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.
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Research Article| February 01 1995
Instability in a spatially periodic open flow
Michael F. Schatz;
Michael F. Schatz, Dwight Barkley, Harry L. Swinney; Instability in a spatially periodic open flow. Physics of Fluids 1 February 1995; 7 (2): 344–358. https://doi.org/10.1063/1.868632
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