Transition to turbulence in duct flow

TL;DR

This paper investigates the transition to turbulence in square duct flow, highlighting the limitations of linear optimal disturbances and emphasizing the role of nonlinear interactions and edge states in the bypass transition process.

Contribution

It demonstrates that nonlinear sub-optimal disturbances trigger turbulence via quadratic interactions, revealing the importance of large-scale vortex patterns in the transition.

Findings

Linear optimal perturbations are ineffective in nonlinear regimes.

Nonlinear interactions lead to large-scale vortex formations.

Flow evolution near the turbulence threshold shows alternating vortex patterns.

Abstract

The transition of the flow in a duct of square cross-section is studied. Like in the similar case of the pipe flow, the motion is linearly stable for all Reynolds numbers; this flow is thus a good candidate to investigate the 'bypass' path to turbulence. Initially the so-called 'linear optimal perturbation problem' is formulated and solved, yielding optimal disturbances in the form of longitudinal vortices. Such optimals, however, fail to elicit a significant response from the system in the nonlinear regime. Thus, streamwise-inhomogeneous, sub-optimal disturbances are focussed upon; nonlinear quadratic interactions are immediately evoked by such initial perturbations and an unstable streamwise-homogeneous large amplitude mode rapidly emerges. The subsequent evolution of the flow, at a value of the Reynolds number at the edge between fully developed turbulence and relaminarization, shows the alternance of patterns with two pairs of large scale vortices near opposing parallel walls. Such edge states bear a resemblance to optimal disturbances