Numerical study of the oscillatory boundary layer over wall-mounted flexible filaments

The oscillatory boundary layer over an array of wall-mounted flexible filaments is studied using a penalty-immersed boundary method to explore the dynamic interaction between the boundary layer and filaments in two-dimensional (2D) flows. It is found that as the bending rigidity increases, the motion of the filament exhibits two patterns, undulation and sway. The undulating pattern exhibits a shorter wavelength of filament displacement and a reduced average filament height. In the swaying pattern, resonance occurs when the flow frequency is close to the natural frequency of the submerged filament, resulting in a peak value in the mechanical energy of the filament. When shifting the profile of the root mean square of the disturbance velocity downwards by the average height of the filament, the curves collapse for different filament parameters and the smooth wall, indicating that the presence of filaments can be modeled as an offset on the bottom boundary. A complex-valued slip length is introduced at the incoming flow frequency to quantify the velocity-velocity gradient relation near the filament tip, which takes into account both the magnitude and the phase lag of the slip velocity. Although this slip length varies dramatically inside the filament layer, it reaches a plateau at the upper edge of the filaments, with a smaller absolute value than those of the smooth wall. This finding provides insight into modeling the effective boundary condition of the filament-attached wall. Furthermore, the three-dimensional (3D) effects of the finite-span filaments are discussed. The 3D vortex has a U shape for very small aspect ratios, which transits to a ring-shape with the increasing aspect ratio. The spanwise range of the 3D effects is approximately one filament length on each side, and these effects can be neglected for the aspect ratios greater than 2, 4, and 8 in swaying, periodic undulating, and aperiodic undulating patterns, respectively.