# Large Eddy Simulation Review¶

## Introduction¶

In M-Star CFD, turbulence is modeled using large eddy simulation (LES) with a Smagorinsky sub-grid closure model. Within this approach, the velocity field is decomposed into (i) a filtered component and (ii) a sub-grid component. The filtered component is represented in the visual and numerical output. The sub-grid component is not shown explicitly, but the effects on the filtered flow field are captured by replacing the molecular viscosity with an effective viscosity within the momentum transport equations. In this sense, the effective viscosity of the fluid, $$\nu^*$$, varies as a function of space and time according to:

$\nu^* = \nu_0+\nu_t$

where $$\nu_0$$ is the molecular viscosity and $$\nu_t$$ is the sub-grid eddy viscosity calculated from:

$\nu_t=\left(C_s \Delta_x \right)^2 \overline{ {S}},$

where $$C_s$$ is an empirically-determined Smagorinsky coefficient, $$\Delta_x$$ is the local lattice spacing, and $$\overline{S}$$ is the norm of the filtered strain rate tensor defined by:

$\overline{\textrm{S}}_{ij}=\frac{1}{2}\left( \partial_j \overline{u}_i+ \partial_i \overline{u}_j \right),$

where $$\overline{u}$$ is the filtered velocity and $$\overline{\textrm{S}}_{ij}$$ is the characteristic filtered rate of strain. The value of $$C_s$$ is set by comparing LES simulation predictions to those obtained from DNS simulation. The default value used in M-Star CFD is 0.1, a selection based on results from the literature. [1]

## Energy Dissipation¶

The energy dissipation rate (EDR) is calculated from:

$\epsilon = 2 \nu \displaystyle \sum_{i,j} \overline{\textrm{S}}_{ij} \overline{\textrm{S}}_{ij},$

where $$\epsilon$$ is measured in W/kg and $$\nu$$ is the kinematic viscosity (momentum diffusivity) measured in m^2/s.

For systems with an LES filter, this expression becomes:

$\epsilon = 2 \left(\nu_0+\nu_t\right) \displaystyle \sum_{i,j} \overline{\textrm{S}}_{ij} \overline{\textrm{S}}_{ij},$

such that $$\epsilon$$ can be decomposed into resolved and unresolved components:

$\epsilon_{\textrm{res}} = 2 \nu_0 \displaystyle \sum_{i,j} \overline{\textrm{S}}_{ij} \overline{\textrm{S}}_{ij}$

and

$\epsilon_{\textrm{unres}} = 2 \nu_t \displaystyle \sum_{i,j} \overline{\textrm{S}}_{ij} \overline{\textrm{S}}_{ij}.$

Note that $$\epsilon_{\textrm{unres}}$$ is related to the third power of $$\overline{\textrm{S}}_{ij}$$ through the additional terms defining $$\nu_t$$

In most LES models, the resolved energy dissipation is order-of-magnitude smaller than the resolved energy dissipation. That is, a majority of the energy that is thermalized by the fluid exits through unresolved, sub-grid fluid eddies. Although the magnitude of this energy dissipation is known, the nature of the eddies within the sub-grid model is unspecified. [2]

Note that, for laminar and transitional simulations, where the LES filter is not necessary, the unresolved energy dissipation will be zero. For turbulent systems, if the LES filter is deactivated by manually setting $$C_s$$ to zero, the unresolved energy dissipation will be zero but the resolved energy will likely fall short of the reported energy input. Within this latter case, however, the reported energy dissipation will approach the input power as the simulation resolution approaches that required to perform direct numerical simulations.

Also note that, within a lattice-Boltzmann simulation, the local strain rate can be computed directly from the non-equilibrium components of the probability density function at each point. Within the context of an LES filter, this approach provides a more self-consistent description of strain than explicit finite differencing across the filtered velocity field.

The resolved and unresolved EDR computed from finite difference approaches can be extracted from the filtered velocity field and compared to those reported directly from the solver. Although the results are likely to be order-of-magnitude consistent, those reported directly from the solver are more accurate.

## References¶

[1] DNS and LES of decaying isotropic turbulence with and without frame rotation using lattice Boltzmann method.Y. Huidan, S.. Girimaji, L. Luo, Journal of Computational Physics 2005, 209:2, 599-616

[2] Turbulent Flows, S. Pope, Cambridge University Press 2000, Cambridge