# Kinetic energy subgrid-scale model

(Difference between revisions)
 Revision as of 23:43, 18 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 12:21, 8 May 2006 (view source)Salva (Talk | contribs) mNewer edit → Line 4: Line 4: The subgrid-scale stress can then be written as
The subgrid-scale stress can then be written as
- $\tau_{ij} - \frac{2}{3} k_{\rm sgs} \delta_{ij} =-2 C_k k_{\rm sgs}^{1/2} \Delta_f \overline{S}_{ij}$
+ $\tau_{ij} - \frac{2}{3} k_{\rm sgs} \delta_{ij} =-2 C_k k_{\rm sgs}^{1/2} \Delta \overline{S}_{ij}$
this gives us the transport equation for subgrid-scale kinetic energy
this gives us the transport equation for subgrid-scale kinetic energy
- $\frac{\partial \overline k_{\rm sgs}}{\partial t} + \frac{\partial \overline u_{j} \overline k_{sgs}} {\partial x_{j}} = - \tau_{ij} \frac{\partial \overline u_{i}}{\partial x_{j}} - C_{\varepsilon} \frac{k_{\rm sgs}^{3/2}}{\Delta_f} + \frac{\partial}{\partial x_{j}} \left( \frac{\mu_t}{\sigma_k} \frac{\partial k_{\rm sgs}}{\partial x_{j}} \right)$ + $\frac{\partial \overline k_{\rm sgs}}{\partial t} + \frac{\partial \overline u_{j} \overline k_{sgs}} {\partial x_{j}} = - \tau_{ij} \frac{\partial \overline u_{i}}{\partial x_{j}} - C_{\varepsilon} \frac{k_{\rm sgs}^{3/2}}{\Delta} + \frac{\partial}{\partial x_{j}} \left( \frac{\mu_t}{\sigma_k} \frac{\partial k_{\rm sgs}}{\partial x_{j}} \right)$ Line 13: Line 13: - $\mu_{t} = C_k k_{\rm sgs}^{1/2} \Delta_f$ + $\mu_{t} = C_k k_{\rm sgs}^{1/2} \Delta$ - + - Where the filter-size computed from:
+ - $\Delta_f = V^{1/3}$ +

## Revision as of 12:21, 8 May 2006

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

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