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.