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

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

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

$ilde\{W\}=V\_infty\; e^\{-i\; alpha\}\; +\; frac\{i\; Gamma\}\{2\; pi\; (zeta\; -mu)\}\; -\; frac\{V\_infty\; R^2\; e^\{i\; alpha\{(zeta-mu)^2\}$

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 $R=sqrt\{(1-mu\_x)^2+mu\_y^2\}$
*$Gamma$ is the circulation, found using the Kutta condition, which reduces in this case to

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

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

$W=frac\{\; ilde\{W\{frac\{dz\}\{dzeta\; =frac\{\; ilde\{W\{1-frac\{1\}\{zeta^2$

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 $z=zeta+frac\{1\}\{zeta\}\; =\; S3(S2(S1(zeta)))$, where S3, S2, S1 are:

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

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

References

*cite book
last = Anderson
first = John
year = 1991
title = Fundamentals of Aerodynamics
edition = Second Edition
publisher = McGraw-Hill
location = Toronto
id = ISBN 0-07-001679-8
pages = 195-208
*D.W. Zingg, "Low Mach number Euler computations", 1989, NASA TM-102205

External links

* [http://math.fullerton.edu/mathews/c2003/JoukowskiTransMod.html Joukowski Transform Module by John H. Mathews]