# Large eddy simulation (LES)

## Introduction

Large eddy simulation (LES) is a popular technique for simulating turbulent flows. An implication of Kolmogorov's (1941) theory of self similarity is that the large eddies of the flow are dependant on the geometry while the smaller scales more universal. This feature allows one to explicitly solve for the large eddies in a calculation and implicitly account for the small eddies by using a subgrid-scale model (SGS model).

Mathematically, one may think of separating the velocity field into a resolved and sub-grid part. The resolved part of the field represent the "large" eddies, while the subgrid part of the velocity represent the "small scales" whose effect on the resolved field is included through the subgrid-scale model. Formally, one may think of filtering as the convolution of a function with a filtering kernel $G$:

$\bar{u}_i(\vec{x}) = \int G(\vec{x}-\vec{\xi}) u(\vec{x})d\vec{\xi},$

resulting in

$u_i = \bar{u}_i + u'_i,$

where $\bar{u}_i$ is the resolvable scale part and $u'_i$ is the subgrid-scale part. However, most practical (and commercial) implementations of LES use the grid itself as the filter (the box filter) and perform no explicit filtering. More information about the theory and application of filters is found in the LES filters article.

This page is mainly focused on LES of incompressible flows. For compressible flows, see Favre averaged Navier-Stokes equations.

The filtered equations are developed from the incompressible Navier-Stokes equations of motion:

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

Substituting in the decomposition $u_i = \bar{u}_i + u'_i$ and $u_i = \bar{p} + p'$ and then filtering the resulting equation gives the equations of motion for the resolved field:

$\frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \frac{\partial}{\partial x_j}\left(\nu \frac{\partial \bar{u}_i}{\partial x_j}\right) + \frac {1}{\rho}\frac{\partial \tau_{ij}}{\partial x_j}.$

We have assumed that the filtering operation and the differentiation operation commute, which is not generally the case. It is thought that the errors associated with this assumption are usually small, though filters that commute with differentiation have been developed ("ref?"). The extra term $\frac{\partial \tau_{ij}}{\partial x_j}$ arises from the non-linear advection terms, due to the fact that

$\overline{ u_j \frac{\partial u_i}{\partial x_j} } \ne \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j}$

and hence

$\tau_{ij} = \bar{u}_i \bar{u}_j - \overline{u_i u_j}$

Similar equations can be derived for the subgrid-scale field (i.e. the residual field).

Subgrid-scale turbulence models usually employ the Boussinesq hypothesis, and seek to calculate (the deviatoric part of) the SGS stress using:

$\tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\mu_t \bar S_{ij}$

where $\bar S_{ij}$ is the rate-of-strain tensor for the resolved scale defined by

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

and $\nu_t$ is the subgrid-scale turbulent viscosity. Substituting into the filtered Navier-Stokes equations, we then have

$\frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \frac{\partial}{\partial x_j}\left([\nu+\nu_t]\frac{\partial\bar{u}_i}{\partial x_j}\right),$

where we have used the incompressibility constraint to simplify the equation and the pressure is now modified to include the trace term $\tau _{kk} \delta _{ij}/3$.

## References

• J. Smagorinsky. General circulation experiments with the primitive equations, i. the basic experiment. Monthly Weather Review, 91: 99-164, 1963.
• M. Germano, U. Piomelli, P. Moin, and W. H. Cabot. A dynamic sub-grid scale eddy viscosity model. Physics of Fluids, A(3): 1760-1765, 1991.
• W. Kim and S. Menon. A new dynamic one-equation subgrid-scale model for large eddy simulation. In 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995.
• F. Nicoud and F. Ducros. Subgrid-scale modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62: 183-200, 1999.