Joukowski airfoil

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

The transform is

: $z = \zeta + \frac,$

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 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
z &= x + iy = \zeta + \frac \\
&= \chi + i \eta + \frac \\ pt &= \chi + i \eta + \frac \\ pt &= \chi\left(1 + \frac1\right) + i\eta\left(1 - \frac1\right).
\end

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

:\begin
x &= \chi\left(1 + \frac1\right), \\ pt y &= \eta\left(1 - \frac1\right).
\end

Sample Joukowsky airfoil

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

$, \zeta, = \sqrt = 1,$

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

HOME


Content is Copyleft
Website design, code, and AI is Copyrighted (c) 2014-2017 by Stephen Payne


Consider donating to Wikimedia



As an Amazon Associate I earn from qualifying purchases