Transition to turbulence in duct flow

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 focused upon; nonlinear quadratic interactions are immediately caused 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 boundary 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.