Editing Large eddy simulation (section)

== Numerical methods for LES ==

Large eddy simulation involves the solution to the discrete filtered governing equations using [[computational fluid dynamics]].  LES resolves scales from the domain size <math>L</math> down to the filter size <math>\Delta</math>, and as such a substantial portion of high wave number turbulent fluctuations must be resolved.  This requires either [[High-resolution scheme|high-order numerical schemes]], or fine grid resolution if low-order numerical schemes are used.  Chapter 13 of Pope<ref name="Pope_2000" /> addresses the question of how fine a grid resolution <math>\Delta x</math> is needed to resolve a filtered velocity field <math>\overline{u}(\boldsymbol{x})</math>.  Ghosal<ref name="Ghosal_1996">{{cite journal
|title=An analysis of numerical errors in large-eddy simulations of turbulence
|author=Ghosal, S.
|date=April 1996
|journal=[[Journal of Computational Physics]]
|volume=125
|issue=1
|doi=10.1006/jcph.1996.0088|bibcode = 1996JCoPh.125..187G
|pages=187–206 |doi-access=free
}}</ref> found that for low-order discretization schemes, such as those used in finite volume methods, the truncation error can be the same order as the subfilter scale contributions, unless the filter width <math>\Delta</math> is considerably larger than the grid spacing <math>\Delta x</math>.  While even-order schemes have truncation error, they are non-dissipative,<ref name="Leveque_1992">{{cite book
|title=Numerical Methods for Conservation Laws
|author=Randall J. Leveque
|year=1992
|publisher=Birkhäuser Basel
|edition=2nd
|isbn=978-3-7643-2723-1}}</ref> and because subfilter scale models are dissipative, even-order schemes will not affect the subfilter scale model contributions as strongly as dissipative schemes.

=== Filter implementation ===

The filtering operation in large eddy simulation can be implicit or explicit.  Implicit filtering recognizes that the subfilter scale model will dissipate in the same manner as many numerical schemes.  In this way, the grid, or the numerical discretization scheme, can be assumed to be the LES low-pass filter.  While this takes full advantage of the grid resolution, and eliminates the computational cost of calculating a subfilter scale model term, it is difficult to determine the shape of the LES filter that is associated with some numerical issues.  Additionally, truncation error can also become an issue.<ref name="Grinstein_2007">{{cite book
|title=Implicit large eddy simulation
|author1=Grinstein, Fernando
|author2=Margolin, Len
|author3=Rider, William
|year=2007
|publisher=Cambridge University Press
|isbn=978-0-521-86982-9}}</ref>

In explicit filtering, an [[Filter (large eddy simulation)|LES filter]] is applied to the discretized Navier–Stokes equations, providing a well-defined filter shape and reducing the truncation error.  However, explicit filtering requires a finer grid than implicit filtering, and the computational cost increases with <math>(\Delta x)^4</math>. Chapter 8 of Sagaut (2006) covers LES numerics in greater detail.<ref name="Sagaut_2006" />