## THE JOUKOWSKI TRANSFORMATION

We introduce the conformal transformation due to Joukowski (who is pictured above)

and analyze how a cylinder of radius R defined in the z plane maps into the z' plane:

1. If the circle is centered at (0, 0) and the circle maps into the segment between and lying on the x-axis;
2. If the circle is centered at and , the circle maps in an airfoil that is symmetric with respect to the x'-axis;
3. If the circle is centered at and , the circle maps into a curved segment;
4. If the circle is centered at and , the circle maps into an asymmetric airfoil.

To summarize, moving the center of the circle along the x-axis gives thickness to the airfoil, moving the center of the circle along the y-axis gives camber to the airfoil.

In the following interactive application it is possible to move the center of the circle in the z plane and see the resulting transformed airfoil.

We need to introduce some notations on airfoils.

The generic Joukowski airfoil has a rounded leading edge and a cusp at the trailing edge where the camber line forms an angle with the chord line. In the cylinder plane, is related to the vertical coordinate of the center of the cylinder so that

Usually the angle of attack (sometimes called physical) is defined as the angle that the uniform flow forms with the chord line. More interesting for aerodynamics is the angle

In fact, when the angle is zero, the lift, as will be shown, vanishes. Then the angle is often defined as the effective angle of attack.

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