Transition in Plane Channel Flow with Spatially Periodic Perturbations.

We studied experimentally the primary and secondary instabilities in a plane channel flow perturbed by a streamwise -periodic array of cylinders. Wall-bounded shear flow in plane channels typically undergoes a direct transition from simple laminar behavior to turbulence with complex spatial and temporal intermittency; such behavior is characteristic of open flows, where fluid can advect through the system. However, the spatially perturbed channel flow displays bifurcations to well-ordered stable states, similar to transition exhibited by closed flows (flows confined in a box). The primary transition is a supercritical Hopf bifurcation arising from convective rather than absolute instability. The critical value of Reynolds number R _1 = 130 for the transition is more than an order of magnitude less than that for the unperturbed flow (R_1 = 5772 from linear stability theory). The stable secondary flow, a two-dimensional travelling-wave, resembles Tollmein-Schlichting waves, the linear modes of plane Poiseuille flow. As in the spatially unperturbed case, intentionally imposed, controlled disturbances are required to reveal transition since the bifurcation arises from convective instability. Numerical simulations are in quantitative agreement with the experimental observations. The secondary flow loses stability at R _2~ 160 to a three-dimensional state with a preferred spanwise periodicity. This tertiary flow demonstrates standing-wave behavior as it evolves along the streamwise direction; equivalent behavior results from differing initial disturbances. The flow structure and the strictly periodic spectra resemble the beginning stages of turbulent breakdown typically displayed by unperturbed plane channel flow; however, we observed no evidence in our experiment that the three-dimensional states continue to evolve toward turbulence. For R _sp {~}{>} 200, power spectra from our experiment have broad subharmonics that are also observed in other wall-bounded shear flows. Our work demonstrates experimentally that spatially periodic perturbations in plane channel flow isolates and stabilizes structures that are only fleetingly observed in the unperturbed flow.