Page 177 - Aerodynamics for Engineering Students

160 Aerodynamics for Engineering Students

potential flow theory in its pure form to estimate the lift or drag of wings and thereby
to develop aerodynamic design methods.
Flow separation and d’Alembert’s paradox both result from the subtle effects of
viscosity on flows at high Reynolds number. The necessary understanding and
knowledge of viscous effects came largely from work done during the first two
decades of the twentieth century. It took several more decades, however, before this
knowledge was fully exploited in aerodynamic design. The great German aeronaut-
ical engineer Prandtl and his research team at the University of Gottingen deserve
most of the credit both for explaining these paradoxes and showing how potential
flow theory can be modified to yield useful predictions of the flow around wings and
thus of their aerodynamic characteristics. His boundary-layer theory explained why
flow separation occurs and showed how skin-friction drag could be calculated. This
theory and its later developments are described in Chapter 7 below. He also showed
how a theoretical model based on vortices could be developed for the flow field of
a wing having large aspect ratio. This theory is described in Chapter 5. There it is
shown how a knowledge of the aerodynamic characteristics, principally the lift
coefficient, of a wing of infinite span - an aerofoil - can be adapted to give estimates
of the aerodynamic characteristics of a wing of finite span. This work firmly estab-
lished the relevance of studying the two-dimensional flow around aerofoils that is the
subject of the present chapter.

4.1.1 The Kutta condition
How can potential flow be adapted to provide a reasonable theoretical model for the
flow around an aerofoil that generates lift? The answer lies in drawing an analogy
between the flow around an aerofoil and that around a spinning cylinder (see Section
3.3.10). For the latter it can be shown that when a point vortex is superimposed with
a doublet on a uniform flow, a lifting flow is generated. It was explained in Section
3.3.9 that the doublet and uniform flow alone constitutes a non-circulatory irrota-
tional flow with zero vorticity everywhere. In contrast, when the vortex is present the
vorticity is zero everywhere except at the origin. Thus, although the flow is still
irrotational everywhere save at the origin, the net effect is that the circulation is non-
zero. The generation of lift is always associated with circulation. In fact, it can be
shown (see Eqn 3.52) that for the spinning cylinder the lift is directly proportional to
the circulation. It will be shown below that this important result can also be extended
to aerofoils. The other point to note from Fig. 3.25 is that as the vortex strength, and
therefore circulation, rise both the fore and aft stagnation points move downwards
along the surface of the cylinder.
Now suppose that in some way it is possible to use vortices to generate circulation,
and thereby lift, for the flow around an aerofoil. The result is shown schematically
in Fig. 4.1. Figure 4.la shows the pure non-circulatory potential flow around
an aerofoil at an angle of incidence. If a small amount of circulation is added the
fore and aft stagnation points, SF and SA, move as shown in Fig. 4.lb. In this case
the rear stagnation point remains on the upper surface. On the other hand, if
the circulation is relatively large the rear stagnation point moves to the lower surface,
as shown in Fig. 4.1~. For all three of these cases the flow has to pass around the
trailing edge. For an inviscid flow this implies that the flow speed becomes infinite at
the trailing edge. This is evidently impossible in a real viscous fluid because viscous
effects ensure that such flows cannot be sustained in nature. In fact, the only position
for the rear stagnation point that is sustainable in a real flow is at the trailing edge, as
illustrated in Fig. 4.ld. Only with the rear stagnation point at the trailing edge does