Filter (large eddy simulation)

The low-pass filtering operation used in LES can be applied to a spatial and temporal field, for example ${\displaystyle \phi ({\boldsymbol {x}},t)}$. The LES filter operation may be spatial, temporal, or both. The filtered field, denoted with a bar, is defined as:[1][2]

${\displaystyle {\overline {\phi ({\boldsymbol {x}},t)}}=\displaystyle {\int _{-\infty }^{\infty }}\int _{-\infty }^{\infty }\phi ({\boldsymbol {r}},t^{\prime })G({\boldsymbol {x}}-{\boldsymbol {r}},t-t^{\prime })dt^{\prime }d{\boldsymbol {r}},}$

where ${\displaystyle G}$ is a convolution kernel unique to the filter type used. This can be written as a convolution operation:

${\displaystyle {\overline {\phi }}=G\star \phi .}$

The filter kernel ${\displaystyle G}$ uses cutoff length and time scales, denoted ${\displaystyle \Delta }$ and ${\displaystyle \tau _{c},}$ respectively. Scales smaller than these are eliminated from ${\displaystyle {\overline {\phi }}.}$ Using this definition, any field ${\displaystyle \phi }$ may be split up into a filtered and sub-filtered (denoted with a prime) portion, as

${\displaystyle \phi ={\bar {\phi }}+\phi ^{\prime }.}$

This can also be written as a convolution operation,

${\displaystyle \phi ^{\prime }=\left(1-G\right)\star \phi .}$

The filtering operation removes scales associated with high frequencies, and the operation can accordingly be interpreted in Fourier space. For a scalar field ${\displaystyle \phi ({\boldsymbol {x}},t),}$ the Fourier transform of ${\displaystyle \phi }$ is ${\displaystyle {\hat {\phi }}({\boldsymbol {k}},\omega ),}$ a function of ${\displaystyle {\boldsymbol {k}},}$ the spatial wave number, and ${\displaystyle \omega ,}$ the temporal frequency. ${\displaystyle {\hat {\phi }}}$ can be filtered by the corresponding Fourier transform of the filter kernel, denoted ${\displaystyle {\hat {G}}({\boldsymbol {k}},\omega ):}$

${\displaystyle {\overline {\hat {\phi }}}({\boldsymbol {k}},\omega )={\hat {\phi }}({\boldsymbol {k}},\omega ){\hat {G}}({\boldsymbol {k}},\omega )}$

or,

${\displaystyle {\overline {\hat {\phi }}}={\hat {G}}{\hat {\phi }}.}$

The filter width ${\displaystyle \Delta }$ has an associated cutoff wave number ${\displaystyle k_{c},}$ and the temporal filter width ${\displaystyle \tau _{c}}$ also has an associated cutoff frequency ${\displaystyle \omega _{c}.}$ The unfiltered portion of ${\displaystyle {\hat {\phi }}}$ is:

${\displaystyle {\hat {\phi ^{\prime }}}=(1-{\hat {G}}){\hat {\phi }}.}$

The spectral interpretation of the filtering operation is essential to the filtering operation in large eddy simulation, as the spectra of turbulent flows is central to LES subgrid-scale models, which reconstruct the effect of the sub-filter scales (the highest frequencies). One of the challenges in subgrid modeling is to effectively mimic the cascade of kinetic energy from low to high frequencies. This makes the spectral properties of the implemented LES filter very important to subgrid modeling efforts.

Homogeneous LES filters must satisfy the following set of properties when applied to the Navier-Stokes equations.[1]

1. Conservation of constants

The value of a filtered constant must be equal to the constant,

${\displaystyle {\overline {a}}=a,}$

which implies,

${\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }G({\boldsymbol {\xi }},t^{\prime })d^{3}{\boldsymbol {\xi }}dt^{\prime }=1.}$

2. Linearity

${\displaystyle {\overline {\phi +\psi }}={\overline {\phi }}+{\overline {\psi }}.}$

3. Commutation with derivatives

${\displaystyle {\overline {\frac {\partial \phi }{\partial s}}}={\frac {\partial {\overline {\phi }}}{\partial s}},\qquad s={\boldsymbol {x}},t.}$

If notation is introduced for operator commutation ${\displaystyle [f,g]}$ for two arbitrary operators ${\displaystyle f}$ and ${\displaystyle g}$, where

${\displaystyle [f,g]\phi =f\circ g(\phi )-g\circ f(\phi )=f(g(\phi ))-g(f(\phi )),}$

then this third property can be expressed as

${\displaystyle \left[G\star ,{\frac {\partial }{\partial s}}\right]=0.}$

Filters satisfying these properties are generally not Reynolds operators, meaning, first:

${\displaystyle {\begin{array}{rcl}{\overline {\overline {\phi }}}&\neq &{\overline {\phi }},\\G\star G\star \phi =G^{2}\star \phi &\neq &G\star \phi ,\end{array}}}$

and second,

${\displaystyle {\overline {\phi ^{\prime }}}=G\star (1-G)\star \phi \neq 0.}$

Box filter in physical and spectral space

The filter kernel in physical space is given by:

${\displaystyle G({\boldsymbol {x}}-{\boldsymbol {r}})={\begin{cases}{\frac {1}{\Delta }},&{\text{if}}\left|{\boldsymbol {x}}-{\boldsymbol {r}}\right|\leq {\frac {\Delta }{2}},\\0,&{\text{otherwise}}.\end{cases}}}$

The filter kernel in spectral space is given by:

${\displaystyle {\hat {G}}({\boldsymbol {k}})={\frac {\sin {({\frac {1}{2}}k\Delta )}}{{\frac {1}{2}}k\Delta }}.}$

Gaussian filter in physical and spectral space

The filter kernel in physical space is given by:

${\displaystyle G({\boldsymbol {x}}-{\boldsymbol {r}})=\left({\frac {6}{\pi \Delta ^{2}}}\right)^{\frac {1}{2}}\exp {\left(-{\frac {6({\boldsymbol {x-r}})^{2}}{\Delta ^{2}}}\right)}.}$

The filter kernel in spectral space is given by:

${\displaystyle {\hat {G}}({\boldsymbol {k}})=\exp {\left(-{\frac {{\boldsymbol {k}}^{2}\Delta ^{2}}{24}}\right)}.}$

Sharp spectral filter in physical and spectral space

The filter kernel in physical space is given by:

${\displaystyle G({\boldsymbol {x}}-{\boldsymbol {r}})={\frac {\sin {(\pi ({\boldsymbol {x-r}})/\Delta )}}{\pi ({\boldsymbol {x-r}})}}.}$

The filter kernel in spectral space is given by:

${\displaystyle {\hat {G}}({\boldsymbol {k}})=H\left(k_{c}-\left|k\right|\right),\qquad k_{c}={\frac {\pi }{\Delta }}.}$