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how to implement periodic bc on a staggered grid

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Old   December 20, 2018, 04:29
Default how to implement periodic bc on a staggered grid
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Theo
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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 x-direction. 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(N-1,j) ; u(N,j) = u(2,j)
v(1,j) = vN-1,j) ; v(N,j) = v(2,j)
w(1,j) = w(N-1,j) ; w(N,j) = w(2,j)
p(1,j) = p(N-1,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?
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Old   December 20, 2018, 06:01
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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) (extra-point at L+dx/2)
p(1,1) is linked to p(Nx+1,1) (extra-point at L+dx/2)
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Old   December 20, 2018, 06:32
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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 žIII.B) involving a solution of Poisson's equation for the periodic bc?
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Old   December 20, 2018, 07:00
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Quote:
Originally Posted by holgerbre View Post
thank you once again for your answer.

but do you understand why the people in the paper which I linked do that procedure (see žIII.B) involving a solution of Poisson's equation for the periodic bc?



I see Eq.(16) that is the divergence-free 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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