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Kinetic energy subgrid-scale model

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The subgrid-scale stress can then be written as <br>
The subgrid-scale stress can then be written as <br>
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<math> \tau_{ij} - \frac{2}{3} k_{\rm sgs} \delta_{ij} =-2 C_k k_{\rm sgs}^{1/2} \Delta_f \overline{S}_{ij} </math> <br>
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<math> \tau_{ij} - \frac{2}{3} k_{\rm sgs} \delta_{ij} =-2 C_k k_{\rm sgs}^{1/2} \Delta \overline{S}_{ij} </math> <br>
this gives us the transport equation for subgrid-scale kinetic energy <br>
this gives us the transport equation for subgrid-scale kinetic energy <br>
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<math> \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) </math>
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<math> \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) </math>
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<math> \mu_{t} = C_k k_{\rm sgs}^{1/2} \Delta_f </math>
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<math> \mu_{t} = C_k k_{\rm sgs}^{1/2} \Delta </math>
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Where the filter-size computed from: <br>
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<math> \Delta_f = V^{1/3} </math>
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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

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