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Center for Nonlinear Studies |
In recent years many analytical sub-grid scale models of turbulence were introduced based on the Navier--Stokes-alpha model (also known as a viscous Camassa--Holm equations or the Lagrangian Averaged Navier--Stokes-alpha (LANS-alpha)). Some of these are the Leray-alpha, the modified Leray-alpha, the simplified Bardina-alpha and the Clark-alpha models. In this talk we will show the global well-posedness of these models and provide estimates for the dimension of their global attractors, and relate these estimates to the relevant physical parameters. Furthermore, we will show that up to certain wave number in the inertial range the energy power spectra of these models obey the Kolmogorov -5/3 power law, however, for the rest the inertial range the energy spectra are much steeper. In addition, we will show that by using these alpha models as closure models to the Reynolds averaged equations of the Navier--Stokes one gets very good agreement with empirical and numerical data of turbulent flows in infinite pipes and channels. We also observe that, unlike the three-dimensional Euler equations and other inviscid alpha models, the inviscid simplified Bardina model has global regular solutions for all initial data. Inspired by this observation we introduce new inviscid regularizing schemes for the three-dimensional Euler and Navier--Stokes equations, which does not require, in the Navier--Stokes case, any additional boundary conditions. This same kind of inviscid regularization is also used to regularize the Surface Quasi-Geostrophic model. Finally, and based on the alpha regularization we will present new approximation of vortex sheets dynamics.