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When starting from the Navier-Stokes equation the computation of observables in turbulent pipe flow is a numerically very demanding enterprise [1,2]. We therefore preferentially treat this problem in terms of high-dimensional dynamical systems [3]. In order to explore the role of the one-dimensional spatial extension of the pipe we
propose a Coupled Map Lattice(CML) model, which is discrete both in time and space. Its on-site time evolution is governed by a chaotic tent map, and, in addition, there is an asymmetric forward) coupling of neighboring sites. This simple model mimics the main features of turbulent pipe flow: turbulent puffs, slugs, fully developed turbulence and the transition between these states. Since the numerical simulation of the CML is very fast, we can fully explore the parameter dependence of the transition between these regimes. With that knowledge, we identify basic principles leading to turbulence. Our special interest lies in clarifying the relation of the very long life time of pipe flow to those in spatially extended dynamical systems (cf. [4]), and the characterization of the observed
spatio-temporal pattern in terms of Lyapunov vectors.
[1] R.R. Kerswell, Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18 (2005) R17. [2] B. Eckhardt, T.M. Schneider, B. Hof, and J. Westerweel, Turbulence Transition in Pipe Flow. Annual Review of Fluid Mechanics 39 (2007) 447. [3] J. Vollmer, T.M. Schneider, and B. Eckhardt, Basin boundary, edge of chaos and edge state in a two-dimensional model. New J. Phys. 11 (2009) 013040. [4] T. Tel and Y.-C. Lai, Chaotic transients in spatially extended systems, Phys. Rep. 460 (2008) 245-275. |
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