Lambda-2 criterion

SonyCFD · May 23, 2012, 20:41

Thanks,
I will try it and get back to you.
Cheers,
Sony

upal_arif · May 24, 2012, 01:51

Hi,

I am looking for a freelancer to simulate supersonic flow through C-D nozzle by fluent 6.2.16

Please contact upal_arif@yahoo.com

Quote:

Originally Posted by sbaffini

Dear saeedi,

the lambda 2 criterion simply concerns the definition of the scalar lambda2 and how turbulent structures can be visualized by proper isosurfaces of lambda2 (like for the Q criterion). Hence, the real difference with the scalar Q is how you compute the scalar lambda2.

This is defined as the second (in magnitude) eigenvalue of the matrix:

$S_{ik} S_{kj} + \Omega_{ik} \Omega_{kj}$

where:

$S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)$

$\Omega_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i}\right)$

This requires the construction of the characteristic cubic equation and its resolution in order to obtain lambda2 (that is, lambda2 is the second solution of the cubic characteristic equation).

For coherent structures the matrix above can be related to the opposite of the pressure Hessian matrix. As this matrix is real and symmetric, it has two positive eigenvalues when the pressure is at minimum. As a consequence, the matrix above has two negative eigenvalues and lambda2 is certainly negative (for coherent structures). How much negative you have to pick its isosurfaces is (for what i understand) related to the visual appealing of your images (like for the Q criterion)

I still have to seriously use and check it (so it comes without any warranty) but, some time ago, i made a routine for the computation of lambda2 in Fluent (you find it attached). It is very rudimental but i remember it did the job. Also, it is basically C so you can clearly adapt the core part to your needs (Fluent related parts are very understandable also for non-Fluent users)

Once you got lambda2, you just need to pick some negative isosurface and you're done

Hope it helps

magicbretzel · August 3, 2012, 15:20

Hello,

If I can save some time to the people who downloaded the lambda2.c, there is a small error in the definitionof p, (which has a huge impact on the output, of course)...
Instead of p=b/(3.0*a);

One should read p= -b/(3.0*a);

Hope this can be helpful to someone !

sbaffini · August 3, 2012, 15:38

I knew there was something wrong with it (can't say i didn't write it clear

) but i lost my reference on the cubic equation and put it aside. Thank you very much for your attention, it will help me.

brunntho · May 6, 2013, 05:48

Hi all,

I've just implemented both criteria, Lambda2 and the Q-criterion. However the two methods yield almost the same result, which lead me to the questions:

In which cases the Q-criterion perform better than the lambda2 criterion?
What exactly differs them?

My data set under consideration is a measurement and contains noise. Both methods are sensitive of the gradient computation of the vector field.

Considering noise in data, can we make a statement about which methods works best for measured data sets?

Thank you for your feedback

sbaffini · May 6, 2013, 06:50

The following reference should answer some of your questions:

http://www.aero.polimi.it/~quadrio/papers/1999-ejmb.pdf

brunntho · May 6, 2013, 07:00

Hi,

thank you, in fact most of my questions can be answered with your reference.

bests,
brunntho

Eddieme · June 19, 2013, 21:06

Hey,

Can anybody point out where in Lesieur's book(Turbulence in Fluids) can I find the definition of lamda2 ? esp. one that Filippo used in his script?

Thanks!
~e

brunntho · June 20, 2013, 11:03

maybe following paper will also be helpful for you.

http://journals.cambridge.org/action...ine&aid=353418

Eddieme · June 21, 2013, 02:08

Hey all CFDists,

Firstly, I need correct my last post: the script that "Paolo" (not Filippo) wrote. All Italian names sound the same

.

Secondly, if Paolo or whoever went through the script of lambda2.c
would you please tell me how you calculate the P11, P22, P33, P12, P13, P23?
I can not find any reference for it.

Thirdly, thank you, brunntho, I started writing my code based on the paper of Jeong & Hussain. I agree with you. Although it's a pretty helpful article,I couldn't find any reference for calculating the value of Pij

Cheers!
~e

sbaffini · June 21, 2013, 05:26

Dear Eddieme,

first of all, the reference book is: Lesieur, Metais, Comte - "Large Eddy Simulation of Turbulence" - Cambridge.

There you can find that P_ij is actually S_ij S_jk + W_ij W_jk (sum of squared symmetric and anti-symmetric parts of velocity gradient), which i calculated by hand. Then i used some reference to compute the roots of the cubic equation.

I guess you are right not trusting my computation of P_ij in lambda2.c as i already made a mistake in the cubic equation. But i don't think you will find a reference for it, you need to do it by hand. If you actually find a mistake, however, i would be glad to know.

QBeast · December 22, 2013, 17:47

Great contribution, Thanks sbaffini for this precious piece of code, it's really valuable for me.

msshadloo · April 2, 2014, 04:18

Quote:

Originally Posted by sbaffini

Dear saeedi,

the lambda 2 criterion simply concerns the definition of the scalar lambda2 and how turbulent structures can be visualized by proper isosurfaces of lambda2 (like for the Q criterion). Hence, the real difference with the scalar Q is how you compute the scalar lambda2.

This is defined as the second (in magnitude) eigenvalue of the matrix:

$S_{ik} S_{kj} + \Omega_{ik} \Omega_{kj}$

where:

$S_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)$

$\Omega_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i}\right)$

This requires the construction of the characteristic cubic equation and its resolution in order to obtain lambda2 (that is, lambda2 is the second solution of the cubic characteristic equation).

For coherent structures the matrix above can be related to the opposite of the pressure Hessian matrix. As this matrix is real and symmetric, it has two positive eigenvalues when the pressure is at minimum. As a consequence, the matrix above has two negative eigenvalues and lambda2 is certainly negative (for coherent structures). How much negative you have to pick its isosurfaces is (for what i understand) related to the visual appealing of your images (like for the Q criterion)

I still have to seriously use and check it (so it comes without any warranty) but, some time ago, i made a routine for the computation of lambda2 in Fluent (you find it attached). It is very rudimental but i remember it did the job. Also, it is basically C so you can clearly adapt the core part to your needs (Fluent related parts are very understandable also for non-Fluent users)

Once you got lambda2, you just need to pick some negative isosurface and you're done

Hope it helps

Dear All,

I am new in the field and I work with compressible flows. Does anyone knows if the lambda 2 criterion in the mentioned format will work for my problems (compressible flow)? because in compressible flow this formula does not recover the pressure Hessian matrix.

Best,
Mostafa

sbaffini · April 2, 2014, 13:01

Dear Mostafa,

the short answer is no. In the link i posted above:

http://www.aero.polimi.it/~quadrio/papers/1999-ejmb.pdf

there is also a derivation for the compressible pressure Hessian, which is obviously different from the incompressible one. I don't remember if a solution for compressible flows is also effectively presented.

The first link i got from Google on the matter is instead:

http://www.wseas.us/e-library/confer.../MAFLUH-03.pdf

which seems to discuss some compressible aspects (still, i didn't read it).

msshadloo · April 3, 2014, 03:26

Thank you Paolo.
I think I should go to Delta criteria. I found one paper from Farrell and Martin (JOT 2009) where they showed the Delta criteria. Their results are nice, but they use a lot of filtering!

Best,
Mostafa

Azy · August 25, 2014, 11:56

hello every one,

I actually have a question. I calculated the eigenvalues for the lambda 2, but the range includes the positive numbers as well as negative ones. So my question is did I do my calculation wrongly, and I only should have negative numbers? or I only should choose the negative numbers from the range ?

Thanks

sbaffini · August 25, 2014, 13:05

You should just pick up a reasonable negative number to use as value for the lambda2 iso-surface. By reasonable i mean close to 0 but different enough to produce a nice picture. Obviously, by proper non-dimensionalization of lambda2, it may be possible to pick-up more physically sound values... but your eyes are probably gonna work better than this.

Azy · August 25, 2014, 15:12

Thanks Paolo, Ok so it is alright to have positive numbers, I should just pick up negative ones to plot the iso surfaces. And is it better for my numbers to be as close as possible to zero? I dont know why I think in reverse :/ :/

sbaffini · August 25, 2014, 16:04

In theory, you need a strictly negative value for lambda2 in order to select zones where the pressure has a local minimum. This is all you need to know if you want to use lambda2 as a visualization tool (this might not be true if you need lambda2 for other means).

According to this, i really suggest a trial and error approach in order to find the best value (which might be even dependent on the specific visualization/iso-surface routines).

Azy · August 25, 2014, 17:13

yes I just want it for visualization. Thanks a lot Paolo

August 3, 2012, 15:20	Small error in the code	#23
magicbretzel New Member Guillaume Beardsell Join Date: Aug 2012 Posts: 19 Rep Power: 13	Hello, If I can save some time to the people who downloaded the lambda2.c, there is a small error in the definitionof p, (which has a huge impact on the output, of course)... Instead of p=b/(3.0a); One should read p= -b/(3.0a); Hope this can be helpful to someone !

May 6, 2013, 05:48		#25
brunntho New Member Tom Join Date: May 2013 Posts: 4 Rep Power: 12	Hi all, I've just implemented both criteria, Lambda2 and the Q-criterion. However the two methods yield almost the same result, which lead me to the questions: In which cases the Q-criterion perform better than the lambda2 criterion? What exactly differs them? My data set under consideration is a measurement and contains noise. Both methods are sensitive of the gradient computation of the vector field. Considering noise in data, can we make a statement about which methods works best for measured data sets? Thank you for your feedback

June 19, 2013, 21:06	lambda-2 refernce	#28
Eddieme New Member MonaK. Join Date: Jun 2013 Location: Ein Sof Posts: 2 Rep Power: 0	Hey, Can anybody point out where in Lesieur's book(Turbulence in Fluids) can I find the definition of lamda2 ? esp. one that Filippo used in his script? Thanks! ~e

June 20, 2013, 11:03		#29
brunntho New Member Tom Join Date: May 2013 Posts: 4 Rep Power: 12	maybe following paper will also be helpful for you. http://journals.cambridge.org/action...ine&aid=353418 FMDenaro likes this.

June 21, 2013, 02:08	lambda2- reference	#30
Eddieme New Member MonaK. Join Date: Jun 2013 Location: Ein Sof Posts: 2 Rep Power: 0	Hey all CFDists, Firstly, I need correct my last post: the script that "Paolo" (not Filippo) wrote. All Italian names sound the same. Secondly, if Paolo or whoever went through the script of lambda2.c would you please tell me how you calculate the P11, P22, P33, P12, P13, P23? I can not find any reference for it. Thirdly, thank you, brunntho, I started writing my code based on the paper of Jeong & Hussain. I agree with you. Although it's a pretty helpful article,I couldn't find any reference for calculating the value of Pij Cheers! ~e

May 23, 2012, 20:41		#21
SonyCFD New Member Sony Join Date: May 2010 Posts: 14 Rep Power: 15	Thanks, I will try it and get back to you. Cheers, Sony

August 3, 2012, 15:38		#24
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	I knew there was something wrong with it (can't say i didn't write it clear ) but i lost my reference on the cubic equation and put it aside. Thank you very much for your attention, it will help me.

May 6, 2013, 06:50		#26
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	The following reference should answer some of your questions: http://www.aero.polimi.it/~quadrio/papers/1999-ejmb.pdf

May 6, 2013, 07:00		#27
brunntho New Member Tom Join Date: May 2013 Posts: 4 Rep Power: 12	Hi, thank you, in fact most of my questions can be answered with your reference. bests, brunntho

June 21, 2013, 05:26		#31
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	Dear Eddieme, first of all, the reference book is: Lesieur, Metais, Comte - "Large Eddy Simulation of Turbulence" - Cambridge. There you can find that P_ij is actually S_ij S_jk + W_ij W_jk (sum of squared symmetric and anti-symmetric parts of velocity gradient), which i calculated by hand. Then i used some reference to compute the roots of the cubic equation. I guess you are right not trusting my computation of P_ij in lambda2.c as i already made a mistake in the cubic equation. But i don't think you will find a reference for it, you need to do it by hand. If you actually find a mistake, however, i would be glad to know.

December 22, 2013, 17:47		#32
QBeast New Member Saâd Debbahi Join Date: Oct 2010 Posts: 26 Rep Power: 15	Great contribution, Thanks sbaffini for this precious piece of code, it's really valuable for me.

April 2, 2014, 13:01		#34
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	Dear Mostafa, the short answer is no. In the link i posted above: http://www.aero.polimi.it/~quadrio/papers/1999-ejmb.pdf there is also a derivation for the compressible pressure Hessian, which is obviously different from the incompressible one. I don't remember if a solution for compressible flows is also effectively presented. The first link i got from Google on the matter is instead: http://www.wseas.us/e-library/confer.../MAFLUH-03.pdf which seems to discuss some compressible aspects (still, i didn't read it).

April 3, 2014, 03:26		#35
msshadloo New Member Mostafa Join Date: Apr 2013 Posts: 4 Rep Power: 13	Thank you Paolo. I think I should go to Delta criteria. I found one paper from Farrell and Martin (JOT 2009) where they showed the Delta criteria. Their results are nice, but they use a lot of filtering! Best, Mostafa

August 25, 2014, 11:56	Lambda 2	#36
Azy New Member Azadeh Saeedi Join Date: Mar 2014 Location: Canada Posts: 23 Rep Power: 12	hello every one, I actually have a question. I calculated the eigenvalues for the lambda 2, but the range includes the positive numbers as well as negative ones. So my question is did I do my calculation wrongly, and I only should have negative numbers? or I only should choose the negative numbers from the range ? Thanks Last edited by Azy; August 25, 2014 at 15:06.

August 25, 2014, 13:05		#37
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	You should just pick up a reasonable negative number to use as value for the lambda2 iso-surface. By reasonable i mean close to 0 but different enough to produce a nice picture. Obviously, by proper non-dimensionalization of lambda2, it may be possible to pick-up more physically sound values... but your eyes are probably gonna work better than this. FMDenaro and Azy like this.

August 25, 2014, 15:12		#38
Azy New Member Azadeh Saeedi Join Date: Mar 2014 Location: Canada Posts: 23 Rep Power: 12	Thanks Paolo, Ok so it is alright to have positive numbers, I should just pick up negative ones to plot the iso surfaces. And is it better for my numbers to be as close as possible to zero? I dont know why I think in reverse :/ :/

August 25, 2014, 16:04		#39
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	In theory, you need a strictly negative value for lambda2 in order to select zones where the pressure has a local minimum. This is all you need to know if you want to use lambda2 as a visualization tool (this might not be true if you need lambda2 for other means). According to this, i really suggest a trial and error approach in order to find the best value (which might be even dependent on the specific visualization/iso-surface routines).

August 25, 2014, 17:13		#40
Azy New Member Azadeh Saeedi Join Date: Mar 2014 Location: Canada Posts: 23 Rep Power: 12	yes I just want it for visualization. Thanks a lot Paolo

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