Lifting Line Theory

15.1.2.1 Flow Model

The vortex sheet that is shed by the finite wing is composed of vortex filaments parallel to the x-axis, that induce two velocity components, v and w in its proximity, that vanish at infinity upstream, but not at infinity downstream (Trefftz plane). The kinetic energy of the flow increases along a stream tube. An energy balance indicates that work must be done, for the kinetic energy to increase. This work corresponds to the work of the thrust that is needed to balance the induced drag, in order to maintain the motion. The vortex sheets is a trace of the passage of the wing, that does not disappear in inviscid flow.

The lift and induced drag coefficients are given by

Cl = n ARAi

Cdi = nAR (A2 + 2A2 + •••+ nA’2 + •••)

15.1.2.2 Non Singular Lift Distribution

The combination of modes 1 and 3 reads

= 2Ub (A1 sin t + A3 sin 3t) y (t) = — cos t

Taking the derivative with respect to y gives

dГ dГ dt d A1cost + 3A3cos3t

dy dt dy it sin t

A necessary condition is that the numerator go to zero when t ^ 0. This will be satisfied if A3 = – A1. It is possible to verify that, with this condition, the derivative will go to zero, dry & 16UA1t, as t ^ 0.

See Fig. 15.2.

15.1.2.3 Drag Penalty

For a given CL (at take-off), the coefficient A1 is fixed since A1 = Ar. The induced drag of the wing is given by

The induced drag has been increased by 33 %, which is not negligible. The efficiency factor e = (1 + £)-1 = 4 = 0.75.