The boundary layer development and traveling wave mechanisms during flapping of a flexible foil

We present a numerical study of the self-induced flapping motion of a flexible cantilevered foil in a uniform axial flow. A high-order fluid-structure solver based on fully coupled Navier-Stokes and non-linear structural dynamics equations is employed. We investigate the evolution of the unsteady laminar boundary layer and identify three phases in its periodical development along the flapping foil based on the Blasius scale, η, namely: (i) uniformly decelerating; (ii) accelerating upper boundary layer and (iii) mixed accelerating and decelerating. This allows us to map out the boundary layer regimes in a phase diagram spanned by the Lagrangian abscissa s and nondimensional time t¯. We show that the induced tension within the foil is dominated by pressure effects and only marginally affected by skin friction. The boundary layer thickness is analyzed through the temporal and spatial evolutions of the displacement and momentum thicknesses. Finally, a space-time power spectrum analysis of the traveling mechanisms of kinematic and dynamic data along the foil is performed. From the study of the flapping regimes, we report the co-existence of direct kinematic waves traveling downstream along the structure as well as reverse dynamic waves traveling in the opposite direction to the axial flow.