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Subgrid variance

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An equation for the subgrid variance is
An equation for the subgrid variance is
 +
:<math>
 +
\frac{\partial \overline{\rho} \widetilde{Z_{sgs}''^2} }{\partial t} +
 +
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z_{sgs}''^2} }{\partial x_j}=
 +
\frac{\partial}{\partial x_j}\left(D \frac{\partial \widetilde{Z_{sgs}''^2} }{\partial x_j} \right)
 +
-2 D \widetilde{\frac{\partial Z}{\partial x_j}\frac{\partial Z }{\partial x_j}}
 +
+ 2 D \frac{\partial \tilde{Z}}{\partial x_j}\frac{\partial \tilde{Z}}{\partial x_j} -
 +
\frac{\partial J_j}{\partial x_j}  +
 +
2  \tilde{Z} \frac{\partial}{\partial x_j}
 +
\left(  \widetilde{\rho u_j Z}- \overline{\rho}\tilde{u}_j \tilde{Z} \right)
 +
</math>
 +
 +
where <math> J_j = \widetilde{\rho u_j {Z_{sgs}''^2}}- \overline{\rho} \tilde{u_j} \widetilde{Z_{sgs}''^2} </math> is a subgrid variance flux and is often modeled using a gradient approach with turbulent diffusivity.
Instead of solving the above equation, algebraic models are often used.
Instead of solving the above equation, algebraic models are often used.

Latest revision as of 11:19, 11 January 2006

The subgrid variance of a passive scalar is defined as


\widetilde{Z_{sgs}''^2} = \widetilde{Z^2}- \widetilde{Z}^2

The scalar subgrid variance is also known as the subgrid scalar energy in analogy to the kinetic subgrid energy. An equation for the subgrid variance is


\frac{\partial \overline{\rho} \widetilde{Z_{sgs}''^2} }{\partial t} +
\frac{\partial \overline{\rho} \widetilde{u}_j \widetilde{Z_{sgs}''^2} }{\partial x_j}=
\frac{\partial}{\partial x_j}\left(D \frac{\partial \widetilde{Z_{sgs}''^2} }{\partial x_j} \right)
-2 D \widetilde{\frac{\partial Z}{\partial x_j}\frac{\partial Z }{\partial x_j}}
+ 2 D \frac{\partial \tilde{Z}}{\partial x_j}\frac{\partial \tilde{Z}}{\partial x_j} -
\frac{\partial J_j}{\partial x_j}   +
 2  \tilde{Z} \frac{\partial}{\partial x_j}
 \left(  \widetilde{\rho u_j Z}- \overline{\rho}\tilde{u}_j \tilde{Z} \right)

where  J_j = \widetilde{\rho u_j {Z_{sgs}''^2}}- \overline{\rho} \tilde{u_j} \widetilde{Z_{sgs}''^2} is a subgrid variance flux and is often modeled using a gradient approach with turbulent diffusivity.

Instead of solving the above equation, algebraic models are often used. For dimensional analysis


\widetilde{Z_{sgs}''^2} =
C_Z \Delta^2 \frac{\partial \widetilde{Z} }{\partial x_i} \frac{\partial \widetilde{Z} }{\partial x_i}

where  C_Z can be obtained from the scalar spectra and its value is 0.1-0.2.

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