Prandtl Lifting Line Theory

14.5.2.1 Origin of Induced Drag

Explain how an inviscid flow (no friction) past a finite wing, can produce a drag. (Hint: you can discuss an energy balance).

14.5.2.2 Maximum Lift of a Wing

The maximum lift of a wing would ideally correspond to a constant circulation along the span, Г(y) = rmax, where rmax is the maximum circulation that a profile can sustain, given the flow conditions (Reynolds number, etc.). Such a distribution is given by:

where y(t) = —f cos t, 0 < t < n. It corresponds to a single, finite strength horse­shoe vortex trailing the wing from the wing tips. The problem with this model, as found out by Prandtl, is that the induced drag is infinite. Since this is not feasible, we are going to consider only a few terms of the infinite series.

Consider modes 1 and 3 only (p = 0, 1).

Give the values of A1 and A3.

Sketch the corresponding distribution of circulation and downwash (Use sin 3t = sin t (3 — 4sin21)).

Show that there is no upwash.

14.5.2.3 Induced Drag

Calculate the induced drag and the gain/loss compared to the elliptic loading.
Show that the induced drag due to two finite strength vortices is infinite.