Page 157 - Aerodynamics for Engineering Students

140 Aerodynamics for Engineering Students

It therefore follows from Eqn (3.60) that
%
qR=-=- Q
dR 4rR2
Integration then gives the expression for the velocity potential of a point source as
4=-- Q (3.63)
4rR
In a similar fashion an expression for stream function can be derived using Eqn (3.62)
giving
Q
$ = -- coscp (3.64)
47r

3.4.4 Point source and sink in a uniform axisymmetric flow
Placing a point source and/or sink in a uniform horizontal stream of -U leads to
very similar results as found in Section 3.3.5 for the two-dimensional case with line
sources and sinks.
First the velocity potential and stream function for uniform flow, -U, in the z
direction must be expressed in spherical coordinates. The velocity components qR
and qp are related to -U as follows
qR = -Ucosp and qp = Usinp

Using Eqn (3.60) followed by integration then gives
_- - -Ucoscp 4 4 = -U Rcos cp + f (cp)
dR
-_ - URsincp --f q5 = -URcoscp+g(R)
84
a(p
f(cp) and g(R) are arbitrary functions that take the place of constants of integration
when partial integration is carried out. Plainly in order for the two expressions for q5
derived above to be in agreement f (cp) = g(R) = 0. The required expression for the
velocity potential is thereby given as
+=-URCOS~ (3.65)
Similarly using Eqn (3.62) followed by integration gives
alCl 2 UR2 . U R2
-=
acp -U R cos cpsinp = -- 2 sin2p + $ = - cos2cp + f (R)
4
_- $ = - -
UR2
dR - -U R sin2 cp
2 sin2 cp + g(cp)
Recognizing that cos 2cp = 1 - 2 sin2 cp it can be seen that the two expressions given above
for $ will agree if the arbitrary functions of integration take the values f (R) = - U R2/4
and g(p) = 0. The required expression for the stream function is thereby given as
U RZ
$ = -- sin’p (3.66)
2