Karman-Trefftz transform

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The Karman-Trefftz transform is a conformal map derived from the Joukowsky transform. When the Joukowsky transform is written as a composition of three transformations, one of these can be modified independently, allowing for a third parameter which affects the angle of the trailing edge of the airfoil.

The Joukowsky transform is z=\zeta+\frac{1}{\zeta} = S3(S2(S1(\zeta))), where S3, S2, S1 are:

S3(z) = \frac{2+2z}{1-z}
S2(z) = z^2 \,\!
S1(z) = \frac{z-1}{z+1}

Reducing the exponent in S2(z) by a small amount increases the thickness of the trailing edge to a finite amount.

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