# Structural modeling

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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, which divide the modeled SGS stress into a model for the SGS kinetic energy and a model for the structure of the SGS stress tensor (relative weights of each of the components) (Pomraning and Rutland, 2002; Lu et. al. 2007, 2008; Lu and Porte-Agel, 2010) $\tau_{ij} = 2k_{sgs} \left(\frac{L_{ij}}{L_{kk}}\right)$

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

The transport equation for the SGS kinetic energy can be used to close the model (Pomraning and Rutland, 2002; Lu et. al., 2008) $\frac{\partial k_{sgs}}{\partial t} + \frac{\partial \overline u_{j} k_{sgs}} {\partial x_{j}} = - \tau_{ij} \frac{\partial \overline u_{i}}{\partial x_{j}} - C_{\varepsilon} \frac{k_{sgs}^{3/2}}{\Delta} + \frac{\partial}{\partial x_{j}} \left( \nu_t \frac{\partial k_{sgs}}{\partial x_{j}} \right)$

or use a zero-equation procedure (Lu and Porte-Agel, 2010) to estimate the SGS kinetic energy.