The configuration of multiple flapping foils is studied via numerical simulations. We comprehensively consider the effects of the streamwise distance, vertical spacing, and phase difference on the energy harvesting performance of flapping foils. We divide flapping foil configurations into three categories: tandem, unaligned, and parallel. The foil tandem configuration is optimal for multiple foils if using the existing efficiency formula. However, tandem configurations expand the diffusion range of the wake turbulence. Wake diffusion has a critical effect on multiple foil configurations, and the utilization of the vertical spacing has been neglected. Here, the effective angle of attack and the effective velocity are proposed based on multiple flapping foils, which can well predict various rules of the lift coefficient and guide studies on the optimal configuration of multiple foils. The optimal phase difference for the simple parallel configuration system is 135°, and the energy harvesting efficiency of parallel foils decreases with increasing vertical spacing. The stepwise configuration maximizes the utilization of the vertical spacing, making it optimal for practical engineering applications. The stepwise configuration not only recycles the wake of the upstream foils but also utilizes it to enhance energy harvesting from flapping turbines. In the multiple foil configurations, the energy harvesting efficiency of the downstream foil over the upstream foil is achieved for the first time using a stepwise configuration.

At specified Reynolds numbers Re=ρvc/μ=5×105, this study investigates the power-extraction regime of a flapping-foil turbine executing a figure-eight trajectory. This study thoroughly explored the impacts of trajectory shape, heave and pitch amplitudes, phase difference, and pitch axis location on the power-extraction regime of a flapping turbine. A figure-eight trajectory substantially bolsters the energy harvesting capacity, achieving a peak efficiency of up to 50%. This trajectory capitalizes on the energy generated from the drag encountered by the flapping foil, thereby marking a significant efficiency breakthrough. This could denote a considerable progression for flapping foils tailored for heave and pitch motions since the free stream can be optimally harnessed by the trajectory we have established. In scenarios where the flapping foil undergoes the same maximum effective angle of attack, a wake diffusion spectrum aligns with the Betz limit threshold. Crucially, the closeness to this efficiency frontier suggests a universal maximum effective angle of attack—a consistent law that remains valid for the conventional flapping turbine design. It implies that selections for stroke and pitch amplitudes should be methodical rather than arbitrary. In addition, the positioning of the pitch axis ought to be modified in tandem with phase difference adjustments to bolster the synchronization between heaving motions and resultant lift. The ideal phase difference is variable, contingent on pitch amplitude and pitch axis position. Even with low pitch amplitudes, fine-tuning the phase difference guarantees that the energy harvesting efficiency does not fall below 30%. Such an enhancement would significantly broaden the operational envelope and the parameter space for flapping turbines.

The energy harvesting characteristics of actively pitching flapping foils under a two-degrees-of-freedom (2DOF) system were investigated through numerical simulations. At a Reynolds number of 1100, the effects of the pitching amplitude, reduced frequency, and structural parameters on the energy harvesting performance were compared with the traditional one-degree-of-freedom (1DOF) case. The optimal pitching amplitude (85°), reduced frequency (0.16), and structural parameters (bx*=0.5, kx*=0.7) of the streamwise vibrating flapping foil were determined. The additional velocity generated by streamwise vibrations increased the optimal reduced frequency and pitching amplitude over the traditional case. Streamwise vibrations accelerate the wake propulsion, and the wake vortevx spacing is about 0.8 times the chord length larger than that of the traditional case. Furthermore, the 2DOF case allows the vortex-shedding process of the flapping foil to participate in wake propulsion. The trajectory of the streamwise vibrating flapping foil was observed to be a figure “8” shape. The “8” shape gradually regularizes with an increased streamwise damping coefficient. There is an ideal parameter combination at the optimal reduced frequency that allows the flapping foil to reach the most unstable motion mode. The energy harvesting efficiency of the flapping foil can be increased by up to 25% due solely to vortex-induced vibrations of the 2DOF.

This study investigates the energy-harvesting behavior and configuration effects of two semi-active flapping foils through numerical simulations, considering both tandem and unaligned configurations, as well as the influence of Reynolds number and pitch amplitude. The tandem configuration shows the greatest efficiency improvement, maximizing the utilization of the upstream foil's wake. In contrast, the unaligned configuration allows the downstream foil to surpass the energy-harvesting efficiency of the upstream foil, which is advantageous for clustered systems. The phase difference impacts wake structures, with the downstream foil recapturing the wake, though its efficiency is limited by the lower wake speed compared to the incoming flow velocity. The Reynolds number effect does not appear to influence the peak efficiency ratio, suggesting that the efficiency relationship between the foils remains consistent across different Reynolds numbers, despite potential differences in the optimal phase difference. Furthermore, reducing the pitch amplitude of the upstream foil optimally enhances the efficiency of the downstream foil, a factor that has often been overlooked. Lowering the pitch amplitude increases the wake speed, providing the downstream foil with a larger energy utilization space. This study provides valuable insights into the configuration design of flapping turbines, with the optimal two-foil system achieving an impressive efficiency of up to 46.3%. In the unaligned configuration, the lower the pitch amplitude of the upstream foil, the higher the efficiency ratio, which can reach up to ∼1.8. With different pitch amplitudes, the efficiency ratio consistently exceeds 1, indicating that the downstream foil benefits from the wake of the upstream foil. Combining more foils might further enhance efficiency in the unaligned configuration, as the higher efficiency in the tandem configuration is due to the repeated utilization of the sweep area by the two-foil system.