Large eddy simulation

The idea behind large eddy simulation (LES) is to resolve the "large eddies" while modeling the effect of unresolved sub-grid scale motions. This is done usually be assuming eddy viscosity and eddy diffusivity models and providing an estimate for the eddy viscosity $\nu_e$ and diffusivity $\kappa_e$.

Much of the early work on LES was motivated by the study of atmospheric boundary layer turbulence, being developed by Smagorinsky (1963) and Lilly (1966), then first implemented by Deardorff (1970) and Deardorff (1974).

In the LES framework, the Navier-Stokes equations are averaged in the same way as Reynolds (1895) except that the mean field $\overline{\boldsymbol{v}}$ is obtained via convolution with a filter convolution kernel $G$

\[\overline{\boldsymbol{v}(\boldsymbol{x}, t)} = G \star \boldsymbol{v} = \int_{-\infty}^\infty \int_{-\infty}^\infty \boldsymbol{v}(\boldsymbol{x}^\prime, t) G(\boldsymbol{x} - \boldsymbol{x}^\prime, t - \tau) \, \mathrm{d}\boldsymbol{x}^\prime \, \mathrm{d} \tau \, ,\]

as described by Leonard (1975) who introduced the general filtering formalism.

The $\overline{v_i^\prime v_j^\prime}$ terms are now components of what is called the sub-grid scale (SGS) stress tensor $\tau^\text{SGS}_{ij}$, which looks the same as the Reynolds stress tensor so we will drop the SGS superscript.

It is probably important to note that the large eddy simulation filtering operation does not satisfy the properties of a Reynolds operator (§2.1) (Sagaut and Meneveau, 2006) and that in general, the filtered residual is not zero: $\overline{\boldsymbol{v}^\prime(\boldsymbol{x}, t)} \ne 0$.

§13.2 of Pope (2000) lists a number of popular choices for the filter function $G$. For practical reasons we simply employ the box kernel

\[ \begin{equation} \label{eq:box-kernel} G_\Delta = G(\boldsymbol{x}, t) = \frac{1}{\Delta} H \left( \frac{1}{2}\Delta - |\boldsymbol{x}| \right) \delta(t - t_n) \, , \end{equation}\]

where $H$ is the Heaviside function, $\Delta$ is the grid spacing, and $t_n$ is the current time step. With \eqref{eq:box-kernel} we get back the averaging operator originally used by Deardorff (1970)

\[\overline{\boldsymbol{v}(x, y, z, t)} = \frac{1}{\Delta x \Delta y \Delta z} \int_{x - \frac{1}{2}\Delta x}^{x + \frac{1}{2}\Delta x} \int_{y - \frac{1}{2}\Delta y}^{y + \frac{1}{2}\Delta y} \int_{z - \frac{1}{2}\Delta z}^{z + \frac{1}{2}\Delta z} \boldsymbol{v}(\xi, \eta, \zeta, t) \, \mathrm{d} \xi \, \mathrm{d} \eta \, \mathrm{d} \zeta \, ,\]

which if evaluated at the cell centers just returns the cell averages we already compute in the finite volume method.

Smagorinsky-Lilly model

Smagorinsky (1963) estimated the eddy viscosity $\nu_e$ via a characteristic length scale $\Delta$ times a velocity scale given by $\Delta |\overline{S}|$ where $|\overline{S}| = \sqrt{2\overline{S}_{ij}\overline{S}_{ij}}$. Thus the SGS stress tensor is given by

\[\tau_{ij} = -2 \nu_e \overline{S}_{ij} = -2 (C_s \Delta)^2 |\overline{S}| \overline{S}_{ij} \, ,\]

where $C_s$ is a dimensionless constant. The grid spacing is usually used for the characteristic length scale $\Delta$. The eddy diffusivities are calculated via $\kappa_e = \nu_e / \text{Pr}_t$ where the turbulent Prandtl number $\text{Pr}_t$ is usually chosen to be $\mathcal{O}(1)$ from experimental observations.

Assuming that the SGS energy cascade is equal to the overall dissipation rate $\varepsilon$ from the Kolmogorov (1941) theory, Lilly (1966) was able to derive a value of

\[C_s = \left( \frac{3}{2}C_K\pi^\frac{4}{3} \right)^{-\frac{3}{4}} \approx 0.16 \, ,\]

using an empirical value of $C_K \approx 1.6$ for the Kolmogorov constant. This seems reasonable for isotropic turbulence if the grid spacing $\Delta$ falls in the inertial range. In practice, $C_s$ is a tunable parameter.

Due to the presence of the constant $C_s$, the model is sometimes referred to as the constant Smagorinsky model in contrast to dynamic Smagorinsky models that dynamically compute $C_s$ to account for effects such as buoyant convection.

Anisotropic minimum dissipation models

Minimum-dissipation eddy-viscosity models are a class of LES closures that use the minimum eddy dissipation required to dissipate the energy of sub-grid scale motion. Rozema et al. (2015) proposed the first minimum-dissipation model appropriate for use on anisotropic grids, termed the anisotropic minimum dissipation (AMD) model.

It has a number of desirable properties over Smagorinsky-type closures: it is more cost-effective than dynamic Smagorinsky, it appropriately switches off in laminar and transitional flows, and it is consistent with the exact SGS stress tensor on both isotropic and anisotropic grids. Abkar et al. (2016) extended the AMD model to model SGS scalar fluxes for tracer transport. Abkar and Moin (2017) further extended the model to include a buoyancy term that accounts for the contribution of buoyant forces to the production and suppression of turbulence.

Vreugdenhil and Taylor (2018) derive a modified AMD model by following the requirement suggested by Verstappen (2018), which entail normalising the displacement, the velocity, and the velocity gradient by the filter width to ensure that the resulting eddy dissipation properly counteracts the spurious kinetic energy transferred by convective nonlinearity, to derive a modified AMD model.

The eddy viscosity and diffusivity are defined in terms of eddy viscosity and diffusivity predictors $\nu_e^\dagger$ and $\kappa_e^\dagger$, such that

\[\nu_e = \max \lbrace 0, \nu_e^\dagger \rbrace \quad \text{and} \quad \kappa_e = \max \lbrace 0, \kappa_e^\dagger \rbrace \, ,\]

to ensure that $\nu_e \ge 0$ and $\kappa_e \ge 0$. Leaving out the overlines and understanding that all variables represent the resolved/filtered variables, the eddy viscosity predictor is given by

\[ \begin{equation} \label{eq:nu-dagger} \nu_e^\dagger = -(C\Delta)^2 \frac {\left( \hat{\partial}_k \hat{v}_i \right) \left( \hat{\partial}_k \hat{v}_j \right) \hat{S}_{ij} + C_b\hat{\delta}_{i3} \alpha g \left( \hat{\partial}_k \hat{v_i} \right) \hat{\partial}_k \theta} {\left( \hat{\partial}_l \hat{v}_m \right) \left( \hat{\partial}_l \hat{v}_m \right)} \, , \end{equation}\]

and the eddy diffusivity predictor by

\[ \begin{equation} \kappa_e^\dagger = -(C\Delta)^2 \frac {\left( \hat{\partial}_k \hat{v}_i \right) \left( \hat{\partial}_k \hat{\theta} \right) \hat{\partial}_i \theta} {\left( \hat{\partial}_l \hat{\theta} \right) \left( \hat{\partial}_l \hat{\theta} \right)} \, , \end{equation}\]


\[ \begin{equation} \hat{x}_i = \frac{x_i}{\Delta_i}, \quad \hat{v}_i(\hat{x}, t) = \frac{v_i(x, t)}{\Delta_i}, \quad \hat{\partial}_i \hat{v}_j(\hat{x}, t) = \frac{\Delta_i}{\Delta_j} \partial_i v_j(x, t), \quad \hat{\delta}_{i3} = \frac{\delta_{i3}}{\Delta_3} \, , \end{equation}\]

so that the normalized rate of strain tensor is

\[ \begin{equation} \label{eq:S-hat} \hat{S}_{ij} = \frac{1}{2} \left[ \hat{\partial}_i \hat{v}_j(\hat{x}, t) + \hat{\partial}_j \hat{v}_i(\hat{x}, t) \right] \, . \end{equation}\]

In equations \eqref{eq:nu-dagger}–\eqref{eq:S-hat}, $C$ is a modified Poincaré "constant" that is independent from the filter width $\Delta$ but does depend on the accuracy of the discretization method used. Abkar et al. (2016) cite $C^2 = \frac{1}{12}$ for a spectral method and $C^2 = \frac{1}{3}$ for a second-order accurate scheme. $\Delta_i$ is the filter width in the $x_i$-direction, and $\Delta$ is given by the square root of the harmonic mean of the squares of the filter widths in each direction

\[ \frac{1}{\Delta^2} = \frac{1}{3} \left( \frac{1}{\Delta x^2} + \frac{1}{\Delta y^2} + \frac{1}{\Delta z^2} \right) \, .\]

The term multiplying $C_b$ is the buoyancy modification introduced by Abkar and Moin (2017) and is small for weakly stratified flows. We have introduced the $C_b$ constant so that the buoyancy modification term may be turned on and off.