Courant Number for Finite Volume Method

fimbull · November 17, 2016, 19:07

Does anybody know if there is a difference between the Courant Number equation for the finite difference method and the finite volume method?

A book title with the equation would be enough.

thanks

FMDenaro · November 18, 2016, 03:21

Quote:

Originally Posted by fimbull

Does anybody know if there is a difference between the Courant Number equation for the finite difference method and the finite volume method?

A book title with the equation would be enough.

thanks

https://books.google.it/books/about/...er&redir_esc=y

https://books.google.it/books/about/...ir_esc=y&hl=it

fimbull · November 18, 2016, 19:46

When do i need the folowing equation?

And what does it mean?

is there Delta V the cell volume and Delta S a area from the cell surface?

found it on:

https://openfoamwiki.net/index.php/Co

FMDenaro · November 19, 2016, 03:16

http://www.cfd-online.com/Forums/mai...red-grids.html

fimbull · November 20, 2016, 14:04

Well that means OpenFOAM is calculating the Courant Number as a product
of the ratio of the sum of the flux trough the cell faces per cell volume and the timestep width.

$Co = \frac{\sum \phi \Delta t}{\Delta V} \cdot 0.5$

But what means the factor 0.5.

msahin.ae00 · May 1, 2019, 02:01

Most probably it is trying to obtain minimum edge length from the available control volume surface areas and volumes. The absolute control volume area summation close to walls is mainly dominated by lower and upper control volume surface area values. Therefore, the area summation should be divided by 2. Then minimum edge length will be approximately Volume/(0.5 Area sum).

LuckyTran · May 1, 2019, 12:30

Quote:

Originally Posted by msahin.ae00

Most probably it is trying to obtain minimum edge length from the available control volume surface areas and volumes. The absolute control volume area summation close to walls is mainly dominated by lower and upper control volume surface area values. Therefore, the area summation should be divided by 2. Then minimum edge length will be approximately Volume/(0.5 Area sum).

The 1/2 is OpenFOAM specific and has to do with the way OpenFOAM calculates "the summation over all faces." The way it is done in OpenFOAM, it actually double counts because there is a face/flux belonging to the current cell and a face/flux belonging to the neighboring cell and both get summed in the summation.

msahin.ae00 · May 1, 2019, 14:00

OpenFOAM calculates fluxes using a loop over faces and add/subtract flux vector value to right and left cells, respectively. This would be i and j vertices for a vertex based method FVM. When the loop is finished, all control volume faces are done. This does not change the face flux value. But it significantly reduce the computational cost (almost by half).

msahin.ae00 · September 15, 2023, 03:53

CFL number may be considered as the time required to move out/replace the conserved variable trough control volume surfaces with either positive or negative fluxes (incoming or outgoing) for FVM approximation (CFL=1). The absolute summation gives two times of this flux (for steady state, this flux summation without absolute value is zero). But one can sum negative and positive flux, and then take the maxımum absolute value. Then no need for 1/2.

November 17, 2016, 19:07	Courant Number for Finite Volume Method	#1
fimbull New Member Felix Join Date: Sep 2015 Location: Hannover, Berlin Posts: 23 Rep Power: 10	Does anybody know if there is a difference between the Courant Number equation for the finite difference method and the finite volume method? A book title with the equation would be enough. thanks

November 19, 2016, 03:16		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,771 Rep Power: 71	http://www.cfd-online.com/Forums/mai...red-grids.html fimbull likes this.

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November 18, 2016, 19:46		#3
fimbull New Member Felix Join Date: Sep 2015 Location: Hannover, Berlin Posts: 23 Rep Power: 10	When do i need the folowing equation? And what does it mean? is there Delta V the cell volume and Delta S a area from the cell surface? found it on: https://openfoamwiki.net/index.php/Co

November 20, 2016, 14:04		#5
fimbull New Member Felix Join Date: Sep 2015 Location: Hannover, Berlin Posts: 23 Rep Power: 10	Well that means OpenFOAM is calculating the Courant Number as a product of the ratio of the sum of the flux trough the cell faces per cell volume and the timestep width. $Co = \frac{\sum \phi \Delta t}{\Delta V} \cdot 0.5$ But what means the factor 0.5.

May 1, 2019, 02:01		#6
msahin.ae00 New Member Mehmet SAHIN Join Date: Dec 2011 Posts: 12 Rep Power: 14	Most probably it is trying to obtain minimum edge length from the available control volume surface areas and volumes. The absolute control volume area summation close to walls is mainly dominated by lower and upper control volume surface area values. Therefore, the area summation should be divided by 2. Then minimum edge length will be approximately Volume/(0.5 Area sum).

May 1, 2019, 14:00		#8
msahin.ae00 New Member Mehmet SAHIN Join Date: Dec 2011 Posts: 12 Rep Power: 14	OpenFOAM calculates fluxes using a loop over faces and add/subtract flux vector value to right and left cells, respectively. This would be i and j vertices for a vertex based method FVM. When the loop is finished, all control volume faces are done. This does not change the face flux value. But it significantly reduce the computational cost (almost by half).

September 15, 2023, 03:53		#9
msahin.ae00 New Member Mehmet SAHIN Join Date: Dec 2011 Posts: 12 Rep Power: 14	CFL number may be considered as the time required to move out/replace the conserved variable trough control volume surfaces with either positive or negative fluxes (incoming or outgoing) for FVM approximation (CFL=1). The absolute summation gives two times of this flux (for steady state, this flux summation without absolute value is zero). But one can sum negative and positive flux, and then take the maxımum absolute value. Then no need for 1/2.