CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > Main CFD Forum

Production term in Finite Volumes Schemes

Register Blogs Members List Search Today's Posts Mark Forums Read

Reply
 
LinkBack Thread Tools Display Modes
Old   October 13, 2012, 14:39
Default Production term in Finite Volumes Schemes
  #1
New Member
 
Ivan
Join Date: Aug 2012
Posts: 22
Rep Power: 5
Ivan is on a distinguished road
Dear all,

I have a working scheme for LES for a transport equation of a scalar, let's say Phi, done with Finite Volumes schemes in the code I am using. I need to modify this code by adding a production term which consists substantially of two non-linear terms, let's say k1*Phi^2 and k2*grad(Phi)*grad(Phi), where k1 and k2 are constants.

I want to implement it as an implicit scheme, which should be like:

A_d * Phi = A_nd * Phi + b

where A_d and A_nd are the diagonal matrix and not-diagonal matrix of coefficients and b is the vector of known terms.

I am confused on how to implement these new terms. Does someone have an idea or can suggest me some book where to look for?

Thank you!
Ivan is offline   Reply With Quote

Old   October 13, 2012, 17:53
Default
  #2
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,662
Rep Power: 23
FMDenaro will become famous soon enough
Quote:
Originally Posted by Ivan View Post
Dear all,

I have a working scheme for LES for a transport equation of a scalar, let's say Phi, done with Finite Volumes schemes in the code I am using. I need to modify this code by adding a production term which consists substantially of two non-linear terms, let's say k1*Phi^2 and k2*grad(Phi)*grad(Phi), where k1 and k2 are constants.

I want to implement it as an implicit scheme, which should be like:

A_d * Phi = A_nd * Phi + b

where A_d and A_nd are the diagonal matrix and not-diagonal matrix of coefficients and b is the vector of known terms.

I am confused on how to implement these new terms. Does someone have an idea or can suggest me some book where to look for?

Thank you!

First, you have to consider the accuracy order of your FV discretization.
At second order of accuracy the volume integral can be discretized simply by the mean value formula but I do not suggest that approach since it is not effective in terms of top-hat filtering. Better is the multi-dimensional extension of the trapezoidal rule. If you have an accuracy more than two then your volume integral must be discretized congruently, for example by using shape functions of high degree so that you can integrate them analitically
FMDenaro is offline   Reply With Quote

Old   October 13, 2012, 18:31
Default
  #3
New Member
 
Ivan
Join Date: Aug 2012
Posts: 22
Rep Power: 5
Ivan is on a distinguished road
It is said that the terms in the coefficient matrix are calculated as upwind and then a second order correction is added on the RHS in a deferred way (so that they can use a blending factor I guess)...so the accuracy is second order. Therefore, I'm thinking:

1) On one side, wether using an upwind-like discretization for those terms (plus a second order correction which I would not be sure how to apply) or directly a second order scheme (ex. a central scheme).

2) I'm confused, chosen the scheme, what I should write. For example, since for each time step, a number of subiterations are performed, can I discretize the term, i.e., k1*Phi^2 in cell i as k1*Vol_i*Phi(n-1)_i * Phi(n)_i, where n is the current subiteration? And what about the term grad(Phi)^2
Ivan is offline   Reply With Quote

Old   October 13, 2012, 18:35
Default
  #4
New Member
 
Ivan
Join Date: Aug 2012
Posts: 22
Rep Power: 5
Ivan is on a distinguished road
Quote:
Originally Posted by Ivan View Post
It is said that the terms in the coefficient matrix are calculated as upwind and then a second order correction is added on the RHS in a deferred way (so that they can use a blending factor I guess)...so the accuracy is second order. Therefore, I'm thinking:

1) On one side, wether using an upwind-like discretization for those terms (plus a second order correction which I would not be sure how to apply) or directly a second order scheme (ex. a central scheme).

2) I'm confused, chosen the scheme, what I should write. For example, since for each time step, a number of subiterations are performed, can I discretize the term, i.e., k1*Phi^2 in cell i as k1*Vol_i*Phi(n-1)_i * Phi(n)_i, where n is the current subiteration? And what about the term grad(Phi)^2
I forgot to say that in LES the blending factor should be kept to one, so the scheme is fully second order!
Ivan is offline   Reply With Quote

Old   October 14, 2012, 04:25
Default
  #5
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,662
Rep Power: 23
FMDenaro will become famous soon enough
Quote:
Originally Posted by Ivan View Post
I forgot to say that in LES the blending factor should be kept to one, so the scheme is fully second order!
I think is better avoiding in LES any upwind discretization...More precisely this is when you use an explicit SGS model... otherwise you can use an ILES approach, without using SGS model.

Furthermore, the volume integral discretized with the mean value formula is second order of accuracy but it is not effective as filter, as you can simply see in the wavenumber space
FMDenaro is offline   Reply With Quote

Old   October 14, 2012, 08:54
Default
  #6
New Member
 
Ivan
Join Date: Aug 2012
Posts: 22
Rep Power: 5
Ivan is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
I think is better avoiding in LES any upwind discretization...More precisely this is when you use an explicit SGS model... otherwise you can use an ILES approach, without using SGS model.

Furthermore, the volume integral discretized with the mean value formula is second order of accuracy but it is not effective as filter, as you can simply see in the wavenumber space
Thank you! I'm looking in the code to better understand what is done for the convective and diffusive terms. Effectively the convective term is done by interpolating the Phi at the cell boundary surface with a central interpolation first, solved implicitly, plus a correction term kept on the RHS. I'm looking to the structure of this correction term so that I can adopt an identical structure (central implicit plus correction on RHS) for my production term.

Do you know some book/paper which explain this issue of the Upwind when using sgs model and the issue of the "non-effctiveness" of the filter you were mentioning (I will be required to justify my choices)?
Ivan is offline   Reply With Quote

Old   October 17, 2012, 17:07
Default
  #7
Senior Member
 
Join Date: Aug 2011
Posts: 251
Rep Power: 7
leflix is on a distinguished road
Quote:
Originally Posted by Ivan View Post
Dear all,

let's say k1*Phi^2 and k2*grad(Phi)*grad(Phi), where k1 and k2 are constants.

I want to implement it as an implicit scheme, which should be like:

A_d * Phi = A_nd * Phi + b

where A_d and A_nd are the diagonal matrix and not-diagonal matrix of coefficients and b is the vector of known terms.

I am confused on how to implement these new terms. Does someone have an idea or can suggest me some book where to look for?

Thank you!
In any case, if your schemes are second order, you can always use
k1(Phy_p)^2*Volp +k2*(grad(Phi)_p*grad(Phi)_p)*Volp which is second order too

where Phy_p and grad(Phi)_p are evaluated at the cell center P and Volp is the volume of cell P.

these terms of course are located in rhight hand side, i.e in b

You could also implict the term k1(Phy_p)^2 while adding the quantity k1(Phy_p)*Volp in the main diagonal A_d.
It can add stability if large values of Phy occurs.

for k2*grad(Phi)*grad(Phi) I do not see any other way instead of expliciting this term.

If you have a kind of deferred correction used in the code, it should work fine.
leflix is offline   Reply With Quote

Old   October 17, 2012, 17:19
Default
  #8
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,662
Rep Power: 23
FMDenaro will become famous soon enough
Quote:
Originally Posted by leflix View Post
In any case, if your schemes are second order, you can always use
k1(Phy_p)^2*Volp +k2*(grad(Phi)_p*grad(Phi)_p)*Volp which is second order too

where Phy_p and grad(Phi)_p are evaluated at the cell center P and Volp is the volume of cell P.

these terms of course are located in rhight hand side, i.e in b

You could also implict the term k1(Phy_p)^2 while adding the quantity k1(Phy_p)*Volp in the main diagonal A_d.
It can add stability if large values of Phy occurs.

for k2*grad(Phi)*grad(Phi) I do not see any other way instead of expliciting this term.

If you have a kind of deferred correction used in the code, it should work fine.

Hi leflix,
consider that you are correct in term of accuracy order in physical space but for the aim of an LES, the volume average should be like a top-hat filter. That means, for example in 1D, that the transfer function is in the form sin(kh/2)/kh/2. If you use simply the mean value you will find that the transfer function is the unity function...thus is not effective as filter
FMDenaro is offline   Reply With Quote

Old   October 17, 2012, 17:24
Default
  #9
Senior Member
 
Join Date: Aug 2011
Posts: 251
Rep Power: 7
leflix is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
Hi leflix,
consider that you are correct in term of accuracy order in physical space but for the aim of an LES, the volume average should be like a top-hat filter. That means, for example in 1D, that the transfer function is in the form sin(kh/2)/kh/2. If you use simply the mean value you will find that the transfer function is the unity function...thus is not effective as filter

Hi Filippo,

ok my contribution here was just about discretization principles. From LES concerns I do trust you !
leflix is offline   Reply With Quote

Old   October 17, 2012, 17:36
Default
  #10
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,662
Rep Power: 23
FMDenaro will become famous soon enough
Quote:
Originally Posted by leflix View Post
Hi Filippo,

ok my contribution here was just about discretization principles. From LES concerns I do trust you !
you are right, mean value formula is very common in FV discretization with second order central schemes, however I just suggested Ivan to do differently as he wants a LES-suited discretization for the production term.
Regards
Filippo
FMDenaro is offline   Reply With Quote

Old   October 17, 2012, 17:39
Default
  #11
Senior Member
 
Join Date: Aug 2011
Posts: 251
Rep Power: 7
leflix is on a distinguished road
Quote:
Originally Posted by FMDenaro View Post
I just suggested Ivan to do differently as he wants a LES-suited discretization for the production term.
Regards
Filippo
You were right to point out this serious issue ! I'm sure Ivan will got the point !!
leflix is offline   Reply With Quote

Reply

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are On
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Plotting a contour of the turbulence production term in CFX-Post kellio9 CFX 4 March 19, 2013 12:34
production term on staggered grid behzadmaha Main CFD Forum 0 April 24, 2011 16:30
ATTENTION! Reliability problems in CFX 5.7 Joseph CFX 14 April 20, 2010 15:45
Finite volumes vs. finite elements: applications Lionel S. Main CFD Forum 10 January 22, 2007 11:51
Numerical method for the k-epsilon production term carlslack Main CFD Forum 0 January 23, 2003 17:13


All times are GMT -4. The time now is 14:12.