## Abstract

The main objective of this paper is to review and report on key mathematical issues related to the theory of Large Eddy Simulation of turbulent flows. We review several LES models for which we attempt to provide mathematical justifications. For instance, some filtering techniques and nonlinear viscosity models are found to be regularization techniques that transform the possibly ill-posed Navier-Stokes equation into a well-posed set of PDE's. Spectral eddy-viscosity methods are also considered. We show that these methods are not spectrally accurate, and, being quasi-linear, that they fail to be regularizations of the Navier-Stokes equations. We then propose a new spectral hyper-viscosity model that regularizes the Navier-Stokes equations while being spectrally accurate. We finally review scale-similarity models and two-scale subgrid viscosity models. A new energetically coherent scale-similarity model is proposed for which the filter does not require any commutation property nor solenoidality of the advection field. We also show that two-scale methods are mathematically justified in the sense that, when applied to linear non-coercive PDE's, they actually yield convergence in the graph norm.

Original language | English (US) |
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Pages (from-to) | 194-248 |

Number of pages | 55 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 6 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2004 |

Externally published | Yes |

### Bibliographical note

Funding Information:Acknowledgments. The first author gratefully acknowledges the support of Centre National de la Recherche Scientifique (CNRS) and that of the Texas Institute for Computational and Applied Mathematics under a TICAM Visiting Faculty Fellowship. The support of this work through the Office of Naval Research under grant N00014-95-0401 is gratefully acknowledged. The authors are also indebted to P. Azerad (Univ. Montpellier, France), R. Pasquetti (University of Nice, France) and J.-P. Magnaud (CEA, France) for stimulating discussions about turbulence and LES that certainly improved the presentation of the present work.

## Keywords

- Large eddy simulation
- Navier-Stokes equations
- Turbulence

## ASJC Scopus subject areas

- Mathematical Physics
- Condensed Matter Physics
- Computational Mathematics
- Applied Mathematics