Joukowsky transform

Short description: In mathematics, a type of conformal map

Example of a Joukowsky transform. The circle above is transformed into the Joukowsky airfoil below.

In applied mathematics, the Joukowsky transform (sometimes transliterated Joukovsky, Joukowski or Zhukovsky) is a conformal map historically used to understand some principles of airfoil design. It is named after Nikolai Zhukovsky, who published it in 1910.

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 complex plane ($z$-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 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.

General Joukowsky transform

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

$\begin{array}{rl}z& =x+iy=\zeta +\frac{1}{\zeta }\\ & =\chi +i\eta +\frac{1}{\chi +i\eta }\\ & =\chi +i\eta +\frac{\chi -i\eta }{{\chi }^2+{\eta }^2}\\ & =\chi (1+\frac{1}{{\chi }^2+{\eta }^2})+i\eta (1-\frac{1}{{\chi }^2+{\eta }^2}).\end{array}$

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

$\begin{array}{rl}x& =\chi (1+\frac{1}{{\chi }^2+{\eta }^2}),\\ y& =\eta (1-\frac{1}{{\chi }^2+{\eta }^2}).\end{array}$

Sample Joukowsky airfoil

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

$$|\zeta |=\sqrt{{\chi }^2+{\eta }^2}=1,$$

which gives

$${\chi }^2+{\eta }^2=1.$$

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

Thus the complex unit circle maps to a flat plate on the real-number line from −2 to +2.

Transformations from other circles make a wide range of airfoil shapes.

Velocity field and circulation for the Joukowsky airfoil

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 $\tilde{W}={\tilde{u}}_x-i{\tilde{u}}_y,$ around the circle in the $\zeta$-plane is $$\tilde{W}=V_{\mathrm{\infty }}{e}^{-i\alpha }+\frac{i\mathrm{\Gamma }}{2\pi (\zeta -\mu )}-\frac{V_{\mathrm{\infty }}{R}^2{e}^{i\alpha }}{(\zeta -\mu {)}^2},$$

where

• $\mu ={\mu }_x+i{\mu }_y$ is the complex coordinate of the centre of the circle,
• $V_{\mathrm{\infty }}$ 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 $R=\sqrt{{(1-{\mu }_x)}^2+{\mu }_y^2}$,
• $\mathrm{\Gamma }$ is the circulation, found using the Kutta condition, which reduces in this case to $$\mathrm{\Gamma }=4\pi V_{\mathrm{\infty }}R\sin (\alpha +{\sin }^{-1}\frac{{\mu }_y}{R}).$$

The complex velocity $W$ around the airfoil in the $z$-plane is, according to the rules of conformal mapping and using the Joukowsky transformation, $$W=\frac{\tilde{W}}{\frac{dz}{d\zeta }}=\frac{\tilde{W}}{1-\frac{1}{{\zeta }^2}}.$$

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 and lift per unit of span can be calculated.

Kármán–Trefftz transform

Example of a Kármán–Trefftz transform. The circle above in the $\zeta$-plane is transformed into the Kármán–Trefftz airfoil below, in the $z$-plane. The parameters used are: ${\mu }_x=-0.08,$ ${\mu }_y=+0.08$ and $n=1.94.$ Note that the airfoil in the $z$-plane has been normalised using the chord length.

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 $\zeta$-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 $\alpha .$ This transform is

$z=nb\frac{(\zeta +b{)}^n+(\zeta -b{)}^n}{(\zeta +b{)}^n-(\zeta -b{)}^n},$

( A )

where $b$ is a real constant that determines the positions where $dz/d\zeta =0$, and $n$ is slightly smaller than 2. The angle $\alpha$ between the tangents of the upper and lower airfoil surfaces at the trailing edge is related to $n$ as

$\alpha =2\pi -n\pi ,n=2-\frac{\alpha }{\pi }.$

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

$\frac{dz}{d\zeta }=\frac{4n^2}{{\zeta }^2-1}\frac{{(1+\frac{1}{\zeta })}^n{(1-\frac{1}{\zeta })}^n}{{[{(1+\frac{1}{\zeta })}^n-{(1-\frac{1}{\zeta })}^n]}^2}.$

Background

First, add and subtract 2 from the Joukowsky transform, as given above:

$\begin{array}{rl}z+2& =\zeta +2+\frac{1}{\zeta }=\frac{1}{\zeta }(\zeta +1{)}^2,\\ z-2& =\zeta -2+\frac{1}{\zeta }=\frac{1}{\zeta }(\zeta -1{)}^2.\end{array}$

Dividing the left and right hand sides gives

$\frac{z-2}{z+2}={\left(\frac{\zeta -1}{\zeta +1}\right)}^2.$

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 2 will result in flow around a finite angle. So, by changing the power in the Joukowsky transform to a value slightly less than 2, the result is a finite angle instead of a cusp. Replacing 2 by $n$ in the previous equation gives

$\frac{z-n}{z+n}={\left(\frac{\zeta -1}{\zeta +1}\right)}^n,$

which is the Kármán–Trefftz transform. Solving for $z$ gives it in the form of equation A.

Symmetrical Joukowsky airfoils

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 $\alpha$:

$z=e^{i\alpha }(\zeta -ϵ+\frac{1}{\zeta -ϵ}+\frac{2{ϵ}^2}{a+ϵ}).$

The parameter $ϵ$ yields a flat plate when zero, and a circle when infinite; thus it corresponds to the thickness of the airfoil. Furthermore the radius of the cylinder $a=1+ϵ$.

References

• Anderson, John (1991). Fundamentals of Aerodynamics (Second ed.). Toronto: McGraw–Hill. pp. 195–208. ISBN 0-07-001679-8. 
• Zingg, D. W. (1989). "Low Mach number Euler computations". NASA TM-102205. https://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. 

External links

• Joukowski Transform NASA Applet
• Joukowsky Transform Interactive WebApp