A Formulation of the Three-Dimensional Potential Flow Field around a Lifting Wing by Use of the Surface Velocity Components -an Extension of the Prager-Vandrey-Martensen Procedure to the Three-dimensional Case-

A Fredholm integral equation of the second kind for the surface velocity components of the three-dimensional incompressible potential flow around a body is obtained by extending the procedure explored by W.Prager, F.Vandrey and E.Martensen for the two-dimensional case. The formulation is based on the representation of the velocity potential by a doublet distribution over the body surface, and is realized by replacing the original surface boundary condition of the vanishing normal velocity component with the equivalent condition of the quiescent flow in the region inside the body. The formulation is then generalized to cases where the normal velocity component to the body surface does not necessarily vanish identically so that it can be utilized in the boundary-layer-displacement-model procedure designed to account for viscous effects within the scope of the inviscid flow thoery. This generalization is accomplished by combining the above-mentioned doublet distribution with a source distribution of prescribed strengths. Since our formulation lacks redundancy in variables to take care of the Kutta condition at the trailing edge of a lifting wing, the implication of this condition in our formulation is studied by examining the behaviour of our basic integral equation at the trailing edge of a wing. It is found that the geometry of the trailing vortex sheet and the direction of vortex shedding in the immediate neighbourhood of the trailing edge bear essential relations to the fulfilment of the Kutta condition. The classical Prandtl model of the trailing vortex sheet is, in principle, not adequate for our formulation.