Dynamic Subgrid Scale Model

Recognizing $C_{s}$ variations in space and time, Germano et al. (1991) proposed the dynamic model to compute the value of $C_{s}$ rather than specifying it explicitly.

It is implemented by utilizing two filters: a cutoff filter $Δ$ and a test (coarse) cutoff filter $\tilde{Δ}$ .

The subgrid stress tensor $τ_{i j}^{'}$ with the cutoff filter ( $Δ$ ) is: $τ_{i j}^{'} = ρ \tilde{u_{i} u_{j}} - ρ \tilde{u_{i}} \tilde{u_{j}} = - 2 ρ {(C_{S} Δ)}^{2} | \tilde{S} | \tilde{S_{i j}}$ . Where $| \tilde{S} | = \sqrt{2 \tilde{S_{i j}} \tilde{S_{i j}}}$ 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_{i j}^{'}

Figure 1. Energy Spectrum for LES Using the Cutoff Filter Width ( $Δ$ )

The test subgrid stress tensor $T_{i j}$ with the coarse filter ( $\tilde{Δ}$ ) can be written as

T_{i j} = ρ \tilde{\tilde{u_{i} u_{j}}} - ρ \tilde{\tilde{u_{i}}} \tilde{\tilde{u_{j}}} = - 2 ρ {(C_{S} \tilde{Δ})}^{2} | \tilde{\tilde{S}} | \tilde{\tilde{S_{i j}}}

where

$| \tilde{\tilde{S}} | = \sqrt{2 \tilde{\tilde{S_{i j}}} \tilde{\tilde{S_{i j}}}}$ is the coarse filtered strain rate magnitude.
$\hat{\tilde{S_{i j}}} = \frac{1}{2} (\frac{\partial \hat{\tilde{u_{i}}}}{\partial x_{j}} + \frac{\partial \hat{\tilde{u_{j}}}}{\partial x_{i}})$ 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 ( $\tilde{Δ}$ )

Because of the coarse filtering, the test (coarse) subgrid stress tensor $T_{i j}$ should be a summation of the coarse filtered subgrid stress tensor $\tilde{τ_{i j}^{'}}$ and the Leonard stress tensor $L_{i j}$ .

L_{i j} = T_{i j} - \tilde{τ_{i j}^{'}} = ρ \tilde{\tilde{u_{i} u_{j}}} - ρ \tilde{\tilde{u_{i}}} \tilde{\tilde{u_{j}}} - ρ \tilde{\tilde{u_{i} u_{j}}} + ρ \tilde{\tilde{u_{i}} \tilde{u_{j}}} = ρ \tilde{\tilde{u_{i}} \tilde{u_{j}}} - ρ \tilde{\tilde{u_{i}}} \tilde{\tilde{u_{j}}}

where

$\tilde{τ_{i j}^{'}}$ is the subgrid tensor for the cutoff filter (or grid filtered), then test filtered.

\tilde{τ_{i j}^{'}} = ρ \tilde{\tilde{u_{i} u_{j}}} - ρ \tilde{\tilde{u_{i}} \tilde{u_{j}}} = - 2 ρ {(C_{S} Δ)}^{2} \tilde{| \tilde{S} | \tilde{S_{i j}}}

$L_{i j}$ 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

L_{i j} = - 2 ρ {(C_{S} \hat{Δ})}^{2} | \tilde{\tilde{S}} | \tilde{\tilde{S_{i j}}} + 2 ρ {(C_{S} Δ)}^{2} \tilde{| \tilde{S} | \tilde{S_{i j}}} = 2 ρ C_{S}^{2} Δ^{2} (\tilde{| \tilde{S} | \tilde{S_{i j}}} - α^{2} | \tilde{\tilde{S}} | \tilde{\tilde{S_{i j}}})

where $α = \hat{Δ} / Δ$ .

The Leonard subgrid stress tensor can be rewritten as

L_{i j} = 2 ρ C_{S}^{2} Δ^{2} (\tilde{| \tilde{S} | \tilde{S_{i j}}} - α^{2} | \tilde{\tilde{S}} | \tilde{\tilde{S_{i j}}}) = C_{S}^{2} M_{i j}

where $M_{i j} = 2 ρ Δ^{2} (\tilde{| \tilde{S} | \tilde{S_{i j}}} - α^{2} | \tilde{\tilde{S}} | \tilde{\tilde{S_{i j}}})$ .

Since the above equation is overdetermined a minimum least square error method is used to determine the coefficient $C_{s}$ .

C_{s}^{2} = \frac{L_{i j} M_{i j}}{M_{i j} M_{i j}}

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.

C_{s}^{2} = \frac{L_{i j} M_{i j}}{M_{i j} M_{i j}}