[3D Vortex Particle Method]How to calculate velocity derivatives

wyj216 · April 9, 2015, 11:06

Now, I am trying to write a simple code about 3D Vortex Particle Method. However, when I dealing with the stretching term (( $\omega\cdot\nabla)u$ ), I face a problem. I don't know what's a good way to calculate the velocity derivatives.

I am reading Cottet's book "Vortex Method: theory and practice". This problem is not explained explicitly, only be referred as a distribution way $\sum\nu_{p}u_{p}\partial_{i}\xi_{\epsilon}(x-x_{p})$ . It seems that, this formula regard the velocity in the same way with the vorticity. I mean, firstly concentrate it on particles, and then smooth it.

My idea is to apply the derivative operator directly on the velocity kern. I mean while velocity is got through $u=\sum K_{\epsilon} \times a_{k}$ , can I calculate velocity derivatives by $\nabla u = \sum (\nabla K_{\epsilon}) \times a_{k}$ .

Thanks for any response.

adrin · April 9, 2015, 16:28

differentiating the kernel K directly is what most people in the field do. I haven't read the book, but I suspect the smooth velocity approach (your first summation) might be related to the topic of formulating it to preserve the solenoidal nature of the vorticity transport equation (the original idea actually came from me during a discussion with Cottet). The smooth velocity approach does not work well.

adrin

wyj216 · April 10, 2015, 03:09

Thank you very much.
The smooth velocity approach is actually a method used to preserve the solenoidal nature of the vorticity. And it's used in a version of vorticity-in-cell methods as written in the Book.
It's exciting to know the original idea came from you.

April 9, 2015, 11:06	[3D Vortex Particle Method]How to calculate velocity derivatives	#1
wyj216 New Member Youjiang Wang Join Date: Apr 2015 Location: Hamburg Posts: 22 Rep Power: 11	Now, I am trying to write a simple code about 3D Vortex Particle Method. However, when I dealing with the stretching term (( $\omega\cdot\nabla)u$ ), I face a problem. I don't know what's a good way to calculate the velocity derivatives. I am reading Cottet's book "Vortex Method: theory and practice". This problem is not explained explicitly, only be referred as a distribution way $\sum\nu_{p}u_{p}\partial_{i}\xi_{\epsilon}(x-x_{p})$ . It seems that, this formula regard the velocity in the same way with the vorticity. I mean, firstly concentrate it on particles, and then smooth it. My idea is to apply the derivative operator directly on the velocity kern. I mean while velocity is got through $u=\sum K_{\epsilon} \times a_{k}$ , can I calculate velocity derivatives by $\nabla u = \sum (\nabla K_{\epsilon}) \times a_{k}$ . Thanks for any response.

April 9, 2015, 16:28		#2
adrin Senior Member adrin Join Date: Mar 2009 Posts: 115 Rep Power: 17	differentiating the kernel K directly is what most people in the field do. I haven't read the book, but I suspect the smooth velocity approach (your first summation) might be related to the topic of formulating it to preserve the solenoidal nature of the vorticity transport equation (the original idea actually came from me during a discussion with Cottet). The smooth velocity approach does not work well. adrin wyj216 likes this.

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April 10, 2015, 03:09		#3
wyj216 New Member Youjiang Wang Join Date: Apr 2015 Location: Hamburg Posts: 22 Rep Power: 11	Thank you very much. The smooth velocity approach is actually a method used to preserve the solenoidal nature of the vorticity. And it's used in a version of vorticity-in-cell methods as written in the Book. It's exciting to know the original idea came from you.