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Dynamic subgrid-scale model

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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 model (DSM) of Germano et al (1991). DSM 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.
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