Page 181 - Aerodynamics for Engineering Students

164 Aerodynamics for Engineering Students

V
V+V f-
f-

Fig. 4.4

Consider this by the reverse argument. Look again at Fig. 4.3b. By definition the
velocity potential of C relative to A (&A) must be equal to the velocity potential of
C relative to B (~cB) in a potential flow. The integration continued around ACB gives
i=
= ~CA ~CB 0
=
This is for a potential flow only. Thus, if I' is finite the definition of the velocity
potential breaks down and the curve ACB must contain a region of rotational flow.
If the flow is not potential then Eqn (ii) in Section 3.2 must give a non-zero value for
vorticity.
An alternative equation for I' is found by considering the circuit of integration to
consist of a large number of rectangular elements of side Sx by (e.g. see Section 2.7.7
and Example 2.2). Applying the integral I' = J (u dx + v dy) round abcd, say, which is
the element at P(x, y) where the velocity is u and v, gives (Fig. 4.5).
av sx av sx

The sum of the circulations of all the areas is clearly the circulation of the circuit as
a whole because, as the AI' of each element is added to the AI? of the neighbouring
element, the contributions of the common sides disappear.
Applying this argument from element to neighbouring element throughout the
area, the only sides contributing to the circulation when the AI'S of all areas are
summed together are those sides which actually form the circuit itself. This means
that for the circuit as a whole

over the area round the circuit
and
av
au
__-_ -c
ax ay
This shows explicitly that the circulation is given by the integral of the vorticity
contained in the region enclosed by the circuit.