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

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Old   April 17, 2015, 07:34
Question [3D Vortex Particle Method]The function in diffusion operator
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Youjiang Wang
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For the vorticity diffusion, the Laplacian operator \Delta\vec{\omega} is always replaced by an integral operator . A function \eta(\rho) is used in the integral. Then the diffusion term looks like

\Delta\vec{\omega} = \frac{2}{\sigma^2}\int\eta_{sigma}(\vec{x}-\vec{y})(\vec{\omega}(\vec{y})-\vec{\omega}(\vec{x}))dy

and \eta_{sigma}(\vec{x}-\vec{y}) = \frac{1}{\sigma^3}\eta(\frac{|\vec{x}-\vec{y}|}{\sigma}),for the vortex particle method, the change of particles' strength according to the diffusion would be

\frac{d\vec{\alpha}_k}{dt}=\frac{2\nu}{\sigma^2}\sum_l(vol_k\vec{\alpha}_l-vol_l\vec{\alpha}_k)\eta_{\sigma}(\vec{x}_k-\vec{x}_l)

It is said that the function \eta(\rho) can be got from the vorticity smoothing function \zeta(\rho) as
\eta(\rho)  = -\frac{1}{\rho}\frac{d\zeta(\rho)}{d\rho}
My question is that, there is a 2nd order algebraic smoothing function \frac{15}{8\pi}\frac{1}{(1+\rho^2)^{3.5}} and a 4th order Gaussian smoothing function -\frac{3}{4\pi}e^{-\rho^3}+\frac{3}{\pi}e^{-2*\rho^3}. I have calculated the corresponding integral function for the both smoothing functions, and plotted against \rho. 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 04:39. Reason: correct notation errors and make it more clear
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Old   April 17, 2015, 08:28
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1) Two small comments on notation. It might be a personal preference more than a requirement, but still.
a) \nabla\omega is gradient operator, while Laplacien operator is usually either \nabla^2\omega or \Delta\omega.
b) In 3D vortex formulation, the vorticity is a vector. It is not explicit in your notation.
2) I have not yet worked an applied project with vortex formulation, but the two function plot made me think about classic results from Lamb-Oseen (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.
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Old   April 20, 2015, 04:28
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Quote:
Originally Posted by Alex C. View Post
1) Two small comments on notation. It might be a personal preference more than a requirement, but still.
a) \nabla\omega is gradient operator, while Laplacien operator is usually either \nabla^2\omega or \Delta\omega.
b) In 3D vortex formulation, the vorticity is a vector. It is not explicit in your notation.
2) I have not yet worked an applied project with vortex formulation, but the two function plot made me think about classic results from Lamb-Oseen (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.
Thanks for you advice. Actually, i have used the incorrect notation. Now I have corrected them. It is a blob or ball for vortex particle method, and as i see it, Lamb-Oseen regard the vortex as filament. And the derivative is really w.r.t r or distance from field point to the particle.
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.
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Old   April 20, 2015, 18:06
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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 3-d? So, I mean, what is the vorticity "atom", or the singular element physically?
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Old   April 22, 2015, 05:29
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Originally Posted by serguei View Post
First of all, forget for the moment about math and diffusion also. What is the vortex particle in your model for the 3-d? So, I mean, what is the vorticity "atom", or the singular element physically?
In my opinion, the vortex particle is an approximation of the real physical vorticity field. The vorticity carried by the particle is the total vorticity around it ( of a specified volume, although I don't know whether the volume should be changed with time. )

\alpha_k(t) = \int_{vol_k}\omega(t)dv \approx vol_k\omega(t)

And the position of the particle k is the position of the initial vorticity at time t. So it's a Lagrangian description of the vorticity field.
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Old   April 24, 2015, 10:03
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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 2-d, 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 3-d 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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