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Actual varibles or filtered varibles, what obtained from implicit filter LES?

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Old   August 4, 2020, 09:35
Default Actual varibles or filtered varibles, what obtained from implicit filter LES?
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Hello, everyone!

I'm working on some projects about LES using OpenFOAM in which LES is implemented by implicit filter. Recently, I read the illuminating thread explaining how to carry out the explicit filter LES.
Explicit filtering in LES
But I think that I've found some question beyond this thread itself.
This my question:

As I know, the implicit filter assumes or view the discrete operations like interpolation scheme, integral scheme,gradient scheme and so on as the filtering operation.What you need to do for solve the equation is just to directly discrete your momentum equation with SGS viscosity and solve it by PISO or other method.
In pricinpal, the filtered varibles are used to assemble the discret equation and as a result , the obtained solution should be also filtered.Does it mean that we can only get filtered variables from implicit filter LES?Moreover, if it was ture, when we assemble and solve the momentum equation every time step, we actually filter the varibles which have been filterd in the last time step?

I know that it may sounds wired, but I really hope someone toe clear up my confusion.

Thank you!
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Old   August 4, 2020, 18:01
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Quote:
Originally Posted by lietoflaw View Post
Hello, everyone!

I'm working on some projects about LES using OpenFOAM in which LES is implemented by implicit filter. Recently, I read the illuminating thread explaining how to carry out the explicit filter LES.
Explicit filtering in LES
But I think that I've found some question beyond this thread itself.
This my question:

As I know, the implicit filter assumes or view the discrete operations like interpolation scheme, integral scheme,gradient scheme and so on as the filtering operation.What you need to do for solve the equation is just to directly discrete your momentum equation with SGS viscosity and solve it by PISO or other method.
In pricinpal, the filtered varibles are used to assemble the discret equation and as a result , the obtained solution should be also filtered.Does it mean that we can only get filtered variables from implicit filter LES?Moreover, if it was ture, when we assemble and solve the momentum equation every time step, we actually filter the varibles which have been filterd in the last time step?

I know that it may sounds wired, but I really hope someone toe clear up my confusion.

Thank you!



In principle, any unresolved simulation is "implicitly filtered" by the discretization. Only when the grid is so fine that all the characteristic scales are resolved you have that the filtered components are not relevant to the solution.
The issue that the filtered field is filtered again after a time step was discussed a lot of time ago by Lund. No, you do not cumulate the filtered at each time step. Note that Nyquist is a projective spectral cut-off filter.
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Old   August 5, 2020, 06:22
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Look at it this way: what equation are you actually solving when using an implicit filtering approach? Does it imply double filtering? Turns out that no, it doesn't imply double filtering
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Old   August 5, 2020, 06:39
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This old paper can be useful for the discussion, of course more recent papers are available
https://core.ac.uk/reader/82797562
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Old   August 5, 2020, 07:51
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Thank you so much for your enthusiasitc help! (Even my supervisor hasnt provide any paper for learning,sigh....)
Wish it can solve all my puzzle. If not, I will come back here to discuss it with you.
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Old   August 5, 2020, 07:56
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Yes, you're right. I think that I can use some mathmatic equations to illustrate my question more clearly. But Im not familiar with latex, it has to take some time.
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Old   August 5, 2020, 11:25
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Originally Posted by lietoflaw View Post
Yes, you're right. I think that I can use some mathmatic equations to illustrate my question more clearly. But Im not familiar with latex, it has to take some time.



You can also use Word and attach a pdf
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Old   August 6, 2020, 00:45
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You can also use Word and attach a pdf
I have followed your advise, and for the sake of convience and tidy, I put mathematic equation and descrption together
Attached Files
File Type: pdf mathmatic_description.pdf (23.1 KB, 3 views)
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Old   August 6, 2020, 03:55
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I have followed your advise, and for the sake of convience and tidy, I put mathematic equation and descrption together



The confusion appears when you consider Eq.(1) and (2). First, Eq.(1) is the continuous form, that is the exact PDE.
Now, introduce any discretization of the domain (that is a computational grid) and of the PDE Eq.(1) (FD, FV, SM, etc.).
Express now the resulting discrete equation in terms of a discrete vector field Ud.
The key is that you have to intepret what Ud really mean depending on the the flow problem and grid discretization. The grid introduce the Nyquist cut-off frequency Kc, that is an implicit filtering. If your flow problem has characteristic scales up to the Kolmogorov frequency Ke, you have two possible situations:
1) Kc>=Ke the discrete field Ud resolves all the scales, that is a DNS.
2) Kc<Ke the discrete field Ud does not resolve all the scales, the content of the flow problem behind Kc is implicitly filtered. Hence, Ud is the implicitly filtered variable. That means you are in a LES framework. Now you can add an SGS model to take into account the unresolved part.
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Old   August 6, 2020, 06:46
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Originally Posted by FMDenaro View Post
The confusion appears when you consider Eq.(1) and (2). First, Eq.(1) is the continuous form, that is the exact PDE.
Now, introduce any discretization of the domain (that is a computational grid) and of the PDE Eq.(1) (FD, FV, SM, etc.).
Express now the resulting discrete equation in terms of a discrete vector field Ud.
The key is that you have to intepret what Ud really mean depending on the the flow problem and grid discretization. The grid introduce the Nyquist cut-off frequency Kc, that is an implicit filtering. If your flow problem has characteristic scales up to the Kolmogorov frequency Ke, you have two possible situations:
1) Kc>=Ke the discrete field Ud resolves all the scales, that is a DNS.
2) Kc<Ke the discrete field Ud does not resolve all the scales, the content of the flow problem behind Kc is implicitly filtered. Hence, Ud is the implicitly filtered variable. That means you are in a LES framework. Now you can add an SGS model to take into account the unresolved part.
Excellent! It's really heuristic. Especially, "The grid introduce the Nyquist cut-off frequency Kc".
I will re-express your words by my understanding, and if you find any error, please correct me.

My understanding is that:
Discretion equations can only get filtered solution rather than filter acual solution. It sounds confusing so I'll give a example to explain it.

You have to get a curve from five different points on the plane. And you decide to use a polynomial to do this. No matter what method you use, you can only get a function with remainder term of O(x^4). Even you change the basis like Fourier series, you can't get all information of the original curve due to the limited points.

It's same to what happens in FVM FD SM these methods. When your mesh isn't refined into the dissipation level, you will never get all information of the actual flow. Because of the "coarse" grid you use, you can only get partial information of actual flow, which is viewed as "implicit filterd variable".

And SGS model is added to the original for compensating or mimicing the behavior of the information filtered apart from the actual flow.

That's the truth of "implicit flitered LES". It's rather lossing than filtering in essence. Different discretions will neglect different imformation, which cause the different shape of "implicit filter".

However, depending on the knowledge I read from Pope's book and some papers of dynamic model, there seems to be a closed relation among energy spectral ,filter and SGS model. Many coefficient are determined by this relationship.

So, how to assure that we really apply the corresponding SGS model properly in the simulation?
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Old   August 6, 2020, 07:51
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Yes, the concept of implicit filtering is that the loss of components of the flow due to the discretization are viewed as filtering.

The second part of your question: are you talking about the exact Germano identity? It is deduced by means of the application of a second filter.
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Old   August 6, 2020, 09:30
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Yes, the concept of implicit filtering is that the loss of components of the flow due to the discretization are viewed as filtering.

The second part of your question: are you talking about the exact Germano identity? It is deduced by means of the application of a second filter.
Yes, you're right. But what I want to discuss is something deeper in the theory.
The eqn(13.132) and eqn(13.134) in Pope's book give different value of Cs relying on corresponding filter, which implies a relationship between filter and SGS model.
Moreover, it demonstrates the dilemma between practice and theory of LES model, where we use a specific filter to derive the LES model, but apply it to simulation by implicit filter.

Although the refinement of grid always works in improving the effect of model except special cases, the differences of LES models coming from models themselves and the abuse of filter become ambiguous. In my opinion, it will bring a huge limitation of the analysis and development of LES. After all, LES is the technology based on filtering information and building the model of filtered part.

But, what I said above doesn't means that I opposes to implicit filter. On the contrary, I think that we should implement the implicit filter more carefully to guarantee the consistency between theory and practice while it looks quite hard.
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Old   August 6, 2020, 09:35
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Yes, you're right. But what I want to discuss is something deeper in the theory.
The eqn(13.132) and eqn(13.134) in Pope's book give different value of Cs relying on corresponding filter, which implies a relationship between filter and SGS model.
Moreover, it demonstrates the dilemma between practice and theory of LES model, where we use a specific filter to derive the LES model, but apply it to simulation by implicit filter.

Although the refinement of grid always works in improving the effect of model except special cases, the differences of LES models coming from models themselves and the abuse of filter become ambiguous. In my opinion, it will bring a huge limitation of the analysis and development of LES. After all, LES is the technology based on filtering information and building the model of filtered part.

But, what I said above doesn't means that I opposes to implicit filter. On the contrary, I think that we should implement the implicit filter more carefully to guarantee the consistency between theory and practice while it looks quite hard.

You could find useful the details in my paper:
https://www.researchgate.net/publica...dy_Simulations
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Old   August 6, 2020, 10:16
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Thank you so much! I am going to read the paper carefully and if I come across questions, I'll be back here.
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Old   August 10, 2020, 05:26
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You could find useful the details in my paper:
https://www.researchgate.net/publica...dy_Simulations
Dear professor

I am reading your intriguing paper, but I have a question requiring your help.

When I derived eqn (25), I found that the flux represented by area multipled face-center value which implys a average on the surface worked as a basis of discreted one-dimensional filter, giving an impression that FV method leads to an one-dimensional implicit filter akin to FD method referred by Lund's paper. After that, this one-dimensional filter was compared with the exact three-dimensional filter in Fig 3, making it more weird.

In my opinion, the key difference between FD and FV method is that basic equation of FD was bulit in the nods causing a net-like discrete structure, but for FV, all things were done in the control volumes casuing a block-like discrete structure, so the numerical integral is an essential and indispensable step for FV method.
Moreover, when even the interpolation scheme was considered, the average effect from center integral of flux is not supposed to be negelected, which provides top-hat filter in the other two directions and makes the discrete term in FV method own a three-dimensional filter.

Would you like to be so kind to answer my question:
Why was the effect of numerical integral in FV method negelgected in the analysis of the shape of implicit filter?

Looking forward to your reply.

Liet
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Old   August 10, 2020, 07:10
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Quote:
Originally Posted by lietoflaw View Post
Dear professor

I am reading your intriguing paper, but I have a question requiring your help.

When I derived eqn (25), I found that the flux represented by area multipled face-center value which implys a average on the surface worked as a basis of discreted one-dimensional filter, giving an impression that FV method leads to an one-dimensional implicit filter akin to FD method referred by Lund's paper. After that, this one-dimensional filter was compared with the exact three-dimensional filter in Fig 3, making it more weird.

In my opinion, the key difference between FD and FV method is that basic equation of FD was bulit in the nods causing a net-like discrete structure, but for FV, all things were done in the control volumes casuing a block-like discrete structure, so the numerical integral is an essential and indispensable step for FV method.
Moreover, when even the interpolation scheme was considered, the average effect from center integral of flux is not supposed to be negelected, which provides top-hat filter in the other two directions and makes the discrete term in FV method own a three-dimensional filter.

Would you like to be so kind to answer my question:
Why was the effect of numerical integral in FV method negelgected in the analysis of the shape of implicit filter?

Looking forward to your reply.

Liet
The effect is not neglected at all in FVM.
Let me start the discussion from the continuos formulation of the equations.
The differential form has no filtering but the integral-based form has really that owing to the volume averaging. That happens in general, not only in terms of the LES equations. The FD discretization Introduces implicit 1d filtering but the FV discretization of the integral form is 3d by definition. Thus, while the FD-based implicit filtering resulting from the discretization has a unknown shape, the FV-based implicit filtering must result an approximation of the shape of the 3d top-hat filter. That is what I highlighted in my paper.
Lund discussed the implicit filtering for the FD discretization, highlighting the implicit 1d actions
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Old   August 10, 2020, 07:29
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So, why was only the filter caused by discreted derivative compared with the exact filter in Figure 3, where I think we should add the effect of integral. It seems more reasonable to show the shape of implicit filter coming from FV method in this way.

Thank you for replying to me so soon!
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Old   August 10, 2020, 07:38
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So, why was only the filter caused by discreted derivative compared with the exact filter in Figure 3, where I think we should add the effect of integral. It seems more reasonable to show the shape of implicit filter coming from FV method in this way.

Thank you for replying to me so soon!
If you are talking about FIGs 3 in my paper, the filter shapes are 3D. They consider the integral.
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Old   August 10, 2020, 08:06
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If you are talking about FIGs 3 in my paper, the filter shapes are 3D. They consider the integral.
I want to discuss what happened in the first step in eqn (25):

The exact flux expression;
Flux=\int_{z-h/2}^{z+h/2}\int_{y-h/2}^{y+h/2}\phi dydx

The discreted flux expression with central integral;
Flux_{D}=\phi_{C} h^2

It's easy to show that:

\hat{\phi_{C}}=\hat{G(k)}\cdot\hat{\phi}
where G(k) is the top-hat function with filter width h

However, \phi replaced \phi_{C} and the effect of the numerical integral didn't appear in the final equation.
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Old   August 10, 2020, 08:13
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I want to discuss what happened in the first step in eqn (25):

The exact flux expression;
Flux=\int_{z-h/2}^{z+h/2}\int_{y-h/2}^{y+h/2}\phi dydx

The discreted flux expression with central integral;
Flux_{D}=\phi_{C} h^2

It's easy to show that:

\hat{\phi_{C}}=\hat{G(k)}\cdot\hat{\phi}
where G(k) is the top-hat function with filter width h

However, \phi replaced \phi_{C} and the effect of the numerical integral didn't appear in the final equation.

Be careful, you are discussing one of the possibile FV flux reconstruction. The mean value formula is indeed not really useful as an implicit filtering, exactly as you observed. Different discretizations are more effective.
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