A numerical study on flapping of a flexible foil

Abstract

A numerical study of the self-induced flapping motion of a flexible cantilevered foil in a uniform axial flow is presented. A high-order fluid-structure solver based on fully coupled Navier-Stokes and non-linear structural dynamics equations is employed. The evolution of the unsteady laminar boundary layer is investigated and three phases in its periodical development along the flapping foil are identified, based on the Blasius scale, eta , namely: (i) uniformly decelerating; (ii) accelerating upper boundary layer and (iii) mixed accelerating and decelerat- ing. Consequently, the spatial distribution of the boundary layer is studied and boundary layer regimes are mapped out in a phase diagram spanned by the Lagrangian abscissa and nondimensional time. The boundary layer is thus fully characterized based on the tip displacement of the foil. Induced tension within the foil is shown to be 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, the traveling mechanisms of kinematic and dynamic data along the foil are investigated using Complex Empirical Orthogonal Functions and Space-Time Power Spectrum analyses. From the study of the flapping regimes, 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 are reported.