A simple transformation with remarkable properties was used by Nikolai Zhukovsky around 1910 to study the flow around aircraft wings. It is defined by

and is usually called the *Joukowsky Map*. We begin with a discussion of the theory of fluid flow in two dimensions. Readers familiar with 2D potential flow may skip to the section *Joukowsky Airfoil*.

**Analytic Functions and Harmonic Functions**

Complex variable theory is a powerful tool in modelling fluid flow in two dimensions. The independent complex variable is . If a function is *analytic*, its derivative does not depend on the direction of the increment . Therefore

Writing , this leads to , which immediately yields the Cauchy-Riemann equations:

It then follows that , so that ; and similarly . Thus, both and satisfy the 2D Laplace equation; they are *harmonic functions*. This is remarkable: the real and imaginary parts of *any analytic function* are harmonic functions.

** The Complex Potential **

We denote the velocity by . To keep things simple, we will consider steady-state fluid flow () that is both irrotational and incompressible:

The flow can be represented by a velocity potential ,

or by a streamfunction ,

It follows from that and are harmonic functions

and comprise the real and imaginary parts of an analytical function, the *complex potential*:

** Examples of Potential Flow **

**Example I:** The complex potential corresponds to

yielding a uniform eastward flow

so that . This is shown in the Figure below (left panel).

**Example II:**

The potential has velocity potential and streamfunction

with corresponding velocity components

This is a dipole flow, shown in the FIgure above (right panel).

**Example III:** The potential can be expressed as

and corresponds to velocity potential and streamfunction

with radial and tangential flow components

which is flow counterclockwise about the origin. Note that the magnitude of decreases with radius by precisely the amount to ensure

so the flow, although circulating, is irrotational.

** The Joukowsky Transform **

We combine the complex potentials and to produce a potential with some remarkable properties, the Joukowsky potential

This mapping is also called the Zhukovsky transform (Nikolai Zhukovsky studied it around 1910). The velocity potential and streamfunction corresponding to are

These functions are plotted below. We note that the unit circle corresponds to constant : there is no flow across this curve and the fluid flows around the unit circle or, in three dimensions, around an infinite cylinder.

It is obvious from the definition that . If we restrict the mapping to the exterior of the unit circle , it is one-to-one. Rewriting the map as , we have two solutions

Since , one of these is inside and the other is outside.

Letting the mapping may be written

Now writing we easily show that

where and (so ). Thus, circles centered at the origin with transform to ellipses

with semi-axes and . Moreover,

so lines through the origin are mapped to hyperbolae.

Under the Joukowsky transform, the unit circle maps to the line segment (Figure below, left panel). When the centre of the circle is moved to but still passes through , the image curve represents an airfoil (right panel).

The mapping that converts a circle into an airfoil *also maps the flow* around the circle or, in 3D, the cylinder to flow around the airfoil. In the figure below we show a single streamline (left panel) and the corresponding streamline around the airfoil (right panel).

** Adding Circulation **

We saw in Example III above that a potential implies circulation around the wing. This results in lift. In the figure below, the flow without circulation (left panel, ) and with circulation (right panel, ) are shown.

A more comprehensive solution for a Joukowsky airfoil is shown in the final figure. This solution was generated using code on the Mathematica StackExchange site referenced below.

** Sources **

Complex-analysis.com: Website with visualizations: URL

Mathematica StackExchange site: URL

Spiegel, Murray R., 1964: *Schaums Outline Series of Complex Variables*. Schaum Publishing Co., New York, 313pp.

Strang, Gilbert, 1986: *Introduction to Applied Mathematics.* Wellesley-Cambridge Press, 758pp. ISBN: 0-961-40880-4.