# Subgrid-Scale Advection

Large-eddy simulation (LES) simulates the dynamics of a spatially-filtered quantity $q = \widetilde{Q} = \int G(\bm{r}, \bm{x}) Q(\bm{x}−\bm{r}) \mathrm{d}\bm{r}$, where $G$ defines the filtering operation applied to the “real”, unfiltered quantity $Q$.

## Momentum Advection

If the filtering operation is defined as a projection (i.e. $\widetilde{\widetilde{U_i}} = \widetilde{U_i}$) that commutes with derivation in time and space, applying the filter to the momentum and continuity equations gives equations for the filtered velocity $\widetilde{U_i}$. Linear processes retain the same mathematical form, but the non-linear advection term produces additional contributions, i.e.

\[\begin{aligned} \frac{∂\widetilde{U_i}}{∂t} &= … − \frac{∂ \widetilde{U_i U_j}}{∂x_j} \\ &= … − \frac{∂ \widetilde{U_i} \widetilde{U_j}}{∂x_j} − \frac{∂ τ_{ij}^\mathrm{R}}{∂x_j} \quad &&\text{with} \quad τ_{ij}^\mathrm{R} = \widetilde{U_i U_j} − \widetilde{U_i} \widetilde{U_j} \\ &= … − \frac{∂ \widetilde{U_i} \widetilde{U_j}}{∂x_j} − \frac{∂ τ_{ij}^\mathrm{SGS}}{∂x_j} − \frac{∂}{∂x_i} \frac{τ_{jj}^\mathrm{R}}{3} \quad &&\text{with} \quad τ_{ij}^\mathrm{SGS} = τ_{ij}^\mathrm{R} − \frac{1}{3} τ_{ii}^\mathrm{R} δ_{ij} \,. \end{aligned}\]

If we set $u_i = \widetilde{U_i}$, the first contribution can be handled with the regular advection term and the last term can be included in a modified pressure term. The second term, however, needs to be modeled based on the resolved velocity field. This is the purpose of the subgrid-scale advection process, which adds a contribution

\[\frac{∂u_i}{∂t} = … − \frac{∂ τ_{ij}^\mathrm{SGS}}{∂x_j}\]

to the simulated momentum dynamics. Currently the `StaticSmagorinskyModel`

is the only implemented model for $τ_{ij}^\mathrm{SGS}$.

## Boundary Conditions

The subgrid-scale transport requires vertical boundary conditions for $τ_{13}^\mathrm{SGS}$ and $τ_{23}^\mathrm{SGS}$.

- For a
`FreeSlipBoundary`

, the subgrid-scale fluxes are assumed to vanish at the boundary.- For a
`RoughWall`

boundary, a local equilibrium layer is assumed in the near-wall region and the wall-stress is estimated with $τ_{i3}^\mathrm{w} = κ² \sqrt{u₁²+u₂²} \, u_i / \log²(x₃/z₀)$, evaluated at the first grid point at $ζ=½/N₃$. The roughness length $z₀$ and the magnitude of the von Kármán constant $κ$ are model parameters. See Algebraic Equilibrium Rough-Wall Model for more details.

- For a
- The
`CustomBoundary`

is currently not supported for subgrid-scale advection, as it is not clear how the subgrid-scale fluxes should be evaluated at such a boundary.

## Static Smagorinsky Model

The static Smagorinsky subgrid-scale model relies on the approximation

\[τ_{ij}^\mathrm{SGS} = − 2 \, l_\mathrm{S}^2 \, \mathcal{S} \, S_{ij} \quad \text{with} \quad l_\mathrm{S} = C_\mathrm{S} \, \left(Δx₁Δx₂Δx₃\right)^{1/3} \,,\]

where the Smagorinsky coefficient $C_\mathrm{S}$ is typically set to a value around $C_\mathrm{S} ≈ 0.1$. The resolved strain rate $S_{ij}$ and the total strain rate $\mathcal{S}$ are defined as

\[S_{ij} ≡ \frac{1}{2} \left( \frac{∂u_i}{∂x_j} + \frac{∂u_j}{∂x_i} \right) \quad \text{and} \quad \mathcal{S} ≡ \sqrt{2 S_{ij} S_{ij}} \,.\]

The computation of $S_{33}$ near the boundary relies on the homogeneous Dirichlet boundary conditions for $u₃$, which are assumed for all supported domain boundaries. Additionally, the computation of $\mathcal{S}$ on the first $ζ_C$-nodes relies on the near-wall behavior of $S_{13}$ and $S_{23}$, which is handled differently depending on the type of boundary:

- For free-slip boundaries, the strain rates are assumed to vanish at the boundary and can be interpolated from the values at the first $ζ_I$-nodes.
- For rough-wall boundaries, the values are derived from the wall model as $S_{i3} = u_i / (2 x₃ \log(x₃/z₀))$ (evaluated at the first $ζ_C$-node, analogous for upper boundary).

Since the model relies on non-linear expressions, those are computed in the physical domain.

`BoundaryLayerDynamics.Processes.StaticSmagorinskyModel`

— Type`StaticSmagorinskyModel(; kwargs...)`

Subgrid-scale advective transport of resolved momentum, approximated with the static Smagorinsky model.

**Keywords**

`Cs = 0.1`

: The Smagorinsky coefficient $C_\mathrm{S}$.`dealiasing`

: The level of dealiasing, determining the physical-domain resolution at nonlinear operations are performed. The default`nothing`

gives a physical-domain resolution that matches the frequency-domain resolution (rounded up to an even value). Alternatively,`dealiasing`

can be set to`:quadratic`

for a resolution based on the “3/2-rule” or to a tuple of two integers to set the resolution manually. Aliasing errors should be reduced for larger resolutions but they can be expected at any resolution.`wall_damping = true`

: Adjust the Smagorinsky lenghtscale towards the expected value of $κ x₃$ towards the wall.`wall_damping_exponent = 2`

: Exponent of the near-wall adjustment.

## Contributions to Budget Equations

Contribution to the instantaneous momentum equation:

\[\frac{\partial}{\partial t} u_i = … − \frac{\partial \tau_{ij}^\mathrm{SGS}}{\partial x_j}\]

Contribution to the mean momentum equation:

\[\frac{\partial}{\partial t} \overline{u_i} = … − \frac{\partial \overline{\tau_{ij}^\mathrm{SGS}}}{\partial x_j}\]

Contribution to turbulent momentum equation:

\[\frac{\partial}{\partial t} u_i^\prime = … − \frac{\partial {\tau_{ij}^\mathrm{SGS}}^\prime}{\partial x_j}\]

Contribution to the mean kinetic energy equation:

\[\frac{\partial}{\partial t} \frac{\overline{u_i}^2}{2} = … + \frac{\partial}{\partial x_j} \left( − \overline{u_i} \overline{\tau_{ij}^\mathrm{SGS}} \right) + \overline{\tau_{ij}^\mathrm{SGS}} \frac{\partial \overline{u_i}}{\partial x_j}\]

Contribution to the turbulent kinetic energy equation (could use $S_{ij}^\prime$ for the last term, since the SGS-tensor is symmetric):

\[\frac{\partial}{\partial t} \frac{\overline{u_i^\prime u_i^\prime}}{2} = … + \frac{\partial}{\partial x_j} \left( − \overline{u_i^\prime {\tau_{ij}^\mathrm{SGS}}^\prime} \right) + \overline{ {\tau_{ij}^\mathrm{SGS}}^\prime \frac{\partial u_i^\prime}{\partial x_j} }\]