Page 159 - Aerodynamics for Engineering Students

142 Aerodynamics for Engineering Students

If this source-sink pair is placed in a uniform stream -U in the z direction it
generates the flow around a body of revolution known as a Rankine body. The shape
is very similar to the two-dimensional Rankine oval shown in Fig. 3.18 and described
in Section 3.3.7.

3.4.5 The point doublet and the potential flow around
a sphere
A point doublet is produced when the source-sink pair in Fig. 3.30 become infinitely
close together. This is closely analogous to line doublet described in Section 3.3.8.
Mathematically the expressions for the velocity potential and stream function for a
point doublet can be derived from Eqns (3.70) and (3.71) respectively by allowing
a + 0 keeping p = 2Qa fixed. The latter quantity is known as the strength of the
doublet.
If a is very small a2 may be neglected compared to 2Ra cos cp in Eqn (3.70) then it
can be written as
1
{ R2 cos2 cp + R2 sin2 cp + 2aR cos cp} 'I2
1
- 1
{ R2 cos2 cp + R2 sin2 cp - 2aR cos cp}lI2

1 1 1 (3.72)
{ 1 + 2(a/R) cos cp}'I2 - { 1 - 2(a/R) cos cp}l/'
On expanding
1
-- 1
&E - lTF-X+ **.
2
Therefore as a --t 0 Eqn (3.72) reduces to
Q
~=-(l--coscp-1--coscp
a
4sR R R
Qa
-
- -- coscp = -- coscp (3.73)
2rR2 4sR2
In a similar way write

a a
=coscp~-~-cos2cp
R R
Thus as a + 0 Eqn (3.71) reduces to
$=- Qa (1 - cos cp) = - cp (3.74)
2
P
2
sin
21rR 41rR
The streamline patterns corresponding to the point doublet are similar to those
depicted in Fig. 3.20. It is apparent from this streamline pattern and from the form
of Eqn (3.74) that, unlike the point source, the flow field for the doublet is not