# How is the Joukowsky Transform used to calculate the Flow of an Airfoil?

As I read in The Road to Reality by Roger Penrose, the Joukowsky transform $$w(z) = \frac12\left( z + \frac1z \right)$$ after Nikolai Zhukovsky (transcribed in several versions from Никола́й Его́рович Жуко́вский) can be used to calculate the flow of a non-viscious, incompressible and irrotational flow around an airfoil.

This can be done since the solution of a potential flow around a cylinder is known in full analyticity and the given transform conformally maps a circle on an airfoil-like geometry. I don't understand this argumentation, so:

### How is the Joukowsky Transform used to calculate the Flow of an Airfoil?

An example of such a transformation is given in the mentioned Wikipedia article:

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The crux of the argument is that we can treat complex analytic (holomorphic) functions as functions in 2D, and their real and imaginary parts (separately) are solutions of Laplace's equation ($\nabla^2 \psi = 0$), due to the Cauchy-Riemann condition. Conformal maps such as the one you cite map analytic functions to analytic functions, i.e generate new solutions from old ones. Thus, by knowing a trivial solution (such as around the cylinder), we can generate the flow around a new object by finding a conformal map to it.

Relevant details:

• Laplace's equation solves potential flow problems: incompressible, inviscid, curl-free flow (though we are allowed rotational flow around finite objects --- the resulting singularity is technically outside of the domain).

• We can somewhat relax the need for fully holomorphic functions.

• Such mappings tend to mess up your boundary conditions at infinity --- so it may be quite hard in general to find such mappings.

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@Robert Filter: assuming you can following this lead, do you see how the Joukowski mapping affects the cylinder? –  Gerben Feb 9 '11 at 10:29
@Gerben: I am not sure if I got you correctly. I have to put some parametrization of a circle for $z$ such that I get the airfoil like thing or what did you mean? –  Robert Filter Feb 9 '11 at 15:23
you're right, its pretty much that conformal maps map Laplace-solutions onto each other - would be nice to know about more difficult boundary conditions than Neumann or Dirichlet, though. Greets –  Robert Filter Feb 9 '11 at 15:24
@Robert Filter: I think we're talking about the same thing. You plug in a parametrization of the circle, and Joukowski should give you a model of an airfoil. (In real life, engineers of course don't use the $w(z)$ you've stated above, but that's another story.) If you're looking for more details (it's might be non-intuitive), open any hydrodynamics/... textbook where it should be done explicitly. –  Gerben Feb 9 '11 at 20:25

Check this reference: http://www.grc.nasa.gov/WWW/K-12/airplane/map.html

Be sure that your browser supports java -- worth it.

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Thank you for the link :) –  Robert Filter Feb 9 '11 at 15:25