Publication

State Space Properties of Transitional Pipe Flow

Tobias Schneider
2007
Non-EPFL thesis
Abstract

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.

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Related concepts (34)
Laminar flow
In fluid dynamics, laminar flow (ˈlæmənər) is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross-currents perpendicular to the direction of flow, nor eddies or swirls of fluids. In laminar flow, the motion of the particles of the fluid is very orderly with particles close to a solid surface moving in straight lines parallel to that surface.
Turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent.
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In fluid mechanics, the Reynolds number (Re) is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers, flows tend to be turbulent. The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow (eddy currents).
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