On the connection between the viscous Camassa-Holm equations (Navier-Stokes-alpha model) and turbulence theory
E. S. Titi (UC Irvine)
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 arguments
showing 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.
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