In applied mathematics, the **Joukowsky transform**, named after Nikolai Zhukovsky, is a conformal map historically used to understand some principles of airfoil design.

The transform is

- ,

where is a complex variable in the new space and is a complex variable in the original space. This transform is also called the **Joukowsky transformation**, the **Joukowski transform**, the **Zhukovsky transform** and other variations.

In aerodynamics, the transform is used to solve for the two-dimensional potential flow around a class of airfoils known as Joukowsky airfoils. A **Joukowsky airfoil** is generated in the *z* plane by applying the Joukowsky transform to a circle in the plane. The coordinates of the centre of the circle are variables, and varying them modifies the shape of the resulting airfoil. The circle encloses the point = −1 (where the derivative is zero) and intersects the point = 1. This can be achieved for any allowable centre position by varying the radius of the circle.

Joukowsky airfoils have a cusp at their trailing edge. A closely related conformal mapping, the **Kármán–Trefftz transform**, generates the much broader class of Kármán–Trefftz airfoils by controlling the trailing edge angle. When a trailing edge angle of zero is specified, the Kármán–Trefftz transform reduces to the Joukowsky transform.

## Contents

## General Joukowsky transform[edit]

The Joukowsky transform of any complex number to is as follows

So the real (*x*) and imaginary (*y*) components are:

### Sample Joukowsky airfoil[edit]

The transformation of all complex numbers on the unit circle is a special case.

So the real component becomes and the imaginary component becomes

Thus the complex unit circle maps to a flat plate on the real number line from −2 to +2.

Transformation from other circles make a wide range of airfoil shapes.

## Velocity field and circulation for the Joukowsky airfoil[edit]

The solution to potential flow around a circular cylinder is analytic and well known. It is the superposition of uniform flow, a doublet, and a vortex.

The complex velocity around the circle in the plane is

where

- is the complex coordinate of the centre of the circle
- is the freestream velocity of the fluid
- is the angle of attack of the airfoil with respect to the freestream flow
- R is the radius of the circle, calculated using
- is the circulation, found using the Kutta condition, which reduces in this case to

The complex velocity *W* around the airfoil in the *z* plane is, according to the rules of conformal mapping and using the Joukowsky transformation:

Here with and the velocity components in the and directions, respectively ( with and real-valued). From this velocity, other properties of interest of the flow, such as the coefficient of pressure or lift can be calculated.

A Joukowsky airfoil has a cusp at the trailing edge.

The transformation is named after Russian scientist Nikolai Zhukovsky. His name has historically been romanized in a number of ways, thus the variation in spelling of the transform.

## Kármán–Trefftz transform[edit]

The **Kármán–Trefftz transform** is a conformal map closely related to the Joukowsky transform. While a Joukowsky airfoil has a cusped trailing edge, a **Kármán–Trefftz airfoil**—which is the result of the transform of a circle in the *ς*-plane to the physical *z*-plane, analogue to the definition of the Joukowsky airfoil—has a non-zero angle at the trailing edge, between the upper and lower airfoil surface. The Kármán–Trefftz transform therefore requires an additional parameter: the trailing-edge angle *α*. This transform is equal to:^{[1]}

- (A)

with *n* slightly smaller than 2. The angle *α*, between the tangents of the upper and lower airfoil surface, at the trailing edge is related to *n* by:^{[1]}

The derivative , required to compute the velocity field, is equal to:

### Background[edit]

First, add and subtract two from the Joukowsky transform, as given above:

Dividing the left and right hand sides gives:

The right hand side contains (as a factor) the simple second-power law from potential flow theory, applied at the trailing edge near From conformal mapping theory this quadratic map is known to change a half plane in the -space into potential flow around a semi-infinite straight line. Further, values of the power less than two will result in flow around a finite angle. So, by changing the power in the Joukowsky transform—to a value slightly less than two—the result is a finite angle instead of a cusp. Replacing 2 by *n* in the previous equation gives:^{[1]}

which is the Kármán–Trefftz transform. Solving for *z* gives it in the form of equation (A).

## Notes[edit]

- ^
^{a}^{b}^{c}Milne-Thomson, Louis M. (1973).*Theoretical aerodynamics*(4th ed.). Dover Publ. pp. 128–131. ISBN 0-486-61980-X.

## References[edit]

- Anderson, John (1991).
*Fundamentals of Aerodynamics*(Second ed.). Toronto: McGraw–Hill. pp. 195–208. ISBN 0-07-001679-8. - Zingg, D.W. (1989). "Low Mach number Euler computations". NASA TM-102205.

## External links[edit]

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