Pressure-driven fluid flow through a circular pipe has for a long time been the intriguing stability problem in fluid mechanics. Ever since the original experiments of Reynolds (1883), it has been known that the steady laminar flow observed at low flow rates can undergo transition to turbulence when disturbed sufficiently strongly at high enough flow rates. Despite numerous numerical and theoretical investigations, an understanding of the transition mechanism remains far from complete. Very recently there has been a substantial breakthrough in this problem.
Two groups (Faisst and Eckhardt 2003, Wedin and Kerswell 2004) have discovered travelling wave solutions in pipe flow. These solutions, which have now been observed in the laboratory (Hof et al. 2004), are important because they provide for the first time a solid theoretical foundation on which the transitional dynamics can be explored. This proposal aims at finding the relevance of these newly discovered solutions to the transitional process, by first studying numerically their stability.
This study requires the writing of a direct numerical simulation code specific to cylindrical pipe flow. This numerical code will then be used to find the possible connections between these finite-amplitude solutions from a dynamical systems point of view, with attention focused on the search for periodic orbits of the system in phase space. This proposal is extremely timely as there is a strong confluence of research activity. The results should lay down the theoretic al foundation for a significant and immediate advancement in the understanding of this intriguing problem, which has wide application in industry and Nature.
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