Page 256 - Aerodynamics for Engineering Students

Finite wing theory 239

by the wing system and that in fact is lost to the wing by being left behind. This
constant expenditure of energy appears to the wing as the induced drag. In what
follows, a third explanation of this important consequence of downwash will be of
use. Figure 5.29 shows the two velocity components of the apparent oncoming flow
superimposed on the circulation produced by the wing. The forward flow velocity
produces the lift and the downwash produces the vortex drag per unit span.
Thus the lift per unit span of a finite wing (I) (or the load grading) is by the Kutta-
Zhukovsky theorem:
I = pvr
the total lift being

L = /:pVTdz (5.34)

The induced drag per unit span (d,), or the induced drag grading, again by the
Kutta-Zhukovsky theorem is
d, = pwr (5.35)
and by similar integration over the span

D, = /:pwrdz (5.36)

This expression for D, shows conclusively that if w is zero all along the span then D,
is zero also. Clearly, if there is no trailing vorticity then there will be no induced drag.
This condition arises when a wing is working under two-dimensional conditions, or if
all sections are producing zero lift.
As a consequence of the trailing vortex system, which is produced by the basic
lifting action of a (finite span) wing, the wing characteristics are considerably modi-
fied, almost always adversely, from those of the equivalent two-dimensional wing of
the same section. Equally, a wing with flow systems that more nearly approach the
two-dimensional case will have better aerodynamic characteristics than one where

I =pvr d, =pwr

L= f spl/rdz
-S

Fig. 5.29 Circulation superimposed on forward wind velocity and downwash to give lift and vortex drag
(induced drag) respectively