Conformal mapping is a mathematical technique used to
convert (or map) one mathematical problem and solution into
another. It involves the study of complex variables.
Complex variables are combinations of real and imaginary
numbers, which is taught in secondary schools. The use of
complex variables to perform a conformal mapping is taught in
college. Under some very
restrictive conditions, we can define a complex mapping function that
will take every point in one complex plane and map it onto another
complex plane. The mapping is represented by the red lines in the
Many years ago, the Russian mathematician Joukowski
developed a mapping function that converts a circular cylinder
into a family of airfoil shapes.
If points in the cylinder plane are represented by the
complex coordinates x for the horizontal and y for the
vertical, then every point z is specified by:
z = x + i y
Similarly, in the airfoil plane, the horizontal coordinate is B and
the vertical coordinate is C, and every point A is specified
A = B + i C
Then Joukowski's mapping function that relates points in the airfoil plane
to points in the cylinder plane is given in the red box:
A = z + 1 / z
The mapping function also converts
the entire flow field around the cylinder into the flow field around
the airfoil. We know the velocity and pressures in the plane
containing the cylinder.
The mapping function gives us the velocity and pressures
around the airfoil. Knowing the pressure
around the airfoil, we can then compute the lift.
The computations are difficult to perform by hand, but can be solved
quickly on a computer.
Here is a Java simulator which solves for Joukowski's transformation.
On the left side of the simulator is flow around a cylinder with
circulation. On the right side is the mapped geometry and flow for
a Joukowski airfoil.
You can vary the shape and inclination of the airfoil by using the sliders
below the view window on the right,
or by backspacing over an input box, typing in your new value and
hitting the Enter key on the keyboard. The
is a measure of the curvature of the airfoil,
and the thickness is the maximum height between the upper and lower
surface of the airfoil. Angle sets the
angle of attack
of the airfoil.
As you change the airfoil shape, the cylinder moves to a new
location and the amount of circulation around the cylinder is changed.
You can also vary conditions in the cylinder plane by using the
sliders and input boxes on the left side of the simulator.
Radius is the size of the cylinder, X-val is the horizontal
location and Y-val is the vertical location of the center
of the cylinder.
Circulation is related to the strength of the
vortex that is located at the center of the cylinder.
As part of the Joukowski analysis method, the
Kutta condition specifies that the airfoil generates enough circulation
to move the rear stagnation point on the airfoil to the trailing edge.
Using the menu button at the bottom of
the right input panel, you can turn off the Kutta condition to
study its effects.
The Joukowski transformation has two poles in the cylinder
plane where the transformation is undefined. On the blue horizontal
axis, the poles occur at x = 1 and x = -1 and are noted by a small "*".
If both poles remain inside the cylinder, a closed body is formed
in the airfoil plane. If the surface of the cylinder touches a
pole, a sharp edge is formed on the airfoil.
If only one pole is inside the cylinder, a very non-physical body
is mapped into the airfoil plane. Moving the generating cylinder
up and down creates camber in the airfoil. Moving the cylinder left and right
changes the thickness distribution on the airfoil.
Joukowski's transformation and the Kutta condition are used in the
the FoilSim computer program. A
describing the details of the method used in FoilSim is also