Instability in a spatially periodic open flow

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Abstract
Laboratory experiments and numerical computations are conducted for plane channel flow with a streamwise‐periodic array of cylinders. Well‐ordered, globally stable flow states emerge from primary and secondary instabilities, in contrast with other wall‐bounded shear flows, where instability generally leads directly to turbulence. A two‐dimensional flow resembling Tollmien–Schlichting waves arises from a primary instability at a critical value of the Reynolds number, R1=130, more than 40 times smaller than for plane Poiseuille flow. The primary transition is shown to be a supercritical Hopf bifurcation arising from a convective instability. A numerical linear stability analysis is in quantitative agreement with the experimental observations, and a simple one‐dimensional model captures essential features of the primary transition. The secondary flow loses stability at R2≊160 to a tertiary flow, with a standing wave structure along the streamwise direction and a preferred wave number in the spanwise direction. This three‐dimensional flow remains stable for a range of R, even though the structures resemble the initial stages of the breakdown to turbulence typically displayed by wall‐bounded shear flow. The results of a Floquet stability analysis for the onset of three‐dimensional flow are in partial agreement with experiment.