Decomposition into Symmetric and Lifting Problems

The small disturbance assumption has allowed for the complete linearization of the problem of the flow past a thin airfoil. It is therefore possible to use linear superposition of solutions. The general problem of the flow past a thin cambered airfoil at incidence can be viewed as the sum of a symmetric problem corresponding to the flow past a symmetric profile having the same thickness distribution at zero incidence, and a lifting problem corresponding to the flow past a thin cambered plate having the same camberline and incidence. This is depicted in Fig. 3.7.

The symmetric problem will be solved by distributing sources and sinks on a cut along the x-axis, from zero to c. Due to the symmetry of the problem, there will be no net force in the z-direction and no net moment. The drag will also be zero (d’Alembert paradox).

The lifting problem will be handled by a distribution of vortices on a cut along the x-axis, from zero to c. In general, the flow will produce lift and moment, but again zero drag.