# Dynamic subgrid-scale model

## Introduction

The limitations of the Smagorinsky model have lead to the formulation of more general subgrid-scale models. Perhaps the best known of these newer models is the dynamic subgrid-scale (DSGS) model of Germano et al (1991). The DSGS model may be viewed as a modification of the Smagorinsky model, as the dynamic model allows the Smagorinsky constant $C_S$ to vary in space and time. $C_S$ is calculated locally in each timestep based upon two filterings of the flow variables, which we will denote by superscript $r$ and superscript $t$. These filters are called the grid filter and the test filter, respectively, and the test filter width is assumed to be larger the grid filter width.

## Original model

Filtering with the grid filter results in the normal LES equations, with $\tau_{ij}$ given by

$\tau_{ij}=(u_iu_j)^r-u_i^ru_j^r.$

Filtering again with the test filter results in a similar set of equations, but with a different subgrid-scale stress term, given by

$T_{ij}= (u_iu_j)^{rt}-u_i^{rt}u_j^{rt},$

where the superscript $rt$ indicates grid filtering followed by test filtering. The two subgrid-scale stress terms are related by the Germano identity:

$\mathcal{L}_{ij}=T_{ij}-\tau_{ij}^t,$

where

$\mathcal{L}_{ij}=(u_i^ru_j^r)^t-u_i^{rt}u_j^{rt}$

is the resolved turbulent stress. The Germano identity is used to calculate dynamic local values for $C_S$ by applying the Smagorinsky model to both $T_{ij}$ and $\tau_{ij}$. The anisotropic part of $\mathcal{L}_{ij}$ is the represented as

$\mathcal{L}_{ij}-\delta_{ij}\mathcal{L}_{kk}/3 = -2C_S M_{ij},$

where

$M_{ij}=(\Delta^t)^2|S^{rt}|S^{rt}_{ij} - (\Delta^r)^2 \left(|S^{r}|S^{r}_{ij}\right)^t.$

$C_S$ may now be computed as

$C_S^2=-\frac{1}{2}\frac{\mathcal{L}_{kl}S^r_{kl}}{M_{mn}S^r_{mn}}.$

In practice, DSM requires stabilization. Often, this has been done by averaging $C_S$ in a homogeneous direction. In cases where this is not possible, local averaging has been used in place of an average in a homogenous direction.

## Alternate solution (Lilly)

Lilly (1991) proposed a least squares procedure that is generally preferred to the original calculation of $C_S$:

$C_S^2=-\frac{1}{2}\frac{\mathcal{L}_{kl}M_{kl}}{M_{mn}M_{mn}}.$

Stabilization must also be employed here as well.

## References

• Germano, M., Piomelli, U., Moin, P. and Cabot, W. H. (1991), "A Dynamic Subgrid-Scale Eddy Viscosity Model", Physics of Fluids A, Vol. 3, No. 7, pp. 1760-1765.
• Lilly, D. K. (1991), "A Proposed Modification of the Germano Subgrid-Scale Closure Method", Physics of Fluids A, Vol. 4, No. 3, pp. 633-635.