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Large eddy simulation (LES)

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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).
+
==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 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 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 [[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]] <math>G</math>:
 +
 
 +
:<math>
 +
\bar{u}_i(\vec{x}) = \int G(\vec{x}-\vec{\xi}) u(\vec{\xi})d\vec{\xi},</math>
 +
 
 +
resulting in
 +
 
 +
:<math>u_i = \bar{u}_i + u'_i,</math>
 +
 
 +
where <math>\bar{u}_i</math> is the resolvable scale part and <math>u'_i</math> 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]].  
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Typically, one would begin with the incompressible [[Navier-Stokes equations]] of motion, <br>
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The filtered equations are developed from the incompressible [[Navier-Stokes equations]] of motion:
 +
 
:<math>
:<math>
-
\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} + \frac{\partial}{\partial x_j}\left(\nu \frac{\partial u_i}{\partial x_j}\right).
</math>
</math>
-
and by the application of a filtering kernel, derive the equations of motion for the resolved field,
+
Substituting in the decomposition <math>u_i = \bar{u}_i + u'_i</math> and <math>p = \bar{p} + p'</math> and then filtering the resulting equation gives the equations of motion for the resolved field:
:<math>
:<math>
-
 
+
\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{\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}{\partial x_j}\left(\nu \frac{\partial \bar{u}_i}{\partial x_j}\right)
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+ \frac{\partial \tau_{ij}}{\partial x_j}
+
+ \frac {1}{\rho}\frac{\partial \tau_{ij}}{\partial x_j}.
</math>
</math>
-
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 <math> \frac{\partial \tau_{ij}}{\partial x_j} </math> arises from the non-linear advection terms, due to the fact that
+
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 <math> \frac{\partial \tau_{ij}}{\partial x_j} </math> arises from the non-linear advection terms, due to the fact that
 +
 
:<math>
:<math>
\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}
</math>
</math>
and hence  
and hence  
:<math>
:<math>
-
\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}
</math>
</math>
-
Subgrid-scale turbulence models usually employ the [[Boussinesq hypothesis]], and seek to calculate (the deviatoric part of) the SGS stress using: <br>
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Similar equations can be derived for the subgrid-scale field (i.e. the residual field).
 +
 
 +
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: <br>
:<math>
:<math>
-
\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\mu_t \bar S_{ij}  
</math>
</math>
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</math>
</math>
<br>
<br>
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and <math>  \mu _\tau </math> is the subgrid-scale turbulent viscosity.
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and <math>  \nu_t </math> is the subgrid-scale turbulent viscosity. Substituting into the filtered Navier-Stokes equations, we then have
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== Sub-grid Scale models ==
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:<math>
 +
\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),
 +
</math>
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*[[Smagorinsky's model]] (Smagorinsky, 1963)
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where we have used the incompressibility constraint to simplify the equation and the pressure is now modified to include the trace term <math>\tau _{kk} \delta _{ij}/3</math>.
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*[[Algebraic Dynamic model]] (Germano, et. al., 1991)
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*[[Localized Dynamic model]] (Kim & Menon, 1993)
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== Subgrid-scale models ==
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*[[WALE (Wall-Adapting Local Eddy-viscosity) model]] (Nicoud and Ducros, 1999)
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*[[Smagorinsky-Lilly model|Smagorinsky model]] (Smagorinsky, 1963)
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*[[Dynamic subgrid-scale model|Algebraic Dynamic model]] (Germano, et. al., 1991)
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*[[Dynamic global-coefficient subgrid-scale model|Dynamic Global-Coefficient model]] (You & Moin, 2007)
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*[[Kinetic energy subgrid-scale model|Localized Dynamic model]] (Kim & Menon, 1993)
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*[[Wall-adapting local eddy-viscosity (WALE) model|WALE (Wall-Adapting Local Eddy-viscosity) model]] (Nicoud and Ducros, 1999)
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*[[RNG-LES model]]
== References ==
== References ==
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*<b>J. Smagorinsky.</b> General circulation experiments with the primitive equations, i. the basic experiment. Monthly Weather Review, 91: 99-164, 1963.
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*{{reference-paper|author=Germano, M., Piomelli, U., Moin, P. and Cabot, W. H.|year=1991|title=A dynamic sub-grid scale eddy viscosity model|rest=Physics of Fluids, A(3): pp 1760-1765, 1991}}
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*<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.
+
*{{reference-paper|author=You, D. and Moin, P.|year=2007|title=A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries|rest=Physics of Fluids, 19(6): 065110, 2007}}
-
*<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.
+
*{{reference-paper|author=Kim, W and Menon, S.|year=1995|title=A new dynamic one-equation subgrid-scale model for large eddy simulation|rest=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.
+
*{{reference-paper|author=Nicoud, F. and Ducros, F.|year=1999|title=Subgrid-scale modelling based on the square of the velocity gradient tensor|rest=Flow, Turbulence and Combustion, 62: pp- 183-200, 1999}}
 +
*{{reference-paper|author=Smagorinsky, J|year=1963|title=General circulation experiments with the primitive equations, i. the basic experiment. Monthly Weather Review|rest=91: pp 99-164, 1963}}
 +
 
 +
[[Category:Turbulence models]]

Revision as of 10:06, 17 December 2008

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{\xi})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 p = \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.

Subgrid-scale models

References

  • Germano, M., Piomelli, U., Moin, P. and Cabot, W. H. (1991), "A dynamic sub-grid scale eddy viscosity model", Physics of Fluids, A(3): pp 1760-1765, 1991.
  • You, D. and Moin, P. (2007), "A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries", Physics of Fluids, 19(6): 065110, 2007.
  • Kim, W and Menon, S. (1995), "A new dynamic one-equation subgrid-scale model for large eddy simulation", In 33rd Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995.
  • Nicoud, F. and Ducros, F. (1999), "Subgrid-scale modelling based on the square of the velocity gradient tensor", Flow, Turbulence and Combustion, 62: pp- 183-200, 1999.
  • Smagorinsky, J (1963), "General circulation experiments with the primitive equations, i. the basic experiment. Monthly Weather Review", 91: pp 99-164, 1963.
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