21st Century COE Lecture

Edriss S. Titi

    Date : March 25 (Thu), 2004, 15:00-16:30
    Place : Room 115, RIMS
    Lecturer: Edriss S. Titi (Dept. of Math., Univ. of California, Irvine)
    Title : The Navier-Stokes-alpha models and Turbulence Theory
    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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Research Institute for Mathematical Sciences Department of Mathematics
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Last modified: March 12, 2004