A brief history of simple invariant solutions in turbulence

TL;DR

This paper reviews the historical development of invariant solutions in turbulence, highlighting how computational dynamical systems techniques have advanced understanding of complex fluid flows from the 1980s to today.

Contribution

It provides a comprehensive historical overview of the development and application of invariant solutions in turbulence research over four decades.

Findings

Development of bifurcation detection techniques in fluid flows

Application of computational dynamical systems to Navier-Stokes equations

Progress from simple models to complex turbulent flow analysis

Abstract

When studying fluid mechanics in terms of instability, bifurcation and invariant solutions one quickly finds out how little can be done by pen and paper. For flows on sufficiently simple domains and under sufficiently simple boundary conditions, one may be able to predict the parameter values at which the base flow becomes unstable and the basic properties of the secondary flow. On more complicated domains and under more realistic boundary conditions, such questions can usually only be addressed by numerical means. Moreover, for a wide class of elementary parallel shear flows the base flow remains stable in the presence of sustained turbulent motion. In such flows, secondary solutions often appear with finite amplitude and completely unconnected to the base flow. Only using techniques from computational dynamical systems can such behaviour be explained. Many of these techniques, such as for the detection and classification of bifurcations and for the continuation in parameters of equilibria and time-periodic solutions, were developed in the late 1970s for dynamical systems with few degrees of freedom. The application to fluid dynamics or, to be more precise, to spatially discretized Navier-Stokes flow, is far from straightforward. In this historical review chapter, we follow the development of this field of research from the valiant naivety of the early 1980s to the open challenges of today.