The subcritical transition of wall-bounded shear flows is a long-standing but evolving problem in fluid dynamics. Linear stability theory predicts the growth of infinitesimal disturbances, but the actual threshold for finite disturbances that sustains turbulence, even in localized or intermittent forms, is less clear. Pipe flow is a classical case, where localized turbulence appears as puffs that repeatedly split and decay. The balance of these processes defines the global critical Reynolds number. In planar flows such as Couette and/or Poiseuille flows, turbulence often organizes into oblique bands or stripes, resembling spiral turbulence in Taylor–Couette flow and demonstrating the coexistence of laminar and turbulent phases. In three-dimensional wall-bounded shear flows, i.e., Taylor–Couette–Poiseuille flow, ring-shaped turbulence can also emerge, adding further variety to transitional structures. A key feature of these striped/banded/ring-shaped states is the secondary large-scale flow that develops alongside localized turbulence. This flow should play a stabilizing role for turbulence localization and sustenance, but when disrupted by surface roughness, confinement, or other effects, the characteristic patterns fail to form, and the transition follows alternative routes with higher critical Reynolds numbers. Such sensitivity emphasizes the importance of secondary flows in the organization of subcritical turbulence.
The engineering relevance is evident in systems requiring reliable turbulent heat transfer. High-temperature gas-cooled reactors, for instance, employ helium coolant in an annular channel between coaxial cylinders. This annular Poiseuille flow lies between pipe and planar geometries, and depending on the radius ratio, localized turbulence may resemble puffs or bands. Intermediate regimes are more complex: the inner cylinder can alter puff splitting, while curvature and azimuthal confinement affect band persistence. These features raise fundamental questions regarding the structural characteristics of localized turbulence, its impact on heat transfer, and the dependence of secondary large-scale flows on geometry. Furthermore, laminar–turbulence coexistence near the global critical Reynolds number raises broader issues of non-equilibrium critical phenomena. In particular, the transition dynamics may exhibit properties analogous to directed percolation in absorbing state systems.
This work summarizes recent efforts to investigate these challenges through direct numerical simulations (and some experiments) across canonical planar shear flows and annular geometries. By clarifying similarities and differences, new insights are gained into the physics of subcritical transition, its critical dynamics, and its engineering implications for turbulence control and thermal management.

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