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how to implement periodic bc on a staggered grid 

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December 20, 2018, 03:29 
how to implement periodic bc on a staggered grid

#1 
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Theo
Join Date: Mar 2009
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this is a question related to the question I asked here a few days ago (are periodic boundary conditions exact?). I solve an incompressible flow, so I have 4 variables, u, v, w, and p. p is the fluctuating pressure, i.e. what I receive from solving the NSE/Poisson equation. I use a staggered grid where p is stored in the cell center and the velocity components on the faces.
Now I want to implement periodix bc in xdirection. For that, I can find 2 different approaches: 1) very simple, as described e.g. here: Periodic boundary conditions for solving Navier Stokes Equations on a Staggered Grid . To sum up, on a grid (i=1..N) this results in u(1,j) = u(N1,j) ; u(N,j) = u(2,j) v(1,j) = vN1,j) ; v(N,j) = v(2,j) w(1,j) = w(N1,j) ; w(N,j) = w(2,j) p(1,j) = p(N1,j) ; p(N,j) = p(2,j) 2) more complicated, additional equations are solved such as the Poisson equation for the velocity components as described in https://www.researchgate.net/publica...e_annular_duct . Also Filippo's comment in the above linked discussion indicates something like that. What is the difference between both approaches? Is approach 2) required or is 1) sufficient? 

December 20, 2018, 05:01 

#2 
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Filippo Maria Denaro
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Given the periodicity lenght L = Nx*dx and using the staggering grid you have
Given for example the pressure node p(1,1) in the center of the cell at x=dx/2,y=dy/2 you have u(1,1) is staggered at x=0,y=dy/2 and is linked by periodicity to u(Nx+1,1) v(1,1) is staggered at x=dx/2,y=0 and is linked to v(Nx+1,1) (extrapoint at L+dx/2) p(1,1) is linked to p(Nx+1,1) (extrapoint at L+dx/2) 

December 20, 2018, 05:32 

#3 
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Theo
Join Date: Mar 2009
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thank you once again for your answer.
but do you understand why the people in the paper which I linked do that procedure (see §III.B) involving a solution of Poisson's equation for the periodic bc? 

December 20, 2018, 06:00 

#4  
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Filippo Maria Denaro
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Quote:
I see Eq.(16) that is the divergencefree constraint and then it is rewritten in terms of the pressure equation by substituting the Hodge decomposition. What is exactly your doubt? 

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