STEVE SHKOLLER
Department of Mathematics
University of California Davis
Davis, CA 95616-8633, USA
http://www.math.ucdavis.edu/~shkoller
shkoller@math.ucdavis.edu
Abstract:
Vortex methods provide convenient algorithms for the simulation of inviscid
or high Reynolds number flows. They consist in concentrating the vorticity
field on a discrete number of particles which evolve with the local velocity
of the flow computed in a self-consistent way. For inviscid flows, their
main characteristics is that they do not introduce numerical diffusion and are
quite robust in the sense that they do not suffer time step limitations usually
found in grid-based discretizations of advection problems.
These methods are thus a natural tool to investigate the main features of
turbulence, in particular the mechanisms through which small scales organize
themselves into large eddies.
In a surprising twist, we shall show that the PDEs of non-Newtonian fluids which model polymer flow, are in fact, exactly the vortex numerical method for integrating the Euler equations. This vortex method PDE is also the averaged Euler equations. The fact that a PDE governs the vortex methods is founded on rigorous mathematical well-posedness results for weak solutions to the equations of non-Newtonian fluids, and fit into a new framework for analyzing all incompressible hydrodynamical systems. We shall describe all of these new developments, and indicate the directions in which this program is moving.