[juliom]

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!!!