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

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

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

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

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

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

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.

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.

- 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

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Last updated on Friday August 15, 2008 at 06:24:40 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday August 15, 2008 at 06:24:40 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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