Resolved TKE budget terms calculation from LES Simulations

Agavi · May 29, 2021, 14:33

Hello everyone,

I am running LES and need to plot the turbulent kinetic energy budget of my resolved flow field. I therefore need the equation of the TKE that comes from the square of the $u'_{resolved} = \tilde{u}-\bar{u}$ components.

TKE Budget equations that I have found online are in the physical sense (ie the one in Pope's book) where the filtering operation is obviously ignored. A DNS simulation could use those equations for the calculation of the TKE budget terms.

I'm looking for the derivation of the resolved TKE equation. This probably should come from a similar derivation to the TKE equation but also including the SGS tensor.

-- This old thread discusses the matter but didn't reach a conclusion - That's why I started this new thread.

TKE budget equation in LES

-- I am probably looking for the derivation of equation (1) in this paper. It accounts for the effect of $\tau_{ij}^{SGS}$ to the resolved TKE balance.

https://www.osti.gov/servlets/purl/1500073

Please let me know if I'm understanding something wrong here!

Thanks

Agave

FMDenaro · May 29, 2021, 15:33

Quote:

Originally Posted by Agavi

Hello everyone,

I am running LES and need to plot the turbulent kinetic energy budget of my resolved flow field. I therefore need the equation of the TKE that comes from the square of the $u'_{resolved} = \tilde{u}-\bar{u}$ components.

TKE Budget equations that I have found online are in the physical sense (ie the one in Pope's book) where the filtering operation is obviously ignored. A DNS simulation could use those equations for the calculation of the TKE budget terms.

I'm looking for the derivation of the resolved TKE equation. This probably should come from a similar derivation to the TKE equation but also including the SGS tensor.

-- This old thread discusses the matter but didn't reach a conclusion - That's why I started this new thread.

TKE budget equation in LES

-- Also in this OpenFOAM code the SGS stress tensor is not taken into account for the calculation of the resolved TKE budget and I am not sure why

https://github.com/syavash20/TurbLab.../LESpimpleFoam

-- I am probably looking for the derivation of equation (1) in this paper. It accounts for the effect of $\tau_{ij}^{SGS}$ to the resolved TKE balance.

https://www.osti.gov/servlets/purl/1500073

Please let me know if I'm understanding something wrong here!

Thanks

Agave

What are you searching for, the equation for vbar.vbar or the equation for (v'.v')bar ?

Agavi · May 29, 2021, 15:38

Hi FMDenaro,

Thanks for your reply.

I'm searching for the equation for the resolved by the LES turbulent kinetic energy, so $(u_{resloved}'u_{resolved}')_{bar}$ .

Best

Agave

FMDenaro · May 29, 2021, 15:45

Well, if I understand you, you need first to evaluate the statistical averaging of the resolved field, that is <vbar> (<*> is a time averaging) which is a steady field. Then you want to compute the fluctuation field vbar-<vbar> which is an unsteady field representing the part of the fluctuations you can resolve (of course these are not the DNS-based one).

In practice you need to sample the LES velocity for several time-unit and then compute the statistical mean.

Agavi · May 29, 2021, 16:05

Thanks for your reply!

I do not only need the components (u'u')_bar. I need the whole transport equation of the resolved turbulent kinetic energy (so production, dissipation and diffusion terms - what is explained in post #1 in the thread I referenced TKE budget equation in LES).

FMDenaro · May 29, 2021, 16:40

Quote:

Originally Posted by Agavi

Thanks for your reply!

I do not only need the components (u'u')_bar. I need the whole transport equation of the resolved turbulent kinetic energy (so production, dissipation and diffusion terms - what is explained in post #1 in the thread I referenced TKE budget equation in LES).

Have you read also the textbook of Sagaut? You will find the equations

Agavi · May 29, 2021, 17:04

I looked at the equations and from my understanding eqn. 3.30 is the total resolvet kinetic energy (so it comes from UTilde). I'm not sure I understand equation 3.33. What is filtered subgrid kinetic energy?

If our real physical flow field is

$U = \bar{U}+ U'_{resolved}+U'_{modelled}$

Then it is my understanding that subgrid kinetic energy has to come from $(U'_{modelled}U'_{modelled})_{bar}$ . However I need the equation for the transport of $(U'_{resolved}U'_{resolved})_{bar}$ .

What am I understanding wrong?

FMDenaro · May 29, 2021, 17:19

Quote:

Originally Posted by Agavi

I looked at the equations and from my understanding eqn. 3.30 is the total resolvet kinetic energy (so it comes from UTilde). I'm not sure I understand equation 3.33. What is filtered subgrid kinetic energy?

If our real physical flow field is

$U = \bar{U}+ U'_{resolved}+U'_{modelled}$

Then it is my understanding that subgrid kinetic energy has to come from $(U'_{modelled}U'_{modelled})_{bar}$ . However I need the equation for the transport of $(U'_{resolved}U'_{resolved})_{bar}$ .

What am I understanding wrong?

Have a look to this slide. Remember that the fluctuations in LES are always modelled. What you can evaluate are the fluctuations with respect to the statistical mean.

Agavi · June 2, 2021, 15:19

Quote:

Originally Posted by FMDenaro

Remember that the fluctuations in LES are always modelled.

Hi Fillipo,

I'm not sure I understand that. What I know is that in LES some of the turbulent fluctuations are resolved and the rest of them are modelled (subgrid scale fluctuations).

Best

Agave

FMDenaro · June 2, 2021, 15:26

Quote:

Originally Posted by Agavi

Hi Fillipo,

I'm not sure I understand that. What I know is that in LES some of the turbulent fluctuations are resolved and the rest of them are modelled (subgrid scale fluctuations).

Best

Agave

In LES you assume a decomposion in resolved (filtered) and unresolved components. By definition, the fluctuations of the velocity (with respect to the filtered velocity) are only in the unresolved components.
The resolved components are unsteady and 3d as same as the fluctuations.

Could you address the reference you have read?

Agavi · June 2, 2021, 15:41

Oh OK. I thought that by 'fluctuations' you meant wrt to the mean flow. Now I get you mean the fluctuations wrt to the filtered field that is only modelled in LES.

Now that we have established the definitions, I'm looking for the tke transport equation for the resolved LES fluctuations with respect to the mean flow.
( basically in your post here TKE budget equation in LES looking for the transport equation of $0.5 <(u'_{RANS}-u'_{LES})^2>$ . ) Therefore I think that the equations in Sagaut and in the slides you attached can't work for what I need because they are for the kinetic energy in different regions of the spectrum.

Maybe Im wrong and such an equation cannot be derived. However I'm pretty sure this is what is used in the paper I attached in #1 of this thread and I'm looking for the derivation of it

FMDenaro · June 2, 2021, 17:13

u'RANS(x,t) = u(x,t) -<u>(x)

u'LES(x,t) = u(x,t) -u_bar(x,t)

hence

u'RANS(x,t) -u'LES(x,t) =u_bar(x,t)-<u>(x)

If you want to write an equation for the LHS it would simply be equal do du_bar/dt.

The term you asked for is 0.5*<(u_bar-<u>)^2> which is, by definition of <>, not depending on time.
Is that what you asked for?

May 29, 2021, 14:33	Resolved TKE budget terms calculation from LES Simulations	#1
Agavi New Member Join Date: Aug 2020 Posts: 20 Rep Power: 6	Hello everyone, I am running LES and need to plot the turbulent kinetic energy budget of my resolved flow field. I therefore need the equation of the TKE that comes from the square of the $u'_{resolved} = \tilde{u}-\bar{u}$ components. TKE Budget equations that I have found online are in the physical sense (ie the one in Pope's book) where the filtering operation is obviously ignored. A DNS simulation could use those equations for the calculation of the TKE budget terms. I'm looking for the derivation of the *resolved TKE* equation. This probably should come from a similar derivation to the TKE equation but also including the SGS tensor. -- This old thread discusses the matter but didn't reach a conclusion - That's why I started this new thread. TKE budget equation in LES -- I am probably looking for the derivation of equation (1) in this paper. It accounts for the effect of $\tau_{ij}^{SGS}$ to the resolved TKE balance. https://www.osti.gov/servlets/purl/1500073 Please let me know if I'm understanding something wrong here! Thanks Agave Last edited by Agavi; May 29, 2021 at 15:49.

June 2, 2021, 17:13		#12
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,896 Rep Power: 73	u'RANS(x,t) = u(x,t) -<u>(x) u'LES(x,t) = u(x,t) -u_bar(x,t) hence u'RANS(x,t) -u'LES(x,t) =u_bar(x,t)-<u>(x) If you want to write an equation for the LHS it would simply be equal do du_bar/dt. The term you asked for is 0.5<(u_bar-<u>)^2> which is, by definition of <>, not depending on time. Is that what you asked for? Last edited by FMDenaro; June 2, 2021 at 18:11. Reason: I edited the equation that I've written before to correct them.*

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May 29, 2021, 15:38		#3
Agavi New Member Join Date: Aug 2020 Posts: 20 Rep Power: 6	Hi FMDenaro, Thanks for your reply. I'm searching for the equation for the resolved by the LES turbulent kinetic energy, so $(u_{resloved}'u_{resolved}')_{bar}$ . Best Agave

May 29, 2021, 15:45		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,896 Rep Power: 73	Well, if I understand you, you need first to evaluate the statistical averaging of the resolved field, that is <vbar> (<*> is a time averaging) which is a steady field. Then you want to compute the fluctuation field vbar-<vbar> which is an unsteady field representing the part of the fluctuations you can resolve (of course these are not the DNS-based one). In practice you need to sample the LES velocity for several time-unit and then compute the statistical mean.

May 29, 2021, 16:05		#5
Agavi New Member Join Date: Aug 2020 Posts: 20 Rep Power: 6	Thanks for your reply! I do not only need the components (u'u')_bar. I need the whole transport equation of the resolved turbulent kinetic energy (so production, dissipation and diffusion terms - what is explained in post #1 in the thread I referenced TKE budget equation in LES).

May 29, 2021, 17:04		#7
Agavi New Member Join Date: Aug 2020 Posts: 20 Rep Power: 6	I looked at the equations and from my understanding eqn. 3.30 is the total resolvet kinetic energy (so it comes from UTilde). I'm not sure I understand equation 3.33. What is filtered subgrid kinetic energy? If our real physical flow field is $U = \bar{U}+ U'_{resolved}+U'_{modelled}$ Then it is my understanding that subgrid kinetic energy has to come from $(U'_{modelled}U'_{modelled})_{bar}$ . However I need the equation for the transport of $(U'_{resolved}U'_{resolved})_{bar}$ . What am I understanding wrong?

June 2, 2021, 15:41		#11
Agavi New Member Join Date: Aug 2020 Posts: 20 Rep Power: 6	Oh OK. I thought that by 'fluctuations' you meant wrt to the mean flow. Now I get you mean the fluctuations wrt to the filtered field that is only modelled in LES. Now that we have established the definitions, I'm looking for the tke transport equation for the resolved LES fluctuations with respect to the mean flow. ( basically in your post here TKE budget equation in LES looking for the transport equation of $0.5 <(u'_{RANS}-u'_{LES})^2>$ . ) Therefore I think that the equations in Sagaut and in the slides you attached can't work for what I need because they are for the kinetic energy in different regions of the spectrum. Maybe Im wrong and such an equation cannot be derived. However I'm pretty sure this is what is used in the paper I attached in #1 of this thread and I'm looking for the derivation of it