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I am trying to implement Large eddy simulation for solving air flow simulation with large Reynolds number using Smagorinsky sub-grid stress model. I read that the filter size should be calculated based on the grid size: $\Delta=(\Delta_x\Delta_y\Delta_z)^{1/3} $

Could you please let me know that shall I calculate this $\Delta$ for every cell of the mesh? Or is it ok to calculate only for the smallest cell and use delta as a constant?

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In general the filter size is characteristic of the cell, hence, in case of implicit filtering (as I suppose it is your case), you should compute it for each cell.

Another remark is also that it depends on the mesh that you have. In case of an homogeneous isotropic mesh for example, the cell size is the same for each cell, hence you can compute it for one cell and use it as a constant. For complex geometry case, the mesh are rarely homogeneous and isotropic, hence you have to compute it for all the cells.

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Absolutely you need to calculate it for every cell.

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