# Large Eddy Simulation

Large eddy simulation (LES) is a numerical technique used to solve the partial differential equations governing turbulent fluid flow. It was formulated in the late 1960s and became popular in later years. It was first used by Joseph Smagorinsky to simulate atmospheric air currents, and its primary use at that time was for meteorological calculations and predictions. During the 1980s and 1990s LES became widely used in the field of engineering.

LES requires less computational effort than direct numerical simulation (DNS) but more effort than those methods that solve the Reynolds-averaged Navier-Stokes equations (RANS). The computational demands also increase significantly in the vicinity of walls, and simulating such flows usually exceeds the limits of modern supercomputers today. For this reason, zonal approaches are often adopted, with RANS or other empirically-based models replacing LES in the wall region.

The main advantage of LES over computationally cheaper RANS approaches is the increased level of detail it can deliver. While RANS methods provide averaged results, LES is able to predict instantaneous flow characteristics and resolve turbulent flow structures. This is important, for example, in simulations involving chemical reactions, such as the combustion of fuel in an engine. While the averaged concentration of chemical species may be too low to trigger a reaction, instantaneously there can be localized areas of high concentration in which reactions will occur. LES also offers significantly more accurate results over RANS for flows involving flow separation or acoustic prediction.

## Details

The LES equations are derived from filtering the Navier-Stokes equations. The numerical solution of these equations is usually intractable in turbulent flow, due to the large range of scales of motion. To reduce this range, the equations are filtered. The solution to the equations is now a filtered velocity field, with the smaller scales of motion filtered out of the solution field. With the smaller scales eliminated from direct solution, a wider grid spacing may be used, thus lowering the computational costs.

The effect of the smaller scales on the large scales cannot be ignored. A deduction of Kolmogorov's theory of self similarity is that large eddies of the flow are dependent on the flow geometry, while smaller eddies are self similar and have a universal character. For this reason, it is reasonable to model the effect of the smaller and more universal eddies on the larger ones. Thus, in LES the large scale motions of the flow are calculated, while the effect of the smaller universal scales (the sub-grid scales) are modeled using a sub-grid scale (SGS) model. In practical implementations, one is required to solve the filtered Navier-Stokes equations with an additional sub-grid scale stress term.

In symbols, the LES equations for incompressible flow are

$\frac{\partial \bar{u_i}}{\partial t} + \frac{\partial \bar{u_i}\bar{u_j}}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \frac{\partial^2 \bar{u_i}}{\partial x_j \partial x_j} - \frac{\partial\tau_{ij}}{\partial x_j}$

where ui is the velocity field, p is the pressure, ρ is the density, and ν is the viscosity. The bar represents filtering. The τij term represents the SGS stress that must be modeled.

## Modeling sub-grid scale stress

Modeling the SGS stress term τij is one of the central problems in LES. The most commonly used SGS models are the Smagorinsky model and its variants. They compensate for the unresolved turbulent scales through the addition of an eddy viscosity into the governing equations. The basic formulation of the Smagorinsky model is

$\tau_{ij} = -2\nu_T\bar{S_{ij}}$

where

$\bar{S_{ij}} = \frac{1}{2} \left( \frac{\partial \bar{u_i}}{\partial x_j}+\frac{\partial \bar{u_j}}{\partial x_i} \right)$

is an entry of the strain rate tensor and the eddy viscosity νT is calculated as

$\nu_T = (C_s\Delta_g)^2\sqrt{2\bar{S_{ij}}\bar{S_{ij}}}$

where Δg is the grid size and Cs is a constant.

Many techniques have been developed to calculate Cs. Some models use a static value for Cs, often calculated from empirical experiments of similar flows to those being modeled. Other models dynamically calculate Cs as a function of space and time.

## Derivation

In Einstein notation, the Navier-Stokes equations for an incompressible fluid are

$\frac{\partial u_i}{\partial x_i} = 0$
$\frac{\partial u_i}{\partial t} + \frac{\partial u_iu_j}{\partial x_j} = - \frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \frac{\partial^2 u_i}{\partial x_j \partial x_j}.$

Filtering the momentum equation results in

$\overline{\frac{\partial u_i}{\partial t}} + \overline{\frac{\partial u_iu_j}{\partial x_j}} = - \overline{\frac{1}{\rho} \frac{\partial p}{\partial x_i}} + \overline{\nu \frac{\partial^2 u_i}{\partial x_j \partial x_j}}.$

If we assume that filtering and differentiation commute, then

$\frac{\partial \bar{u_i}}{\partial t} + \overline{\frac{\partial u_iu_j}{\partial x_j}} = - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \frac{\partial^2 \bar{u_i}}{\partial x_j \partial x_j}.$

This equation models the changes in time of the filtered variables $\bar{u_i}$. Since the unfiltered variables ui are not known, it is impossible to directly calculate $\overline{\frac{\partial u_iu_j}{\partial x_j}}$. However, the quantity $\frac{\partial \bar{u_i}\bar{u_j}}{\partial x_j}$ is known. A substitution is made:

$\frac{\partial \bar{u_i}}{\partial t} + \frac{\partial \bar{u_i}\bar{u_j}}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \frac{\partial^2 \bar{u_i}}{\partial x_j \partial x_j} - \left(\overline{ \frac{\partial u_iu_j}{\partial x_j}} - \frac{\partial \bar{u_i}\bar{u_j}}{\partial x_j}\right).$

Let $\tau_{ij} = \overline{u_i u_j} - \bar{u_i} \bar{u_j}$. The resulting set of equations are the LES equations:

$\frac{\partial \bar{u_i}}{\partial t} + \bar{u_j} \frac{\partial \bar{u_i}}{\partial x_j} = - \frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \frac{\partial^2 \bar{u_i}}{\partial x_j \partial x_j} - \frac{\partial\tau_{ij}}{\partial x_j}.$

A few recent applications of the large eddy simulation technique include some geophysical flows including breaking waves and tidal bores. (Lubin, 2010) Large eddy simulation has also been used to study turbulence in the planetary boundary layer.(Moeng, 1984), (Moeng, 2002)

## References

Lubin, P., Glockner, S., and Chanson, H. (2010). Numerical Simulation of a Weak Breaking Tidal Bore. Mechanics Research Communications, Vol. 37, No. 1, pp. 119-121 (DOI: 10.1016/j.mechrescom.2009.09.008).

Moeng, C.H. (1984). "A Large-Eddy Simulation Model for the Study of Planetary Boundary-Layer turbulence.". J. Atmos. Sci. 41: 2052-2062.

Moeng, C.H.; Sullivan, P.P. (2002). "Large Eddy Simulation". Encyclopedia of Atmospheric Sciences. pp. 1140-1150.

http://en.wikipedia.org/wiki/Large_eddy_simulation - wikipedia.com