LES Filter in Smagorinsky model on inhomogenous grids
I'm doing LES with a Dynamic Smagorinsky model. On the book from Pope I have read that the implicit filter implied by the model is very close to a Gaussian filter, which has a filter width delta that is usually taken as cell_colume^(1/3). The point is: if my grid is inhomogenous (the cell size changes across the domain), will the filter width implied by the model change too, that is to say the filter will be inhomogenous??? Or does the filter width stay constant and proportional to the cell size, so that just the constant of proportionality changes?
If the filter is inhomogenous, then I couldn't pass the filter inside the derivatives when I obtain the filtered equations so...how the Smagorinsky model would fulfill this incongruency??
All this is because I need to build a flamelet model which consists in having an additional equation for a "progress variable" with a source term which is closed by looking at values generated externally by laminar calculation. These values must be filtered first, so I do not know which filter size I should use!
In implicit filtering, the real shape of the filter is induced by the discretization of both domain and operators in the equations. Usually this leads to a smooth filter in the wavenumber space. The filter is therefore non homogeneous if the grid size changes...
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