Edriss S. TITI

The Navier-Stokes-alpha model and Turbulence Theory

(Abstract of a talk at the conference "PDE and applicatios")

In this talk we will show the global well-posedness of the three dimensional Navier-Stokes-alpha model (also known as a viscous Camassa-Holm equations). The dimension of its global attractor will be estimated and shown to be comparable with the number of degrees of freedom suggested by classical theory of turbulence. We will present semi-rigorous arguments showing that up to a certain wave number, in the inertial range, the translational energy power spectrum obeys the Kolmogorov power law for the energy decay of the three dimensional turbulent flow. However for the rest the inertial range the energy spectrum of this model obeys the Kraichnan power law for the energy decay of the two dimensional turbulent follows. This observation makes the Navier-Stokes-alpha model more computable than the Navier-Stokes equations. Furthermore, we will show that by using the Navier-Stokes-alpha model as a closure model 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.