Example of Wing Loading with Upwash

When considering Fig. 6.16, one is tempted to design a wing loading with “upwash” on a part of the wing, to locally create a “thrust” or “negative drag” to mitigate the induced drag on the rest of the wing. This can be achieved by combining some odd modes of the Fourier series for the circulation. As seen previously, the downwash in the Trefftz plane is given by

For mode 1 this is a constant downwash

W1T = — 2UA1

For mode 3 this is a parabolic downwash

A combination for which A3 = —A1 /6, with all the other modes suppressed, gives the results for the circulation and the downwash of Fig. 6.23.

The thin lines are for the elliptic loading and constant downwash. The thicker lines show the combined 1 and 3 modes. The last 10 % of the wing span sees an upwash and a thrust. Nevertheless, since it was proved that the elliptic loading is the optimum solution, the induced drag of the wing is increased according to the formula

Fig. 6.23 Wing with upwash near the tips
CDi	(6.88)

a drag increase of 8 %.