# Prandtl Lifting Line Theory

14.6.2.1 Downwash Evolution

For a large aspect ratio (AR) wing without sweep (straight lifting line) how does the downwash vary with distance from the lifting line from upstream infinity to downstream infinity near the x-axis and in particular what is the relationship between ww(y) and wT(y), — b < У < 2, where the former is the downwash at the lifting line and the latter the downwash in the Trefftz plane, far downstream. Explain with a simple argument.

Consider an aircraft wing with ideal loading (minimum induced drag) at a cruise lift coefficient (Cl)cruise.

Calculate the maximum value of the circulation (root circulation) Г0 and the downwash at the lifting line ww(y).

14.6.2.3 Induced Downwash in Manoeuvre

The aircraft needs to roll to initiate a left turn. Given that the rolling moment about the aircraft axis is given in terms of the Fourier coefficients by

n

Cm,0x = —44 AR A2

sketch the circulation distribution and corresponding downwash that will allow such a manoeuvre. Hint: use the simplest Fourier distribution that needs to be added to the ideal loading.

14.6.2.4 Induced Drag in Manoeuvre

Calculate the induced drag and the gain/loss compared to the elliptic loading, assum­ing that the (Cl)cruise remains unchanged and that A2 = – A1/10.