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Subgrid variance

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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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