[FMDenaro]

Originally Posted by vesp

Thanks for the paper, I will have to read it later. I understand that FD filters can be ill defined. What I wonder, though is: different reconstruction procedures under the integral will still give different coarse scale solution, and thus need different sgs models - is this correct?

The key is that:


1) the grid resolution introduces the Nyquist cut-off, no matter FD, FV, FEM or other methods.
2) The flux reconstruction has different resolution depending on the accuracy order but the reconstruction has to approximate only a specific filter shape, the top-hat (or volume average) filter. Just as an example, one can introduce a spectral polynomial for the flux reconstruction, therefore you have a spectral resolution. But, when inserted in to the integral formulation, the implicitly filtered variable is the top-hat one. Clearly, if your reconstrution is second order accurate your approximation of the expected theoretical top-hat filtered variable is different.

3) no need to chang the SGS model depending on the flux reconstruction. In the dynamic SGS model you have to take into accout of the real filter-to-grid width ratio.


If you read carefully the papers, you will find some sections where this topics are covered.

[vesp]

Agree with 1)
Question about 2) : if the filter is approximate, then the subfilter terms are also different. Maybe slightly, but not identical. Is this not a problem for the sgs model?

And a basic question: is your work only valid for schumanns approach? So must the filter be on local control volumes?


Another related question: the top hat filter has zeros, so some modes are set to zero. How can this filter be inverted then? My understanding of ADM is that this inversion of the top hat is tried. How can this ever work then?

Thank you for your input. You should write an LES book!

[FMDenaro]

Originally Posted by vesp

Agree with 1)
Question about 2) : if the filter is approximate, then the subfilter terms are also different. Maybe slightly, but not identical. Is this not a problem for the sgs model?

Another related question: the top hat filter has zeros, so some modes are set to zero. How can this filter be inverted then? My understanding of ADM is that this inversion of the top hat is tried. How can this ever work then?

Thank you for your input. You should write an LES book!

2) see Sec.3 in my JCP paper, your question is answered. Have also a look to Fig.1b.



The approach is based on the integral formulation of the equations, therefore could be used also in a FEM framework.



Your last question should be clarified with a more extended answer.

The exact (continuous) top-hat filter has a transfer function like sin x/x and cannot be inverted due to the infinite number of zeros. But we have to consider only the numerically resolved components of the filtered field, that is all the components contained beore the first zero. This way the transfer function has an approximate inverse function. This is the basis of the Approximate Deconvolution Method.
See these papers (and the cited references) for more details about this topic



https://www.researchgate.net/publica...-uniform_grids



https://www.researchgate.net/publica...-uniform_grids




PS: thanks for your suggestion but there are already published textbooks about LES, written by researchers much more renowned than me. I don't feel is really required that I write a textbook. You can find on my RG page some materials written for students, but I suggest always to use the published textbook and papers.