# Prandtl Lifting Line Theory

The WWII Spitfire was designed with an elliptic wing of span b = 11.2m and wing area S = 22.5 m2. It is equipped with a NACA2209.4 profile of 2 % relative camber (d/c = const = 0.02). The top speed in cruise is V = 170 m/s (378mph) with a take-off mass of M = 3,000 kg.

14.9.2.1 Vortex Sheet

Explain succinctly the key features of the vortex sheet (physically and mathematically) and the effect it has on the flow past a large aspect ratio wing (lifting line).

14.9.2.2 Lift Coefficient

Given the above data, calculate the lift coefficient CL at top speed in cruise (take p = 1.2kg/m3 and g = 9.81 m/s2).

14.9.2.3 Elliptic Loading

Assuming an elliptic planform, a constant relative camber and zero twist, calculate the induced drag CDi and the first mode amplitude A i in the Fourier Series expansion of the circulation

Г[y(ff)] = 2Ub ~1 An sin nd

y(Q) = —§ cos в, 0 < в < п

Write the equation for the lift coefficient CL in terms of aspect ratio AR, geometric incidence a and relative camber d/c.

Find a, the geometric incidence in cruise.

14.9.2.4 Added Twist

The designer of the Spitfire added “washout” (—2.5° of twist between root and tip) so that the local lift coefficient is larger at the root than at the tip, to avoid tip stall. This corresponds to adding only the third mode with an amplitude A3 = —0.002. (A2 = A4 = A5 = … An = 0, n > 4).

Calculate the induced drag (CDi )waShout of the wing with twist.

What is the percent change in induced drag?

What is the corresponding Efficiency factor e?

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