# 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 (378 mph) with a take-off mass of M = 3,000 kg.

15.9.2.1 Vortex Sheet

The vortex sheet is a surface of discontinuity for the velocity, that is generated at the sharp trailing edge of a wing, where a Kutta-Joukowski condition holds. In small disturbance theory and for a large aspect ratio wing, one assumes that the vortex sheet is in the base surface generated by straight lines touching the lifting line and parallel to the incoming velocity vector. The rolling-up of the sheet edges is neglected. The u-component of velocity is continuous (< u >= 0) across the vortex sheet, since the pressure is continuous and Cp = —2u/U. The w-component is also continuous (< w >= 0) since the vortex sheet is a stream surface of zero thickness and the fluid is tangent to it (tangency condition imposes w). Only the v-component has a jump (< v > = 0) at the vortex sheet. At a point, this jump is interpreted as the jump due to a vortex filament passing through that point. The vortex sheet is made of vortex filaments that induce a down-wash (or normal wash), responsible for the induced drag. The fundamental effect of the vortex sheet is to produce an inviscid drag, balanced by a propulsive force, whose work adds irreversible kinetic energy to the flow field.

15.9.2.2 Lift Coefficient

By definition, the lift coefficient CL is given by

W 3,000 9.81

Cl = t———- =——- :——— ;—— = 0.075

L 2 pU2 S 0.5 1.2 170222.5

15.9.2.3 Elliptic Loading

The aspect ratio of the wing is AR = b2/S = 5.58. The induced drag CDi is

c L

= 0.00032

n AR

The first mode amplitude A1 in the Fourier Series expansion is

The equation for the lift coefficient Cl in terms of aspect ratio AR, geometric incidence a and relative camber d/c reads

In cruise, the incidence a will satisfy the above equation: a = -0.0238 rd = -1.36°

15.9.2.4 Added Twist

Adding the third mode only with amplitude A3 = -0.002 increases the induced drag according to

Cot == (Cm)elliptic {1 + 0.655} = 0.00053

The percent change in induced drag is 65.5 %.

The corresponding Efficiency factor e = 1/(1 + 0.655) = 0.604

## Leave a reply