Localization in a spanwise-extended model of plane Couette flow

TL;DR

This study uses a simplified 9-PDE model of plane Couette flow to investigate how flow structures localize in the spanwise direction, revealing localized solutions arising from saddle node bifurcations rather than modulational instabilities.

Contribution

It demonstrates that localized flow states can emerge in a reduced PDE model through saddle node bifurcations, challenging the idea that modulational instabilities are necessary for localization.

Findings

Localized solutions arise via saddle node bifurcations.

Global states can be smoothly continued to spanwise-localized states.

Localization can occur without modulational instability.

Abstract

We consider a 9-PDE (1-space and 1-time) model of plane Couette flow in which the degrees of freedom are severely restricted in the streamwise and cross-stream directions to study spanwise localisation in detail. Of the many steady Eckhaus (spanwise modulational) instabilities identified of global steady states, none lead to a localized state. Localized periodic solutions were found instead which arise in saddle node bifurcations in the Reynolds number. These solutions appear global (domain filling) in narrow (small spanwise) domains yet can be smoothly continued out to fully spanwise-localised states in very wide domains. This smooth localisation behaviour, which has also been seen in fully-resolved duct flow (Okino 2011), indicates that an apparently global flow structure needn't have to suffer a modulational instability to localize in wide domains.