On the Connection Between the Viscous Camassa-Holm Equations (Navier-Stokes-alpha model) and Turbulence Theory

Abstract

In this talk we will show the global well-posedness of the three dimensional viscous Camassa-Holm equations, also known as the Navier-Stokes-alpha model. The dimension of their global attractor will be esitmated and shown to be  comparable with the number of degrees of freedom suggested by classical theory of turbulence.  We will present semi-rigorous argumentsshowing that up to a certain wave number, in the inertial range, the translational energy power specturm 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 Camassa-Holm equations (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.