On the Trailing Edge Singularity and Kutta Condition for 3D Airfoils

The steady incompressible inviscid flow past a 3D airfoil with a sharp trailing edge TE is not uniquely determined unless the vorticity released into the wake at TE is prescribed. In the physical flow, this must be done in such a way that the flow velocity u is finite at TE (Kutta condition). In this paper, the explicit singular behavior of u at the trailing edge is determined using recent analytical results by Kondrat'ev and Oleinik, and the Kutta condition is enforced by removing the divergent part. This process yields a functional equation along the trailing edge that determines the vorticity released into the wake (and hence the circulation about any airfoil section) in terms of the normal component of u on the airfoil surface. A discrete form of this equation is also introduced, and solved for a simple 3D flow past a flat wing of semi-elliptic planform, using a linearized description of the wake. The present analysis gives theoretical support and a rational explanation to most of the procedures adopted in 3D aerodynamics on the basis of reasonable assumptions. For instance, it implies as a straightforward consequence the Kutta condition for potential flow, i.e. the continuity of the jump of the potential at the trailing edge.