LES Subgrid Coupling

When the fluid simulation runs in LES mode, the unresolved scalar flux contributes a subgrid diffusivity that must be added to the molecular value:

\[D_{\text{total}} = D_0 + D_{\text{SGS}}, \qquad D_{\text{SGS}} = \frac{\nu_{\text{SGS}}}{Sc_t}.\]

The turbulent Schmidt number \(Sc_t\) is a configurable constant; the default value is \(Sc_t = 0.7\), the standard choice in atmospheric LES. The same SGS constant plays the role of the turbulent Prandtl number \(Pr_t\) when the transported scalar is temperature: the subgrid thermal diffusivity is \(\kappa_{\text{SGS}} = \nu_{\text{SGS}} / Pr_t\), with \(Pr_t = Sc_t\) entering the identical closure (see Thermal Extension (Boussinesq)).

Key idea: the scalar inherits its subgrid diffusivity from the fluid

The scalar does not run its own subgrid model. It takes the fluid’s Smagorinsky eddy viscosity \(\nu_{\text{SGS}}\) and converts it with a single constant, \(D_{\text{SGS}} = \nu_{\text{SGS}} / Sc_t\). One closure, one tunable constant: the unresolved scalar mixing is tied directly to the unresolved momentum mixing the fluid already computes.

The node-local scalar relaxation rate is then

\[\tau_g = \frac{D_{\text{total}}}{c_{s,\phi}^2} + \frac{1}{2}, \qquad \omega_\phi = \tau_g^{-1}.\]

The subgrid viscosity \(\nu_{\text{SGS}}\) is taken directly from the fluid Smagorinsky model (see Large-Eddy Simulation): the scalar reads the fluid’s stored node-local relaxation \(\omega_{\text{LES}}\) and reconstructs \(\nu_{\text{SGS}} = \nu(\omega_{\text{LES}}) - \nu_0\), so fluid and scalar see identical subgrid values without storing \(\nu_{\text{SGS}}\) as a separate macroscopic. The same node-local \(\tau_g\) enters the multiblock level transfer of \(q^{neq}_\alpha\) through the flux conversion \(J = q^{neq} (1 - \omega_\phi/2)\) described in Coupling with Other Modules, mirroring the fluid’s per-node viscosity treatment in the stress transfer.

In lattice units both \(\nu_{\text{SGS}}\) and \(D_{\text{SGS}}\) have the same dimension \([\Delta x^2 / \Delta t]\), so the closure \(D_{\text{SGS}} = \nu_{\text{SGS}} / Sc_t\) applies directly with no conversion factor between the fluid lattice and the scalar lattice.