A Brief History of Simple Invariant Solutions in Turbulence

Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 50)


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.


Navier-Stokes Flow Time-periodic Solutions Generalized Minimal Residual (GMRES) Plane Couette Flow GMRES Iterations 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I would like to thank Sebastian Altmeyer, Andrew Hazel, Björn Hof, Genta Kawahara, Rich Kerswell, Masato Nagata and Fabian Waleffe for sharing their ideas and memories.


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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of Ontario Institute of TechnologyOntarioCanada

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