# Structural modeling

1. Those that use the physical hypothesis of scale similarity (Bardina et al., 1980)

$\tau_{ij} = L_{ij} = \widetilde{\bar{u}_i} \widetilde{\bar{u}_j} - \widetilde{\bar{u}_i \bar{u}_j}$

2. Those derived by formal series expansions (Clark et al., 1979)

$\tau_{ij} = G_{ij} = \frac{\Delta^2}{12} \frac{\partial \bar{u}_i}{\partial x_{k}} \frac{\partial \bar{u}_j}{\partial x_{k}}$

3. Mixed models, which are based on linear combinations of the eddy-viscosity and structural types

$\tau_{ij} = G_{ij}-2\nu_{sgs} S_{ij}$

or

$\tau_{ij} = L_{ij}-2\nu_{sgs} S_{ij}$

4. Dynamic structure models (non-viscosity version)

$\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right)$

or

$\tau_{ij} = 2k_{sgs} \left(\frac{G_{ij}}{G_{kk}}\right)$

## References

• J. Bardina and J. H. Ferziger and W. C. Reynolds (1980), "Improved subgrid scale models for large eddy simulation", AIAA Paper No. 80-1357.
• R. A. Clark and J. H. Ferziger and W. C. Reynolds (1979), "Evaluation of subgrid-scale models using an accurately simulated turbulent flow", J. Fluid Mech..