The transition to turbulence in parallel flows: transition to turbulence or to regular structures

TL;DR

This paper explores the formation of localized turbulent spots in transition flows, showing that long wavelength fluctuations arise naturally from the subcritical transition without added noise, and analyzes the dynamics near the Benjamin-Feir instability threshold.

Contribution

It derives the instability threshold for a generalized complex Ginzburg-Landau equation and elucidates the phase-driven dynamics near the Benjamin-Feir threshold, linking turbulence to pattern formation.

Findings

Long wavelength fluctuations occur without external noise beyond the Benjamin-Feir threshold.

Phase dynamics dominate near the Benjamin-Feir threshold, governed by Kuramoto-Sivashinsky equation.

Below the threshold, transition becomes mean-field like, leading to pulse patterns.

Abstract

We propose a scenario for the formation of localized turbulent spots in transition flows, which is known as resulting from the subcritical character of the transition. We show that it is not necessary to add 'by hand" a term of random noise in the equations, in order to describe the existence of long wavelength fluctuations as soon as the bifurcated state is beyond the Benjamin-Feir instability threshold. We derive the instability threshold for generalized complex Ginzburg-Landau equation which displays subcriticality. Beyond and close to the Benjamin-Feir threshold we show that the dynamics is mainly driven by the phase of the complex amplitude which obeys Kuramoto-Sivashinsky equation while the fluctuations of the modulus are smaller and slaved to the phase (as already proved for the supercritical case). On the opposite, below the Benjamin-Feir instability threshold, the bifurcated state does loose the randomness associated to turbulence so that the transition becomes of the mean-field type as in noiseless reaction-diffusion systems and leads to pulse-like patterns.