# Prandtl Lifting Line Theory

15.4.2.1 Vortex Sheet

The vortex sheet is a stream surface originating at the sharp trailing edge of a finite wing. It is a surface of discontinuity of V. It has the following properties:

< u > = 0, no jump in u since pressure is continuous at the vortex sheet,

< w >= 0, no jump in w because the surface a zero thickness and the fluid is tangent to it (tangency condition)

< v >= 0, which can be related to the fact that the surface is made of vortex filaments of equation y = const., z = 0, 0 < x <ro.

15.4.2.2 Designing for Tip Vortices

Г[y(t)] = 2Ub{A1 sin t + A3 sin 3t}, where y(t) = -0.5b cos t, 0 < t < n.

Computing Г’ = (dГ/dt) / (dy/dt) shows that as, say, t ^ 0 the fraction is of the form 0/0. However, if dГ/dt ^ 0 faster than sin t a t, the result will be obtained.

dГ/dt = 2Ub{A1 cos t + 3A3 cos 3t}, thus, as t ^ 0 the necessary condition is A1 + 3A3 = 0. With this result, dГ/dt = 2UbA1{cos t – cos3t} a UbAtf2 as t ^ 0. The result is confirmed.

One finds A1 = (CL)t-o / (nAR) and A3 = – (CL)t-o / (3nAR).

The downwash is given by ww [y(t)] = —U {A1 +3 A3 sin3t / sin t }=-2UA1(1- 2 cos21). In terms of y this is ww(y) = -2UA1(1 – 2 (2y/b)2), a parabolic distribution. See Fig. 15.12.

The induced velocity ww is negative near the root, but there is upwash near the wing tips.

15.4.2.3 Induced Drag

The induced drag is given by Cm = nAR{A2 + 3A2} = (CDi)elliptic U + 332} = 4 (CDi )eiiiptic. There is a 33 % loss compared to the elliptic loading.

Fig. 15.12 Circulation and downwash for the wing with weak tip vortices

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