# Prandtl Lifting Line Theory

14.7.2.1 Vortex Sheet Characteristics

Explain why the perturbation velocity components u and w are continuous across the vortex sheet behind a finite wing, i. e. < u >=< w >= 0.

How do you explain the existence of induced drag for a wing in incompressible, inviscid flow?

14.7.2.2 Circulation Representation

The circulation is represented by a Fourier series

■ Г[y(t)] = 2Ub Z“i An sin nt

y(t) = —§ cos t, 0 < t < n

Sketch the first three modes with unit coefficients A5 = A2 = A3 = 1.

14.7.2.3 Efficiency Factor

Give the definition of the Efficiency Factor e in terms of the Fourier coefficients.

The AMAT09 wing has been designed with a rectangular planform, b = 3.1m, c = 0.55 m, and is equipped with the SS1707-0723 double-element airfoil with high lift (Cl)max = 2.7. Prandtl Lifting Line theory indicates that the Efficiency Factor is e = 0.95 for all phases of flight (take-off, top speed, power-off descent). Find all the coefficients A1, A2,…, An (not to be confused with their 2-D counterparts) for the following three phases of flight (symmetrical)

• take-off: a = 18°, Cl = 2.5

• top speed: a = -3°, Cl = 1.0

• power-off descent: a = 4°, Cl = 1.6.

Here we assume that the rectangular wing is the “simplest” wing with e = 0.95, i. e., with the least number of non-zero coefficients.