A traveling wave bifurcation analysis of turbulent pipe flow

TL;DR

This paper rigorously analyzes a PDE model of turbulent pipe flow, revealing bifurcations and traveling waves that explain the transition to turbulence using dynamical systems theory.

Contribution

It introduces a novel mathematical framework applying geometric singular perturbation theory to identify bifurcations and traveling waves in a reduced turbulence model.

Findings

Existence of heteroclinic loop between turbulent and laminar states

Cascade of bifurcations leading to complex traveling wave dynamics

Onset of chaos at intermediate Reynolds numbers

Abstract

Using various techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al., Nature, 526:550-553, 2015], which describes the rise of turbulent pipe flow via a PDE system of reduced complexity. The fast evolution of turbulence is governed by reaction-diffusion dynamics coupled to the centerline velocity, which evolves with advection of Burgers' type and a slow relaminarization term. Applying to this model a spatial dynamics ansatz and geometric singular perturbation theory, we prove the existence of a heteroclinic loop between a turbulent and a laminar steady state and establish a cascade of bifurcations of various traveling waves mediating the transition to turbulence. The most complicated behaviour can be found in an intermediate Reynolds number regime, where the traveling waves exhibit arbitrarily long periodic-like dynamics indicating the onset of chaos. Our analysis provides a systematic mathematical approach to identifying the transition to spatio-temporal turbulent structures that may also be applicable to other models arising in fluid dynamics.