21st Century COE Lecture
Edriss S. Titi
| Date : |
March 25 (Thu), 2004, 15:00-16:30
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| Place : |
Room 115, RIMS
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| Lecturer: |
Edriss S. Titi
(Dept. of Math., Univ. of California, Irvine)
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| Title : |
The Navier-Stokes-alpha models and Turbulence Theory
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| Abstract: |
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
or the Lagrangian Averaged Navier--Stokes equations (LANS)).
The dimension of its 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
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. We observe that similar results hold also to other
turbulence models: the Leray-alpha and the Clark models.
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