Joukowsky transform

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

The Joukowsky transform, also called the Joukowsky transformation, the Joukowski transform, the Zhukovsky transform and other variations, is a conformal map historically used to understand some principles of airfoil design.

The transform is

z=\zeta+\frac{1}{\zeta}

where z = x + iy is a complex variable in the new space and ζ = χ + iη 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 z plane by applying the Joukowsky transform to a circle in the ζ 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 origin (where the conformal map has a singularity) and intersects the point z=1. This can be achieved for any allowable centre position by varying the radius of the circle.

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 velocity \tilde{W} around the circle in the ζ plane is

\tilde{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

  • μ = μx + iμy is the complex coordinate of the centre of the circle
  • V_\infty is the freestream velocity of the fluid
  • α 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}
  • Γ 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} \left( \frac{\mu_y}{R} \right)\right).

The complex velocity W around the airfoil in the z plane is, according the rules of conformal mapping,

W=\frac{\tilde{W}}{\frac{dz}{d\zeta}} =\frac{\tilde{W}}{1-\frac{1}{\zeta^2}}

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. A similar transform, the Karman-Trefftz transform, generates an airfoil with a finite trailing edge. The Karman-Trefftz transform requires an additional parameter: the trailing edge angle.

The transformation is named after Russian scientist Nikolai Zhukovsky. His name has historically been romanized in a number of ways, thus the variation in spelling of the transform.

[edit] References

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