Smagorinsky model details

Bernhard · July 12, 2011, 11:31

Hi all,

I have some questions about this same issue with respect to the dynamic Smagorinsky model. I think I am making a mistake somewhere, but please correct me if I am wrong. I've tried to look at other locations for the same issue, but couldn't find it.

I compare the dynamic model in OpenFOAM (not caring about the domain averages coefficient) with the dynamic model (described in Lilly and Pope).

I think we agree on the following (D defined in OpenFOAM, S common definition)
$\begin{array}{rcl} \overline{S}_{ij} &=& \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right) \\ \overline{D}_{ij} &=& \overline{S}_{ij} - \frac{1}{3} \delta_{ij}\overline{S}_{kk} = \overline{S}_{ij} (incompressible) \\ |\overline{D}| &=& |\overline{S}|/\sqrt{2} \end{array}$

Now, looking at the code (dynSmagorinsky 1.7, homogeneousDynSmagorinsky 2.0), in the .C file at line 57, 62 respectively:
$M_{ij} = \Delta^2 \left[ \widehat{|\overline{D}|\overline{D}_{ij}} - 4 |\widehat{\overline{D}}_{ij}\widehat{\overline{D}}_{ij}|\right]$
as compared to the original of Lilly (in Pope this expression
$M_{ij} = \widehat{\Delta}^2\widehat{|\overline{S}|}\widehat{\overline{S}}_{ij}-\Delta^2\widehat{|\overline{S}|\overline{S}_{ij}}$

If D_ij=S_ij, but |S|=sqrt(2) |D|, then the second term in the OF implementation is off by a factor sqrt(2). (Where the factor 4 is from the double width of the test filter, and the minus sine is also present in L)

Somewhere I must have skipped a step, I hope one of you can point it out for me. Thanks in advance!

Bernhard · July 15, 2011, 02:35

Anybody that can give a comment on this?

Bernhard · July 18, 2011, 03:47

Mistake is this:

Quote:

Originally Posted by Bernhard

$M_{ij} = \Delta^2 \left[ \widehat{|\overline{D}|\overline{D}_{ij}} - 4 |\widehat{\overline{D}}_{ij}\widehat{\overline{D}}_{ij}|\right]$

$M_{ij} = \Delta^2 \left[ \widehat{|\overline{D}|\overline{D}_{ij}} - 4 |\widehat{\overline{D}}|\widehat{\overline{D}}_{ij}\right]$

gregor · August 26, 2011, 06:39

Quote:

Originally Posted by lakeat

Question 1: Why using magSqr(dev(symm(gradU))) instead of symm(gradU) && symm(gradU) to get ${{\bf{\bar S}}{\rm{:}}{\bf{\bar S}}}$ ????
Thank you

I think the reason is to have a consistent formulation for incompressible an compressible LES models :

let symm(gradU) be S, then dev(S) = S - 1/3 trace(S)I

however in a incompressible case 1/3 trace(S)I = 0, since trace(S) is the continuity eq. . Therefore in an incrompessible case it doesnt matter whether you take dev(S) or not , but consider a compressible case then 1/3 trace(S) doesn't vanish.

In a compressible or variable density case the solver calls divDevRhoBeff to compute the source term due to SGS stress B = 2/3k I - 2 nu_t S_D. (See Fureby http://pof.aip.org/resource/1/phfle6/v9/i5/p1416_s1 Eq. 3) There you have the deviatoric part of D. But i guess openFoam uses nu_t = c Delta^2 ||S_D|| and B = 2/3k I - 2 nu_t S. So it takes S_D for the turbulent viscosity and S for the SGS stress tensor B.

hz283 · January 24, 2013, 10:40

Quote:

Originally Posted by MaximeIST

Hello

I may keep on confusing people, but the way it is coded, if I am not doing mistake is
Cs=sqrt(ck*sqrt(2*ck/ce))
in the incompressible Smagorinsky.H line 114.
There is a factor 2 added in the root-mean squared.

And in the case where Ce=1.048 and Ck=0.094, and with this factor 2, we obtain Cs=0.1995.

May be I miss something?

Maxime

Hi All,

For the compressible Smagorinsky model, the parameters for ck=0.02, ce=1.048

Following the following line:
Cs=sqrt(ck*sqrt(ck/ce))
=> Cs=0.0525...

Does anybody know the references for these specification of the ck and ce for compressible Smagorinsky model?

Sunxing · March 11, 2013, 21:25

Hi Yingkun,

As you mentioned, in the incompressible solvers Cs=sqrt(Ck*sqrt(Ck/Ce)). So if I want to set Cs=1, Do I just need to modify Ck and Ce in the LESProperties？Or are there other rules I must obey?

Code:

SmagorinskyCoeffs
{
ce 1.05;
ck 0.0472;
}

regards

itchy · February 28, 2014, 03:03

Hi Bernhard and everybody,

I compare the dynamic model in OpenFOAM (homoDynSmag OF 2.2.2) with the dynamic model (described in Lilly and Pope) and I have the same Problem. To reanimate the discussion:

In Pope (F(.) means filtered):
[1] nu_SGS = cS * delta^2 * sqrt(2 * S_ij S_ij)
[2] cs = (M_ij L_ij)/(M_kl M_kl)
where
[3] S_ij = 0.5 (ui,j + uj,i)
[4] M_ij = 2 * delta^2 * (F(sqrt(2 * S_ij S_ij) S_ij) - F(sqrt(2 * S_ij S_ij)) F(S_ij))
[5] L_ij = F(ui uj) - F(ui) F(uj)

In OF 2.2.2 (homogeneousDySmagorinsky, <.> means averaged):
[6] nu_SGS = cD * delta^2 * sqrt(S_ij S_ij)
[7] cD = 0.5 (<L_ij M_ij>)/(<M_kl M_kl>)
[8] S_ij = D_ij = S_ij Pope
[9] M_ij = delta^2 * (F(sqrt(S_kl S_kl) Sij) - 4 * sqrt(<S_kl> <S_kl>) <S_ij>)
[10] L_ij = F(ui uj) - F(ui) F(uj)

I marked the differences of these models.

- First difference is the factor 0.5 in Eq[7] in comparison to Eq[2]. This comes from the factor 2 in Eq[4]. If we put this in Eq[2] we get 0.5 (ok)

- Second difference is the different filtering in M_ij. What effect does this have???? (x)

- Third difference is the factor 4 in Eq[9] in comparison to Eq[4]. Bernhard:
here the factor 4 is from the double width of the test filter
can you please explain that?? Why we don`t get a fector in the other filtered terms??? (x)

- Fourth difference is the factor 2 in mag(S_ij). (x)

In my opinion these models are different or I`m not able to bring the OF-model in the form of pope-model. If it is possible, anyone can please give some advice?

kind regards
Florian

wenxu · October 28, 2014, 01:39

Quote:

Originally Posted by lakeat

Sorry, I do not understand, I saw in "Smagorinsky.H",

Code:

tmp<volScalarField> k(const tmp<volTensorField>& gradU) const
{
    return (2.0*ck_/ce_)*sqr(delta())*magSqr(dev(symm(gradU)));
}

As I remember:

$\begin{array}{l} {\nu _{SGS}} = {\left( {{C_S}\Delta } \right)^2}\left| {{\bf{\bar S}}} \right| \\ K = {\left( {{C_I}\Delta } \right)^2}{\left| {{\bf{\bar S}}} \right|^2} \\ \left| {{\bf{\bar S}}} \right| = {\left( {{\bf{\bar S}}{\rm{:}}{\bf{\bar S}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \\ \end{array}$

Question 1: Why using magSqr(dev(symm(gradU))) instead of symm(gradU) && symm(gradU) to get ${{\bf{\bar S}}{\rm{:}}{\bf{\bar S}}}$ ????

Question 2: If magSqr(dev(symm(gradU))) = symm(gradU) && symm(gradU) = ${{\bf{\bar S}}{\rm{:}}{\bf{\bar S}}}$ , then

$K = \frac{{2{C_K}}}{{{C_\varepsilon }}}{\Delta ^2}{\left| {{\bf{\bar S}}} \right|^2}$

But I saw in "Smagorinsky.C"

Code:

nuSgs_ = ck_*delta()*sqrt(k(gradU));

Which means

${\nu _{SGS}} = {C_K}\Delta \sqrt K$

Then, replace K with $K = \frac{{2{C_K}}}{{{C_\varepsilon }}}{\Delta ^2}{\left| {{\bf{\bar S}}} \right|^2}$

${\nu _{SGS}} = {C_K}\Delta \sqrt K = {C_K}\Delta \sqrt {\frac{{2{C_K}}}{{{C_\varepsilon }}}{\Delta ^2}{{\left| {{\bf{\bar S}}} \right|}^2}} = {C_K}\sqrt {\frac{{2{C_K}}}{{{C_\varepsilon }}}} {\Delta ^2}\left| {{\bf{\bar S}}} \right|$

Compare with ${\nu _{SGS}} = {\left( {{C_S}\Delta } \right)^2}\left| {{\bf{\bar S}}} \right|$

We'll get

${\left( {{C_S}} \right)^2} = {C_K}\sqrt {\frac{{2{C_K}}}{{{C_\varepsilon }}}}$

But I heard somone said ${\left( {{C_S}} \right)^2} = {C_K}\sqrt {\frac{{{C_K}}}{{{C_\varepsilon }}}}$

So, I'm puzzled, I wonder if it was a mistake, that k should be written as

Code:

tmp<volScalarField> k(const tmp<volTensorField>& gradU) const
{
    return (ck_/ce_)*sqr(delta())*magSqr(dev(symm(gradU)));
}

Thank you

$\begin{array}{l} {\nu _{SGS}} = {\left( {{C_S}\Delta } \right)^2}\left| {{\bf{\bar S}}} \right| \\ K = {\left( {{C_I}\Delta } \right)^2}{\left| {{\bf{\bar S}}} \right|^2} \\ \left| {{\bf{\bar S}}} \right| = {\left( {{\bf{\bar S}}{\rm{:}}{\bf{\bar S}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \\ \end{array}$

the third one shoub be sqrt(2*Sij*Sij)

babakflame · December 11, 2014, 05:40

Greetings All

I have performed a Smagorinsky-based compressible LES simulation with the Coefficients as follows:

HTML Code:

{
  ce = 1.048;
  ck= 0.02;
}

According to the relation:

${\left( {{C_S}} \right)^2} = {C_K}\sqrt {\frac{{{C_K}}}{{{C_\varepsilon}}}}$

I will get

.
My case is a reacting non-premixed combustion with a bluff-body separating fuel and oxidizer streams.

I want to change the

value into 0.13. which value between

and $C_\varepsilon$ should be modified to retain the nature of the problem?

Best,
Bobi

openfoammaofnepo · March 21, 2015, 18:27

Actually in this paper, we cannot find the information about how the model constants c_k=0.094 and c_{\epsilon}=1.048 come out. So which one is correct reference when I use these two constants? Thanks.

Quote:

Originally Posted by alberto

Reference: http://pof.aip.org/resource/1/phfle6/v9/i5/p1416_s1

lyne00 · September 21, 2015, 10:11

Hi Lakeat,now I want to write the Scalar SikSkj,but how to write that in openFoam, Thanks.

Elyas_Mosibat · January 15, 2016, 12:56

hi friends
How can I change the smagorinsky Coeff (Cs) in OF?I want to use Cs=0.1...
Regard

Jingxue Wang · December 26, 2017, 21:16

I read that all the previous posts and reply.

May I understand that Cs=sqrt(Ck*sqrt(2Ck/Ce)) is correct in OpenFoam?

Thanks a lot

manafaero · March 26, 2019, 05:15

dear madam
As you know the smagorinsky constant Cs in OpenFOAM is defined as a function of Ce and Ck. But when i do the simulation for various combinations of Ck and Ce, that gives the same Cs, I am getting different results. So could you please tell Me where else Ck and Ce is used other than in smagorinsky model. My problem is incompressible LES with standard smagorinsky model and vandriest delta.

please help

my mailId is manafaero@gmail.com

wyldckat · March 31, 2019, 15:24

Quote:

Originally Posted by manafaero

But when i do the simulation for various combinations of Ck and Ce, that gives the same Cs, I am getting different results. So could you please tell Me where else Ck and Ce is used other than in smagorinsky model.

"Ce" is loaded here: https://github.com/OpenFOAM/OpenFOAM...ddyViscosity.C
"Ck" is loaded and used here; "Ce" is also used here: https://github.com/OpenFOAM/OpenFOAM.../Smagorinsky.C

The nut, k and epsilon fields are created by these classes, but I 'm not familiar with LES modelling. Either way, if you look for "Ce_" and "Ck_" in those files, you will see how they are used.

manafaero · April 3, 2019, 08:54

I would like to share some experience of mine regarding the discussion. I also verified the equations given in the discussion. But when i run the simulation for various ck and ce combinations I am getting different results. the nu sgs is not the same in all cases. so eventhough the formula is correct the openFoam is not calculating as per the discussion. Note that I had removed the dependence of k in calculating shear stress B. but still the results differ. So be careful before using this. Also I am a beginner to OpenFOAM and C++. If any body is interested and any comments please post. my mail ID is manafaero@gmail.com.

jorkolino · September 16, 2020, 11:30

did you check this page with OF implementation https://caefn.com/openfoam/smagorinsky-sgs-model

July 12, 2011, 11:31	How about dynamic Smagorinsky?	#21
Bernhard Senior Member Bernhard Join Date: Sep 2009 Location: Delft Posts: 790 Rep Power: 21	Hi all, I have some questions about this same issue with respect to the dynamic Smagorinsky model. I think I am making a mistake somewhere, but please correct me if I am wrong. I've tried to look at other locations for the same issue, but couldn't find it. I compare the dynamic model in OpenFOAM (not caring about the domain averages coefficient) with the dynamic model (described in Lilly and Pope). I think we agree on the following (D defined in OpenFOAM, S common definition) $\begin{array}{rcl} \overline{S}_{ij} &=& \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right) \\ \overline{D}_{ij} &=& \overline{S}_{ij} - \frac{1}{3} \delta_{ij}\overline{S}_{kk} = \overline{S}_{ij} (incompressible) \\ \|\overline{D}\| &=& \|\overline{S}\|/\sqrt{2} \end{array}$ Now, looking at the code (dynSmagorinsky 1.7, homogeneousDynSmagorinsky 2.0), in the .C file at line 57, 62 respectively: $M_{ij} = \Delta^2 \left[ \widehat{\|\overline{D}\|\overline{D}_{ij}} - 4 \|\widehat{\overline{D}}_{ij}\widehat{\overline{D}}_{ij}\|\right]$ as compared to the original of Lilly (in Pope this expression $M_{ij} = \widehat{\Delta}^2\widehat{\|\overline{S}\|}\widehat{\overline{S}}_{ij}-\Delta^2\widehat{\|\overline{S}\|\overline{S}_{ij}}$ If D_ij=S_ij, but \|S\|=sqrt(2) \|D\|, then the second term in the OF implementation is off by a factor sqrt(2). (Where the factor 4 is from the double width of the test filter, and the minus sine is also present in L) Somewhere I must have skipped a step, I hope one of you can point it out for me. Thanks in advance! bmikuz likes this.

March 11, 2013, 21:25		#26
Sunxing Member sqing Join Date: Sep 2012 Location: Dalian Posts: 77 Rep Power: 13	Hi Yingkun, As you mentioned, in the incompressible solvers Cs=sqrt(Ck*sqrt(Ck/Ce)). So if I want to set Cs=1, Do I just need to modify Ck and Ce in the LESProperties？Or are there other rules I must obey? Code: SmagorinskyCoeffs { ce 1.05; ck 0.0472; } regards

February 28, 2014, 03:03		#27
itchy Member Florian Ries Join Date: Feb 2014 Location: Darmstadt, Germany Posts: 88 Rep Power: 12	Hi Bernhard and everybody, I compare the dynamic model in OpenFOAM (homoDynSmag OF 2.2.2) with the dynamic model (described in Lilly and Pope) and I have the same Problem. To reanimate the discussion: In Pope (F(.) means filtered): [1] nu_SGS = cS * delta^2 * sqrt(2 * S_ij S_ij) [2] cs = (M_ij L_ij)/(M_kl M_kl) where [3] S_ij = 0.5 (ui,j + uj,i) [4] M_ij = 2 * delta^2 * (F(sqrt(2 * S_ij S_ij) S_ij) - F(sqrt(2 * S_ij S_ij)) F(S_ij)) [5] L_ij = F(ui uj) - F(ui) F(uj) In OF 2.2.2 (homogeneousDySmagorinsky, <.> means averaged): [6] nu_SGS = cD * delta^2 * sqrt(S_ij S_ij) [7] cD = 0.5 (<L_ij M_ij>)/(<M_kl M_kl>) [8] S_ij = D_ij = S_ij Pope [9] M_ij = delta^2 * (F(sqrt(S_kl S_kl) Sij) - 4 * sqrt(<S_kl> <S_kl>) <S_ij>) [10] L_ij = F(ui uj) - F(ui) F(uj) I marked the differences of these models. - First difference is the factor 0.5 in Eq[7] in comparison to Eq[2]. This comes from the factor 2 in Eq[4]. If we put this in Eq[2] we get 0.5 (ok) - Second difference is the different filtering in M_ij. What effect does this have???? (x) - Third difference is the factor 4 in Eq[9] in comparison to Eq[4]. Bernhard: here the factor 4 is from the double width of the test filter can you please explain that?? Why we don`t get a fector in the other filtered terms??? (x) - Fourth difference is the factor 2 in mag(S_ij). (x) In my opinion these models are different or I`m not able to bring the OF-model in the form of pope-model. If it is possible, anyone can please give some advice? kind regards Florian bmikuz likes this.

December 11, 2014, 05:40	How to change ck and ce in Smagorinsky approach	#29
babakflame Senior Member Bobby Join Date: Oct 2012 Location: Michigan Posts: 454 Rep Power: 15	Greetings All I have performed a Smagorinsky-based compressible LES simulation with the Coefficients as follows: HTML Code: { ce = 1.048; ck= 0.02; } According to the relation: ${\left( {{C_S}} \right)^2} = {C_K}\sqrt {\frac{{{C_K}}}{{{C_\varepsilon}}}}$ I will get . My case is a reacting non-premixed combustion with a bluff-body separating fuel and oxidizer streams. I want to change the value into 0.13. which value between and $C_\varepsilon$ should be modified to retain the nature of the problem? Best, Bobi Last edited by babakflame; December 11, 2014 at 06:45.

September 21, 2015, 10:11	Openfoam	#31
lyne00 New Member Liuyue Join Date: Sep 2015 Posts: 1 Rep Power: 0	Hi Lakeat,now I want to write the Scalar SikSkj,but how to write that in openFoam, Thanks.

July 15, 2011, 02:35		#22
Bernhard Senior Member Bernhard Join Date: Sep 2009 Location: Delft Posts: 790 Rep Power: 21	Anybody that can give a comment on this?

January 15, 2016, 12:56	Change the smagorinsky coeff	#32
Elyas_Mosibat New Member Elyas_Kermani Join Date: Dec 2015 Location: Iran Posts: 12 Rep Power: 10	hi friends How can I change the smagorinsky Coeff (Cs) in OF?I want to use Cs=0.1... Regard

December 26, 2017, 21:16		#33
Jingxue Wang Member Jingxue Wang Join Date: Sep 2017 Posts: 58 Rep Power: 8	I read that all the previous posts and reply. May I understand that Cs=sqrt(Ck*sqrt(2Ck/Ce)) is correct in OpenFoam? Thanks a lot

March 26, 2019, 05:15		#34
manafaero New Member Manaf Muhammed Join Date: Oct 2018 Posts: 20 Rep Power: 7	dear madam As you know the smagorinsky constant Cs in OpenFOAM is defined as a function of Ce and Ck. But when i do the simulation for various combinations of Ck and Ce, that gives the same Cs, I am getting different results. So could you please tell Me where else Ck and Ce is used other than in smagorinsky model. My problem is incompressible LES with standard smagorinsky model and vandriest delta. please help my mailId is manafaero@gmail.com

April 3, 2019, 08:54	OpenFoam	#36
manafaero New Member Manaf Muhammed Join Date: Oct 2018 Posts: 20 Rep Power: 7	I would like to share some experience of mine regarding the discussion. I also verified the equations given in the discussion. But when i run the simulation for various ck and ce combinations I am getting different results. the nu sgs is not the same in all cases. so eventhough the formula is correct the openFoam is not calculating as per the discussion. Note that I had removed the dependence of k in calculating shear stress B. but still the results differ. So be careful before using this. Also I am a beginner to OpenFOAM and C++. If any body is interested and any comments please post. my mail ID is manafaero@gmail.com.

September 16, 2020, 11:30		#37
jorkolino Member George Pichurov Join Date: Jul 2010 Posts: 52 Rep Power: 15	did you check this page with OF implementation https://caefn.com/openfoam/smagorinsky-sgs-model

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