Hydrodynamics of flapping foils and flags: scaling analysis and experiments

Abstract

In the present research, we conducted the fluid dynamic study of flapping foils for bio locomotion of swimming and flying animals and the fluttering flags adjacent to rigid plate. The scaling laws for the force produced by the flapping foils are constructed by considering the momentum imparted to the vortical structures generated by the foil motion. The novel concept of the wind energy generator is developed using the contact electrification of the fluttering flag and the plate. We first studied the hydrodynamics of the angularly reciprocating plate without a free stream velocity as an elementary mode of the flapping locomotion. We visualize the flow field around the flat plate to find that two independent vortical structures are formed per half-cycle, resulting in the separation of two distinct vortex pairs at sharp edges. Based on our observations, we derive a scaling law to predict the thrust of the flapping plate considering the momentum imparted to the vortical structures. The scaling law is in good agreement with the experimental observations. The study on the angularly reciprocating plate is extended to find a physical explanation for the relatively stubby fins of small aquatic animals. The thrust and flow field around the angularly reciprocating plate is examined varying the tail shapes and aspect ratio over two order of magnitudes. The thrust of a flapping tail can be predicted based on a universal scaling law regardless of the tail shapes, which considers the momentum imparted to the surrounding flows. The thrust in the given tail area and kinematics is maximized at the low aspect ratio of 0.7, whereas the efficiency continuously increases along with the aspect ratio. Combination of the current mathematical modeling and biological observations suggests that small aquatic animals selected a caudal fin with a low aspect ratio for the purpose of added thrust but at the expense of efficiency. We next construct a scaling law for the lift of hovering insects that elucidates a unified mechanism enabling the insects to be airborne through relatively simple scaling arguments of the strength of the leading edge vortex and the momentum induced by the vortical structure. Comparison of our theory with the measurement data of 33 species of insects confirms that the scaling law captures the essential physics of lift generation of hovering insects. Our results offer a simple yet powerful guideline for biologists who seek the evolutionary direction of the shape and kinematics of insect wings, and for engineers who design flapping-based micro air vehicles. Next, the dynamics and energy transfer of fluttering motions of flags placed adjacent to a plate are investigated systematically by varying the inter-space distance, material, and dimensions of the flag, and incoming velocity. The stability condition of the flag adjacent to the plate is almost identical to that in the case of when the flag is not placed adjacent to a plate. However, when flutter occurs, it exhibits asymmetric dynamic behavior with respect to the centreline because of the influence of the plate, even when no contact is made. The energy transfer process is characterized by three distinct stages for each half cycle: energy absorption at the middle of the flag, transition, and energy release at the end part. Flag-plate contact modes, which can be classified as single contact, double contact, contact-propagation, and chaotic contact, are illustrated in their regime plot based on the mass ratio and non-dimensional velocity. Furthermore, the reduced frequency and the Strouhal number of the flag flutter adjacent to the plate are examined by varying the flag length and incoming velocity. Reduced frequency is confined in the narrow range from 0.3 to 0.6, which corresponds to the single flag flutter. The Strouhal number is shown to converge to the optimal value of 0.3 as the incoming velocity increases, implying self-optimization. Based on the study of the fluttering dynamics of the fluttering flag adjacent to the plate, we developed a novel and powerful wind energy harvesting system, namely, flutter-driven triboelectric generator (FTEG) using the self-sustained oscillation of flags as a mechanical energy, which has the potential use for wireless electronics. The flutter-driven triboelectric generator having dual plate configuration with dimensions of 7.5 cm × 5 cm under the incident air flow velocity of 15 m/s can produce outstanding power performances such as the instantaneous output voltage and current of 200 V and 60 μA, respectively, especially with the extremely high frequency of 158 Hz. Our novel system can fully charge a 100 μF capacitor within 245 seconds, corresponding to an average power density of approximately 0.86 mW. In addition, the flutter-driven triboelectric generator can be easily scaled up toward large energy generating system by employing the parallel devices in wind-rich place. We expect that the flutter-driven triboelectric generator can not only be an ideal source for the small scale energy harvesting system for wireless electronics, but also it can provide a novel pathway to the electrical energy generation using wind.