Page 130 - Aerodynamics for Engineering Students

Potential flow 1 13

Consider a vortex located at the origin of a polar system of coordinates. But the
flow is irrotational, so the vorticity everywhere is zero. Recalling that the streamlines
are concentric circles, centred on the origin, so that qe = 0, it therefore follows from
Eqn (2.79), that

So d(rq,)/dr = 0 and integration gives

rq, = C
where C is a constant. Now, recall Eqn (2.83) which is one of the two equivalent
definitions of circulation, namely

In the present example, 4'. t'= qr and ds = rde, so
r = 2rrq, = 2rC.
Thus C = r/(2r) and

dlCI
qt = --=-
dr 2rr
and
+= J--dr r

2rr
Integrating along the most convenient boundary from radius ro to P(r, 6') which in
this case is any radial line (Fig. 3.8):
'r
+ = - J -dr (ro = radius of streamline, + = 01
ro 2rr

(3.10)

Circulation is a measure of how fast the flow circulates the origin. (It is introduced
and defined in Section 2.7.7.) Here the circulation is denoted by r and, by convention,
is positive when anti-clockwise.

Fig. 3.8