Page 186 - Aerodynamics for Engineering Students

Two-dimensional wing theory 169
The mass flow through the elemental area of the boundary is given by pVr cos 8 SO.
This mass flow has a vertical velocity increase of v cos 8, and therefore the rate of
change of downward momentum through the element is -pVvr cos2 O SO; therefore by
integrating round the boundary, the inertial contribution to the lift, li, is

2n
li =+I pVvrcos20d0
Jo
= pVvr.ir

Thus the total lift is:
I = 2pVvm

From Eqn (4.5):

giving, finally, for the lift per unit span, 1:
1 = pvr (4.10)

This expression can be obtained without consideration of the behaviour of air in
a boundary circuit, by integrating pressures on the surface of the aerofoil directly.
It can be shown that this lift force is theoretically independent of the shape of the
aerofoil section, the main effect of which is to produce a pitching moment in
potential flow, plus a drag in the practical case of motion in a real viscous fluid.

4.2 The development of aerofoil theory

The first successful aerofoil theory was developed by Zhukovsky." This was based on
a very elegant mathematical concept - the conformal transformation - that exploits
the theory of complex variables. Any two-dimensional potential flow can be repre-
sented by an analytical function of a complex variable. The basic idea behind
Zhukovsky's theory is to take a circle in the complex < = (5 + iv) plane (noting that
here ( does not denote vorticity) and map (or transform) it into an aerofoil-shaped
contour. This is illustrated in Fig. 4.8.
A potential flow can be represented by a complex potential defined by = 4 + i+
where, as previously, 4 and $ are the velocity potential and stream function respect-
ively. The same Zhukovsky mapping (or transformation), expressed mathematically as

(where C is a parameter), would then map the complex potential flow around the
circle in the <-plane to the corresponding flow around the aerofoil in the z-plane. This
makes it possible to use the results for the cylinder with circulation (see Section
3.3.10) to calculate the flow around an aerofoil. The magnitude of the circulation is
chosen so as to satisfy the Kutta condition in the z-plane.
From a practical point of view Zhukovsky's theory suffered an important draw-
back. It only applied to a particular family of aerofoil shapes. Moreover, all the

* see footnote on page 161.