Oberseminar Numerische Mathematik / Scientific Computing
Edriss S. Titi
Weizmann Institute of Science and
University of California - Irvine
Alpha Sub-grid Scale Models of Turbulence and Inviscid Regularization
Abstract:
In recent years many analytical sub-grid scale
models of turbulence were introduced based on the
Navier--Stokes-alpha model (also known as a viscous
Camassa--Holm equations or the Lagrangian Averaged
Navier--Stokes-alpha (LANS-alpha)). Some of these
are the Leray-alpha, the modified Leray-alpha, the
simplified Bardina-alpha and the Clark-alpha
models. In this talk we will show the global
well-posedness of these models and provide
estimates for the dimension of their global
attractors, and relate these estimates to the
relevant physical parameters. Furthermore, we will
show that up to certain wave number in the inertial
range the energy power spectra of these models
obey the Kolmogorov -5/3 power law, however, for
the rest the inertial range the energy spectra are
much steeper.
In addition, we will show that by using these alpha
models as closure models to the Reynolds averaged
equations of the Navier--Stokes one gets very good
agreement with empirical and numerical data of
turbulent flows for a wide range of huge Reynolds
numbers in infinite pipes and channels.
It will also be observed that, unlike the
three-dimensional Euler equations and other
inviscid alpha models, the inviscid simplified
Bardina model has global regular solutions for all
initial data. Inspired by this observation we will
introduce new inviscid regularizing schemes for the
three-dimensional Euler, Navier--Stokes and MHD
equations, which does not require, in the viscous
case, any additional boundary conditions. This same
kind of inviscid regularization is also used to
regularize the Surface Quasi-Geostrophic model.
Finally, and based on the alpha regularization we
will present, if time allows, some error estimates
for the rate of convergence of the alpha models to
the Navier-Stokes equations, and will also present
new approximation of vortex sheets dynamics.
| Datum: | 22.11.10 | Zeit: | 17:00 Uhr | Ort: | FU Berlin, Institut für Mathematik, Arnimallee 6, 14195 Berlin. | Raum: | 031 im Erdgeschoss |