### Joukowski airfoil

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

- $z=\backslash zeta+\backslash frac\{1\}\{\backslash zeta\}$

where $z=x+iy$ is a complex variable in the new space and $\backslash zeta=\backslash chi\; +\; i\; \backslash eta$ 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 $\backslash zeta$ 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 $\backslash zeta$ = −1 (where the derivative is zero) and intersects the point $\backslash zeta$ = 1. This can be achieved for any allowable centre position $\backslash mu\_x\; +\; i\backslash mu\_y$ 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

The Joukowsky transform of any complex number $\backslash zeta$ to $z$ is as follows

- $$

\begin{align}

z &= x + iy =\zeta+\frac{1}{\zeta}

\\

&= \chi + i \eta + \frac{1}{\chi + i \eta}

\\

&= \chi + i \eta + \frac{(\chi - i \eta)}{\chi^2 + \eta^2}

\\

&= \frac{\chi (\chi^2 + \eta^2 + 1)}{\chi^2 + \eta^2} + i\frac{\eta (\chi^2 + \eta^2 - 1)}{\chi^2 + \eta^2}.

\end{align}

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

- $$

\begin{align}

x &= \frac{\chi (\chi^2 + \eta^2 + 1)}{\chi^2 + \eta^2} \qquad \text{and} \\ y &= \frac{\eta (\chi^2 + \eta^2 - 1)}{\chi^2 + \eta^2}.

\end{align}

### Sample Joukowsky airfoil

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

- $|\backslash zeta|\; =\; \backslash sqrt\{\backslash chi^2+\backslash eta^2\}\; =\; 1\; \backslash quad\; \backslash text\{which\; gives\}\; \backslash quad\; \backslash chi^2+\backslash eta^2\; =\; 1.$

So the real component becomes $x\; =\; \backslash frac\{\backslash chi\; (1\; +\; 1)\}\{1\}\; =\; 2\backslash chi$ and the imaginary component becomes $y\; =\; \backslash frac\{\backslash eta\; (1\; -\; 1)\}\{1\}\; =\; 0.$

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

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 $\backslash tilde\{W\}$ around the circle in the $\backslash zeta$ plane is

- $\backslash tilde\{W\}=V\_\backslash infty\; e^\{-i\; \backslash alpha\}\; +\; \backslash frac\{i\; \backslash Gamma\}\{2\; \backslash pi\; (\backslash zeta\; -\backslash mu)\}\; -\; \backslash frac\{V\_\backslash infty\; R^2\; e^\{i\; \backslash alpha\}\}\{(\backslash zeta-\backslash mu)^2\}$

where

- $\backslash mu=\backslash mu\_x+i\; \backslash mu\_y$ is the complex coordinate of the centre of the circle
- $V\_\backslash infty$ is the freestream velocity of the fluid
- $\backslash alpha$ is the angle of attack of the airfoil with respect to the freestream flow
- R is the radius of the circle, calculated using $R=\backslash sqrt\{(1-\backslash mu\_x)^2+\backslash mu\_y^2\}$
- $\backslash Gamma$ is the circulation, found using the Kutta condition, which reduces in this case to

- $\backslash Gamma=4\backslash pi\; V\_\backslash infty\; R\; \backslash sin\; \backslash left(\backslash \; \backslash alpha\; +\; \backslash sin^\{-1\}\; \backslash left(\; \backslash frac\{\backslash mu\_y\}\{R\}\; \backslash right)\backslash right).$

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

- $W=\backslash frac\{\backslash tilde\{W\}\}\{\backslash frac\{dz\}\{d\backslash zeta\}\}\; =\backslash frac\{\backslash tilde\{W\}\}\{1-\backslash frac\{1\}\{\backslash zeta^2\}\}.$

Here $W=u\_x\; -\; i\; u\_y,$ with $u\_x$ and $u\_y$ the velocity components in the $x$ and $y$ directions, respectively ($z=x+iy,$ with $x$ and $y$ 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

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]}

- $$

z = n \frac{\left(1+\frac{1}{\zeta}\right)^n+\left(1-\frac{1}{\zeta}\right)^n} {\left(1+\frac{1}{\zeta}\right)^n-\left(1-\frac{1}{\zeta}\right)^n},

(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]}

- $\backslash alpha\; =\; 2\backslash pi\backslash ,\; -\backslash ,\; n\backslash pi\; \backslash quad\; \backslash text\{\; and\; \}\; \backslash quad\; n=2-\backslash frac\{\backslash alpha\}\{\backslash pi\}.$

The derivative $dz/d\backslash zeta$, required to compute the velocity field, is equal to:

- $$

\frac{dz}{d\zeta} = \frac{4n^2}{\zeta^2-1} \frac{\left(1+\frac{1}{\zeta}\right)^n \left(1-\frac{1}{\zeta}\right)^n} {\left[ \left(1+\frac{1}{\zeta}\right)^n - \left(1-\frac{1}{\zeta}\right)^n \right]^2}.

### Background

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

- $$

\begin{align}

z + 2 &= \zeta + 2 + \frac{1}{\zeta}\, = \frac{1}{\zeta} \left( \zeta + 1 \right)^2, \\ z - 2 &= \zeta - 2 + \frac{1}{\zeta}\, = \frac{1}{\zeta} \left( \zeta - 1 \right)^2.

\end{align}

Dividing the left and right hand sides gives:

- $$

\frac{z-2}{z+2} = \left( \frac{\zeta-1}{\zeta+1} \right)^2.

The right hand side contains (as a factor) the simple second-power law from potential flow theory, applied at the trailing edge near $\backslash zeta=+1.$ From conformal mapping theory this quadratic map is known to change a half plane in the $\backslash zeta$-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]}

- $$

\frac{z-n}{z+n} = \left( \frac{\zeta-1}{\zeta+1} \right)^n,

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

## Notes

## References

## External links

- Joukowski Transform Module by John H. Mathews
- Joukowski Transform NASA Applet