# Large eddy simulation (LES)

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Discoganya (Talk | contribs) (Added some introduction and showed the derivation of the main equations) |
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\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 \nabla^2 \bar{u_i} | \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 \nabla^2 \bar{u_i} | ||

+ \frac{\partial \tau_{ij}}{\partial x_j} | + \frac{\partial \tau_{ij}}{\partial x_j} | ||

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<br> | <br> | ||

and <math> \mu _\tau </math> is the subgrid-scale turbulent viscosity. | and <math> \mu _\tau </math> is the subgrid-scale turbulent viscosity. | ||

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+ | == Sub-grid Scale models == | ||

+ | |||

+ | *[[Smagorinsky's model]] (Smagorinsky, 1963) | ||

+ | *[[Algebraic Dynamic model]] (Germano, et. al., 1991) | ||

+ | *[[Localized Dynamic model]] (Kim & Menon, 1993) | ||

+ | *[[WALE (Wall-Adapting Local Eddy-viscosity) model]] (Nicoud and Ducros, 1999) | ||

+ | |||

+ | == References == | ||

+ | |||

+ | *<b>J. Smagorinsky.</b> General circulation experiments with the primitive equations, i. the basic experiment. Monthly Weather Review, 91: 99-164, 1963. | ||

+ | *<b>M. Germano, U. Piomelli, P. Moin, and W. H. Cabot.</b> A dynamic sub-grid scale eddy viscosity model. Physics of Fluids, A(3): 1760-1765, 1991. | ||

+ | *<b>W. Kim and S. Menon.</b> A new dynamic one-equation subgrid-scale model for large eddy simulation. In 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995. | ||

+ | *<b>F. Nicoud and F. Ducros.</b> Subgrid-scale modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62: 183-200, 1999. |

## Revision as of 17:49, 18 September 2005

Large eddy simulation (LES) is a popular technique for simulating turbulent flows. A common deduction 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 sub-grid 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 sub-grid part of the velocity represent the "small scales" whose effect on the resolved field is included through the sub-grid scale model. Formally, one may think of filtering as the convolution of a function with a filtering kernel. However, most practical (and commercial) implimentations of LES, use the grid itself as the filter, and perform no explicit filtering. More information about the theory and application of filters is found here.

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

Typically, one would begin with the incompressible Navier-Stokes equations of motion,

and by the application of a filtering kernel, derive the equations of motion for the resolved field,

Velocities and pressures with an overbar denote the resolved field after the application of the filtering operation. Similar equations can be derived for the sub-grid scale field (i.e. the residual field). An extra term arises from the non-linear advection terms, due to the fact that

and hence

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

where is the rate-of-strain tensor for the resolved scale defined by

and is the subgrid-scale turbulent viscosity.

## Sub-grid Scale models

- Smagorinsky's model (Smagorinsky, 1963)
- Algebraic Dynamic model (Germano, et. al., 1991)
- Localized Dynamic model (Kim & Menon, 1993)
- WALE (Wall-Adapting Local Eddy-viscosity) model (Nicoud and Ducros, 1999)

## 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.