W. G. Pritchard Lab Seminar: 4-5 PM, 116 McAllister Building

	       **Tuesday February 12, 2002**
	
A New Approach to Modeling Turbulent Fluids: Lagrangian Averaging

Steve Shkoller

Department of Mathematics
University of California at Davis

Abstract:

Obtaining a closed system of evolution equations for the mean
velocity of a turbulent fluid has been one of the major open problems
in fluid dynamics.  Guessing the correct "turbulence closure," the
functional relationship between the mean velocity and the average of
the square of the fluctuations, has yielded numerous ad hoc models
which usually over dissipate the system in order to remove the small
(unresolvable) spatial scales, and as such destroy the basic
structure of turbulence - intermittency.  

We shall describe a new approach, which we call Lagrangian averaging,
that decomposes the Lagrangian coordinates (volume-preserving
diffeomorphisms), rather than the Eulerian coordinates (the spatial
velocity field), into mean and fluctuating parts.  In this new
framework, we are able to employ G.I. Taylor's 1938 Frozen Turbulence
Hypothesis as a turbulence closure, and derive a closed system of
Lagrangian averaged Navier-Stokes equations (LANS).   The equations
come in two flavors: isotropic and anisotropic. The former are
appropriate for isotropic turbulence in a periodic box, while the
latter are the ideal model of turbulence on bounded domains such as
channels and pipes. 

The isotropic equations have wonderful analytical properties,
including global well-posedness of classical solutions in
three-dimensions, and interesting geometric properties such as being
geodesics on the volume-preserving diffeomorphism group with an H^1
right-invariant metric.  The anisotropic equations are much more
delicate and require estimates involving elliptic operators which
degenerate on the boundary.  We shall review the theory, state the
main theorems on the subject, and present numerical results of
large-scale simulations of turbulent flow.