Zhukovsky transform

Joukowsky transform

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 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 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 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 zeta 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

  • 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 R=sqrt{(1-mu_x)^2+mu_y^2}
  • Gamma is the circulation, found using the Kutta condition, which reduces in this case to

Gamma=4pi 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 to the rules of conformal mapping,

W=frac{tilde{W}}{frac{dz}{dzeta}} =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.

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.

Karman-Trefftz transform

The Karman-Trefftz transform is a conformal map derived from the Joukowsky transform. While a Joukowsky airfoil has a cusped trailing edge, a Karman-Trefftz airfoil has a finite trailing edge. The Karman-Trefftz transform therefore requires an additional parameter: the trailing edge angle.

When the Joukowsky transform is written as a composition of three transformations, one of these can be modified independently. The Joukowsky transform is z=zeta+frac{1}{zeta} = S3(S2(S1(zeta))), where S3, S2, S1 are:

S3(z) = frac{2+2z}{1-z}
S2(z) = z^2 ,!
S1(z) = frac{z-1}{z+1}

Reducing the exponent in S2(z) by a small amount increases the thickness of the trailing edge to a finite amount.

References

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

External links

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