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Filtering DNS solutions Vs projecting filtered DNS solution

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Old   May 19, 2016, 09:46
Default Filtering DNS solutions Vs projecting filtered DNS solution
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Julio Mendez
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Dear community;
Every time we think about filtering we think about reducing the degree of freedom; which is true. However, we usually think about filtering as projecting a fine solution (let's say DNS) onto a coarser mesh (usually a LES mesh).

Therefore, we implicitly relate the cut-off width to dx. In other words we "hard wire" dx with delta. However, we "know" (I should have said; "I think") that this is not completely true. For example, imagine we use a spectral cut - off where we can assume that our filter is 100% efficient due to its transfer function. Hence, our Nyquist wavenumber is defined by dx and the cut-off by delta. Definitely, we define delta at any point in the spectrum. Here, we see that dx is defined by numerical reasons (maximum resolution of the numerical scheme) and delta by physical reasons (the location of the cut-off in the spectrum).

After this cumbersome introduction, I want to introduce my question.
Is the projection technique from the filtering operation related to the filtering it self?... I will be more specific.
Imagine we have a DNS solution with n_DNS = 128 and we want to filter the DNS solution to "compare" to a LES solution. On the other hand, the LES solution is n_DNS / 2 (h_les = 2h_dns). Meaning that n_LES = 64.
We usually proceeds as follow:
We filter the DNS solution defining delta= 2*h_LES and we perform the convolution integral. The solution is obtained in the DNS mesh. Then, we project this filtered solution to the LES mesh.

Now, can we filter the DNS solution using different filter widths in the same DNS mesh?. If this is true, then it means that dx is not hardwired to delta. And we can have a DNS solution (with n_DNS) filtered with different delta on the same DNS mesh (with n_DNS).

I know that this questions is very cumbersome, but I want to make sure that I properly understood the filtering definition. I have seen that we usually think about filtering and projection are related yielding to mix projection onto a coarser mesh with filtering the DNS solution as the same procedure.

Thank you very much for your valuable time!!!
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Old   May 19, 2016, 14:57
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Filippo Maria Denaro
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This question has not a simple and unique answer....
Starting from an example in literature (http://doc.utwente.nl/66859/2/9008.pdf) you can see that a simple cut-off is adopted on the DNS dataset.

The key is if that can or not represent a good approximation of the LES field.

My opinion is that the main goal is to identify first the type of filtering really in use in the LES code. If a spectral code is used, then I think that is acceptable using a simple cut-off on the DNS dataset.

On the other hand, if FD or FV methods are used for the LES and an implicit filtering is in effect, the procedure to filtering the DNS dataset can be more complex and not univocally implied by the LES solution:

1) the DNS solution is first restricted on the LES grid and then numerically filtered to best match the implicit filter in the LES
2) the DNS solution can be filtered on each node of the original DNS grid but using the width of the LES (and always a numerical filter that best matches the implicit filter in the LES). Then, the filtered DNS is restricted on the LES grid.

I suggest trying some test to check both procedures.
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Old   May 19, 2016, 15:13
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Julio Mendez
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Thank you very much professor; I am glad there are professors like you!!!...
Thank you very much for the literature; I am going to read it thoroughly.

The main issue is with the fact about "Filtering always reduces the number of nodes". I am stressing here, because there are a lot of practitioners in LES claiming that it is impossible to filter a field without reducing the number of nodes. I think this is wrong, otherwise the spectral cut-off would not exists. !!!
Here is the first issue. In the ideal case (spectra cut-off) "dx" is defined by numerical consideration. To fully capture all resolved scale in space. Whereas "delta" is chosen by physical consideration (cut-off in the spectrum). Therefore, dx is independent from delta. Maybe I am wrong with this statement.

At this stage I do not want consider LES yet. Therefore, If I have a DNS dataset (n=n_DNS), can I apply explicit filter on that data set, varying only "delta" and keeping "dx=cte". ? In this case, I think that the Nyquist wavenumber will be the same for all the filtered DNS solution and the cut-off will be defined by "delta". Therefore, in the spectrum plot I will see only one (Nyquist = pi/dx) and different curves (one for each filter width). However, since I am applying the explicit filter upon the DNS solution, the filtered solution will be represented by the same number of nodes from the original solution. Is it right?

If we want to bring LES now, is just to project this nodes into the coarser grid, which is just picking up the specific nodes (depending on the physical location of each node)?.

I fully agree with your opinion; there are plenty of fact supporting this issue in your papers..

Thanks !
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Old   May 19, 2016, 15:34
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Filippo Maria Denaro
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Let me add an example...

consider any continuos function f(x), in principle you can define:

f_bar(x) = (1/h) Int [x-h/2; x+h/2] f(x') dx'

that, according to the definition of top-hat filter in physical space, is a filtered function still continuous. But that is only a "smoothing" on the original function, does not introduce any cut-off. The transfer function is continuous and has an infinite number of zeros, vanishing only asymptotically for k->+Inf.

The reduction of degree of freedom is effective when a cut-off is introduced. That can be done analytically or it is in action in an implicit way when you work in the discrete sense on a computational grid of size dx. Implicitly, that is the born of the Nyquist frequency pi/dx.
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Old   May 19, 2016, 15:47
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professor you have raised a very important fact

Quote:
Originally Posted by FMDenaro View Post
"But that is only a "smoothing" on the original function, does not introduce any cut-off." .
Finally; applying the equation you provided "top hat" upon a DNS solution with n = n_DNS. Does the filter solution have the same number of nodes from the original DNS. No matter how big or small is delta; the filtered DNS has always the same number of nodes (n_DNS). ? The bigger delta is the wider the stencil for the integral. right?

Thanks
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Old   May 19, 2016, 16:06
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Quote:
Originally Posted by juliom View Post
professor you have raised a very important fact



Finally; applying the equation you provided "top hat" upon a DNS solution with n = n_DNS. Does the filter solution have the same number of nodes from the original DNS. No matter how big or small is delta; the filtered DNS has always the same number of nodes (n_DNS). ? The bigger delta is the wider the stencil for the integral. right?

Thanks

yes, given n_DNS nodes on a domain of lenght L, you have h=L/(n_DNS-1) and a Nyquist frequency kc=pi/h. Now, on each of the DNS nodes you can compute the numerically filtered function having width delta= q*h (q>1). Formally, the spectrum extend up to kc but you will see the effect of the numerical transfer function producing the smoothing.
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