We study numerically the propulsive wakes produced by a flapping foil. Both pure pitching and pure heaving motions are considered, respectively, at a fixed Reynolds number of Re = 1700. As the major innovation of this paper, we find an interesting coincidence that the efficiency maximum agrees well with the 2D-3D transition boundary, by plotting the contours of propulsive efficiency in the frequency-amplitude parametric space and comparing to the transition boundaries. Although there is a lack of direct 3D simulations, it is reasonable to conjecture that the propulsive efficiency increases with Strouhal number until the wake transits from a 2D state to a 3D state. By comparing between the pure pitching motion and the pure heaving motion, we find that the 2D-3D transition occurs earlier for the pure heaving foil than that of the pure pitching foil. Consequently, the efficiency for the pure heaving foil peaks more closely to the wake deflection boundary than that of the pure pitching foil. Furthermore, since we have drawn the maps on the same parametric space with the same Reynolds number, it is possible to make a direct comparison in the propulsive efficiency between a pure pitching foil and a pure heaving foil. We note that the maximum efficiency for a pure pitching foil is 15.6%, and that of a pure heaving foil is 17%, indicating that the pure heaving foil has a slightly better propulsive performance than that of the pure pitching foil for the currently studied Reynolds number.

a b s t r a c tNumerical simulations are used to study the energy harvester based on a semi-active flapping foil, in which the profile of the pitching motion is prescribed and the heaving motion is activated by the vertical hydrodynamic force. We consider a two-dimensional NACA0015 airfoil with the Reynolds number Re ¼ 1000. First, for the sinusoidal pitching, an optimal combination of the parameters of pitching amplitude q 0 ¼ 75 and reduced frequency f * ¼ 0.16 is identified, with the highest energy harvesting efficiency of 32% being recorded. Then we study non-sinusoidal pitching, with a gradual change from a sinusoid to a square wave as b is increased from one. We find that its effect of efficiency enhancement is limited for the parameters approaching their optimal values, and the upper boundary of the efficiency appears not to be increased. In detail, we report that when the pitching amplitude is small, nonsinusoidal pitching motions can indeed improve the performance of the system. However, when both the pitching amplitude and the flapping frequency are close to their optimal values, non-sinusoidal pitching motions contribute negatively to the harvesting efficiency. We suggest that a non-sinusoidal profile, at least a simple trapezoidal-like one is ineffective in the semi-active system reported by the current study.

The inertia plays a significant role in the response of a system undergoing flowinduced vibrations, which has been extensively investigated by previous researchers. However, the inertial effects of an energy harvester employing the mechanism of flow-induced vibrations have attracted little attention. This paper concentrates on a semi-passive energy extraction system considering its inertial effects. The incompressible Navier-Stokes equations are solved using a finite-volume based numerical solver with a moving grid technique. A partitioned method is used to couple the fluid and structure motions with the sub-iteration technique and an Aitken relaxation, which guarantees a strong fluid-structure coupling. In addition, a fictitious mass is added to resolve the numerical instability aroused by low density ratios. First, at a fixed mass ratio of r = 1, we identify an optimal set of parameters, at which a maximum efficiency of η = 34% is achieved. Further studies with r ranging from 0.125 to 100 are performed around the optimal parameters. The results show that for the semi-passive flapping energy harvester, the energy harvesting efficiency decreases monotonically with increasing mass ratio. We also notice that the total power extraction stays at a high level with little variation for r < 10; therefore, if we concern more about the amount of power extraction rather than its efficiency, the inertial effects can be neglectable for r < 10. Moreover, since one degree of freedom is released for the semi-passive system, it is possible for the system to automatically determine its optimal operational parameters. We note that the optimal phase difference φ = 82 • has been well determined, which leads to a good timing of vortex-foil interactions. We note two different trends on phase difference for the effects of reduced frequency and mass ratio, respectively. By varying the reduced frequency f * , an optimal f * is identified, at which the minimum phase difference is achieved. While the relationship between phase difference and mass ratio is monotonic, a maximum phase difference is achieved at the nearly zero mass ratio. Nevertheless, both trends point to the same optimal phase difference, i.e., φ = 82 • at θ 0 = 75 • . Furthermore, the relationship between the leading edge vortex and the phase difference is systematically investigated, accounting for the physical reason of existence of the optimal phase difference. C 2015 AIP Publishing LLC. [http://dx.

We propose a natural model to probe in a controlled fashion the instability of interacting vortex rings shed from the edge of an oblate spheroid disk of major diameter c, undergoing oscillations of frequency f_{0} and amplitude A. We perform a Floquet stability analysis to determine the characteristics of the instability modes, which depend strongly on the azimuthal (integer) wave number m. We vary two key control parameters, the Keulegan-Carpenter number K_{C}=2πA/c and the Stokes number β=f_{0}c^{2}/ν, where ν is the kinematic viscosity of the fluid. We observe two distinct flow regimes. First, for sufficiently small β, and hence low frequency of oscillation corresponding to relatively weak interaction between sequentially shedding vortex rings, symmetry breaking occurs directly to a single unstable mode with m=1. Second, for sufficiently large yet fixed values of β, corresponding to a higher oscillation frequency and hence stronger ring-ring interaction, the onset of asymmetry is predicted to occur due to two branches of high m instabilities as the amplitude is increased, with m=1 structures being dominant only for sufficiently large values of K_{C}. These two branches can be distinguished by the phase properties of the vortical structures above and below the disk. The region in (K_{C},β) parameter space where these two high m instability branches arise can be described accurately in terms of naturally defined Reynolds numbers, using appropriately chosen characteristic length scales. We subsequently carry out direct numerical simulations of the fully three-dimensional flow to verify the principal characteristics of the Floquet analysis, in particular demonstrating that high wave-number symmetry-breaking generically occurs when vortex rings sequentially interact sufficiently strongly.

No abstract