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Kinetic energy subgrid-scale model

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The subgrid-scale kinetic energy is defined as

  k_{\rm sgs} = \frac{1}{2}\left(\overline{u_k^2} - \overline{u}_k^2 \right)

The subgrid-scale stress can then be written as
 \tau_{ij} - \frac{2}{3} k_{\rm sgs} \delta_{ij} =-2 C_k k_{\rm sgs}^{1/2} \Delta \overline{S}_{ij}
this gives us the transport equation for subgrid-scale kinetic energy
 \frac{\partial \overline k_{\rm sgs}}{\partial t} + \frac{\partial \overline u_{j} \overline k_{sgs}} {\partial x_{j}} = - \tau_{ij} \frac{\partial \overline u_{i}}{\partial x_{j}}     - C_{\varepsilon} \frac{k_{\rm sgs}^{3/2}}{\Delta} + \frac{\partial}{\partial x_{j}} \left( \frac{\mu_t}{\sigma_k} \frac{\partial k_{\rm sgs}}{\partial x_{j}}  \right)

The subgrid-scale eddy viscosity,  \mu_{t} , is computed using   k_{\rm sgs} as

 \mu_{t} = C_k k_{\rm sgs}^{1/2} \Delta

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