Page 184 - Aerodynamics for Engineering Students

Two-dimensional wing theory 167

But, by Eqn (4.5),

(iii) Radial line CD As for AB, there is no contribution to the circulation from this
part of the circuit.
(iv) Circular arc DA Here the path of integration is from D to A, while the direction
of velocity is from A to D. Therefore a = 180", cosa = -1. Then

Therefore the total circulation round the complete circuit ABCD is

Thus the total circulation round this circuit, that does not enclose the core of the
vortex, is zero. Now any circuit can be split into infinitely short circular arcs joined
by infinitely short radial lines. Applying the above process to such a circuit would
lead to the result that the circulation round a circuit of any shape that does not
enclose the core of a vortex is zero. This is in accordance with the notion that
potential flow is irrotational (see Section 3.1).
4.1.3 Circulation and lift (Kutta-Zhukovsky theorem)

In Eqn (3.52) it was shown that the lift l per unit span and the circulation r of
a spinning circular cylinder are simply related by
1=pm
where p is the fluid density and Vis the speed of the flow approaching the cylinder. In
fact, as demonstrated independently by Kutta* and Zhukovskyt, the Russian physi-
cist, at the beginning of the twentieth century, this result applies equally well to a
cylinder of any shape and, in particular, applies to aerofoils. This powerful and useful
result is accordingly usually known as the KutteZhukovsky Theorem. Its validity is
demonstrated below.
The lift on any aerofoil moving relative to a bulk of fluid can be derived by direct
analysis. Consider the aerofoil in Fig. 4.7 generating a circulation of l-' when in a stream
of velocity V, density p, and static pressure PO. The lift produced by the aerofoil must
be sustained by any boundary (imaginary or real) surrounding the aerofoil.
For a circuit of radius r, that is very large compared to the aerofoil, the lift of the
aerofoil upwards must be equal to the sum of the pressure force on the whole
periphery of the circuit and the reaction to the rate of change of downward momen-
tum of the air through the periphery. At this distance the effects of the aerofoil
thickness distribution may be ignored, and the aerofoil represented only by the
circulation it generates.

* see footnote on page 161.
' N. Zhukovsky 'On the shape of the lifting surfaces of kites' (in German), Z. Flugtech. Motorluftschiffahrt,
1; 281 (1910) and 3, 81 (1912).