Some Generalizations of the Zhukovskii Lift Theorem. Induced Drag

Abstract

The theorem of N. E. Zhukovskii concerning the lift of a wing, which he proved in 1905 and published in 1906 [1], has played an extremely important role in the development of aerodynamics. Zhukovskii was the first to give an explanation of the mechanism of the formation of lift. He showed that the lift on a cylindrical body in a plane-parallel flow is due to a circulation Γ of the velocity about a closed contour containing a wing section. He also gave an explanation of the Euler-d’Alembert paradox concerning the lack of any reaction due to the flow of perfect incompressible fluid, on a body in steady uniform motion. It was shown that this reaction would actually be zero if the circulation Γ were zero. For nonzero circulation, however, even for plane-parallel motion of a cylindrical body through a perfect incompressible fluid, there will be a lift

$$Y = \rho {U_0}\Gamma l$$

where l is the span of the body.