# Contents

## 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:

• Large eddy simulation, Wikipedia

• Berselli, Iliescu, Layton: Mathematics of large eddy simulation of turbulent flows. (Springer 2006, ZMATH)

• Pierre Sagaut: Large eddy simulation for incompressible flows. (Springer 2006, ZMATH)

• Eric Garnier, Nikolaus Adams, Pierre Sagaut: Large eddy simulation for compressible flows (ZMATH)

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)

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)