State Space Properties of Transitional Pipe Flow

Transition to turbulence in pipe flow has puzzled scientists since the studies of Hagen, Poiseuille and, most prominently, Osborne Reynolds in the nineteenth century. Much of the difficulty in understanding the transition is connected with the linear stability of the laminar flow, which implies that a fully nonlinear analysis is required. In this work we apply methods from dynamical systems theory and nonlinear dynamics to explore the system's state space close to the transition. We analyze lifetime distributions of turbulent signals in domains of different lengths and study their variation with Reynolds number Re. Lifetimes are found to follow the exponential distribution typical of a chaotic saddle, with a characteristic time that increases rapidly with Re. The absence of a divergence in the lifetimes suggests that turbulence remains transient even at high flow velocities. The coherent states which appear transiently during the turbulent evolution are characterized. Correlation functions for their detection are introduced and their statistical properties extracted. They can be detected during more than 20% of the time. Finally,the stability border between laminar and turbulent dynamics is studied. Using a specially tailored tracking algorithm the dynamics between laminar and turbulent motion can be followed and an invariant dynamical object whose stable manifold separates the laminar from turbulent dynamics is identified. This object should provide useful for further studies on triggering turbulence or relaminarization.