Transition to turbulence in the Stokes Boundary Layer: Edge States and Unsteady Self-Sustained Process (USSP)*

The Stokes boundary layer is an oscillatory flow above an infinite plate, with oscillations driven either by (1) a transverse sinusoidal motion of the plate or (2) a sinusoidal applied pressure gradient. Beyond a critical Reynolds number of 2511, the laminar solution of the Stokes boundary layer is susceptible to linear instability. However, this instability is subcritical given that turbulence is observed for Reynolds numbers above approximately 700 despite the flow being linearly stable in this range. 

The state space of a subcritical flow consists of two basins of attraction: that of the laminar flow and that of turbulence. A saddle point separates these basins, termed the ‘edge state’, and its codimension-one stable manifold termed the ‘edge manifold’, or simply ‘edge’. The edge states may be found by ‘edge tracking’, an iterative procedure in which the trajectories of initial conditions on either side of the edge are computed and bisected.

Edge states in the Stokes boundary layer are composed of vortical structures of the same nature as canonical shear flows, namely streaks, rolls and waves. For non-oscillating shear flows, these structures are known to coexist and mutually balance through a mechanism known as the Self-Sustained Process. However, in the Stokes boundary layer, these structures are inherently periodic and utilise the oscillating base flow in a novel way. Structures migrate upwards to align with the location of the maximum shear of the laminar velocity profile, and large-scale rolls form to periodically create new streaks near the wall. This talk will present these edge states in the Stokes boundary layer, compare them to their known non-oscillatory counterparts, and provide insights about their effects on mass and momentum transport near the wall.