Numerical calculation of stable three-dimensional tertiary states in grooved-channel flow

Numerical simulations of the early transition process in periodic grooved-channel flow are presented. For Reynolds numbers, R Rc,2 >Rc,1 the wavy two-dimensional flow is unstable to a classical linear three-dimensional secondary instability; and for some range of Reynolds number above Rc,2 the secondary instability saturates in a steady-periodic, three-dimensional, low-order equilibrium. The three-dimensional equilibria owe their existence and stability to the narrow band nature of grooved-channel-flow secondary instability, which in turn reflects the low-Reynolds-number supercritical form of the grooved-channel-flow primary bifurcation. The contrast between the low-order, weak transition in ‘‘inflectional’’ complex-geometry channels and the abrupt, snap-through transition in (subcritical-primary broadband-secondary) planar channels illustrates the important role of primary criticality in the early transition process.