Lifting Line Theory
18.104.22.168 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 + •••)
22.214.171.124 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.
126.96.36.199 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.