[mrbm]

Hello,

I am writing a 3D panel method code using constant strength source/doublet panels.

I how a question, At section 10.4.1 and 10.4.2, of "Katz, Low speed aerodynamics" (here) the component of velocity (u,v,w) induced from constant strength doublet and source quad, are given. Can I calculate the velocity on each panel by summation of this induced velocities (from all panels on center of desired panel) instead of calculation the gradient of mu, over the surface?

(The reason is for complex geometries, finding neighbor elements and calculation the gradient of mu is not a robust way.)



I tried this but not succeed yet, Is there anything I am doing wrong?

[sbaffini]

My experience is with vortex rings only (thus, constant doublets). In this case you can sum all the velocites of course but, the self induced velocity of the ring is not correct and needs the gradient to be added separately. Without such gradient my local velocities were a factor 1/2 lower.

You can give a look at how it is done here:
http://www.3dpanelmethod.com/home.html

[adrin]

>> the self induced velocity of the ring is not correct and needs the gradient to be added separately. Without such gradient my local velocities were a factor 1/2 lower.

I think you've identified the problem, exactly! But, that factor 1/2 difference comes from the fact that, for the self-induced velocity, you are most likely using the (general) integral for the velocity field, which is valid only away from the boundary, for the whole 4pi volume around the point! For the velocity on the boundary you have to consider the semi-infinite domain, or, equivalently, the 2pi volume. The same logic that is applied to the self-induced velocity for setting up the matrix applies to post-processing of velocity at the wall.

Please try this and let us know if it works!

adrin

[sbaffini]

Originally Posted by adrin

>> the self induced velocity of the ring is not correct and needs the gradient to be added separately. Without such gradient my local velocities were a factor 1/2 lower.

I think you've identified the problem, exactly! But, that factor 1/2 difference comes from the fact that, for the self-induced velocity, you are most likely using the (general) integral for the velocity field, which is valid only away from the boundary, for the whole 4pi volume around the point! For the velocity on the boundary you have to consider the semi-infinite domain, or, equivalently, the 2pi volume. The same logic that is applied to the self-induced velocity for setting up the matrix applies to post-processing of velocity at the wall.

Please try this and let us know if it works!

adrin

Dear Adrin,

maybe you remember that it was you, helping me, around 12 years ago, on this exact matter, on this very forum.

Not working on that anymore, yet I obviously remember

Cheers

[ichamail]

I'm also writing a 3D panel method code using constant strength source/doublet panels.
I have alreay calculate the velocity at each panel's control point calculating the gradient of μ using information from neighbor panels.

I also want to calculate the velocity at control points by summation of the induced velocities from each panel. My results are a bit off (about a factor of 1/2 as Sbaffini mentioned). I'm reading the comments but i can't fully understand the reason. Can someone give me more details?

[gregcabral]

I tried this but not succeed yet.
yahoo mail login

[ichamail]

Originally Posted by gregcabral

I tried this but not succeed yet.
yahoo mail login

I'm still trying to work this out. If I manage to make some progress I'll post more details here

[sbaffini]

My direct experience is only with a 3D vortex ring method, which however is equivalent to a constant doublet one. In my case I had to add a surface gradient contribution, which I took from a slide of an aero course of prof. Davenport at Virginia Tech (if you search my posts here on the subject you will eventually find also a link to the slides).

I am not very versed in the deep theory of the method but, in general, the reason for the need of such additional contribution to the velocity is (or should be) that the self induced velocity of the panel is wrong if computed naively, it must be added a principal value component, which is what the gradient does.

But Adrin is certainly more into it. For example, his work here, in equation (2), clearly highlights how, on the boundary, the potential equation (and thus his gradient) is different than in the actual domain.

If I had to give a rough reason, the surface is where you put singularities, it makes sense that you cannot evaluate the equation there but need to take its principal value.

[ichamail]

Thank you very much for your immediate response. I have already saw your posts about these slides. Unfortunately only one of the pdfs with the slides of prof. Davenport are available for everyone (I think you must be registered to his class). Nevertheless, I have access through my Institution to Adrin's Gharakhani work that you shared, so I can study.

Generaly I thought that the method involving the numerical calculation of doublet strenghts using information from the neigbour panels, was the least accurate but much much faster approach in comparison with the one that requires the calculations of the influnces of all panels to every panel's control point. I want to also do the second approach only to satisfy my compulsion :') and of course compare the results.

Katz and Plotkin underline the fact that u and v components of panel's induced velocity vector V (V = u*i + v*j + w*k), are defined everywhere, except at its own edges. But in fact, they give analytical expressions (from calculation of limits) for the induced velocity at panel's control points (centroids).

I've already came across with some typos (missprints) in equations of Katz & Plotkin, maybe there is something more that I'm missing. If a make any progress I will post again.

Thank you for your valuable informations and sorry for my bad english

[sbaffini]

Originally Posted by ichamail

Thank you very much for your immediate response. I have already saw your posts about these slides. Unfortunately only one of the pdfs with the slides of prof. Davenport are available for everyone (I think you must be registered to his class). Nevertheless, I have access through my Institution to Adrin's Gharakhani work that you shared, so I can study.

Generaly I thought that the method involving the numerical calculation of doublet strenghts using information from the neigbour panels, was the least accurate but much much faster approach in comparison with the one that requires the calculations of the influnces of all panels to every panel's control point. I want to also do the second approach only to satisfy my compulsion :') and of course compare the results.

Katz and Plotkin underline the fact that u and v components of panel's induced velocity vector V (V = u*i + v*j + w*k), are defined everywhere, except at its own edges. But in fact, they give analytical expressions (from calculation of limits) for the induced velocity at panel's control points (centroids).

I've already came across with some typos (missprints) in equations of Katz & Plotkin, maybe there is something more that I'm missing. If a make any progress I will post again.

Thank you for your valuable informations and sorry for my bad english

I can only speak for the vortex ring case but, few things are worth mentioning, as I don't know exactly how this translate to other approaches:

1) The issue for this method is just about the TANGENTIAL velocity computation. So, it is a postprocessing issue, if we can say so, that doesn't affect the computation of the gamma intensities, but only the superficial tangential velocity computation. The method itself is based on the neumann condition, fixing the normal velocity at control points to 0.

2) The gradient part is IN ADDITION to the regular part computed by the influence of all the vortex rings. That is, a first tangential velocity contribution is computed by summing the influence of all the vortex rings, then each vortex ring adds the self influence (so only for its control point) trough the gradient part. Obviously, it is trivial the fact that a vortex ring won't induce a tangential velocity on its control point when computed normally.

This is from my experience side.

If I try to translate this into something more rigorous that is of some help to you, I think that relevant examples in Katz&Plotkin (2nd edition) are, respectively, section 11.2.2 2D Constant strength doublet method with neumann condition (as a reference for my 3D vortex ring case) and section 11.3.1 2D combined constant source and doublet method with dirichlet condition (as reference for a 3D source and doublet method).

There I notice that:

- for the doublet only with neumann condition, neglecting the normal velocity self influence part (which enters in the coefficient determination and that vortex rings already take into account), the tangential velocity computation is analogous to what I do in my 3D vortex ring code, that is eq. 11.37 PLUS eq. 11.38

- for the source/doublet case with dirichlet conditions things are different and ONLY the gradient part is used for the tangential velocity computation, see eqs. 11.74-11.76

Honestly, this difference between the two methods is, as of now, beyond my competence, but I understand that the gradient part in the two cases can be computed, in theory, with the same method (which is relevant for 3D cases, especially with unstructured surface grids). However, I understand it's not about the doublet vs doublet/source, but explicitly due to the different bc conditions. Or maybe I miss some simple detail. I don't know.

For your case, I guess, it means that you can't simply use one method or the other. You can only use the local surface differentiation

[ichamail]

Originally Posted by sbaffini

I can only speak for the vortex ring case but, few things are worth mentioning, as I don't know exactly how this translate to other approaches:

1) The issue for this method is just about the TANGENTIAL velocity computation. So, it is a postprocessing issue, if we can say so, that doesn't affect the computation of the gamma intensities, but only the superficial tangential velocity computation. The method itself is based on the neumann condition, fixing the normal velocity at control points to 0.

2) The gradient part is IN ADDITION to the regular part computed by the influence of all the vortex rings. That is, a first tangential velocity contribution is computed by summing the influence of all the vortex rings, then each vortex ring adds the self influence (so only for its control point) trough the gradient part. Obviously, it is trivial the fact that a vortex ring won't induce a tangential velocity on its control point when computed normally.

This is from my experience side.

If I try to translate this into something more rigorous that is of some help to you, I think that relevant examples in Katz&Plotkin (2nd edition) are, respectively, section 11.2.2 2D Constant strength doublet method with neumann condition (as a reference for my 3D vortex ring case) and section 11.3.1 2D combined constant source and doublet method with dirichlet condition (as reference for a 3D source and doublet method).

There I notice that:

- for the doublet only with neumann condition, neglecting the normal velocity self influence part (which enters in the coefficient determination and that vortex rings already take into account), the tangential velocity computation is analogous to what I do in my 3D vortex ring code, that is eq. 11.37 PLUS eq. 11.38

- for the source/doublet case with dirichlet conditions things are different and ONLY the gradient part is used for the tangential velocity computation, see eqs. 11.74-11.76

Honestly, this difference between the two methods is, as of now, beyond my competence, but I understand that the gradient part in the two cases can be computed, in theory, with the same method (which is relevant for 3D cases, especially with unstructured surface grids). However, I understand it's not about the doublet vs doublet/source, but explicitly due to the different bc conditions. Or maybe I miss some simple detail. I don't know.

For your case, I guess, it means that you can't simply use one method or the other. You can only use the local surface differentiation

Thank you again Sbaffini. I think if I invest a llitle bit of time to study your suggestions and the references that you proposed I'll figure it out. Best regards.

[blackjack]

The driving motivation for finding the gradient of the potential at the control points instead of summing the velocity influence of all the panels is the gradient calculation involves approximately 10 surrounding panels, whereas the influence method requires all the panels.

In 3-D, panel models of 10,000 are common. Calculating the velocities by brute force at all the panels requires evaluating 200,000,000 influence coefficients (2 tangential velocities). The gradient matrix has only 100,000 entries, a factor of 2000 less.

Multi-pole expansion may reduce the disparity.

[ichamail]

Originally Posted by blackjack

The driving motivation for finding the gradient of the potential at the control points instead of summing the velocity influence of all the panels is the gradient calculation involves approximately 10 surrounding panels, whereas the influence method requires all the panels.

In 3-D, panel models of 10,000 are common. Calculating the velocities by brute force at all the panels requires evaluating 200,000,000 influence coefficients (2 tangential velocities). The gradient matrix has only 100,000 entries, a factor of 2000 less.

Multi-pole expansion may reduce the disparity.

I strongly agree with you. As I said, finding the gradient is much much faster. Ιt leads to N (N is the number of panels) overdefined linear systems Ax=b (that you can approximately solve with the least squares method). The number of rows of matrix A is the number of neighbor panels. As you said the number of surrounding panels is very small.

Calculating the velocity at each control point with the summation of induced velocities from every panel is for sure computationally expensive.

I'm just obsessed with this approach because seems so straight forward and I can't have logical results. I'm not saying i'm going to use this one for a large number of panels. I just want to make it work for a low number of panels and see if the results with the first method matches.

[sbaffini]

Originally Posted by blackjack

The driving motivation for finding the gradient of the potential at the control points instead of summing the velocity influence of all the panels is the gradient calculation involves approximately 10 surrounding panels, whereas the influence method requires all the panels.

In 3-D, panel models of 10,000 are common. Calculating the velocities by brute force at all the panels requires evaluating 200,000,000 influence coefficients (2 tangential velocities). The gradient matrix has only 100,000 entries, a factor of 2000 less.

Multi-pole expansion may reduce the disparity.

Dear James, I see what you mean, but I have a doubt.

From what you say, I understand that in the source-doublet method with Dirichlet boundary conditions on the potential, using the local gradient or the sum of all the panels velocity contributions should give similar results. You either use one or the other (reasonably, the gradient for performances), but not both.

This seems in line with the Katz&Plotkin example, as they simply use the surface gradient both in the 2D and 3D example for the dirichlet approach with sources and doublets.

So far so good.

The doubt, however, comes from the example with doublets only and neumann boundary conditions. The Katz&Plotkin example (and my experience with a 3D vortex ring code confirms that) says that, in this case, we need to use BOTH: the contribution from all the panels and, to properly take the panel self-influence into account, a surface gradient term. See section 11.2.2 part f (calculation of pressures and loads) and the equations 11.37 and 11.38.

Now, there is a sign and a 0.5 factor difference between the doublet-neumann surface gradient and the source-doublet-dirichlet surface gradient that appear in the K&P examples. So, is it just this? In one example they use the full gradient, so they don't need any other contribution, and in the other one instead they simply use a different approach, implicitly recognizing that half the gradient is, indeed, the self induced term and the other half is the effect of all the other panels?

Time has passed since the last time I've read the full book, but this kind of confuses me. Of course, I'm fine as long as I follow the book examples, but I would like to understand.

For what concerns the surface gradient, ichamail, note that you don't actually need to solve the system for it, there are closed form solutions to it, either using the QR algorithm or the Normal Equations approach.

[blackjack]

I agree that if you limit yourself by using only doublets (how do you do intakes, exhaust?)
then you need to not only take the gradient of the doublets to get the vorticity (difference in velocity from upper to lower side of panel), you also need to take the velocity influence of all the panels to determine the mean velocity on the panel so you can determine the velocity & pressure on the side of interest.

One of the advantages of using sources=d(phi)/dn and doublets=phi is that the derivative of phi is the (perturbation) velocity, just add the reference velocity to get total velocity. I don't see any advantage to not using source terms.

[blackjack]

And to clarify, I do both.

Thin (zero thickness) surfaces, like vortex lattice, modeled without sources, are allowed in the same model as thick surfaces. For example, in my paper of 2004 Applied Aero Conference, I showed a thick wing with zero-thickness endplates.

[sbaffini]

Originally Posted by blackjack

And to clarify, I do both.

Thin (zero thickness) surfaces, like vortex lattice, modeled without sources, are allowed in the same model as thick surfaces. For example, in my paper of 2004 Applied Aero Conference, I showed a thick wing with zero-thickness endplates.

Thanks for the clarification.

To give some context on why I used this method, consider that this was my first ever panel code, made for a university course.

Back then, I just tought that I would have got a lot of bang for the buck with a 3D vortex ring approach. In particular, both closed and open surfaces for external aero and a single vortex segment as routine for the influence with no reference frame issues. Very simple and attractive for someone who had no idea on how to write a code in the first place.

Unfortunately, as I knew very little on the subject, I ended up spending more time on it than if I went the more common route.

However, I was not even remotely considering any complex scenario, just external aero cases with wake shedding.

[ichamail]

Equations 10.107 - 10.109 from ''Low Speed Aerodynamics" for Velocity influence of a quadrialteral constant strength doublet, have some typos in my edition.

I found at least one minus sign that should be a plus sign. Maybe there is something more than that

Katz and Plotkin state that they were taken from Hess and Smith's technical reports. I searched at many technical reports published by them but I can only find the equations for induced velocity by constant strength source and quadratic doublet.

Does anyone know the paper of Hess and Smith that contains the derivation of Constant strength doublet equations?

Maybe the velocity problem I have comes from some typo in the near field formulas.