The advent of the digital computer enabled the development of solution methods for finite wings that are superior to methods that represent the wing by a lifting line. These solution methods are an extension of the airfoil-panel methods discussed in Chapter 5, with singularities here being distributed over the wing planform. The singularities may be sources, doublets, vortices, or certain combinations of these. A source distribution splits the streamlines and thus represents wing thickness, whereas a doublet or vortex distribution gives rise to lift. Vortex elements are discussed here for a wing with an arbitrary planform. Appeal is made to the tangency (i. e., “no-flow”) boundary condition, as in Chapter 5, and a system of simultaneous linear-algebraic equations is developed by applying the tangency-boundary condition at numerous points on the wing surface. The wing is assumed to be at a small angle of attack relative to the freestream flow because the theory is inviscid and irrotational and large areas of separation cannot be accommodated. Unlike liftingline theory, panel methods require no restrictions on wing sweep or AR, but they do rely on the principles of superposition.
The two panel methods described herein, the vortex panel method (VPM) and the vortex lattice method (VLM), can be used to model inviscid flow over wings with airfoil sections of large or small thickness ratios. The VPM is illustrated for a wing of arbitrary-thickness ratio, and the surface-pressure distribution on the wing is found. The VLM is discussed for a thin wing, that is, the wing thickness is assumed to be
small and its effect is neglected. When the wing thickness is neglected, such panel methods often are called lifting-surface methods. Lifting-surface methods apply the tangency-boundary condition on the camber surface of the wing rather than on the surface of the wing. The advantage of a lifting-surface analysis is that it is easier to program because far fewer panels are needed and the influence equations are simpler while satisfactory accuracy is maintained, as it was in thin-airfoil theory. The disadvantage of a lifting-surface analysis is that it provides the pressure difference (i. e., pressure loading) across the camber surface rather than the pressure distribution on the wing surface. The pressure distribution on the wing surface is needed as input for any related boundary-layer solution. However, lifting-surface methods provide force and moment data.
In both of the panel methods described herein, the wing surface must be subdivided into a suitable number of small quadrilateral panels that reflect the wing shape and planform. These panels need not be of the same size, usually being smallest in regions of rapidly varying flow properties. Thus, given wing surface must be discretized into panels as a preliminary step in the calculation. This grid generation is itself an important area of study. Each panel has vorticity distributed over the surface. In the following discussion, the vorticity is combined with vortex elements for convenience. Higher-order solutions with curved panels and distributed vorticity are found in the literature.