*********
Filtering
*********

The filter operation is a concept to remove small-scale motion. 
In the figure below (:footcite:t:`rodi2013large`), the smooth curves comes from filtering the original one, with the filter width :math:`\Delta_{1}` or :math:`\Delta_{2}`.

.. figure:: /_static/img/theory/LES/les_filtering.svg
    :width: 90%
    :align: center

A local quantity can be split into a resolved :math:`\phi^{\sim}` and a fluctuation :math:`\phi^{'}`:

.. math::
    \phi = \phi^{\sim} + \phi^{'}
    :label: les_filter

In practice, the removal of the small-scale motion and hence the averaging is performed mostly by the numerical grid, as on a given grid only motions with scales larger than the mesh size can be resolved.
The others fall through the mesh.

.. include:: filtering.nse.rst.inc

.. footbibliography::