Large eddy simulation in The Azimuth Project

[[!redirects large eddy simulation]]
# Contents
* Automatic table of contents
{: toc}

## Idea ##

**Large eddy simulations (LES)** are numerical approximations to the [[Navier-Stokes equations]] for flows that exhibit [[turbulence]], where only length scales above a certain threshold are resolved, in contrast to [[direct numerical simulations]]. LES can nevertheless be used to simulate turbulent flow structures and instantaneous flow characteristics that the [[Reynolds-averaged Navier–Stokes equations]] cannot resolve.

## Details ##

### Basic Idea ###
The basic idea of LES is the replacement of the flow vector field $u(t,x)$ or other fields that one is interested in, with a field that is spatially smeared. Smearing means the convolution with a test function $G(x, x')$ of localized support, like Gaussians or other cutoff fuctions. The smeared field is usually denoted with a bar:
$$
\bar u(x, t) := \int G(x-x', t-t') \; u(x', t') \; d t' \; d x'
$$
This convolution product is commonly denoted symbolically
$$
\bar = G \star u
$$
The Fourier transform turns the convolution product into a normal product:
$$
\bar \hat u(k, \omega) = \hat G(k, \omega) \; \hat u(k, \omega)
$$

### Basic Filter Properties ###
The "Filter" is a mechanism that filters resp. suppresses high frequency/small scale phenomena, for example by replacing the exact solution $u$ by a function $\bar u$ that is convoluted with an appropriate Filter function $G$, $\bar = G \star u$. We will list some properties that the filter operation should have. These properties will help in the formulation and manipulation of the filtered Navier-Stokes equations.

1. Conservation of constants: $\bar a = a$ for all constants $a$. If the filter operation is a convolution, the necessary and sufficient condition is
$$
\int G(x, t) \; d t \; d x = 1
$$

2. Linearity: $\overline{u + v} = \bar u + \bar v$ for all functions $u, v$. If the filter operation is a convolution, then this property is trivially satisfied.

3. Commutation with derivation.

### Differential Filters ###
If the filter function $G$ is the Green's function of a linear differential operator $F$, so that
$$
u = F(G \star u)
$$ 
then the filter is called a **differential filter**.
### Closure Problem ###
The **closure problem** of LES is the problem of how to model the processes at short length scales. LES depends on a good **sub grid scale (SGS)** model.

### Near Wall Model ###
The **near wall model (NWM)** is a special aspect of the closure problem: 

## References ##

* [Large eddy simulation](http://en.wikipedia.org/wiki/Large_eddy_simulation), Wikipedia

* Berselli, Iliescu, Layton: _Mathematics of large eddy simulation of turbulent flows._ (Springer 2006, [ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1089.76002&format=complete))

* Pierre Sagaut: _Large eddy simulation for incompressible flows._ (Springer 2006, [ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1091.76001&format=complete))

* Eric Garnier, Nikolaus Adams, Pierre Sagaut: _Large eddy simulation for compressible flows_ ([ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1179.76005&format=complete)) 

The following book is a guided tour to the specialized literature:

* Xi Jiang, Choi-Hong Lai: _Numerical techniques for direct and large eddy simulations._ (Chapman & Hall 2009, [ZMATH](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1185.76003&format=complete))

Recently there has been research about the use of [[wavelets]] as test- or smearing functions:

* Marta de la Llave Plata and Stewart Cant: _On the Application of Wavelets to LES Sub-grid Modelling_ (in Pierre Sagaut, Bernard J. Geurts, Johan Meyers (editors):  _Quality and Reliability of Large-Eddy Simulations_, Springer 2008)

[[!redirects LES]]
[[!redirects large eddy simulation]]
[[!redirects large eddy simulations]]

category:computational methods