Vortex-Sheet Representation of an Airfoil

An airfoil may be represented in an inviscid flow by wrapping a vortex sheet around its surface, as depicted in Fig. 5.11. This vortex sheet is of a variable strength, y ds. The strength can be found by applying the boundary condition that there is no flow through the solid surface of the airfoil. The boundary condition at infinity already is satisfied by the vortex-sheet property that at a large distance from a vortex, the induced velocity goes to zero lim (l/r) = 0. If the airfoil is lifting, the Kutta condition

r——^

also must be imposed.

Once the variable-sheet strength, y (s), is found, then:

Г = Х yAs — J Yds (5.3)

and the circulation around the airfoil can be determined. The value of the lift per unit span, L’ = pV^r, follows immediately from the Kutta-Joukowski theorem.

This representation of an airfoil by a vortex sheet (i. e., a collection of singu­larities, each of which is a solution to the linear Laplace’s Equation) is physically appealing as well as an application of the superposition principle for inviscid flows. Recall that in a real (viscous) flow, there is a thin boundary layer on the surface of

the airfoil, which is a region of high vorticity. The vorticity of the vortex sheet at the surface of the inviscid-flow model may be thought of as generated there by the boundary layer that would be present in the actual flow field.