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Numerical investigation of the finite-amplitude solutions in cylindrical pipe flow

Final Activity Report Summary - PIPETRANS (Numerical investigation of the finite-amplitude solutions in cylindrical pipe flow)

Understanding turbulence as a state of fluid flow is the outstanding question in continuum mechanics today. A natural and significant step towards this goal is appreciating how a flow undergoes the transition from a steady smooth laminar state to a turbulent flow. This project was aimed at elucidating this process within the context of pressure-driven flow through a long circular pipe, a classical problem dating back to at least Osborne Reynolds' famous experiments in 1883. The objective was to discover new solutions to the governing equations which influence whether a given disturbance in the flow will either lead to the flow relaminarising or becoming turbulent. To achieve this, the Fellow developed a sophisticated numerical program to find periodic orbits in pipe flow as originally proposed and with it achieved three notable results.

1) He found and numerically continued two quasi-periodic orbits which bifurcate off an important branch of travelling waves found recently (Pringle & Kerswell, Phys. Rev. Lett. 2007). These are the first quasi-periodic solutions found in the pipe flow problem. This work is published as Duguet, Pringle & Kerswell Physics Fluids 20, 114102, 2008.

2) By identifying flow trajectories which momentarily look periodic and taking these flows as initial guesses, he was able to identify new travelling wave exact solutions in pipe flow. His work strongly indicates that these new states are embedded in a separating hypersurface in phase space which divides initial conditions which relaminarise and those which lead to turbulence (the laminar-turbulent boundary). This work is now published as Duguet, Willis & Kerswell, Journal of Fluid Mechanics, 613, 255 2008.

3) The travelling waves discovered in 2) had a notably different structure to those already known so the Fellow explored whether further such waves existed. The picture which emerged is that these new waves (christened `Highly Symmetric' as they have more symmetries than those already known) come into existence at lower values of the flow rate and hence are more fundamental than the waves already know (which actually bifurcate off the highly symmetric ones). This work is published as Pringle, Duguet & Kerswell Phil. Trans. Roy. Soc. A 367, 457 2009.

The project therefore made important discoveries about new travelling waves and new quasi-periodic orbits in pipe flow. Since these were shown to sit in the laminar-turbulent boundary, they and their stable manifolds are important for determining whether the fluid will be laminar or turbulent when disturbed.