Page 182 - Aerodynamics for Engineering Students

Two-dimensional wing theoly 165

Fig. 4.5

If the strength of the circulation remains constant whilst the circuit shrinks to
encompass an ever smaller area, i.e. until it shrinks to an area the size of a rectangular
element, then:
I? = C x SxSy = 5 x area of element
Therefore,
r
vorticity = lim (4.3)
area-0 area of circuit
Here the (potential) line vortex introduced in Section 3.3.2 will be re-visited and the
definition (4.2) of circulation will now be applied to two particular circuits around
a point (Fig. 4.6). One of these is a circle, of radius r1, centred at the centre of the
vortex. The second circuit is ABCD, composed of two circular arcs of radii r1 and r2
and two radial lines subtending the angle ,6 at the centre of the vortex. For the
concentric circuit, the velocity is constant at the value

where C is the constant value of qr.