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

[math]\displaystyle{ z = \zeta + \frac{1}{\zeta}, }[/math]

where [math]\displaystyle{ z = x + iy }[/math] is a complex variable in the new space and [math]\displaystyle{ \zeta = \chi + i \eta }[/math] 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 ([math]\displaystyle{ z }[/math]-plane) by applying the Joukowsky transform to a circle in the [math]\displaystyle{ \zeta }[/math]-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 [math]\displaystyle{ \zeta = -1 }[/math] (where the derivative is zero) and intersects the point [math]\displaystyle{ \zeta = 1. }[/math] This can be achieved for any allowable centre position [math]\displaystyle{ \mu_x + i\mu_y }[/math] 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 [math]\displaystyle{ \zeta }[/math] to [math]\displaystyle{ z }[/math] is as follows:

[math]\displaystyle{ \begin{align} 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 + \frac1{\chi^2 + \eta^2}\right) + i\eta\left(1 - \frac1{\chi^2 + \eta^2}\right). \end{align} }[/math]

So the real ([math]\displaystyle{ x }[/math]) and imaginary ([math]\displaystyle{ y }[/math]) components are:

[math]\displaystyle{ \begin{align} x &= \chi\left(1 + \frac1{\chi^2 + \eta^2}\right), \\[2pt] y &= \eta\left(1 - \frac1{\chi^2 + \eta^2}\right). \end{align} }[/math]

Sample Joukowsky airfoil

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

[math]\displaystyle{ |\zeta| = \sqrt{\chi^2 + \eta^2} = 1, }[/math]

which gives

[math]\displaystyle{ \chi^2 + \eta^2 = 1. }[/math]

So the real component becomes [math]\displaystyle{ x = \chi (1 + 1) = 2\chi }[/math] and the imaginary component becomes [math]\displaystyle{ y = \eta (1 - 1) = 0 }[/math].

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 [math]\displaystyle{ \widetilde{W} = \widetilde{u}_x - i\widetilde{u}_y, }[/math] around the circle in the [math]\displaystyle{ \zeta }[/math]-plane is [math]\displaystyle{ \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}, }[/math]

where

• [math]\displaystyle{ \mu = \mu_x + i \mu_y }[/math] is the complex coordinate of the centre of the circle,
• [math]\displaystyle{ V_\infty }[/math] is the freestream velocity of the fluid,

[math]\displaystyle{ \alpha }[/math] is the angle of attack of the airfoil with respect to the freestream flow,

• [math]\displaystyle{ R }[/math] is the radius of the circle, calculated using [math]\displaystyle{ R = \sqrt{\left(1 - \mu_x\right)^2 + \mu_y^2} }[/math],
• [math]\displaystyle{ \Gamma }[/math] is the circulation, found using the Kutta condition, which reduces in this case to [math]\displaystyle{ \Gamma = 4\pi V_\infty R\sin\left(\alpha + \sin^{-1}\frac{\mu_y}{R}\right). }[/math]

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

Here [math]\displaystyle{ W = u_x - i u_y, }[/math] with [math]\displaystyle{ u_x }[/math] and [math]\displaystyle{ u_y }[/math] the velocity components in the [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] directions respectively ([math]\displaystyle{ z = x + iy, }[/math] with [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] 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 [math]\displaystyle{ \zeta }[/math]-plane is transformed into the Kármán–Trefftz airfoil below, in the [math]\displaystyle{ z }[/math]-plane. The parameters used are: [math]\displaystyle{ \mu_x = -0.08, }[/math] [math]\displaystyle{ \mu_y = +0.08 }[/math] and [math]\displaystyle{ n = 1.94. }[/math] Note that the airfoil in the [math]\displaystyle{ z }[/math]-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 [math]\displaystyle{ \zeta }[/math]-plane to the physical [math]\displaystyle{ z }[/math]-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 [math]\displaystyle{ \alpha. }[/math] This transform is

[math]\displaystyle{ z = nb \frac{(\zeta + b)^n + (\zeta - b)^n}{(\zeta + b)^n - (\zeta - b)^n}, }[/math]

( A )

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

[math]\displaystyle{ \alpha = 2\pi - n\pi, \quad n = 2 - \frac{\alpha}{\pi}. }[/math]

The derivative [math]\displaystyle{ dz/d\zeta }[/math], required to compute the velocity field, is

[math]\displaystyle{ \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}. }[/math]

Background

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

[math]\displaystyle{ \begin{align} 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{align} }[/math]

Dividing the left and right hand sides gives

[math]\displaystyle{ \frac{z - 2}{z + 2} = \left( \frac{\zeta - 1}{\zeta + 1} \right)^2. }[/math]

The right hand side contains (as a factor) the simple second-power law from potential flow theory, applied at the trailing edge near [math]\displaystyle{ \zeta = +1. }[/math] From conformal mapping theory, this quadratic map is known to change a half plane in the [math]\displaystyle{ \zeta }[/math]-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 [math]\displaystyle{ n }[/math] in the previous equation gives

[math]\displaystyle{ \frac{z - n}{z + n} = \left( \frac{\zeta - 1}{\zeta + 1} \right)^n, }[/math]

which is the Kármán–Trefftz transform. Solving for [math]\displaystyle{ z }[/math] gives it in the form of equation A.

Symmetrical Joukowsky airfoils

In 1943 Hsue-shen Tsien published a transform of a circle of radius [math]\displaystyle{ a }[/math] into a symmetrical airfoil that depends on parameter [math]\displaystyle{ \epsilon }[/math] and angle of inclination [math]\displaystyle{ \alpha }[/math]:

[math]\displaystyle{ z = e^{i\alpha} \left(\zeta - \epsilon + \frac{1}{\zeta - \epsilon} + \frac{2\epsilon^2}{a + \epsilon}\right). }[/math]

The parameter [math]\displaystyle{ \epsilon }[/math] 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 [math]\displaystyle{ a=1+\epsilon }[/math].

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