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

pfguo · April 5, 2023, 00:44

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.

arjun · April 5, 2023, 05:19

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.

FMDenaro · April 5, 2023, 08:43

Quote:

Originally Posted by pfguo

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

pfguo · April 5, 2023, 10:01

Quote:

Originally Posted by FMDenaro

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?

pfguo · April 5, 2023, 10:03

Quote:

Originally Posted by arjun

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.

FMDenaro · April 5, 2023, 13:02

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

April 5, 2023, 00:44	Is it possible to use divergence theorem to compute the pressure gradient on a face?	#1
pfguo New Member pengfeiguo Join Date: Nov 2017 Posts: 10 Rep Power: 8	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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April 5, 2023, 05:19		#2
arjun Senior Member Arjun Join Date: Mar 2009 Location: Nurenberg, Germany Posts: 1,278 Rep Power: 34	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.

April 5, 2023, 13:02		#6
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,813 Rep Power: 73	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