Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows

The main objective of this paper is to review and report onkey mathematical issues related to thetheory of Large Eddy Simulation of turbulent flows.We review several LES models for which weattempt to provide mathematical justifications.For instance, some filtering techniques and nonlinear viscositymodels are found to be regularization techniques that transform the possiblyill-posed Navier-Stokes equation into a well-posed set of PDEs.Spectral eddy-viscosity methods are also considered.We show that these methods are not spectrally accurate, and,being quasi-linear, that they fail to beregularizations of the Navier-Stokes equations.We then propose a new spectral hyper-viscosity model thatregularizes the Navier-Stokes equations while being spectrally accurate.We finally review scale-similarity models and two-scalesubgrid viscosity models. A new energetically coherentscale-similarity model is proposedfor which the filter does not require any commutation propertynor solenoidality of the advection field.We also show that two-scale methods are mathematically justifiedin the sense that, when applied to linear non-coercive PDEs, theyactually yield convergence in the graph norm.