## Joukowsky transform +Search for Videos

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

: $z=\zeta+\frac1\zeta$,

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.
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 $\zeta$ 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 $\zeta$ = −1 (where the derivative is zero) and intersects the point $\zeta$ = 1. This can be achieved for any allowable centre position $\mu_x + i\mu_y$ 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.

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

:
\beginalign
z &= x + iy =\zeta+\frac1\zeta
\\
&= \chi + i \eta + \frac1\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.
\endalign

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

:
\beginalign
x &= \frac\chi (\chi^2 + \eta^2 + 1)\chi^2 + \eta^2
\\
y &= \frac\eta (\chi^2 + \eta^2 - 1)\chi^2 + \eta^2.
\endalign

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

:$\sqrt\chi^2+\eta^2 = 1 \quad \textwhich gives \quad \chi^2+\eta^2 = 1.$

So the real component becomes $x = \frac\chi \left(1 + 1\right)1 = 2\chi$
and the imaginary component becomes $y = \frac\eta \left(1 - 1\right)1 = 0.$

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.

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 conjugate velocity $\tildeW= \tildeu_x - i \tildeu_y,$ around the circle in the $\zeta$ plane is

: $\tildeW=V_\infty e^-i \alpha + \fraci \Gamma2 \pi \left(\zeta -\mu\right) - \fracV_\infty R^2 e^i \alpha\left(\zeta-\mu\right)^2$

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 $R=\sqrt\left(1-\mu_x\right)^2+\mu_y^2$
*$\Gamma$ is the circulation+, found using the Kutta condition+, which reduces in this case to

:: $\Gamma=4\pi V_\infty R \sin \left\left(\ \alpha + \sin^-1 \left\left( \frac\mu_yR \right\right)\right\right).$

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

: $W=\frac\tildeW\fracdzd\zeta =\frac\tildeW1-\frac1\zeta^2.$

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 Russia+n scientist Nikolai Zhukovsky+. His name has historically been romanized in a number of ways, thus the variation in spelling of the 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:

:
z = n \frac\left(1+\frac1\zeta\right)^n+\left(1-\frac1\zeta\right)^n
\left(1+\frac1\zeta\right)^n-\left(1-\frac1\zeta\right)^n,
(A)

with ''n'' slightly smaller than 2. The angle ''α'', between the tangent+s of the upper and lower airfoil surface, at the trailing edge is related to ''n'' by:

:$\alpha = 2\pi\, -\, n\pi \quad \text and \quad n=2-\frac\alpha\pi.$

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

:
\fracdzd\zeta = \frac4n^2\zeta^2-1 \frac\left(1+\frac1\zeta\right)^n \left(1-\frac1\zeta\right)^n
\left[ \left(1+\frac1\zeta\right)^n - \left(1-\frac1\zeta\right)^n \right]^2.

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

:
\beginalign
z + 2 &= \zeta + 2 + \frac1\zeta\, = \frac1\zeta \left( \zeta + 1 \right)^2, \\
z - 2 &= \zeta - 2 + \frac1\zeta\, = \frac1\zeta \left( \zeta - 1 \right)^2.
\endalign

Dividing the left and right hand sides gives:

:
\fracz-2z+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 $\zeta=+1.$ From conformal mapping theory this quadratic map is known to change a half plane in the $\zeta$-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:

:
\fracz-nz+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).

reflist:

* book
Anderson
John
1991
Fundamentals of Aerodynamics
Second
McGraw–Hill
Toronto
0-07-001679-8
195–208

* web
| first=D.W.
| last=Zingg
| title=Low Mach number Euler computations
| year=1989
| publisher=NASA+ TM-102205
| url=http://ntrs.nasa.gov/search.jsp?R=19940000533&hterms=Zingg+Low+Mach+Low+Mach&qs=Ntx%3Dmode%2520matchall%7Cmode%2520matchall%26Ntk%3DTitle%7CAuthor-Name%26N%3D0%26Ntt%3DLow%2520Mach%7CZingg

*
*

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