Prandtl Lifting Line Theory
220.127.116.11 Lifting Line Theory
The downwash results from the presence of a vortex sheet, downstream of the sharp trailing edge of a finite wing. The sheet is made of line vortices which induce a negative vertical velocity component when the lift is positive, as required from a balance of momentum.
Downwash is associated with induced drag, whereas upwash will produce thrust (negative drag).
There is no net drag benefit to design a wing with upwash. The wing with minimum induced drag for a given lift has elliptic circulation and constant downwash. Any other induced velocity distribution, including with upwash, will produce more induced drag.
18.104.22.168 Design of a Wing
The circulation distribution is represented by the Fourier series
■ Г[y(t)] = 2Ub Z“=1 An sin nt
y(t) = -§ cos t, 0 < t < n
Combining modes 1 and 3 and taking the derivative w. r.t. t yields
2Ub (A1 cos t + 3A3 cos 3t) = 0, at t = 0 and n
A1 + 3 A3 = 0
Using the identity: cos 3t = cos t 4 cos2 t – 3 one finds d Г [ y (t)]
and the derivative w. r.t. y reads
The requirement is satisfied for zero strength tip vortices. With this choice, the circulation distribution becomes
where we used the identity: sin 3t = sin t (4 cos21 — 1). The circulation distribution Г[y(t)] is shown in Fig. 15.38.