Page 147 - Aerodynamics for Engineering Students

130 Aerodynamics for Engineering Students

It should be noted that the terms in the stream functions must be opposite in sign to
obtain the useful results below. Here again the source must be upstream of the sink in
the flow system. Equation (3.39) converted to rectangular coordinates gives:

IJ
$=--- uy (3.40)
2TX2 + y2
and for the streamline $ = 0

i.e.
y=o or x2+y2=- IJ
2nu
This shows the streamline - 0 to consist of the Ox axis together with a circle,
centre 0, of radius d&l a (say).
Alternatively by converting Eqn (3.39) to polar coordinates:
$2-
8
8
sin
Ur
sin
-
2nr
Therefore
giving
sin8=O so 8=0 or fn
or
’
_- ur =O giving r =
2~r /&=a
the two solutions as before.
The streamline $ = 0 thus consists of a circle and a straight line on a diameter
produced (Fig. 3.22). Again in this case the streamline $ = 0 separates the flow into
two distinct patterns: that outside the circle coming from the undisturbed flow a long

Fig. 3.22 Streamlines due to a doublet in a uniform stream