# Large eddy simulation (LES)

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 Revision as of 19:36, 8 May 2006 (view source)Jasond (Talk | contribs)m← Older edit Revision as of 20:22, 8 May 2006 (view source)Jasond (Talk | contribs) (→Introduction)Newer edit → Line 2: Line 2: 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 models|subgrid-scale model]] (SGS model). 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 models|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 [[LES filters|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 [[LES filters|here]]. + 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 [[LES filters|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 [[LES filters#Box filter|box filter]]) and perform no explicit filtering. More information about the theory and application of filters is found in the [[LES filters|LES filters article]]. This page is mainly focused on LES of incompressible flows. For compressible flows, see [[Favre averaged Navier-Stokes equations]]. 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,
+ Typically, one would begin with the incompressible [[Navier-Stokes equations]] of motion, + :$:[itex] \frac{\partial{u_i}}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \nabla^2 u_i \frac{\partial{u_i}}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i} + \nu \nabla^2 u_i Line 14: Line 24: :[itex] :[itex] - + \frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \nabla^2 \bar{u}_i - \frac{\partial{\bar{u_i}}}{\partial t} + \bar{u_j} \frac{\partial \bar{u_i}}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \nabla^2 \bar{u_i} + + \frac{\partial \tau_{ij}}{\partial x_j} + \frac{\partial \tau_{ij}}{\partial x_j}$ [/itex] Line 22: Line 31: :$:[itex] \overline{ u_j \frac{\partial u_i}{\partial x_j} } \ne \overline{ u_j \frac{\partial u_i}{\partial x_j} } \ne - \bar{u_j} \frac{\partial \bar{u_i}}{\partial x_j} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j}$ [/itex] and hence and hence :$:[itex] - \tau_{ij} = \bar{u_i} \bar{u_j} - \overline{u_i u_j} + \tau_{ij} = \bar{u}_i \bar{u}_j - \overline{u_i u_j}$ [/itex] Subgrid-scale turbulence models usually employ the [[Boussinesq eddy viscosity assumption|Boussinesq hypothesis]], and seek to calculate (the deviatoric part of) the SGS stress using:
Subgrid-scale turbulence models usually employ the [[Boussinesq eddy viscosity assumption|Boussinesq hypothesis]], and seek to calculate (the deviatoric part of) the SGS stress using:
:$:[itex] - \tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\mu _\tau \bar S_{ij} + \tau _{ij} - \frac{1}{3}\tau _{kk} \delta _{ij} = - 2\nu_t \bar S_{ij}$ [/itex] Line 43: Line 52: [/itex] [/itex]

- and $\mu _\tau$ is the subgrid-scale turbulent viscosity. + and $\nu_t$ is the subgrid-scale turbulent viscosity.  Substituting into the filtered Navier-Stokes equations, we then have + + :$+ \frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nabla\cdot\left([\nu+\nu_t]\nabla\bar{u}_i\right), +$ + + where we have used the incompressibility constraint to simplify the equation and the pressure is now modified to include the trace term $\tau _{kk} \delta _{ij}/3$. == Subgrid-scale models == == Subgrid-scale models ==

## 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.

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

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

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

$\frac{\partial{\bar{u}_i}}{\partial t} + \bar{u}_j \frac{\partial \bar{u}_i}{\partial x_j} = -\frac{1}{\rho} \frac{\partial \bar{p}}{\partial x_i} + \nu \nabla^2 \bar{u}_i + \frac{\partial \tau_{ij}}{\partial x_j}$

Velocities and pressures with an overbar denote the resolved field after the application of the filtering operation. Similar equations can be derived for the subgrid-scale field (i.e. the residual field). An 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

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

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

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

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

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

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.