boundedness of TVD in 3D unstructured grids

Lydia · July 4, 2014, 06:08

Hello Foamers,

I have a question concerning the TVD limiting in OpenFOAM:

Does the formulation of the TVD limiter (according to Darwish and Moukalled, 2003) guarantee boundedness for 3D unstructured grids?

From the paper (see ftp://ftp.demec.ufpr.br/disciplinas/...alled_2003.pdf) I understand that the formulation of r results in the traditional formulation when it is applied to 1d structured cases. But I am not sure, whether the formulation for unstructured grids ensures boundedness in the same way.

I am asking that, also because it might help me with my question if MULES is necessary, if I use TVD for the flux calculation. I posted this here:
http://http://www.cfd-online.com/For...-why-both.html

Looking forward to your comments.

Regards,
Lydia

HakikiCanakkaleli · July 6, 2014, 18:38

Yes,

Both formulations are essentially the same. Darwish and Moukalled derived this form because location of upstream node is required in the original r formulation and this is not the case for an unstructured grid. For the OpenFOAM implementation, please look at Jasak's thesis. (Although he derived the NVD counterpart, the procedure is the same for a TVD scheme.).

Lydia · July 7, 2014, 03:52

Thanks for your comment!

So I can conclude, even with 3D unstructured grids, the TVD schemes guarantee a bounded solution?

Any different opinions?

Regards,
Lydia

Lydia · July 9, 2014, 05:15

Hello again,

one more question concerning TVD:

Are TVD methods generally restricted to a small Courant Number?

From some tests with scalarTransportFoam, we saw that most TVD schemes get unstable, when Courant Numbers >1 are used. From where does this Courant number restriction result?

Any comments are highly appreciated!

Regards,
Lydia

HakikiCanakkaleli · July 9, 2014, 17:31

Hi,

== 1 ==
Even though any scheme is Courant number dependent to some extent, TVD schemes are infamous for their dependency on Courant number. The reasoning behind is kind of an essay question/answer. Please consult with Jasak's thesis for further information regarding this. He states Courant number dependency of TVD schemes as you also observed.

== 2 ==
NVD schemes, therefore, are generally recommended over TVD schemes. I personally recommend you to try an NVD scheme, especially Gamma ( preferably Gamma 0.2 ).

== 3 ==
Nothing (e.g. boundedness of a TVD scheme in OF) is guaranteed with OpenFOAM. You need to go through the source code line by line to 'fetch' the details of the implementations.

Hope this helps.

July 4, 2014, 06:08	boundedness of TVD in 3D unstructured grids	#1
Lydia New Member Lydia Schulze Join Date: Jan 2012 Location: Karlsruhe, Germany Posts: 20 Rep Power: 14	Hello Foamers, I have a question concerning the TVD limiting in OpenFOAM: Does the formulation of the TVD limiter (according to Darwish and Moukalled, 2003) guarantee boundedness for 3D unstructured grids? From the paper (see ftp://ftp.demec.ufpr.br/disciplinas/...alled_2003.pdf) I understand that the formulation of r results in the traditional formulation when it is applied to 1d structured cases. But I am not sure, whether the formulation for unstructured grids ensures boundedness in the same way. I am asking that, also because it might help me with my question if MULES is necessary, if I use TVD for the flux calculation. I posted this here: http://http://www.cfd-online.com/For...-why-both.html Looking forward to your comments. Regards, Lydia

July 9, 2014, 05:15	Courant number restriction in TVD?	#4
Lydia New Member Lydia Schulze Join Date: Jan 2012 Location: Karlsruhe, Germany Posts: 20 Rep Power: 14	Hello again, one more question concerning TVD: Are TVD methods generally restricted to a small Courant Number? From some tests with scalarTransportFoam, we saw that most TVD schemes get unstable, when Courant Numbers >1 are used. From where does this Courant number restriction result? Any comments are highly appreciated! Regards, Lydia

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July 6, 2014, 18:38		#2
HakikiCanakkaleli Senior Member Canakkale Dardanelspor Join Date: Aug 2012 Posts: 135 Rep Power: 14	Yes, Both formulations are essentially the same. Darwish and Moukalled derived this form because location of upstream node is required in the original r formulation and this is not the case for an unstructured grid. For the OpenFOAM implementation, please look at Jasak's thesis. (Although he derived the NVD counterpart, the procedure is the same for a TVD scheme.).

July 7, 2014, 03:52		#3
Lydia New Member Lydia Schulze Join Date: Jan 2012 Location: Karlsruhe, Germany Posts: 20 Rep Power: 14	Thanks for your comment! So I can conclude, even with 3D unstructured grids, the TVD schemes guarantee a bounded solution? Any different opinions? Regards, Lydia

July 9, 2014, 17:31		#5
HakikiCanakkaleli Senior Member Canakkale Dardanelspor Join Date: Aug 2012 Posts: 135 Rep Power: 14	Hi, == 1 == Even though any scheme is Courant number dependent to some extent, TVD schemes are infamous for their dependency on Courant number. The reasoning behind is kind of an essay question/answer. Please consult with Jasak's thesis for further information regarding this. He states Courant number dependency of TVD schemes as you also observed. == 2 == NVD schemes, therefore, are generally recommended over TVD schemes. I personally recommend you to try an NVD scheme, especially Gamma ( preferably Gamma 0.2 ). == 3 == Nothing (e.g. boundedness of a TVD scheme in OF) is guaranteed with OpenFOAM. You need to go through the source code line by line to 'fetch' the details of the implementations. Hope this helps.