Vortex Theory for Bodies Moving in Water 
Equation (8.3) expresses the d'Alembert paradox. If free 
vortices are shed by the body, then (8.3) is no longer verified. Na 
has two components ; one of them is a lift and the other one is the 
"induced resistance". 
KUTTA-JOUKOWSKI'S THEOREM - 
The first version of this theorem concerned wing profiles in 
a uniform motion of translation, with Vo = 0. The wing is an infinite 
cylinder and its profile € is its intersection by a plane normal to 
the generatrices. The problem can be considered as the limiting case 
of that of a wing, when the aspect ratio of which tends to infinity. 
When the aspect ratio is finite, the relative velocities on the two 
sides camel P pe of the wing near the trailing edge are equal and 
opposite. This follows from the continuity of the flow between Ves 
and and between 2, and va . Hence, in the case of a wing 
profile, one must have v2 = 0 at the trailing edge B. This is the 
Kutta condition which determines the density of the vortex sheet on 
> a taat 1s the ratio “I. on the contour © of the profile. 
The Kutta condition holds when the motion is unsteady. 
The theory developed in the preceding Sections applies to 
wing profiles. But because one deals with two-dimensional motions, 
the concept of complex velocity potential can be used and leads to 
considerable simplifications. In particular, one can associate a vor- 
tex distribution and a source distribution on the skeletton of the hull 
to obtain the desired profile shape (1) : 
(1) The determination of the exact distribution of the velocity at the 
leading edge requires some care [3] : 
1253 
