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A Brief History of Simple Invariant Solutions in Turbulence

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Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics

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

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.

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Notes

  1. 1.

    Some remarks can be found in the 2011 Woods Hole Lecture Notes [22]. See also later work by Chantry et al. [23] on the validity of Waleffe’s flow setup as a model for the inner region of channel flow.

  2. 2.

    This presentation is, in fact, an anachronism. The original paper by Sánchez et al. [28] focuses on periodic solutions. The unified presentation, with the phase condition differentiating between periodic and equilibrium solutions, can be found in a recent review by Sánchez and Net [33].

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Acknowledgements

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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  1. University of Ontario Institute of Technology, Oshawa, Ontario, Canada

    Lennaert van Veen

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Correspondence to Lennaert van Veen .

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  1. School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel

    Alexander Gelfgat

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van Veen, L. (2019). A Brief History of Simple Invariant Solutions in Turbulence. In: Gelfgat, A. (eds) Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics. Computational Methods in Applied Sciences, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-319-91494-7_7

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