Dynamics, Geometry, and Topology of Fluid Turbulence

Abstract

Dramatic progress in understanding fluid turbulence, especially at moderate Reynolds numbers, has been made in the past decade usi! ng a deterministic framework based on the state space geometry of unstable solutions of the Navier-Stokes equation. Initial results obtained by restricting attention to minimal flow units capable of sustaining turbulence and imposing unphysical (e.g., spatially periodic) boundary conditions seemed to suggest that fluid turbulence is in many ways similar to low-dimensional chaos, with unstable periodic solutions forming the geometric skeleton for dynamics.

However, extending these results to larger flow domains with physical boundary conditions both proved very challenging and produced a number of surprises. In particular, our experimental and numerical studies have shown that unstable equilibria, quasiperiodic states, and heteroclinic connections can play an equally important role. We have also demonstrated that unstable solutions can be used for forecasting the evolution of experimental turbulent flows.