Emergence of spatio-temporal dynamics from exact coherent solutions in pipe flow

Paul Ritter,Marc Avila,Fernando Mellibovsky

doi:10.1088/1367-2630/18/8/083031

Abstract

Turbulent-laminar patterns are ubiquitous near transition in wall-bounded shear flows. Despite recent progress in describing their dynamics in analogy to non-equilibrium phase transitions, there is no theory explaining their emergence. Dynamical-system approaches suggest that invariant solutions to the Navier–Stokes equations, such as traveling waves and relative periodic orbits in pipe flow, act as building blocks of the disordered dynamics. While recent studies have shown how transient chaos arises from such solutions, the ensuing dynamics lacks the strong fluctuations in size, shape and speed of the turbulent spots observed in experiments. We here show that chaotic spots with distinct dynamical and kinematic properties merge in phase space and give rise to the enhanced spatio-temporal patterns observed in pipe flow. This paves the way for a dynamical-system foundation to the phenomenology of turbulent-laminar patterns in wall-bounded extended shear flows.

Highlights

Despite the ubiquity of turbulence, a theory explaining the emergence of its complexity has remained elusive even though the governing equations have been known for nearly two centuries
We further reduced the Reynolds number in an attempt to detect the initiation of the merger, but this resulted in the sudden disappearance of B at around Re ∼ 1450, suggesting that, unlike what happens with A and B at higher Re, the dynamics associated with the independent B1 and B2 saddles is highly unstable and only becomes detectable upon their merging to form B
Several routes to chaos were proposed in the seventies to account for the emergence of disordered dynamics in systems far from equilibrium [2, 4, 38]

Summary

12 August 2016

Any further distribution of Abstract this work must maintain Turbulent-laminar patterns are ubiquitous near transition in wall-bounded shear flows. While recent studies have shown how transient chaos arises from such solutions, the ensuing dynamics lacks the strong fluctuations in size, shape and speed of the turbulent spots observed in experiments. We here show that chaotic spots with distinct dynamical and kinematic properties merge in phase space and give rise to the enhanced spatio-temporal patterns observed in pipe flow. This paves the way for a dynamical-system foundation to the phenomenology of turbulent-laminar patterns in wall-bounded extended shear flows

Introduction

Methods

Boundary crisis to transient chaos

Onset of spatio-temporal fluctuations via saddle merger

Origin of chaotic saddles

Conclusion