Joukowsky transform

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.

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

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:

${\begin{aligned}z&=x+iy=\zeta +{\frac {1}{\zeta }}\\&=\chi +i\eta +{\frac {1}{\chi +i\eta }}\\[2pt]&=\chi +i\eta +{\frac {\chi -i\eta }{\chi ^{2}+\eta ^{2}}}\\[2pt]&=\chi \left(1+{\frac {1}{\chi ^{2}+\eta ^{2}}}\right)+i\eta \left(1-{\frac {1}{\chi ^{2}+\eta ^{2}}}\right).\end{aligned}}$

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

${\begin{aligned}x&=\chi \left(1+{\frac {1}{\chi ^{2}+\eta ^{2}}}\right),\\[2pt]y&=\eta \left(1-{\frac {1}{\chi ^{2}+\eta ^{2}}}\right).\end{aligned}}$

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

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 ${\widetilde {W}}={\widetilde {u}}_{x}-i{\widetilde {u}}_{y},$ around the circle in the $\zeta$-plane is $${\widetilde {W}}=V_{\infty }e^{-i\alpha }+{\frac {i\Gamma }{2\pi (\zeta -\mu )}}-{\frac {V_{\infty }R^{2}e^{i\alpha }}{(\zeta -\mu )^{2}}},$$

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 ${\textstyle R={\sqrt {\left(1-\mu _{x}\right)^{2}+\mu _{y}^{2}}}}$,
• $\Gamma$ is the circulation, found using the Kutta condition, which reduces in this case to $$\Gamma =4\pi V_{\infty }R\sin \left(\alpha +\sin ^{-1}{\frac {\mu _{y}}{R}}\right).$$

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

Here $W=u_{x}-iu_{y},$ 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

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

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

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

Background

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

${\begin{aligned}z+2&=\zeta +2+{\frac {1}{\zeta }}={\frac {1}{\zeta }}(\zeta +1)^{2},\\[3pt]z-2&=\zeta -2+{\frac {1}{\zeta }}={\frac {1}{\zeta }}(\zeta -1)^{2}.\end{aligned}}$

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 $\epsilon$ and angle of inclination $\alpha$:

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

The parameter $\epsilon$ 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 $a=1+\epsilon$.

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.

External links

• Joukowski Transform NASA Applet
• Joukowsky Transform Interactive WebApp