# Prandtl Lifting Line Theory

14.8.2.1 Circulation Representation

A paper airplane wing is made with manila folder by cutting an ellipse of large aspect ratio (b, c0). Then the wing is given some small camber such that the relative camber is constant (d/c = const) along the span, and there is no twist, t (y) = 0. The circulation for an arbitrary wing is represented by a Fourier series

Г[y(0)] = 2Ub “=1 An sin n0
y(0) = -—b cos 0, 0 < 0 < n

The chord distribution of the wing can be expressed as c[y(0)] = c0sin0 with the above change of variable.

Show that the circulation is given by the first mode only (Hint: use Prandtl Integro- differential equation to prove it).

What is the relationship between the root circulation Г0, the incidence a, the constant induced incidence a and the relative camber d/c?

14.8.2.2 Ideal Angle of Attack

From the previous result, calculate the lift coefficient, CL = Ci that corresponds to the geometric incidence, aideal, that will make the effective incidence zero, i. e. aeff = aideal + ai = 0. (Hint: first find Г0 in this case and use the relationship between Г0 and Cl to eliminate Г0).

Find the corresponding value of aideal in term of the wing aspect ratio, AR, and d/c, given that the lift of this ideal wing is Make a sketch of the flow in a cross section of the wing (for example the wing root).

14.8.2.3 Including Twist

The paper wing is actually slightly warped with a linear twist t (y) = —2tx y/b, where tx is a small positive number that represents the tip twist.

Show that this linear twist is represented by the second mode in the Fourier series. Use Prandtl Integro-differential equation to find A2 in terms of tx. (Hint: use the identity sin(20) = 2sin0cos0).

14.8.2.4 Efficiency Factor

The warped wing loading is represented by the first two modes in the Fourier series.

Given that A2 = A1 /10, calculate the induced drag of the warped wing and compare it with that of the ideal (untwisted) wing.

What is the percentage of increase of the induced drag?

Calculate the efficiency factor e.