Question

The incorrect statement regarding the downwash for a single horseshoe vortex in Prandtl’s lifting line theory is____

(a) Downwash has contribution from trailing vortices

(b) Downwash becomes infinite at the tips

(c) Downwash is given by w=\(\frac {-\Gamma}{4\pi } \frac {b}{(\frac {b}{2})^2-y^2}\)

(d) The contribution of two semi-infinite trailing vortices is same as an infinite vortex

This question was addressed to me in class test.

Question is taken from Prandtl’s Classical Lifting-Line Theory topic in portion Flow over Airfoils over Finite Wing of Aerodynamics

Answer 1

Right choice is (d) The contribution of two semi-infinite trailing vortices is same as an infinite vortex

The explanation is: The downwash along the bound vortex is induced by the two semi-infinite trailing vortices, which is not same as that of an infinite trailing vortex. It acts downwards on the wing. When calculated using the Biot-Savart law, it comes equal to w=\(\frac {-\Gamma}{4\pi } \frac {b}{(\frac {b}{2})^2-y^2}\), which becomes infinite at the tips (±b/2) if we consider only a single horseshoe vortex.