Dynamic Subgrid Scale Model

Recognizing CsCs variations in space and time, Germano et al. (1991) proposed the dynamic model to compute the value of Cs rather than specifying it explicitly.

It is implemented by utilizing two filters: a cutoff filter Δ and a test (coarse) cutoff filter Δ .

The subgrid stress tensor τ'ij with the cutoff filter ( Δ ) is: τ'ij=ρ˜uiujρ˜ui˜uj=2ρ(CSΔ)2|˜S|˜Sij . Where |˜S|=2˜Sij˜Sij is the strain rate magnitude.

Figure 1 shows a resolved turbulence region utilizing Large Eddy Simulation (LES) and a modeled region assuming the subgrid tensor τ'ij .


Figure 1. Energy Spectrum for LES Using the Cutoff Filter Width ( Δ )
The test subgrid stress tensor Tij with the coarse filter ( Δ ) can be written as (1)
Tij=ρ˜˜uiujρ˜˜ui˜˜uj=2ρ(CSΔ)2|˜˜S|˜˜Sij
where
  • |˜˜S|=2˜˜Sij˜˜Sij is the coarse filtered strain rate magnitude.
  • ^˜Sij=12(^˜uixj+^˜ujxi) is the filtered strain rate tensor, using the coarse cutoff filter.
Figure 2 shows a resolved LES region and a corresponding subgrid modelled region (T) when the coarse filter is employed.


Figure 2. Energy Spectrum for LES Using the Test Cutoff Filter Width ( Δ )
Because of the coarse filtering, the test (coarse) subgrid stress tensor Tij should be a summation of the coarse filtered subgrid stress tensor ˜τ'ij and the Leonard stress tensor Lij . (2)
Lij=Tij˜τ'ij= ρ˜˜uiujρ˜˜ui˜˜ujρ˜˜uiuj+ρ˜˜ui˜uj=ρ˜˜ui˜ujρ˜˜ui˜˜uj
where
  • ˜τ'ij is the subgrid tensor for the cutoff filter (or grid filtered), then test filtered.
(3)
˜τ'ij=ρ˜˜uiujρ˜˜ui˜uj=2ρ(CSΔ)2˜|˜S|˜Sij
  • Lij is the Leonard subgrid stress tensor, representing the contribution to the subgrid stresses by turbulence length scales smaller than the test filter but larger than the cutoff filter.
The Leonard subgrid stress tensor can be arranged as (4)
Lij=2ρ(CSˆΔ)2|˜˜S|˜˜Sij+2ρ(CSΔ)2˜|˜S|˜Sij=2ρCS2Δ2(˜|˜S|˜Sijα2|˜˜S|˜˜Sij)

where α=ˆΔ/Δ .

The Leonard subgrid stress tensor can be rewritten as (5)
Lij=2ρCS2Δ2(˜|˜S|˜Sijα2|˜˜S|˜˜Sij)=CS2Mij

where Mij=2ρΔ2(˜|˜S|˜Sijα2|˜˜S|˜˜Sij) .

Since the above equation is overdetermined a minimum least square error method is used to determine the coefficient Cs . (6)
Cs2=LijMijMijMij
In order to avoid numerical instabilities associated with the above equation, as the numerator could become negative, averaging of the error in the minimization is employed. (7)
Cs2=LijMijMijMij