ABSTRACT
The simplest external disk transformation (Subsection 33.9.1) has two diametrically opposite critical points, and maps the circle into a flat plate (Section 34.1); since the flow past a flat plate without incidence remains uniform, the inverse transformation is a third way to specify the flow past a circle. Thus the flow past a circular cylinder can be obtained by four methods, viz.: (i) by the circle theorem (Section 24.7); (ii) as a limit of the Rankine oval (Section 28.5); (iii) through the insertion of a dipole (Section 28.6); (iv) via the inversion of the Schwartz-Christoffel transformation (Section 34.1). The simplest case of external disk mapping is the Joukowski transformation, and when applied to a circle it leads to: (i) an ellipse if the center is the origin, viz. the potential flow past an elliptic cylinder (Section 34.2), including the flat plate (circular cylinder) as the particular case of zero smaller half-axis (equal half-axis); (ii) a circular arc if the center is on the imaginary axis; (iii) if the center is on the real axis, a symmetric airfoil with rounded leading-edge and sharp trailing-edge; (iv) if the center is off both axes, the airfoil (v) with the circular arc as midline yields a cambered or unsymmetric airfoil. These Joukowski airfoils (Section 34.3) have a sharp trailing edge, where the velocity is made finite by coincidence with a stagnation point; this Kutta condition of specifies the circulation, and thus the lift on the airfoil, regardless of whether it is: (i) a flat plate (Section 34.1); (ii) a Joukowski airfoil (Sections 34.2-34.4). The trailing edge need not be sharp, and the Kutta conditions may not apply Section 34.5 to: (iii) other parametric families of airfoils, for example, von Karman-Trefftz or von Mises; (iv) an arbitrary airfoil.
