Calculating Flow Around Elliptical Cylinder w/ Point Charges

[meldraft]

Hey all,

I am using the Joukowski transform to calculate the electrostatic potential and streamlines around an elliptical cylinder. The flow is caused by two point charges (one positive, one negative). My idea was, of course, to first solve the problem for a circular cylinder, and then transform everything using the Joukowski transform.

I am getting some weird results, so I would be grateful is anyone can point out the mistake in my calculations!

The equation for the complex potential of a point charge located at point z0=a is:

$$φ_1=\frac{Λ}{2 \pi}ln(z-a)$$

where Λ is the strength of the source (or sink), and z,a are complex numbers.

Also, the potential to simulate a circular cylinder originating at (0,0), is that of a doublet:

$$φ_c=-\frac{μ}{z}$$

where μ is a real constant. Therefore, the potential function for my problem should be:

$$Φ=\frac{Λ}{2 \pi}ln(z-a)-\frac{Λ}{2 \pi}ln(z+a)-\frac{μ}{z}$$

The field looks realistic at this point.

Then, I do the Joukowski transform:

$$J=z+\frac{λ^2}{z}$$

$$λ=\sqrt{a*R-R^2}$$

$$R=(a+b)/2$$

where a,b are the 2 semi-axes of the ellipse and R is the radius of the original circular cylinder. I know that this is correct, because i can see the ellipse in the plot. My problem is that the field around it is not what I would expect:

1. The potential lines are no longer always perpendicular to the boundary of the ellipse

2. The conformal map is causing the source and sink to come closer to point (0,0), because of the distortion caused by the growing ellipse.

I have no idea why either of them is happening, as I figured the transform should continue to satisfy the original boundary conditions.

The way that I plot the potential is the following:

contour(real(J),imag(J),-real(f))

so, basically, the original potential corresponds to the transformed mesh. I have also set the axes to always be equal to each other, so what I see is not distorted due to axis scaling.

I have been trying to figure out where the mistake is for a few days now, and I really am stuck. Can anyone spot what I have done wrong?

P.S. I am also attaching two pictures, one for the cylinder, where everything is normal, and one for the problematic transform. You can see how the source and sink have moved away from their origins at (-4,0) and (4,0), and that the potential lines are no longer perpendicular to the boundary of the ellipse.

 

[marcusl]

Here are a few points to keep in mind:
1. 2D conformal maps are not appropriate to finding fields from point charges (I assume you know this). The problem you are working is ok for sources that are uniformly charged lines perpendicular to the z plane.
2. Regarding motion of the sources, you need to predistort their positions so they end up in the right place in the transformed plane.
3. You don't say want the cylinder is. I assume it's perfectly conducting?
4. How do you do the transform? (Did you invert to find z in terms of J)?

 

[meldraft]

Thanks for your response!

1. Actually, I've only ever used conformal maps for fluid mechanics problems (so for uniform flows of the form z=v*exp(iθ)). I was not aware that they are not appropriate for this type of modelling. Can you elaborate? I am very curious since all of my books describe the uniform flow situation with added sources or sinks, and never a dipole (like my problem) on its own. They just don't mention it, however, I never read that it was not suitable (although I'm kind of inferring it from experience).

Btw, the straight perpendicular lines you saw at (-4,0),(4,0) are auxiliary lines I ploted to keep track of how off the sources are from their original position. They are not part of the problem.

2. Thanks, I'll do that.

3. The cylinder in this case is a perfect insulator. I want the flow to go completely around it. Once I have this, I want to try out the problem with partial conductivity.

4. I calculated J(z) through the Joukowski equation, and then plotted the original potentials in the new mesh ( so newX=real(J(z)) and newY=imag(J(z)) ).

 

[marcusl]

1. Conformal maps are, by definition, two dimensional. The assumption is that everything looks the same in and out of the paper (call it the x direction), from x = - to + infinity. Point charges are, obviously, not uniform along x and cannot be analyzed by conformal mapping. Whether you intended to or not, you are analyzing two uniform infinitely long lines of charge next to an infinitely long cylinder.

3. You don't mean perfect insulator. Vacuum is a perfect insulator, and it corresponds to no cylinder at all! You have the choice of perfect conductor or infinite dielectric constant. They'll give the same field contours outside--the field lines will terminate normally on the cylinder and the equipotential lines will look like the fluid flow contours you are visualizing--but keep in mind that nothing is actually flowing here. (A minor point: a conductor has no field inside while the dielectric will conduct the field lines through the cylinder).

Something more--I wonder if you have properly described your problem. If I examine the circle
$$z=\lambda \exp(i2\pi\phi),$$
it transforms to a branch cut extending between the foci of the ellipse in the J plane. I think you need to surround this circle with another larger one that maps to the desired elliptic boundary. This means that the dielectric in the z plane is an annulus, and is a uniformly filled ellipse with a branch cut between the foci in the J plane.

In other words, I think the problem is more complicated that what you set up...

 

[meldraft]

1. Yes, I understand what you mean. It was a bad description on my part. I assume that these 'point charges' are actually infinitely long cylinders, whose projection in the complex plane is what I see in the plot, that was why I called them point charges.

3. My physical problem is a hole filled with air, so what I actually meant is that, basically (in polar coordinates):

$$\nabla{Φ(R,\theta)}\cdot \bar{r}=0$$

where R is the radius of the cylinder and $$\bar{r}$$ is the r unit vector (so essentially the intensity vectors are always tangent to the boundary)

I'm not sure that I understand why an annulus would be a better fit. I think you mean that the problem is connected to the branch cuts, but in either case they would still be enclosed by the cylinder correct?

The transform works perfectly for uniform flow, so if I have an ellipse in uniform flow, the field looks as it should. Therefore, shouldn't the problem be connected to the fact that I am using only sources and sinks?

 

[marcusl]

1. Yes, I understand what you mean. It was a bad description on my part. I assume that these 'point charges' are actually infinitely long cylinders, whose projection in the complex plane is what I see in the plot, that was why I called them point charges.

Good, we're on the same page.

3. My physical problem is a hole filled with air, so what I actually meant is that, basically (in polar coordinates):

$$\nabla{Φ(R,\theta)}\cdot \bar{r}=0$$

where R is the radius of the cylinder and $$\bar{r}$$ is the r unit vector (so essentially the intensity vectors are always tangent to the boundary)

I'm confused here. Air (or vacuum) does not affect electric fields/potentials, so they'll pass right through your cylinder. Can you better define the problem you are trying to solve?

I'm not sure that I understand why an annulus would be a better fit. I think you mean that the problem is connected to the branch cuts, but in either case they would still be enclosed by the cylinder correct?

I'll need to stop and think about this--I haven't look at a conformal map since the mid-90's. Won't be able to devote much time until this weekend, however.

 

[meldraft]

Take your time, I am grateful that you spend any time at all to answer my questions

I am modelling a solid conductor with a hole. The conductor is roughly 2kOhms per square centimeter, and the current and voltage are both very small. Thus, the current is not allowed to pass through the hole (air), neither radiate. Realistically, this field is so weak on air that I can neglect it and consider Neumann boundary conditions on the boundary of the hole. That is why I considered it as a perfect insulator.

I have read that both Neumann and Dirichlet BCs are invariant under conformal mappings. Thus, if I impose a Neumann BC on a unit disc, it should be carried over to whatever shape the disc is transformed to. This is why is think I am not applying the theory correctly

 

[marcusl]

Oh, your sources are in a conductor and the ellipse is a hole! I didn't get that at all before--no wonder my comments about dielectric constant made no sense.

I'll try to think about your problem this weekend.

 

[meldraft]

Thank you I'll also post any updates as I work on this.