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Can dynamical systems approach turbulence?

Authors :
Philip Holmes
Source :
Whither Turbulence? Turbulence at the Crossroads ISBN: 9783540525356
Publication Year :
2008
Publisher :
Springer Berlin Heidelberg, 2008.

Abstract

I review some ideas and methods from dynamical systems theory and discuss applications, actual and potential, to the study of fully developed turbulent flows in an open system: the wall region of a boundary layer. After a brief account of applications to a closed flow system, the approach I concentrate on attempts a marriage between statistical methods and deterministic dynamical systems, both orderly and chaotic. Specifically, coherent structures are identified with combinations of certain basis functions using the proper orthogonal decomposition. A relatively low dimensional ordinary differential equation describing the dynamical interactions of a set of these spatially organized structures is then derived by Galerkin projection of the Navier-Stokes equations. The resulting system is optimal in the sense that it retains the greatest turbulent kinetic energy, in a time averaged sense, among all projections of the same dimension. The model is analyzed using the methods of dynamical systems and symmetries are found to play a crucial role. In particular, structurally and asymptotically stable heteroclinic cycles emerge as a common feature in models of various dimensions and orbits attracted to these cycles lead to solutions exhibiting intermittent, violent “events,” which appear to reproduce key features of the bursting process. I speculate on the validity of this approach, the “understanding” of turbulent processes it offers and on how some of the gaps in the procedure might be bridged. I do not suggest that this is the only way in which dynamical systems methods can be used, but it is one which seems worth pursuing.

Details

ISBN :
978-3-540-52535-6
ISBNs :
9783540525356
Database :
OpenAIRE
Journal :
Whither Turbulence? Turbulence at the Crossroads ISBN: 9783540525356
Accession number :
edsair.doi...........20a5712faa920dbd1842fd508fe9bc0d