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February 26, 2013, 06:00 
'SIMPLE' question

#1 
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Philipp
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Hello to everyone!
I have a question regarding the SIMPLEalgorithm for finite volumes. The derivation of the algorithm starts (in different books) with the discretized momentum equation, such as: A_p * u_p + SUM (A_x * u_x) = Q_p  pressure gradient_p I am a bit confused about this equation: I though we need the volume integrated momentum equation to start with, with conservative integrations of all terms  because that is what we are interested in, in FV. Here, the derivation starts with something that looks like the momentum equation itself. Or I can not explain the appearance of the pressure gradient... Can someone clear this up for me?
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February 26, 2013, 07:28 

#2 
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Lefteris
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If my memory is not failing me, this is derived from the integral form of the momentum equation by approximating each of its terms.
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Lefteris 

February 26, 2013, 14:09 

#3 
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Philipp
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Then, I am confused by the Ferziger / Peric book. They say that nearly all commercial codes use the conservative form of integrating the pressure term. Later they explain SIMPLE, SIMPLEC,... using a derivation that starts with the non conservative form.
Anyway, what you say, means that the cell volume "dV" is taken from the "pressure gradient_p" term and put into the coefficients of the other summands, correct?
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February 26, 2013, 15:25 

#4  
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Filippo Maria Denaro
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Quote:
The key is that the integrals are approximated by centred second order formulas (mean value approximation). This often leads to have discrete equations that appears similar both for FV and FD. The confusion vanishes for higher order of accuracy 

February 27, 2013, 14:00 

#5 
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Philipp
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I was talking about conservative and nonconservative formulation. It looks like they use a nonconservative pressure integral, although they previously state that nearly everyone (commercial) uses the conservative form.
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February 27, 2013, 17:20 

#6  
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Filippo Maria Denaro
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Quote:
The fluxintegrated form of the pressure term should write as Int [S] n.(pI) dS but I think that by using the volume average of the pressure gradient, the second order discretization is done with the mean value formula therefore it seems a nonconservative formula ... 

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