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Large eddy simulation (changes)

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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.


Basic Idea

The basic idea of LES is the replacement of the flow vector field u(t,x)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)G(x, x') of localized support, like Gaussians or other cutoff fuctions. The smeared field is usually denoted with a bar:

u¯(x,t):=G(xx,tt)u(x,t)dtdx \bar u(x, t) := \int G(x-x', t-t') \; u(x', t') \; d t' \; d x'

This convolution product is commonly denoted symbolically

=¯Gu \bar = G \star u

The Fourier transform turns the convolution product into a normal product:

u^¯(k,ω)=G^(k,ω)u^(k,ω) \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 uu by a function u¯\bar u that is convoluted with an appropriate Filter function GG, =¯Gu\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: a¯=a\bar a = a for all constants aa. If the filter operation is a convolution, the necessary and sufficient condition is

    G(x,t)dtdx=1 \int G(x, t) \; d t \; d x = 1
  2. Linearity: u+v¯=u¯+v¯\overline{u + v} = \bar u + \bar v for all functions u,vu, v. If the filter operation is a convolution, then this property is trivially satisfied.

  3. Commutation with derivation.

Differential Filters

If the filter function GG is the Green’s function of a linear differential operator FF, so that

u=F(Gu) 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)

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