# Kinetic energy subgrid-scale model

(Difference between revisions)
 Revision as of 23:27, 18 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 23:43, 18 September 2005 (view source)Zxaar (Talk | contribs) Newer edit → Line 1: Line 1: - $k_{\rm sgs} = \frac{1}{2}\left(\overline{u_k^2} - \overline{u}_k^2 \right)$ + 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
+ $\tau_{ij} - \frac{2}{3} k_{\rm sgs} \delta_{ij} =-2 C_k k_{\rm sgs}^{1/2} \Delta_f \overline{S}_{ij}$
+ this gives us the transport equation for subgrid-scale kinetic energy
+ $\frac{\partial \overline k_{\rm sgs}}{\partial t} + \frac{\partial \overline u_{j} \overline k_{sgs}} {\partial x_{j}} = - \tau_{ij} \frac{\partial \overline u_{i}}{\partial x_{j}} - C_{\varepsilon} \frac{k_{\rm sgs}^{3/2}}{\Delta_f} + \frac{\partial}{\partial x_{j}} \left( \frac{\mu_t}{\sigma_k} \frac{\partial k_{\rm sgs}}{\partial x_{j}} \right)$ + + + + The subgrid-scale eddy viscosity,$\mu_{t}$, is computed using $k_{\rm sgs}$ as + + + $\mu_{t} = C_k k_{\rm sgs}^{1/2} \Delta_f$ + + Where the filter-size computed from:
+ $\Delta_f = V^{1/3}$

## Revision as of 23:43, 18 September 2005

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

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

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

Where the filter-size computed from:
$\Delta_f = V^{1/3}$