[LES] Subgrid-scale and grid-scale energy equations

Moreza7 · January 20, 2021, 17:41

Hello,

I'm reading some books about LES, in most of them, there are two equations about grid-scale and subgrid-scale energy equations.
But none of them explain that how did they achieve these equations. And they also didn't refer to a source!
For example:

Can someone please explain to me that how these relations obtained? Or at least give me a reference?

Best regards

sbaffini · January 20, 2021, 17:58

If you know how to obtain the total kinetic energy equation (by multiplying each momentum equation for the respective velocity component and summing them), you can do the same for the resolved kinetic energy equation (using the resolved velocity and momentum equations). Then the sgs kinetic energy equation is obtained by difference between the two

Moreza7 · January 20, 2021, 18:13

Quote:

Originally Posted by sbaffini

If you know how to obtain the total kinetic energy equation (by multiplying each momentum equation for the respective velocity component and summing them), you can do the same for the resolved kinetic energy equation (using the resolved velocity and momentum equations). Then the sgs kinetic energy equation is obtained by difference between the two

Thanks a lot dear pauolo.
But as you can see in rhs of first equation there is filtered(p*u) [second term in parenthesis] I don't think that it can be obtained without modeling (just like filtered(uv))! Then why is it there in the resolved part equations?

sbaffini · January 20, 2021, 18:28

Can't double check now, but looks like an error to me. Which book is that?

Moreza7 · January 20, 2021, 18:34

Quote:

Originally Posted by sbaffini

Can't double check now, but looks like an error to me. Which book is that?

Thank you for the reply.
Here it is:
https://www.springer.com/gp/book/9783319453026

FMDenaro · January 20, 2021, 18:38

Quote:

Originally Posted by Moreza7

Thank you for the reply.
Here it is:
https://www.springer.com/gp/book/9783319453026

It is clearly an error in the equation

Moreza7 · January 20, 2021, 18:43

Hello,
Thanks for your reply.
Yes. That seems to be an error because in this article that problem does not exist.

U. Piomelli and J.R. Chasnov. Large-eddy simulations: Theory and applications.

Moreza7 · January 21, 2021, 07:50

Hello,

I just found something confusing about these equations.

The derived equations are based on: $\mathbf{\widetilde{k}=k_{SGS}+k_{GS}}$

GS: Grid-Scale | SGS: SubGrid-Scale

Where $\mathbf{\widetilde{k}=\widetilde{u_{k}u_{k}}/2}$ and $\mathbf{\widetilde{u_{k}}\widetilde{u_{k}}/2}$

Please note that $\mathbf{\widetilde{x}}$ means filtered value of $\mathbf{x}$ .

The author emphasizes:
"It is often misinterpreted that the sum of the grid-scale and subgrid-scale energy $\mathbf{k_{GS}+k_{SGS}}$ is $\mathbf{k}$ . However it should be noted that this sum should be $\mathbf{\widetilde{k}}$ . The kinetic energy distribution from experiments or DNS should be filtered when LES results are compared with such results."

This means that $\mathbf{\widetilde{k}}$ is the filtered DNS or experimental kinetic energy instead of the total kinetic energy. Then why it's not like the following equations just the same as filtered N-S equations?
$\mathbf{k=k_{SGS}+k_{GS}}$ and $\mathbf{k=u_{k}u_{k}/2}$

Why should the kinetic energy of DNS or experimental data be filtered when comparing with LES? If we filter DNS, then we only have the scales which have the size of the grid cell. While the LES data contains these scales + modeled small scales!

sbaffini · January 21, 2021, 09:02

Ok, my last sentence in the post above [LES] Subgrid-scale and grid-scale energy equations was actually misleading as few details were missing.

Let me recap differently:

1) Kinetic energy of all the scales is $k = \frac{u_i u_i}{2}$ . Its equation is derived from taking the scalar product $u_i \frac{\partial u_i}{\partial t}$ . Also $\widetilde{k} = \frac{\widetilde{u_i u_i}}{2}$

2) The kinetic energy of the grid scales is $\widetilde{k}_{GS} = \frac{\widetilde{u}_i \widetilde{u}_i}{2}$ . Its equation is derived from taking the scalar product $\widetilde{u}_i \frac{\partial \widetilde{u}_i}{\partial t}$

3) The filtered kinetic energy of the subgrid scales is $\widetilde{k}_{SGS} = \frac{\widetilde{u'_i u'_i}}{2}$ . Its equation is derived from the filtered scalar product $\widetilde{u'_i \frac{\partial u'_i}{\partial t}}$ . Where $\frac{\partial u'_i}{\partial t} = \frac{\partial u_i}{\partial t} - \frac{\partial \widetilde{u_i}}{\partial t}$ (and, obviously, $u'_i = u_i - \widetilde{u_i}$ )

4) The whole SGS stress tensor with the so called Leonard triple decomposition is $\tau_{ij} = \widetilde{u_iu_j}-\widetilde{u_i}\widetilde{u_j}$ . Note that $\frac{\tau_{kk}}{2}= \widetilde{k}-\widetilde{k}_{GS}$ is known in LES as the Generalized SGS kinetic energy $\widetilde{k}_{GSGS}$

5) The unsolved kinetic energy is $k-\widetilde{k}_{GS}$

Now:

- The kinetic energy of the book you are referring to is the one in 4 (i.e., $\widetilde{k}_{GSGS}$ )

- The common SGS kinetic energy usually referred to in LES is the one in 3 (see, for example, Sagaut)

- The one I was referring to in my post is the one in 5

Which one you should refer to is really just a matter of definitions. But, obviously, they have different properties.

Moreza7 · January 21, 2021, 11:10

Quote:

Originally Posted by sbaffini

Ok, my last sentence in the post above [LES] Subgrid-scale and grid-scale energy equations was actually misleading as few details were missing.

Let me recap differently:

1) Kinetic energy of all the scales is $k = \frac{u_i u_i}{2}$ . Its equation is derived from taking the scalar product $u_i \frac{\partial u_i}{\partial t}$ . Also $\widetilde{k} = \frac{\widetilde{u_i u_i}}{2}$

2) The kinetic energy of the grid scales is $\widetilde{k}_{GS} = \frac{\widetilde{u}_i \widetilde{u}_i}{2}$ . Its equation is derived from taking the scalar product $\widetilde{u}_i \frac{\partial \widetilde{u}_i}{\partial t}$

3) The filtered kinetic energy of the subgrid scales is $\widetilde{k}_{SGS} = \frac{\widetilde{u'_i u'_i}}{2}$ . Its equation is derived from the filtered scalar product $\widetilde{u'_i \frac{\partial u'_i}{\partial t}}$ . Where $\frac{\partial u'_i}{\partial t} = \frac{\partial u_i}{\partial t} - \frac{\partial \widetilde{u_i}}{\partial t}$ (and, obviously, $u'_i = u_i - \widetilde{u_i}$ )

4) The whole SGS stress tensor with the so called Leonard triple decomposition is $\tau_{ij} = \widetilde{u_iu_j}-\widetilde{u_i}\widetilde{u_j}$ . Note that $\frac{\tau_{kk}}{2}= \widetilde{k}-\widetilde{k}_{GS}$ is known in LES as the Generalized SGS kinetic energy $\widetilde{k}_{GSGS}$

5) The unsolved kinetic energy is $k-\widetilde{k}_{GS}$

Now:

- The kinetic energy of the book you are referring to is the one in 4 (i.e., $\widetilde{k}_{GSGS}$ )

- The common SGS kinetic energy usually referred to in LES is the one in 3 (see, for example, Sagaut)

- The one I was referring to in my post is the one in 5

Which one you should refer to is really just a matter of definitions. But, obviously, they have different properties.

Hello. Thanks a lot for your reply.

Why did you say that kinetic energy of all scales is also $\widetilde{k} = \frac{\widetilde{u_i u_i}}{2}$ ? (Last equation of part 1)
I think it shouldn't be filtered. Because if it is filtered,then we will not have small scales.

sbaffini · January 21, 2021, 11:27

Quote:

Originally Posted by Moreza7

But why did you use the filter(~) for the SGS kinetic energy equation?

Because, as I mentioned, it is a matter of definitions. You can attach whatever definition to whatever label. But, for me, in LES, the reference is Pierre Sagaut. End of the story. This is important when being part of a community which has consolidated nomenclature. Everyone can write whatever he thinks is legit but, most people in LES agree on using the book of Sagaut as general reference.

Why the Sagaut reference has such definition has, of course, the same reason behind it. Because other people used it with that definition.

Now, if you want me to further dig into this and explain why these people used this specific definition, it is a little more complex. The general answer is because that is the quantity that they ended up with in their equations. The unfiltered counterpart of that quantity is of relatively scarce interest because it is a quantity you don't know on scales you don't have access to. The filtered counterpart instead is used by several authors as base equation for 1 equation SGS LES models (in this case it is fundamental that it refers to scales you have access to, i.e., that it is filtered).

Quote:

Originally Posted by Moreza7

And why did you say that kinetic energy of all scales is also Also $\widetilde{k} = \frac{\widetilde{u_i u_i}}{2}$ ? (In case 1)
I think it shouldn't be filtered

That statement was a blind application of the filter to the k definition. What I wrote is:

1) The conventional definition for the kinetic energy of all the scales is $k = \frac{u_i u_i}{2}$ . Its equation is derived from taking the scalar product $u_i \frac{\partial u_i}{\partial t}$ .

2) From 1, filtering both sides of the k defintion, obviously follows that $\widetilde{k} = \frac{\widetilde{u_i u_i}}{2}$

sbaffini · January 21, 2021, 11:39

My personal experience with several CFD books (altough not this one in particular, as I haven't read it) is that turbulence modeling, and LES in particular, is not something that can be trusted on that sources. They basically have to treat it but, trust me, there is no single general CFD book that treats it correctly.

In LES this is worst because even the two books of Sagaut (there is also a compressible version) are not enough to completely describe all the known approaches, and in some spots it has to treat some stuff by blindly reporting only some specific point of view.

Now, if you just want a general grasp of the matter, just play along. But if you want to get into the equations, my suggestion is to first read Turbulent flows by Pope (in all its parts) and then go with Sagaut.

Moreza7 · January 21, 2021, 13:46

Quote:

Originally Posted by sbaffini

My personal experience with several CFD books (altough not this one in particular, as I haven't read it) is that turbulence modeling, and LES in particular, is not something that can be trusted on that sources. They basically have to treat it but, trust me, there is no single general CFD book that treats it correctly.

In LES this is worst because even the two books of Sagaut (there is also a compressible version) are not enough to completely describe all the known approaches, and in some spots it has to treat some stuff by blindly reporting only some specific point of view.

Now, if you just want a general grasp of the matter, just play along. But if you want to get into the equations, my suggestion is to first read Turbulent flows by Pope (in all its parts) and then go with Sagaut.

Thank you very much.
I appreciate your help.

You are right, but Sagaut's book is a hard-to-read book. I thinks it's really hard for a beginner.
And I think this time, found an error in Sagaut's book:

In the term XII, i should be replaced by j because it leads to continuity equation and becomes zero.

By the way. Thanks for your patience and great replies.

sbaffini · January 21, 2021, 14:40

Maybe you mean some other term (but there were no errors last time I did it myself), but XII is certainly correct. It is the diffusion term and there is no continuity involved... moreover, as this is a scalar equation, each term is scalar, so any index must be a repeated one (it must appear twice, meaning summation over that index), and while it is ugly to change the repeated index between different terms, it is actually fully customary which one you pick, might even be a or any other symbol

Moreza7 · January 21, 2021, 15:07

Quote:

Originally Posted by sbaffini

Maybe you mean some other term (but there were no errors last time I did it myself), but XII is certainly correct. It is the diffusion term and there is no continuity involved... moreover, as this is a scalar equation, each term is scalar, so any index must be a repeated one (it must appear twice, meaning summation over that index), and while it is ugly to change the repeated index between different terms, it is actually fully customary which one you pick, might even be a or any other symbol

Yes you are right.
I was reading two books with different inde and symbols at the same time and got confused!

Thanks a lot

Moreza7 · January 21, 2021, 17:09

Quote:

Originally Posted by sbaffini

Maybe you mean some other term (but there were no errors last time I did it myself), but XII is certainly correct. It is the diffusion term and there is no continuity involved... moreover, as this is a scalar equation, each term is scalar, so any index must be a repeated one (it must appear twice, meaning summation over that index), and while it is ugly to change the repeated index between different terms, it is actually fully customary which one you pick, might even be a or any other symbol

Just one more question:
How did you notice that XII term is diffusion? Why didn't you say it's about dissipation? Why do we sort XII as diffusion, but X as dissipation?
The same question stands for IX which is labeled as a dissipative term and XIII which labeled as a production term.
Does it come from the mathematical for of the terms or it's just due to the physical understanding of the terms?

Best Regards

FMDenaro · January 21, 2021, 17:12

Quote:

Originally Posted by Moreza7

Just one more question:
How did you notice that XII term is diffusion? Why didn't you say it's about dissipation? Why do we sort XII as diffusion, but X as dissipation?
The same question stands for IX which is labeled as a dissipative term and XIII which labeled as a production term.
Does it come from the mathematical for of the terms or it's just due to the physical understanding of the terms?

Best Regards

The dissipation of kinetic energy is a well defined term, it is mu*D. It is characterized to be alway positive

Moreza7 · January 21, 2021, 17:24

Quote:

Originally Posted by FMDenaro

The dissipation of kinetic energy is a well-defined term, it is mu*D. It is characterized to be always positive

Thank you. But how about the definition of the diffusive terms? How are they defined?

Apart from definitions, I just wanted to know is there any mathematical technique to sort the terms by their dissipative or diffusive nature?

FMDenaro · January 22, 2021, 05:45

Quote:

Originally Posted by Moreza7

Thank you. But how about the definition of the diffusive terms? How are they defined?

Apart from definitions, I just wanted to know is there any mathematical technique to sort the terms by their dissipative or diffusive nature?

The meaning of a "diffusive term" is much more general than the use done in the KE expressione. You should turn back to the origin of the term as defined in Chap 1 of Kundu

sbaffini · January 22, 2021, 06:22

Quote:

Originally Posted by Moreza7

Thank you. But how about the definition of the diffusive terms? How are they defined?

Apart from definitions, I just wanted to know is there any mathematical technique to sort the terms by their dissipative or diffusive nature?

The diffusion term is the one that, when you heat up a solid in a point, will make that heat diffuse in the rest of the solid. That is, lower the local peak of the temperature and increase the temperature in the surrounding (or viceversa for a local freezing point). It works the same in fluids and, in its simplest form, it is a diffusivity times the Laplacian of some variable (temperature in my example).

Don't take me wrong but, maybe, you should step back from LES in order to first clarify some basic concepts. The major risk here is investing time without a proper return. Also, as LES involves both numerics, fluid dynamics and turbulence, a strong background in all of them is required in order to understand it.

Maybe you can still grasp some general concept, but then I see no point in trying to decode, say, the SGS kinetic energy equation. Just know that it is what it is and go over it.

January 20, 2021, 17:41	[LES] Subgrid-scale and grid-scale energy equations	#1
Moreza7 Senior Member Join Date: Jan 2018 Posts: 121 Rep Power: 8	Hello, I'm reading some books about LES, in most of them, there are two equations about grid-scale and subgrid-scale energy equations. But none of them explain that how did they achieve these equations. And they also didn't refer to a source! For example: Can someone please explain to me that how these relations obtained? Or at least give me a reference? Best regards

January 20, 2021, 17:58		#2
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	If you know how to obtain the total kinetic energy equation (by multiplying each momentum equation for the respective velocity component and summing them), you can do the same for the resolved kinetic energy equation (using the resolved velocity and momentum equations). Then the sgs kinetic energy equation is obtained by difference between the two Moreza7 and aero_head like this.

January 20, 2021, 18:28		#4
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Can't double check now, but looks like an error to me. Which book is that? Moreza7 likes this.

January 21, 2021, 09:02		#9
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Ok, my last sentence in the post above [LES] Subgrid-scale and grid-scale energy equations was actually misleading as few details were missing. Let me recap differently: 1) Kinetic energy of all the scales is $k = \frac{u_i u_i}{2}$ . Its equation is derived from taking the scalar product $u_i \frac{\partial u_i}{\partial t}$ . Also $\widetilde{k} = \frac{\widetilde{u_i u_i}}{2}$ 2) The kinetic energy of the grid scales is $\widetilde{k}_{GS} = \frac{\widetilde{u}_i \widetilde{u}_i}{2}$ . Its equation is derived from taking the scalar product $\widetilde{u}_i \frac{\partial \widetilde{u}_i}{\partial t}$ 3) The filtered kinetic energy of the subgrid scales is $\widetilde{k}_{SGS} = \frac{\widetilde{u'_i u'_i}}{2}$ . Its equation is derived from the filtered scalar product $\widetilde{u'_i \frac{\partial u'_i}{\partial t}}$ . Where $\frac{\partial u'_i}{\partial t} = \frac{\partial u_i}{\partial t} - \frac{\partial \widetilde{u_i}}{\partial t}$ (and, obviously, $u'_i = u_i - \widetilde{u_i}$ ) 4) The whole SGS stress tensor with the so called Leonard triple decomposition is $\tau_{ij} = \widetilde{u_iu_j}-\widetilde{u_i}\widetilde{u_j}$ . Note that $\frac{\tau_{kk}}{2}= \widetilde{k}-\widetilde{k}_{GS}$ is known in LES as the Generalized SGS kinetic energy $\widetilde{k}_{GSGS}$ 5) The unsolved kinetic energy is $k-\widetilde{k}_{GS}$ Now: - The kinetic energy of the book you are referring to is the one in 4 (i.e., $\widetilde{k}_{GSGS}$ ) - The common SGS kinetic energy usually referred to in LES is the one in 3 (see, for example, Sagaut) - The one I was referring to in my post is the one in 5 Which one you should refer to is really just a matter of definitions. But, obviously, they have different properties. FMDenaro, acgnipper, Moreza7 and 1 others like this.

January 21, 2021, 11:39		#12
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	My personal experience with several CFD books (altough not this one in particular, as I haven't read it) is that turbulence modeling, and LES in particular, is not something that can be trusted on that sources. They basically have to treat it but, trust me, there is no single general CFD book that treats it correctly. In LES this is worst because even the two books of Sagaut (there is also a compressible version) are not enough to completely describe all the known approaches, and in some spots it has to treat some stuff by blindly reporting only some specific point of view. Now, if you just want a general grasp of the matter, just play along. But if you want to get into the equations, my suggestion is to first read Turbulent flows by Pope (in all its parts) and then go with Sagaut. Moreza7 likes this.

January 20, 2021, 18:43		#7
Moreza7 Senior Member Join Date: Jan 2018 Posts: 121 Rep Power: 8	Hello, Thanks for your reply. Yes. That seems to be an error because in this article that problem does not exist. U. Piomelli and J.R. Chasnov. Large-eddy simulations: Theory and applications.

January 21, 2021, 07:50		#8
Moreza7 Senior Member Join Date: Jan 2018 Posts: 121 Rep Power: 8	Hello, I just found something confusing about these equations. The derived equations are based on: $\mathbf{\widetilde{k}=k_{SGS}+k_{GS}}$ GS: Grid-Scale \| SGS: SubGrid-Scale Where $\mathbf{\widetilde{k}=\widetilde{u_{k}u_{k}}/2}$ and $\mathbf{\widetilde{u_{k}}\widetilde{u_{k}}/2}$ Please note that $\mathbf{\widetilde{x}}$ means filtered value of $\mathbf{x}$ . The author emphasizes: "It is often misinterpreted that the sum of the grid-scale and subgrid-scale energy $\mathbf{k_{GS}+k_{SGS}}$ is $\mathbf{k}$ . However it should be noted that this sum should be $\mathbf{\widetilde{k}}$ . The kinetic energy distribution from experiments or DNS should be filtered when LES results are compared with such results." This means that $\mathbf{\widetilde{k}}$ is the filtered DNS or experimental kinetic energy instead of the total kinetic energy. Then why it's not like the following equations just the same as filtered N-S equations? $\mathbf{k=k_{SGS}+k_{GS}}$ and $\mathbf{k=u_{k}u_{k}/2}$ Why should the kinetic energy of DNS or experimental data be filtered when comparing with LES? If we filter DNS, then we only have the scales which have the size of the grid cell. While the LES data contains these scales + modeled small scales!

January 21, 2021, 14:40		#14
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Maybe you mean some other term (but there were no errors last time I did it myself), but XII is certainly correct. It is the diffusion term and there is no continuity involved... moreover, as this is a scalar equation, each term is scalar, so any index must be a repeated one (it must appear twice, meaning summation over that index), and while it is ugly to change the repeated index between different terms, it is actually fully customary which one you pick, might even be a or any other symbol Moreza7 likes this.

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