Page 265 - Aerodynamics for Engineering Students

248 Aerodynamics for Engineering Students

Plainly 6 is always a positive quantity because it consists of squared terms that must
always be positive. Co, can be a minimum only when S= 0. That is when
A3 = A5 = A7 = . . . = 0 and the only term remaining in the series is A1 sin 8.

Minimum induced drag condition
Thus comparing Eqn (5.50) with the induced-drag coefficient for the elliptic case
(Eqn (5.43)) it can be seen that modifying the spanwise distribution away from the
elliptic increases the drag coefficient by the fraction S that is always positive. It
follows that for the induced drag to be a minimum S must be zero so that the
distribution for minimum induced drag is the semi-ellipse. It will also be noted that
the minimum drag distribution produces a constant downwash along the span
whereas all other distributions produce a spanwise variation in induced velocity.
This is no coincidence. It is part of the physical explanation of why the elliptic
distribution should have minimum induced drag.
To see this, consider two wings (Fig. 5.33a and b), of equal span with spanwise
distributions in downwash velocity w = wg = constant along (a) and w = f(z) along
(b). Without altering the latter downwash variation it can be expressed as the sum of
two distributions wo and w1 = fl(z) as shown in Fig. 5.33~.
If the lift due to both wings is the same under given conditions, the rate of change
of vertical momentum in the flow is the same for both. Thus for (a)
L 0; 1:mwodz (5.51)

and for (b)

(5.52)

where riz is a representative mass flow meeting unit span. Since L is the same on each
wing

l)lfl(z)dz = 0 (5.53)

Now the energy transfer or rate of change of the kinetic energy of the representative
mass flows is the induced drag (or vortex drag). For (a):

(5.54)

Fig. 5.33 (a) Elliptic distribution gives constant downwash and minimum drag. (b) Non-elliptic distribution
gives varying downwash. (c) Equivalent variation for comparison purposes