Hodograph method for airfoil design - The general case

Many airfoils, including all of the symmetric airfoils, generate hows such that the velocity-contour (i.e., the contour defined on the hodograph plans by the vector velocities of the airfoil's boundary) self-intersects. This self intersection implies that in different points of the space the flow has the same vector velocity. In such cases the classical hodograph theory fails and it is impossible to use the velocity-contours for designing such airfoils, A mathematical extension of the theory is developed to remedy the failure. This extended theory Is then successfully applied to develop a general inverse method for designing airfoils that generate all of the dynamic and geometric requirements imposed in the input velocity-contour. It is also shown that there exists not only one, but an infinite number, of airfoils capable of generating exactly the same input self-intersecting velocity-contour.