well, my opinion is different ... when you write for example the top-hat filter, the transfer function is somehow like sin(eta)/eta, therefore it has an infinite number of zero, vanishing only asymptotically. But that happens in the continuous formulation... If you use a discrete top-hat, the numerical transfer function is always terminated at the Nyquist frequency that is nothing else a cut-off filter introduced by the computational grid. In conclusion, the degrees of freedom are reduced.

Professor; I have been reading about the explicit filter and the papers your recommended me; but now I am more confused due to tress facts:
1.- Lund 2003; he proposed that the explicit filter has to be applied only on the convective term, in order to recover the frequencies to the bar level. However, He said tht the convective term needs to be constructed from a mesh two times finer. This fact is in line with Sagaut book.
2.- Gullbrand 2003; she proposed an explicit technique applied on the entire LES equation, not only on the convective term as Lund 2003. Also the filter was applied ina mesh twice as large as the grid mesh.
3.- Why the mesh needs to be two times smaller in Sagaut and Lund approach?; How these two approaches (section 2 and 1 from this message) are different since both are explicit filter.

Sincerely
JM

Ok, let me do an example in 1D.

Assume a computational mesh size h and an implicit filtering based on a second order central discretization. The resulting filtered velocity is a consequence of the adoption of the FD (transfer function like sin(csi)/csi) and the cut-off kc=pi/h (Nyquist). Therefore, the filtered velocity has no components outside kc but you can also see that the transfer function does not maintain the value 1 for all the resolved frequency (before kc) but it is approximately equal to 1 only for the first one-third of the range.

Now, what is the reason to think that an explicit filtering is better? The idea is that the SGS model uses the information coming from the filtered field and that, ideally, the best resolved field for a goo SGS model is provided by a spectral cut-off that has a transfer function exactly =1 until kc. Therefore, if we want to follow this idea still using FD/FV methods, we need to clear the implicitly filtered field, cancelling the part of the resolved frequencies that deviate from 1. For doing that we need to aplly explicitly a filter using a second computational grid with h_f>h. This way, the Nyquist frequency for the second grid cancel-out the components.

Therefore h_f/h is a ratio defining the portion of the original filtered field we want to cancel. This ratio can be 2 or higher.
However, in a practical computation, we need a good resolved field, therefore h_f must be quite small and consequently h is much smaller.
Hence, in principle h_f/h>2 is better as effective filtering but h_f/h=2 is less computationally expensive.

Professor; thank you very much I really appreciate your kindness. Your thoughts and explanations are very encouraging and important to me. I wished I could have worked under you.

Now, I think I got the gist of it. At the beginning I thought that the reconstruction was the other way around. But, according to your explanation the idea is to start with a fine mesh that yields to Nyquist_1. Then with a coarser mesh (at least twice as larger as the first) the convective term is filtered. The second mesh may have a Nyquist_2 > Nyquist_1. However, I visualize this as running two problems at the same time. Am I right? Because I need the finer grid at each time step to have the information to perform the filter of the coarser one. Otherwise, how could I perform the filter (the convolution integral over the filter width.)

Professor; this assumption that you mentioned at the beginning is what papers calls "scale similarity"?

Respectfully
JM

...be carefull, what you called Nyquist_2 is pi/h_coarse < Nyquist_1.

The computational problem has the cost determined by the finest mesh, hence you compute all the components on the finest grid but your LES solution is determined only until Nyquist_2. For this reason many authors consider more useful to use the implicit filtering approach on a quite fine grid. The advantage of the explicit filtering is that, for a fixed coarse grid size (Nyquist_2= constant=), you can do a computational grid refinement (Nyquist_1-> +Inf) to look for a filtered grid-independent solution. Conversely, the implicit filtering produces a convergence towards the DNS solution.

see this work to understand the role of the filtering in the SGS model:
https://www.researchgate.net/publica...ddy_simulation

thank you professor for the correction; I mixed the symbols. I wanted to type "<" instead of ">".
Professor, your previous explanation sounds to me like spectral method. Based on my understanding; in order to implement a filter I need to integrate over certain points. Also, based on Sagaut "This means that the grid used for composing u_i (bar)u_j(bar) product has to be twice as fine as the one used to represent the velocity field." Hence the velocity field, will be computed on a coarser mesh based on the filtered convective term. Maybe I am being to fuzzy on my statement, but It is difficult for me to visualize a filter with a difference in mesh sizes. That is why, I see this as two problem running at the same time.

Thanks professor.

"that the grid used for composing u_i (bar)u_j(bar) product has to be at least twice as fine"

You must think about two concurrent problems, the discrete equation that provides the implicit filtering and the explicit filtering that must clear the field. Of course, in order to run at each time step the method, the explicit filtering is computed on all nodes of the finest grid.
In principle, the explicit filtering makes sense only for FD/FV, in SM it is used the de-aliasing technique.

thank you professor; thank you very much!!!

Dear Professor;
Currently I am reading/working with the dynamic procedure. After reading the Germano's Paper "Turbulence:the filtering approach" and Sagaut book I ended up mixing concepts and physics.

Last time you explained me about the explicit filtering on the convective term. Where basically I carry out two runs at the same time. The finest mesh defining the smalles Nyquist wavenumber and the coarser one the biggest Nyquist wavenumber. After each time step, I used the velocity field from the finest mesh to filter the solution to the coarser mesh. Once I have the convective term filtered out at the coarser mesh, I assume that I just need to use that velocity field to evolve the solution in time after making sure the velocity field is solenoidal i.e: SIMPLE or projection method.

However, after reading this material I am visualizing the Germano process as an explicit filtering too. Because, I have a F-level filtered values (that I can see as the finest mesh) and and test filter G (as the coarser mesh). I am aware that the goal with the Germano paper (Lilly) is to obtain a variable value of Cs, while in the explicit the idea is to clean the implicitly filtered field.

Very Respectfully
JM

right, in the dynamic procedure you work with the so-called test-filtering...it is necessary to perform a further explicit test-filter having the third Nyquist frequency greater than the other two.

Filtering the non-linear Term
 

I would like to dig up this subject again, beacuse I'm interested in implementing explicit filtering in FVM code.
As mentioned before, filtering the convective term is the main challenge in explicit filtering. When using FVM, the non-linear term is linearised and the volume integral of the convective term turns into a surface integral, which in turns become a sum of fluxes (phi) times velocity (u) over all faces of the cell.
How would you then filter the convective (linearised) term (assuming we are using a smooth Gauss filter) ? I tried the following:

divergence(uiuj) --> divergence(phi*uj) this is the linearisation step.

assuming a smooth filter: u_filtered = u + const.*laplace(u); where const. represents the filter width.

Applying this on the non-linear term:

divergence((uiuj)_f) = divergence(uiuj)+const.*divergence(laplace(uiuj)).

The first term can be treated the same way as in implicit filtering. The treatment of the second term however is not trivial. I tired calculating laplace(uiuj) using values from the previous time step and then taking the divergence of it. The solver has then stability problems, which I assume has something to do with Rhie-Chow interpolation (Im using collocated grid variables).

Any advice on how treating this problem would be appreciated.

Regards

I am not sure of what you are doing...if you work with the FV discretization of the integral form, you get to work with this convective flux term:


1/|V| Int [S] n.v_bar v_bar dS

But I want to highlight that that is equivalent to

[Div ( v_bar v_bar)]_bar

where [ ]_bar is the volume filter (top-hat). Hence you are already applying a filtering on the resolved convective term.
I analysed this type of formulation much deeper, if you are interested in I suggest this paper:

https://www.researchgate.net/publica...dy_Simulations

In your paper, you discuss the implicit filtering. Yes, the FV method does filter the solution (grid filter). What I want is to apply an explicit filter. Lets say I have a very fine grid and a very coarse one. Just by using the grid filter would ofcourse lead to different results, since each grid has its own cutoff wavenumber.

By applying explicit filtering, one can get (almost) identical solutions on both grids, if one sets the cutoff wavenumber of the explicit filter on the fine grid equal to the cutoff wavenumber of the coarse one.

What I want to do is exactly the method used by Lund, Gullbrand, etc. I think the papers were already mentioned in this thread.

As you can read in that papers, they use the differential, not the integral formulation. Therefore, they apply the filter on the term

Div ( v_bar v_bar)

but then, they commute filter and divergence

Div [( v_bar v_bar)_bar]

Which formulation do you want to use? If you use the integral formulation you should consider that the surface integral is by itself a volume filter applied explicitly.

I use the integral formulation. The filter applied by the integral itself is similar to implicit filtering, since I cannot control the filter width of the integral. Therefore I would like to apply the filtering formulation mentioned above (the differential one) on my integral formulation. My goal is actually to get a "coarse" solution on a "fine" grid.

That means you want to work with

1/|V| Int [S] n.[v_bar v_bar]_expl dS

where [ ]_expl is the explicit volume filtering.
Well, this was never used in the literature and deserves much more thinking...
At first, this formulation leads to solve for a very different meaning of the resolved velocity because this way you have to consider that you have implicityly generate a gerarchy of filtering in

[Div ( v_bar v_bar)_expl]_bar

the question is what is the meaning of the velocity you have in the time derivative...the LES equatione must be reformulate properly and I immagin a new unresolved tensor appears.

well I see no big difference between the differential and the integral form. The filter applied through the integral is applied on all terms in the momentum equation (for the time derivative too). Wouldn't be possible to just write the explicitly filtered equations as in the differential form, and apply a volume integral over the whole equation ? The result should be the explicitly filtered equations in integral form. As a result, the stress terms emerging from explicit filtering would be the same as described in the papers above. The only problem would be, as mentioned above, filtering the convective term.

the issue is much more complex...when you applly the filter on the convective terms, you must consider you solve a term like

d/dt [u_bar]_expl

and this is different if you use the differential or the integral formulation

I thought the explicit filtering (denoted by [...]_expl ) was only applied on the convective term and on terms included in the SFS tensor. Why would you expl. filter the time derivative ?

P.S.: by the convective term I mean (duiuj/dxj) and not (Duiuj/Dt) ... so I just mean the non-linear term in the momentum equation.

Yes, you do not apply on the time derivative but it results as a consequence of the application of the explicit filtering on the convective term...if you filter the convective flux, what is the result in terms of the time dependent solution? think about...

I suggest to give a carefull reading to the papers of Lund and the paper of Gullbrand on JFM, You will find the answer.