
m 2+y2+c2-2xc
+=-ln - ux (3.33)
47r x2+y2+$+2xc
The streamline $ = 0 gives a closed oval curve (not an ellipse), that is symmetrical
about the Ox and Oy axes. Flow of stream function $ greater than $ = 0 shows the
flow round such an oval set at zero incidence in a uniform stream. Streamlines can be
obtained by plotting or by superposition of the separate standard flows (Fig. 3.18).
The streamline $ = 0 again separates the flow into two distinct regions. The first is
wholly contained within the closed oval and consists of the flow out of the source and
into the sink. The second is that of the approaching uniform stream which flows
around the oval curve and returns to its uniformity again. Again replacing $ = 0 by a
solid boundary, or indeed a solid body whose shape is given by $ = 0, does not
influence the flow pattern in any way.
Thus the stream function $I of Eqn (3.31) can be used to represent the flow around
a long cylinder of oval section set with its major axis parallel to a steady stream. To
find the stream function representing a flow round such an oval cylinder it must be
possible to obtain m and c (the strengths of the source and sink and distance apart) in
terms of the size of the body and the speed of the incident stream.

Fig. 3.18