Editing Large eddy simulation (section)

== Filter definition and properties ==
{{Main|Filter (large eddy simulation)}}
[[File:DNS Velocity Field.png|thumb|right|300px|A velocity field produced by a [[direct numerical simulation]] (DNS) of [[homogeneous isotropic turbulence|homogeneous decaying turbulence]]. The domain size is <math>L^3</math>.]]
[[File:DNS Filtered Velocity Field Small.png|thumb|right|300px|The same DNS velocity field filtered using a [[Filter (large eddy simulation)#Box filter|box filter]] and <math>\Delta=L/32</math>.]]
[[File:DNS Filtered Velocity Field Large.png|thumb|right|300px|The same DNS velocity field filtered using a [[Filter (large eddy simulation)#Box filter|box filter]] and <math>\Delta=L/16</math>.]]
An [[Filter (large eddy simulation)|LES filter]] can be applied to a spatial and temporal field <math>\phi(\boldsymbol{x},t)</math> and perform a spatial filtering operation, a temporal filtering operation, or both.  The filtered field, denoted with a bar, is defined as:<ref name="Pope_2000">{{cite book|title=Turbulent Flows|year=2000|publisher=Cambridge University Press |author=Pope, S. B.}}</ref><ref name="Sagaut_2006">{{cite book
|author=Sagaut, Pierre
|title=Large Eddy Simulation for Incompressible Flows
|publisher=Springer
|year=2006
|edition=Third
|isbn=978-3-540-26344-9 }}</ref>

:<math>
\overline{\phi(\boldsymbol{x},t)} = \displaystyle{
\int_{-\infty}^{\infty}} \int_{-\infty}^{\infty} \phi(\boldsymbol{r},\tau) G(\boldsymbol{x}-\boldsymbol{r},t - \tau) d\tau d \boldsymbol{r}
</math>

where <math>G</math> is the filter convolution kernel.  This can also be written as:

:<math>
\overline{\phi} = G \star \phi .
</math>

The filter kernel <math>G</math> has an associated cutoff length scale <math>\Delta</math> and cutoff time scale <math>\tau_{c}</math>.  Scales smaller than these are eliminated from <math>\overline{\phi}</math>.  Using the above filter definition, any field <math>\phi</math> may be split up into a filtered and sub-filtered (denoted with a prime) portion, as

:<math>
\phi = \bar{\phi} + \phi^{\prime} .
</math>

It is important to note that the [[Filter (large eddy simulation)|large eddy simulation filtering operation]] does not satisfy the properties of a [[Reynolds operator]].