# Dynamic subgrid-scale model

(Difference between revisions)
 Revision as of 17:09, 16 September 2005 (view source)Jasond (Talk | contribs)m (→Original model)← Older edit Revision as of 18:41, 14 April 2006 (view source)Jasond (Talk | contribs) m (→Introduction)Newer edit → Line 1: Line 1: == Introduction == == Introduction == - The failings of the [[Smagorinsky-Lilly model|Smagorinsky model]] has lead to the formulation of more complicated models that attempt to improve upon the performance of the Smagorinsky model.  Perhaps the best known of these newer models is dynamic subgrid-scale (DSGS) model of [[#References|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 + The limitations of the [[Smagorinsky-Lilly model|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 [[#References|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 $t$.  The 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. + 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 == == Original 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, the DSGS model 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.