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Gerrit Groot September 7, 2007 11:06

Panel Method Doublet Derivatives

In order to get the velocity distribution after having solved for the doublets on the surface in a panel method with constant sources and doublets, you could take the derivative in local coordinates of the doublets and so getting the local velocity at each panel collocation point, right?

Well, this works fine, but I am having numerical problems with these derivatives near the leading edge, possibly because of the panel size overthere.

Does anyone know whether these collocation points can be treated as external flow field points instead????

(This in order to avoid the need for numerical derivatives)

I am asking this because they are formally inside the geometry.

Please don't refer to Katz' book, I got that already.



Harish September 7, 2007 13:05

Re: Panel Method Doublet Derivatives
One thing that you can do is try clustering grid points along the edge to prevent the large panel over there.

Harish September 7, 2007 13:06

Re: Panel Method Doublet Derivatives
You might also want to check

and the references provided therein.

Gerrit Groot September 8, 2007 07:10

Re: Panel Method Doublet Derivatives
Thanks for your help.

There are very few reactions, it seems that no one is working on panel methods anymore these days.

Harish September 9, 2007 14:58

Re: Panel Method Doublet Derivatives
Due to the increase in computing power there is not a lot of work on using panel method these days.But if you look in the field of micro air vehicles, flights of insects and birds and biofluid dynamics there are still a lot of people who use panel methods to calculate the forces and moments.

Peter Attar September 9, 2007 22:12

Re: Panel Method Doublet Derivatives
Which is kind of ironic since it is precisely in the RE regimes in which MAVs fly that ignoring viscous effects is usually not a good idea. For example it has been shown that dynamic stall is the main high lift mechanism which allow small birds and insects to hover like they do...

Gerrit Groot September 10, 2007 12:18

Re: Panel Method Doublet Derivatives
Yes, this is exactly what I was thinking, the viscous effects are dominant in these cases, but okay, a panel method is still much much faster, so maybe it will compensate in come special cases.

gocarts September 11, 2007 15:32

Re: Panel Method Doublet Derivatives
As you say, a panel method alone would not be a good fit for low Reynolds Number flows (see Why Use a Panel Method) - the underlying assumption for potential flow is a thin boundary layer. However, with a panel method coupled to a boundary layer method isn't there a chance such low Reynolds Number simulations are viable away from stall conditions?

Gerrit Groot September 12, 2007 05:16

Re: Panel Method Doublet Derivatives
Yes it is, away from stall conditions. But that means that you could use it coupled with a boundary layer code for steady flying airplanes, submarines, or whatever. However, when it comes to flapping birds the viscous area is so big that it wouldn't compensate in my opinion, maybe for a gliding hawk, but okay, not for something like a sparrow. Insects is another class of flow, this is completely viscous, be honest how would you feel flapping in the yoghurt, because that's how an insect feels when he flies :)

Harish September 12, 2007 12:49

Re: Panel Method Doublet Derivatives
Depending on the speed of the freestream and frequency of hovering, there are range of frequencies and freestream velocities over which the panel method gives a very good match with the Navier-Stokes simulation. Its mostly problem and aerodynamic parameter dependent. You might want to look into the dissertation on Young for more information on this topic.


Peter Attar September 12, 2007 23:22

Re: Panel Method Doublet Derivatives
As I stated above, I think it has been shown (experimentally) that the main high-lift mechanism used by insects in hover is a dynamic stall type mechanism. If you really want to understand what is going on in hover and perching then there is no way panel methods will work. However for birds or insects at "high" forward flight speeds this might be ok.

Adrin Gharakhani September 13, 2007 14:56

Re: Panel Method Doublet Derivatives
It seems that the discussion has diverged from the original question to one of whether panel methods can be used for insect flow analysis.

(1) The exact problem/difficulty was not explained well in the original post. If I understand the question correctly, yes you can use the integral equations themselves to evaluate the velocity anywhere in the field, including very near the boundary. However, accurate evaluation near the boundary requires either very high order cubature schemes or the use of exact integrals (the latter is not a problem). Nevertheless, it is important to appreciate that what you're using is afterall a collocation scheme, meaning that your solution is accurate only at the collocation point - away from the collocation (and on the panel itself) the velocity varies substantially from its collocation value. So, the point you want to use away from the panel is most accurate along the line normal to the panel and passing through the collocation point. I hope I've understood your problem. Obviously, near singularities you'll need higher resolution for better accuracy, but this is true of any method.

(2) As for insect and MAV related flows, while panel methods are not the right tool, vortex methods (for the solution of the vorticity transport equation, including viscous effects) are probably optimal in handling complicated unsteady geometries of this type.


Gerrit Groot September 14, 2007 10:33

Re: Panel Method Doublet Derivatives
Thank you Adrin, you understand my question, I want to use external points just on top of the collocation points inside the fluid, indeed perpendicular to the panel in order to get rid of the finite difference derivatives in adjacent panel direction.

(Katz, calls this dq/deta,dq/dpsi,dq/dn for those who happen to have the book).

As the solution of the local derivatives taking finite differences of the doublets ON the surface is quite unprecise (panels are not all in the same plane) this causes numerical problems.

I thought that by treating the collocation points as external points in the fluid, this may be solved, or not? (I don't care about computational time for now)

If you know any book on 3D panel methods other than Katz' book, your tips are very welcome.



Adrin Gharakhani September 14, 2007 21:39

Re: Panel Method Doublet Derivatives
Yes, theoretically, you can differentiate the boundary integral equation for the potential to get the velocity field anywhere (including the boundary itself). If you're using a piecewise constant variation of the potential the accuracy of your velocity will be very poor no matter how you deal with it. If I were you I'd use a linear variation on the panel, which is computationally not much different from the constant panel case (the number of unknowns can actually be smaller). With a linear variation you can then get piecewise constant velocities _on_ the panel using very simple finite difference formulation.

You can also use "meshless" techniques such the moving least squares method (MLSM) to get the velocities using the data you have as collocation points (it doesn't matter that they're not in the same plane). The accuracy of this will not be any worse than what you're suggesting to do. There are programs on netlib, for example, that you can use to get started with the MLSM approach (for a quick test)


Leo Lazauskas September 14, 2007 21:55

Re: Panel Method Doublet Derivatives
You might like to search for "desingularised methods". These have been used on a variety of problems in ship hydrodynamics. I don't know if they have been used with success on viscous flows.

The PhD thesis of D.C. Scullen at Uni of Adelaide is one recent reference. Others who use the method include Robert Beck and his PhD students at U. Mich.


Gerrit Groot September 15, 2007 15:07

Re: Panel Method Doublet Derivatives
Thank you both Adrin and Leo for your very helpful answers.

Where do you guys get this info?


Leo Lazauskas September 15, 2007 19:37

Re: Panel Method Doublet Derivatives
I work with E. O. Tuck and David Scullen so I know of some of their work and influences.

A very interesting problem to solve using panel methods is the lift of a thin wing with a circular planform. There is an exact solution available for the lift and moment, but panel methods struggle to get the strength of the tip singularity.

See for example the preprint of a J. Ship Research paper by Tuck and myself at:


Gerrit Groot September 17, 2007 09:13

Re: Panel Method Doublet Derivatives
Thanks for the link!

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