Filter (large eddy simulation)

Filtering in the context of large eddy simulation (LES) is a mathematical operation intended to remove a range of small scales from the solution to the Navier-Stokes equations. Because the principal difficulty in simulating turbulent flows comes from the wide range of length and time scales, this operation makes turbulent flow simulation cheaper by reducing the range of scales that must be resolved. The LES filter operation is low-pass, meaning it filters out the scales associated with high frequencies.

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:

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

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

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