conformal transformations
Trying to transform an airfoil into a near circle using the inverse of the joukowski formula x = z + a^2/z where x,z complex , "a" effectively constant and getting stupid results. Any suggestions?

Re: conformal transformations

Re: conformal transformations
Thanks Andy, you seem to have a certain knowledge about this, have you ever come across a reverse solution for the specific problem of the inverse of x= 1/2(z+1/z)?
The site was helpful regardless. Alex 
Re: conformal transformations
I have some experience with the inverse transformation of KarmanTreffz airfoils. I encountered a difficulty in getting a circle for highly cambered airfoil. I found that the mapping is not onetoone inside a certain lense shape. So, if the airfoil is so cambered that a part of it (typically, lower surface, near trailing edge) will be contained in this lense shape, then the points in there will not be mapped back onto the circle from which the airfoil was generated. So, going from circle to airfoil is fine, but its inverse is not necessarily fine.
I don't know how to get around this yet. Do you have a problem for cambered airfoil? 
Re: conformal transformations
Sorry. I last looked at conformal transformation nearly 20 years ago and most things are pretty vague now  if not completely forgotten!,
Andy. 
Re: conformal transformations
Thanks Nishikawa, I haven't come across KarmanTreffz airfoils but I assume the transform for them is similar that of Joukowsky's. What method did you use to transform it to the circle? I'll look into the discontinuities to see if I can give you any input,
Alex 
Re: conformal transformations
KarmanTreffz airfoil is more general than Joukowsky's, it generates airfoils with trailing edge of finite angle. In both cases, a circle is mapped to an airfoil. Inverse transformation will map the airfoil to a circle.
So, in the case of Joukowsky, x = z + a^2/z which transforms a circle in zplane to an airfoil in xplane, if we have a solution for a circle, we get a solution for an airfoil in x. This transformation can be written also as (x2a)/(x+2a)=(za)^2/(z+a)^2. You can solve this for z easily, and get the inverse transformation: z = a [sqrt(x+2a)+sqrt(x2a)]/[sqrt(x+2a)sqrt(x2a)]. This is what I use to transform an airfoil into a circle. It sounds like what you're saying is something different... 
Re: conformal transformations
Nishikawa, I think what you have given me is the solution to what I'm looking for but in practice I cannot get it to produce a circle in the z plane.
Do you have any suggestions as to books I could find this reverse transfrom in to be used in practicle applications? 
Re: conformal transformations
I don't know any books that describe the inverse transformation.
I wonder why you can't produce a circle. What is the problem? What do you get? As I mentioned, it has a problem when the sirfoil has large camber, but it works for others. But of course it does not transform an arbitrary airfoil into a perfect circle. 
Re: conformal transformations
I can produce an airfoil fine but when I reverse transform it it comes out as a smaller shifted version of the original airfoil. To get this I got z in terms of x using a quadratic to solve for z. Is this an incorrect procedure in general?
With your solution I am getting similar results. should I be applying it by replacing x and z with complex numbers in the form of e.g. u+iv? Thanks for you help in the matter 
Re: conformal transformations
Well, I never had such a problem. The inverse transformation formula I wrote down worked for me. I wrote a code to do the transformation in FORTRAN 77 and 90 which handle complex numbers. I don't have to break z=u+iv into u and v. I use the formula as it is.
Good Luck! 
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