# 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

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