Dynamics of spatially localized states in transitional plane Couette flow

TL;DR

This paper investigates the dynamics of spatially localized states in plane Couette flow, revealing how their behavior and transition to turbulence depend on Reynolds number and pattern width, including the presence of long chaotic transients.

Contribution

It characterizes the regimes and phase space structure of localized states in plane Couette flow across different Reynolds numbers and pattern widths, highlighting the role of homoclinic snaking and chaotic transients.

Findings

Localized states are attracted to spatially localized periodic orbits near snaking.

Chaotic transients of variable duration occur at higher Reynolds numbers.

Long chaotic transients (>10^4) are observed at Re around 250.

Abstract

Unsteady spatially localized states such as puffs, slugs or spots play an important role in transition to turbulence. In plane Couette flow, steady versions of these states are found on two intertwined solution branches describing homoclinic snaking (Schneider $\textit{et al.}$ 2010a). These branches can be used to generate a number of spatially localized initial conditions whose transition can be investigated. From the low Reynolds numbers where homoclinic snaking is first observed ($Re < 175$) to transitional ones ($Re \approx 325$), these spatially localized states traverse various regimes where their relaminarisation time and dynamics are affected by the dynamical structure of phase space. These regimes are reported and characterised in this paper for a $4\pi$ periodic domain in the streamwise direction as a function of the two remaining variables: the Reynolds number and the width of the localized pattern. Close to the snaking, localized states are attracted by spatially localized periodic orbits before relaminarising. At larger values of the Reynolds number, the flow enters a chaotic transient of variable duration before relaminarising. Very long chaotic transients ($t > 10^4$) can be observed without difficulty for relatively low values of the Reynolds number ($Re \approx 250$).