Page 183 - Aerodynamics for Engineering Students

166 Aerodynamics for Engineering Students

Fig. 4.6 Two circuits in the flow around a point vortex

Since the flow is, by the definition of a vortex, along the circle, a is everywhere zero
and therefore cos a = 1. Then, from Eqn (4.2)

Now suppose an angle 8 to be measured in the anti-clockwise sense from some
arbitrary axis, such as OAB. Then
ds = rld8
whence

Since C is a constant, it follows that r is also a constant, independent of the radius.
It can be shown that, provided the circuit encloses the centre of the vortex, the
circulation round it is equal to I?, whatever the shape of the circuit. The circulation
I' round a circuit enclosing the centre of a vortex is called the strength of the vortex.
The dimensions pf circulation and vortex strength are, from Eqn (4.2), velocity times
length, Le. L2T- , the units being m2 s-*. Now r = 2nC, and C was defined as equal
to qr; hence
I' = 2nqr
and
r
q=- (4.5)
2nr
Taking now the second circuit ABCD, the contribution towards the circulation from
each part of the circuit is calculated as follows:
(i) Rudiul line AB Since the flow around a vortex is in concentrk circles, the
velocity vector is everywhere perpendicular to the radial line, i.e. a = 90°,
cosa = 0. Thus the tangential velocity component is zero along AB, and there
is therefore no contribution to the circulation.
(ii) Circular arc BC Here a = 0, cos a = 1. Therefore