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Edges in models of shear flow

  • Norman Lebovitz (a1) and Giulio Mariotti (a2)


A characteristic feature of the onset of turbulence in shear flows is the appearance of an ‘edge’, a codimension-one invariant manifold that separates ‘lower’ orbits, which decay directly to the laminar state, from ‘upper’ orbits, which decay more slowly and less directly. The object of this paper is to elucidate the structure of the edge that makes this behaviour possible. To this end we consider a succession of low-dimensional models. In doing this we isolate geometric features that are robust under increase of dimension and are therefore candidates for explaining analogous features in higher dimension. We find that the edge, which is the stable manifold of a ‘lower-branch’ state, winds endlessly around an ‘upper-branch’ state in such a way that upper orbits are able to circumnavigate the edge and return to the laminar state.


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Avila, D., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333, 192196.
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal disturbances in viscous flows. Phys. Fluids A 12, 16371650.
Clever, R. M. & Busse, F. H. 1992 Three-dimensional convection in a horizontal fluid layer subjected to constant shear. J. Fluid Mech. 234, 511527.
Clever, R. M. & Busse, F. H. 1997 Tertiary and quaternary solutions for plane couette flow. J. Fluid Mech. 344, 137153.
Cossu, C. 2005 An optimality condition on the minimum energy threshold in subcritical instabilities. C. R. Mecanique 333, 331336.
Dauchot, O. & Vioujard, N. 2000 Phase space analysis of a dynamical model for the subcritical transition to turbulence in plane couette flow. Eur. Phys. J. B 14, 377381.
Dhooge, A., Govaerts, W. & Kuznetsov, Yu. A. 2003 Matcont: a Matlab package for numerical bifurcation analysis of odes. ACM Trans. Math. Software 29, 141164.
Duguet, Y., Willis, A. P. & Kerswell, R. 2008 Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255274.
Eckhardt, B. 2008 Turbulence transition in pipe flow: some open questions. Nonlinearity 21, T1T11.
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near the onset of chaos in plane couette flow.
Lebovitz, N. 2009 Shear-flow transition: the basin boundary. Nonlinearity 22, 26452655.
Lebovitz, N. 2012 Boundary collapse in models of shear-flow transition. Commun. Nonlin. Sci. Numer. Sim. 17, 20952100.
Mariotti, G. 2011 A low dimensional model for shear turbulence in plane poiseuille flow: an example to understand the edge. In Proceedings of the Program in Geophysical Fluid Dynamics. Woods Hole Oceanographic Institution.
Mullin, T. 2011 Experimental studies of transition to turbulence in a pipe. Annu. Rev. Fluid. Mech. 43, 124.
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.
Vollmer, J., Schneider, T. M. & Eckhardt, B. 2009 Basin boundary, edge of chaos and edge state in a two-dimensional model. New J. Phys. 11, 013040.
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883900.
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Edges in models of shear flow

  • Norman Lebovitz (a1) and Giulio Mariotti (a2)


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