Definitions

# Joukowsky transform

The Joukowsky transform, also called the Joukowsky transformation, the Joukowski transform, the Zhukovsky transform and other variations, is a conformal map historically used to understand some principles of airfoil design.

The transform is

$z=zeta+frac\left\{1\right\}\left\{zeta\right\}$

where $z=x+iy$ is a complex variable in the new space and $zeta=chi + i eta$ is a complex variable in the original space.

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 $zeta$ 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 origin (where the conformal map has a singularity) and intersects the point z=1. This can be achieved for any allowable centre position by varying the radius of the circle.

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

$tilde\left\{W\right\}=V_infty e^\left\{-i alpha\right\} + frac\left\{i Gamma\right\}\left\{2 pi \left(zeta -mu\right)\right\} - frac\left\{V_infty R^2 e^\left\{i alpha\right\}\right\}\left\{\left(zeta-mu\right)^2\right\}$

where

• $mu=mu_x+i mu_y$ is the complex coordinate of the centre of the circle
• $V_infty$ is the freestream velocity of the fluid
• $alpha$ is the angle of attack of the airfoil with respect to the freestream flow
• R is the radius of the circle, calculated using $R=sqrt\left\{\left(1-mu_x\right)^2+mu_y^2\right\}$
• $Gamma$ is the circulation, found using the Kutta condition, which reduces in this case to

$Gamma=4pi V_infty R sin left\left( alpha + sin^\left\{-1\right\} left\left(frac\left\{mu_y\right\}\left\{R\right\} right\right)right\right)$.

The complex velocity W around the airfoil in the z plane is, according to the rules of conformal mapping,

$W=frac\left\{tilde\left\{W\right\}\right\}\left\{frac\left\{dz\right\}\left\{dzeta\right\}\right\} =frac\left\{tilde\left\{W\right\}\right\}\left\{1-frac\left\{1\right\}\left\{zeta^2\right\}\right\}$

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.

## Karman-Trefftz transform

The Karman-Trefftz transform is a conformal map derived from the Joukowsky transform. While a Joukowsky airfoil has a cusped trailing edge, a Karman-Trefftz airfoil has a finite trailing edge. The Karman-Trefftz transform therefore requires an additional parameter: the trailing edge angle.

When the Joukowsky transform is written as a composition of three transformations, one of these can be modified independently. The Joukowsky transform is $z=zeta+frac\left\{1\right\}\left\{zeta\right\} = S3\left(S2\left(S1\left(zeta\right)\right)\right)$, where S3, S2, S1 are:

$S3\left(z\right) = frac\left\{2+2z\right\}\left\{1-z\right\}$
$S2\left(z\right) = z^2 ,!$
$S1\left(z\right) = frac\left\{z-1\right\}\left\{z+1\right\}$

Reducing the exponent in $S2\left(z\right)$ by a small amount increases the thickness of the trailing edge to a finite amount.

## References

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