Edge states and the periodic self-sustaining process in the Stokes boundary layer

TY - JOUR

T1 - Edge states and the periodic self-sustaining process in the Stokes boundary layer

AU - Sandoval, Jorge

AU - Eaves, Tom S.

N1 - Publisher Copyright: © 2025 The Author(s).

PY - 2025/10/29

Y1 - 2025/10/29

N2 - The Stokes boundary layer (SBL) is the oscillating flow above a flat plate. Its laminar flow becomes linearly unstable at a Reynolds number of, where is the amplitude of the oscillation, is the period of oscillation and is the fluid's kinematic viscosity, but turbulence is observed subcritically for. The state space consists of laminar and turbulent basins of attraction, separated by a saddle point (the 'edge state') and its stable manifold (the 'edge'). This work presents the edge trajectories for the transitional regime of the SBL. Despite linear dynamics disallowing the lift-up mechanism in the laminar SBL, edge trajectories are dominated by coherent structures as in other canonical shear flows: streaks, rolls and waves. Stokes boundary layer structures are inherently periodic, interacting with the oscillating flow in a novel way: streaks form near the plate, migrate upward at a speed and dissipate. A streak-roll-wave decomposition reveals a spatiotemporally evolving version of the self-sustaining process (SSP): (i) rolls lift fluid near the plate, generating streaks (via the lift-up mechanism); (ii) streaks can only persist in regions with the same sign of laminar shear as when they were created, defining regions that moves upward at a speed; (iii) the sign of streak production reverses at a roll stagnation point, destroying the streak and generating waves; (iv) trapped waves reinforce the rolls via Reynolds stresses; (v) mass conservation reinforces the rolls. This periodic SSP highlights the role of flow oscillations in sustaining transitional structures in the SBL, providing an alternative picture to 'bypass' transition, which relies on pre-existing free stream turbulence and spanwise vortices.

AB - The Stokes boundary layer (SBL) is the oscillating flow above a flat plate. Its laminar flow becomes linearly unstable at a Reynolds number of, where is the amplitude of the oscillation, is the period of oscillation and is the fluid's kinematic viscosity, but turbulence is observed subcritically for. The state space consists of laminar and turbulent basins of attraction, separated by a saddle point (the 'edge state') and its stable manifold (the 'edge'). This work presents the edge trajectories for the transitional regime of the SBL. Despite linear dynamics disallowing the lift-up mechanism in the laminar SBL, edge trajectories are dominated by coherent structures as in other canonical shear flows: streaks, rolls and waves. Stokes boundary layer structures are inherently periodic, interacting with the oscillating flow in a novel way: streaks form near the plate, migrate upward at a speed and dissipate. A streak-roll-wave decomposition reveals a spatiotemporally evolving version of the self-sustaining process (SSP): (i) rolls lift fluid near the plate, generating streaks (via the lift-up mechanism); (ii) streaks can only persist in regions with the same sign of laminar shear as when they were created, defining regions that moves upward at a speed; (iii) the sign of streak production reverses at a roll stagnation point, destroying the streak and generating waves; (iv) trapped waves reinforce the rolls via Reynolds stresses; (v) mass conservation reinforces the rolls. This periodic SSP highlights the role of flow oscillations in sustaining transitional structures in the SBL, providing an alternative picture to 'bypass' transition, which relies on pre-existing free stream turbulence and spanwise vortices.

KW - nonlinear dynamical systems

KW - transition to turbulence

UR - https://www.scopus.com/pages/publications/105020010559

U2 - 10.1017/jfm.2025.10725

DO - 10.1017/jfm.2025.10725

M3 - Article

SN - 0022-1120

VL - 1022

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

M1 - A10

ER -