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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:

RotatingDisc

Thank you in advance.

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3 Answers 3

up vote 4 down vote accepted

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
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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
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A classic textbook on this topic from the perspective of Complex Analysis is

Churchill: Complex Variables and Applications.

Chapter 8 of that book discusses Fluid Flow (amongst other applications). The aerofoil transform itself is introduced in some Exercises, although the earlier theory describes the assumptions. Of course the basic assumption is that the fluid plane x-y describes all the relevant properties, since it is a 2-D model.

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Thank you for this reference. It seems like this is the reference concerning such calculations. Greets –  Robert Filter Feb 9 '11 at 15:26
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