# Prandtl Lifting Line Theory

14.10.2.1 Lifting Line Theory

In Lifting Line theory, explain the origin of downwash.

In terms of drag, what is the difference between downwash and upwash?

Is there a net drag benefit to design a wing with upwash?

14.10.2.2 Design of a Wing

The circulation distribution is represented by the Fourier series

Г[y(t)] = 2U&£~1 An sin nt b

Design the simplest wing (meaning the wing with the least number of non-zero Fourier coefficients) with zero strength tip vortices, in order to minimize risks for airplanes caught in the wake of the wing. To achieve this goal, combine modes 1 and 3 in such a way that at y = ±­7 2

Note: you can impose

d Г [y(t)] n _ n л = 0, at t = 0 and n dt   to find a relation between A1 and A3(verify your result as the rest depends on it), then verify that

Use the following identity: cos 3t = cos t (4 cos21 — 3).

Sketch the circulation distribution Г[y(t)] in terms of y or t. Use the following identity: sin3t = sin t (4 cos21 — 1).