Revealing the state space of turbulent wall-bounded shear flows

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Cvitanovic DMS-1211827 A large conceptual gap separates the theory of low-dimensional chaotic dynamics from the infinite-dimensional nonlinear dynamics of turbulence. Advances in experimental imaging, computational methods, and dynamical systems theory suggest a way to bridge this gap and make a fundamental breakthrough in our understanding of turbulence. It has recently been discovered that recurrent coherent structures observed in wall-bounded shear flows (such as pipes and boundary layers) result from close passes to weakly unstable invariant solutions of the Navier-Stokes equations. These 3D, fully nonlinear solutions (equilibria, traveling waves, and periodic orbits) structure the state space of turbulent flows and provide a skeleton for analyzing their dynamics. The investigator calculates a hierarchy of invariant solutions for a canonical wall-bounded shear flow and uses these solutions to develop a geometrical and quantitative description of the flow's turbulent dynamics. He uses a combination of novel and proven numerical and analytical techniques, such as periodic orbit theory, group representation theory, nonlinear search methods, variational solvers, and computational fluid dynamics. The project is conducted with collaborators in Japan, Germany, the UK, and the US, and all results and numerical software are disseminated through the investigator's collaborative e-book www.ChaosBook.org. Turbulence is not only the great unsolved fundamental problem of classical physics, but a problem of great practical importance for technology and engineering, where turbulence is exhibited by cardiac systems, electromagnetic plasmas, oceans, and the atmosphere. A mathematical description of turbulent phenomena is of necessity high-dimensional, and accurate prediction requires solving millions of equations. Modern advances in computation and experimental data analysis have brought such computations within reach. Even more recent is the discovery of "recurrent coherent structures," whorls that one observes over and over in structures such as clouds. Such structures enable us to classify types of turbulent states in a mathematically precise manner and to predict their evolution without resorting to further large-scale computations. The project aims to deploy these "coherent structures" as a platform from which to chart the extremely high-dimensional world of all possible turbulent states, create an atlas over the important ones, and give a detailed predictive (as opposed to statistical) description of turbulent motions. Any progress in fundamental understanding of turbulence affects applications that range from the suppression of plasma instabilities in magnetic confinement fusion reactors, to weather prediction, to the key challenge of reducing turbulent drag. Even an incremental reduction of, for instance, turbulent drag by insights so achieved might have a significant economic impact, because drag is responsible for a significant part of the fuel consumed in transportation.