Solving Conformal Mapping Flow Problem

[ChrisHarvey]

Hi everyone,

Let me set the scene. I'm writing a program to model the flow of an ideal fluid around various singularities using the complex potential and then using conformal transformations to map boundaries into new shapes. It's very nearly done but one of the transformations (what appears to be the easiest one as well) is giving me grief.

Here we go:

w = Complex Number
z = Complex Number
U = Constant
Zeta = Complex Number

Let w = Uz

This is simply uniform flow.

I want to map z |-> z ^ (2/3) which should give me flow around a corner.

Doing this I get W = Uz^(2/3)

To get the velocity vector I take the conjugate of dw/dz

i.e. u - iv = 2/3 * U * z^(-1/3)

If I hard code this directly into my program, I get the right flow pattern.

However, because my program has to cope with lots of different transformations not all so simple, it must work using the chain rule.

For this I introduced the complex number zeta, which is simply z after it has been mapped.

i.e. zeta = z ^ (2/3)

The velocity is now given by dw/d(zeta). Using the chain rule...

dw/d(zeta) = dw/dz * dz/d(zeta)

If w = Uz, dw/dz = U
& If zeta = z ^ (2/3), d(zeta)/dz = (2/3) * z ^ (-1/3)

therefore dz/d(zeta) = 3/2 * z ^(1/3)

& dw/d(zeta) = 3U/2 * z ^ (1/3)

which does of course give a different velocity to the one calculated the other way. It seems that d(zeta)/dz is the inverse of what is required.

Strangely though, this same method works for other transformations such as zeta = z + 1/z.

I have spent about 1 and a half days now tracking the problem down to this and trying to work out what's wrong.

Is there anybody there who can help?? Please.

 

[ChrisHarvey]

I may have just solved it (many hours {and a Tesco shift!} later). I think the problem may be this: yes, the transformation to transform uniform flow into flow around a corner is z |-> z ^ (2/3), but if I'm going to use the zeta method, z = zeta ^ (2/3), NOT (as I put above) zeta = z ^ (2/3). Then I progress as before and my 2 answers match up. I've just edited the code, compiled & run and I seem to be getting the right flow patterns. If I am correct it would explain why it seems to work for all other transformations, but not for this one. I will work through a couple of test environments tomorrow by hand and see the program gives the same answers.

 

[ravenprp]

Hi there,

It sounds like you have put in a lot of hard work and effort into solving this conformal mapping flow problem! I can understand your frustration when the program is not producing the desired results. From what you have described, it seems like the issue lies in the chain rule and the incorrect inverse. Have you tried checking your calculations and code to ensure that the inverse is being calculated correctly? It might also be helpful to double-check if the complex number zeta is being mapped correctly as well.

Another suggestion would be to reach out to other experts or forums in the field of fluid dynamics or complex analysis for assistance. Sometimes having a fresh perspective can help identify any errors or provide new insights on how to solve the problem. Don't give up and keep persevering, I'm sure you will find a solution soon. Best of luck!