Geometry of state space in plane Couette flow

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

A large conceptual gap separates the theory of low-dimensional chaotic dynamics from the infinite-dimensional nonlinear dynamics of turbulence. Recent 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. We propose to calculate a hierarchy of invariant solutions for a canonical wall-bounded shear flow and to use these solutions (1) to develop a quantitative description of the flow's turbulent dynamics, and (2) to predict, directly from the fundamental equations, physical quantities such as bulk flow rate and mean wall drag. We will use a combination of novel and proven numerical and analytical techniques, such as periodic orbit theory, group representation theory, nonlinear search methods and variational solvers, and computational fluid dynamics. The proposed research will be conducted with collaborators in Japan, Germany, the UK, and the US, and all results and numerical software will be disseminated through through our group's collaborative e-book www.ChaosBook.org and open-source CFD software and invariant solution database www.Channelflow.org. Turbulence is not only the great unsolved fundamental problem of classical physics, but a problem of great practical importance. Any progress in our fundamental understanding of turbulence impacts technology and engineering. Applications range from the suppression of plasma instabilities in magnetic confinement fusion reactors to the key challenge of reducing turbulent drag. Drag is responsible for a significant part of the fuel consumed in automotive, aviation and shipping industry as well as in transport of fluids. Even an incremental reduction of drag by improving flow control methodology would have a significant economic impact across a broad range of industries.