A primary goal of the research that is supported with this award is to develop an asymptotic theory for `lower branch coherent states', a recently discovered class of 3D traveling wave solutions of the Navier-Stokes equations for shear flows (plane Couette, channel and pipe flows). Numerical calculations reveal that these solutions are unstable but with very low dimensional unstable manifolds (1 or 2 dimensional) and that they control transition to turbulence. Numerical calculations also reveal that the solutions have a striking asymptotic structure involving a critical layer. An asymptotic theory is needed to clarify, solidify and extend the numerical results to large Reynolds number. Such an asymptotic theory would have interesting mathematical features and implications about the high Reynolds number limit of solutions to the Navier-Stokes equations because of the nonlinear self-interaction of a singular critical layer structure. This research project will continue to develop such a theory. A second aspect of this research program is to continue the numerical study of the various coherent states and their connection with turbulent flows. Parts of this research involve US and international collaborations, in particular with P. Cvitanovic, J. Gibson, D. Viswanath and M. Graham (USA), R. Kerswell (UK), G. Kawahara, S. Toh, S. Kida and M. Nagata (Japan) and B. Eckhardt (Germany).
We have walked on the moon but we still do not understand flow of water down a pipe. For low speeds, the water flows in an orderly manner (`laminar flow'), but at higher speeds the flow becomes very disordered (`turbulent flow'). Turbulence is an ubiquitous fluid phenomenon that also occurs for flow of oil in pipelines, air around cars, airplanes and buildings, as well as in plasmas inside tokamaks, the sun and stars. The energetic and environmental effects of turbulence are major. Turbulent flows consume a lot more energy and mix things much better than laminar flows. Turbulence has been actively studied for over 120 years and is widely considered as the major unsolved problem of classical physics. A main emphasis of that research has been to developed semi-empirical models of turbulence for engineering calculations. These models are invariably based upon various presumptions about the nature of turbulence. This research program is one of an handful of pioneering programs around the world that have uncovered a series of previously unsuspected other possible states of fluid flow, intermediate between laminar and turbulent states. These new flow states consists of equilibria and periodic states that are unstable but appear to control onset of turbulence as well as fully developed turbulent flows. The discovery of these states forces a fundamental rethinking of the nature of turbulence. These discoveries were suggested and made possible by advances in experimental visualization methods as well as computers and computer calculation methods.
