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

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

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.

The Joukowsky transform of any complex number $\backslash zeta$ 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

\qquad \textand

\\

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.

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

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.

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 $\backslash tildeW=\; \backslash tildeu\_x\; -\; i\; \backslash tildeu\_y,$ around the circle in the $\backslash zeta$ plane is

: $\backslash tildeW=V\_\backslash infty\; e^-i\; \backslash alpha\; +\; \backslash fraci\; \backslash Gamma2\; \backslash pi\; (\backslash zeta\; -\backslash mu)\; -\; \backslash fracV\_\backslash infty\; R^2\; e^i\; \backslash alpha(\backslash zeta-\backslash mu)^2$

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

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

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

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

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 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:

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

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

:$$

\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 $\backslash zeta=+1.$ 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:

:$$

\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).

In 1943 Hsue-shen Tsien+ published a transform of a circle of radius ''a'' into a symmetrical airfoil that depends on parameter ε and angle of inclination α:

:$z\; =\; e^i\; \backslash alpha\; (\backslash zeta\; -\; \backslash epsilon\; +\; \backslash frac1\backslash zeta\; -\; \backslash epsilon\; +\; \backslash frac2\; \backslash epsilon^2a\; +\; \backslash epsilon).$

The parameter ε yields a flat plate when zero, and a circle when infinite; thus it corresponds to the thickness of the airfoil.

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. |