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Large eddy simulation

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)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:

References

  • 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)