Section ii. general approach

In section II Kutta sketched his mathematical method and introduced some of the basic formulas. His aim was to use a number of conformal transfor­mations. The strategy was to exploit the known flow around a circular cylin­der by transforming the cylinder into the arc representing the cross section of Lilienthal’s wing. The streamlines around the cylinder would be trans­formed into the streamlines around the wing. What is more, any circulation that is ascribed to the flow around the cylinder would be transformed into a circulation around the wing. The two steps in the procedure are therefore (1) describing the flow around a circular cylinder and (2) finding the trans­formation to turn a circular cylinder into a shallow circular arc.

Here is the formula that Kutta gave to describe the most general flow round a circular cylinder, where the circle is inscribed on the Z – (zeta-) plane:

W ^z+Zj – і-c2 jV-Zj + і■c-logC-

Each of the three parts on the right-hand side of the formula characterized one aspect of the flow around the cylinder. The term q specified the component of flow along the horizontal axis of the Z-plane, while c2 gave the component of flow in the direction of the vertical axis. The term c toward the end of the formula (without a subscript) was the constant associated with the circulatory component of the flow, that is, the component of flow in concentric circles around the cylinder. Kutta did not use exactly the same formula as Rayleigh. Whereas Rayleigh worked with the stream function for the flow around a circular cylinder, Kutta worked with the more general complex-variable for­mula which captured both the streamlines and the potential lines.

The problem was to find the right transformation to apply to the circle. How could the circle be turned into a shape resembling Lilienthal’s wing pro­file? There are no rules for finding the right transformation, and Kutta did not spot any direct way to do this. He therefore had to proceed in a piecemeal way. He constructed the required transformation by combining known for­mulas that could act as intermediate steps and whose combination had the desired outcome. He built a mathematical bridge between the simple case of the circle and the difficult case of a winglike shape, but he did so by starting at both ends and meeting in the middle. In one direction he went from a circle represented on the Z-plane (which he called the “transcendental plane”) to another shape on the t-plane. In the other direction he went from a simpli­fied geometrical description of Lilienthal’s wing, on the z-plane (sometimes, today, called the “physical plane”), to another shape on an intermediate plane called the z’-plane, and finally from the z’-plane to the t-plane. The two procedures therefore mapped their respective starting points onto the same shape in the same plane, the t-plane. This was where they met. Kutta then had the required connection between the circle and the wing.

The insight that allowed Kutta to construct this transformational bridge was that he knew a transformation that would turn an arc of a circle into a straight line and another that would turn the exterior of a circle into the top half of an entire plane. As we saw in chapter 2, a straight line counts as a poly­gon, so once Kutta had a straight line he could use the Schwarz-Christoffel theorem to link the flow to the simple and basic case of the flow along a straight boundary.