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[3D Vortex Particle Method]The function in diffusion operator 

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April 17, 2015, 08:34 
[3D Vortex Particle Method]The function in diffusion operator

#1 
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Youjiang Wang
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For the vorticity diffusion, the Laplacian operator is always replaced by an integral operator . A function is used in the integral. Then the diffusion term looks like
and ,for the vortex particle method, the change of particles' strength according to the diffusion would be It is said that the function can be got from the vorticity smoothing function as My question is that, there is a 2nd order algebraic smoothing function and a 4th order Gaussian smoothing function . I have calculated the corresponding integral function for the both smoothing functions, and plotted against . However, as i see it, the properties are really different . tmp.jpg Which is correct? Or both can be used? Last edited by wyj216; April 20, 2015 at 05:39. Reason: correct notation errors and make it more clear 

April 17, 2015, 09:28 

#2 
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1) Two small comments on notation. It might be a personal preference more than a requirement, but still.
a) is gradient operator, while Laplacien operator is usually either or .2) I have not yet worked an applied project with vortex formulation, but the two function plot made me think about classic results from LambOseen (Vorticity distribution and Velocity distribution). Your 2nd order algebraic looks like it's more or less the cylindrical derivative about r or your 4th order Gaussian. I might be wrong, but maybe you have mixed some things you shouldn't mix. 

April 20, 2015, 05:28 

#3 
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Youjiang Wang
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Quote:
I have just began to do some testing code for the 3D particle method. Thus i'm not clear with some concept and equations. Sometimes my question may seem to be quite strange, and i'm sorry for that. 

April 20, 2015, 19:06 

#4 
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Serguei
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First of all, forget for the moment about math and diffusion also. What is the vortex particle in your model for the 3d? So, I mean, what is the vorticity "atom", or the singular element physically?


April 22, 2015, 06:29 

#5  
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Youjiang Wang
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Quote:
And the position of the particle is the position of the initial vorticity at time . So it's a Lagrangian description of the vorticity field. 

April 24, 2015, 11:03 

#6 
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Serguei
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Actually, there two way to describe the vorticity fields: contentious and discrete. When you are talking about particles "atoms", you assume some discrete set of the elements, which induct the VELOCITY NOT THE VORTICITY between themselves and interact with each other. That is it. After understanding that, you should start to think about nature of those vorticity particles "atoms". For 2d, that is easy. They are just points, or some pieces of distributed by some way vorticity inside. (Inside each of them, not outside !). They are strait and infinite in the perpendicular direction. For 3d everything become much complicate on this stage. The vorticity lines should either go to infinity or be closed. Each of those way of representation is a very challenging.


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3d vortex particle method, diffusion, smoothing function 
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