The Big Idea

My target audience is probably familiar with the Joukowsky transformation:

\[n = c + {{a^2}\over{c}}\]
Note that I have replaced the traditional variable names (ok, avoided picking from among the many traditional variable names) with my own: \(c\) represents "cylinder space", where calculating inviscid flows are easy, and \(n\) represents "normal space", where "user space" has been rotated and translated to place the center of the airfoil on the origin, and the chord in line with the x axis. For the remainder of this article, the above relation will be referred to as \(J(c)\). This transformation is well known for transforming a cylinder of radius \(a\) centered on the origin onto a flat plate extending from \(-2a\) to \(2a\), and for transforming cylinders centered near the origin, and passing through the point \(c=a\), into airfoil like shapes.

Abbott and Van Daenhoff, in "Theory of Wing Sections", demonstrated that the Joukowsky transformation could be extended as a Laurent series to create a transformation that can describe an arbitrary airfoil:
\[T(c) = c + \sum_{i=1}^\infty {{a_i} \over {c^i}}\]
This transform is nifty in that it is analytic, and formulas exist to calculate all those coefficients, but there are all those cooefficients, and the series is not typically at all well behaved (I know, I’ve tried brute force, and when that failed, pondered the significance of large negative powers of small numbers). So calulating \(T\) directly is out, but we know that an anlalytic \(T\) exists, and that is enough. Consider
\[T(c) = J(c) + K(c)\]
\(K\) is my "correcting" transform. Since \(T\) and \(J\) are analytic, so must \(K\) be. Assuming c describes a suitable cylinder in "cylinder space", \(T(c)\) describes our airfoil, \(J(c)\) describes a similar Joukowski airfoil, and \(K(c)\) is the difference between the two.

It’s easy to describe the flow around the cylinder. It’s easy to transform that into the flow about the corresponding Joukowski airfoil. It’s not too hard to calculate \(K\)(we have a set of points from the user corresponding to \(T\), and we have our wholly analytic function \(J\)), and \(K\)’s derivative (via somewhat fancy finite difference methods ). With exact knowledge of \({dJ}\over{dc}\), and good knowledge of \({dK}\over{dc}\), we have a good estimate of \({dT}\over{dc}\), which with our exact knowledge of the flow about a cylinder via the complex potential (\({d\Phi}\over{dc}\)), we have a good estimate of the flow about our arbitrary airfoil:
\[{{d\Phi}\over{dT}} = {{{d\Phi}/{dc}}\over{dT/dc}} = {{{d\Phi}/dc}\over{{d\over{dc}}(J+K)}} = {{{d\Phi}/dc}\over{dJ/dc+dK/dc}}\]
This is the big picture. I am left with two smaller sub problems:

• Find an appropriate circle(minimize the magnitude of K)
• Find the derivative of K (the method described in the link above is still not all I'd hoped for)

The better my choice of a circle, the smaller my K function, and the less important the accuracy of K’s derivative. At the same time, the more accurate the derivative, the less important the choice of circle. Art remains.