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:

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

INDEX