filter

Homogeneous filters

Definition in physical space

The low-pass filtering operation used in LES can be applied to a spatial and temporal field, for example

ϕ

(

x

,

t

)

{\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:

ϕ

(

x

,

t

)

¯

=

∫

−

∞

∞

∫

−

∞

∞

ϕ

(

r

,

t

′

)

G

Definition in spectral space

The filtering operation removes scales associated with high frequencies, and the operation can accordingly be interpreted in Fourier space. For a scalar field

ϕ

(

x

,

t

)

,

{\displaystyle \phi ({\boldsymbol {x}},t),}

the Fourier transform of

ϕ

{\displaystyle \phi }

is

ϕ

^

(

k

,

ω

)

,

{\displaystyle {\hat {\phi }}({\boldsymbol {k}},\omega ),}

a function of

k

,

{\displaystyle {\boldsymbol {k}},}

the spatial wave number, and

ω

,

{\displaystyle \omega ,}

Homogeneous filter properties

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

1. Conservation of constants

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

a

¯

=

a

,

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

which implies,

∫

−

∞

∞

∫

−

∞

∞

G

(

ξ

,

t

′

)

d

3

ξ

d

t

′

=

1.

Inhomogeneous filters

Implementations of filtering operations for all but the simplest flows are inhomogeneous filter operations. This means that the flow either has non-periodic boundaries, causing problems with certain types of filters, or has a non-constant filter width

Δ

{\displaystyle \Delta }

, or both. This prevents the filter from commuting with derivatives, and the commutation operation leads to several additional error terms:

[

∂

∂

x

,

G

⋆

]

ϕ

=

∂

∂

x

(

G

⋆

ϕ

)

−

G

⋆

∂

ϕ

∂

x

=

∂

Classic large eddy simulation filters

There are three filters ordinarily used for spatial filtering in large eddy simulation. The definition of

G

(

x

,

t

)

{\displaystyle G({\boldsymbol {x}},t)}

and

G

^

(

k

,

ω

)

,

{\displaystyle {\hat {G}}({\boldsymbol {k}},\omega ),}

and a discussion of important properties, is given.

Box filter

The filter kernel in physical space is given by:

G

(

x

−

r

)

=

{

1

Δ

,

if

|

x

−

r

|

≤

Δ

2

,

0

,

otherwise

.

{\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:

G

^

(

k

)

Gaussian filter

The filter kernel in physical space is given by:

G

(

x

−

r

)

=

(

6

π

Δ

2

)

1

2

exp

⁡

(

−

6

(

x

−

r

)

2

Δ

2

)

.

{\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:

Sharp spectral filter

The filter kernel in physical space is given by:

G

(

x

−

r

)

=

sin

⁡

(

π

(

x

−

r

)

/

Δ

)

π

(

x

−

r

)

.

{\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:

G

^

(

k

)