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Is it possible to use divergence theorem to compute the pressure gradient on a face?

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Old   April 5, 2023, 00:44
Default Is it possible to use divergence theorem to compute the pressure gradient on a face?
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pengfeiguo
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I was reading this really great paper on momentrum weight interplation (Bartholomew P, Denner F, Abdol-Azis M H, et al. Unified formulation of the momentum-weighted interpolation for collocated variable arrangements[J]. (Journal of Computational Physics, 2018, 375: 177-208.).

It has been enjoyable and rewarding until I came across Eq. (71). How is it possible to apply the divergence theorem on one surface? Is this a typo?

A screenshot of the formula is here :

http://www.aliyundrive.com/s/F7HCnF6u6RH

Maybe it is a typo?

Any suggestions are welcome.

Thanks.

Last edited by pfguo; April 5, 2023 at 02:34. Reason: correct divergence
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Old   April 5, 2023, 05:19
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Don't have access to the paper nor have read it but usually for this type of integration the single cv is considered that is union of the two cvs on either sides of face.
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Old   April 5, 2023, 08:43
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Filippo Maria Denaro
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Quote:
Originally Posted by pfguo View Post
I was reading this really great paper on momentrum weight interplation (Bartholomew P, Denner F, Abdol-Azis M H, et al. Unified formulation of the momentum-weighted interpolation for collocated variable arrangements[J]. (Journal of Computational Physics, 2018, 375: 177-208.).

It has been enjoyable and rewarding until I came across Eq. (71). How is it possible to apply the divergence theorem on one surface? Is this a typo?

A screenshot of the formula is here :

http://www.aliyundrive.com/s/F7HCnF6u6RH

Maybe it is a typo?

Any suggestions are welcome.

Thanks.
There is also the approximation of the mean value in the formula
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Old   April 5, 2023, 10:01
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Originally Posted by FMDenaro View Post
There is also the approximation of the mean value in the formula
Thanks for your attentions.

Can you please explain in more detail about the first approximately equal sign in that equation?
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Old   April 5, 2023, 10:03
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Originally Posted by arjun View Post
Don't have access to the paper nor have read it but usually for this type of integration the single cv is considered that is union of the two cvs on either sides of face.
Yes, two cell centers P and F both appear in the equation, I am really confused.

Last edited by pfguo; April 5, 2023 at 10:04. Reason: correct english words
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Old   April 5, 2023, 13:02
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I cannot read now the paper, however I mean that


Grad p = div (Ip) -> Int[V] div (Ip) dV = Int[S] np dS


Grad p = lim |V| ->0 1/|V| Int[S] np dS
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