Resonant response and optimal energy harvesting of an elastically mounted pitching and heaving hydrofoil

The aeroelastic response and energy harvesting performance of an elastically mounted hydrofoil subject to a prescribed pitching motion are experimentally studied using a cyber-physical force-feedback control system in a uniform flow. By taking advantage of this cyber-physical system, we systematically sweep through the parameter space of the elastic support (stiffness, damping, and mass) for various frequencies of the prescribed pitching motion. It is found that the flow-induced heave amplitude and the energy harvesting performance are both strongly affected by the frequency ratio between the prescribed pitching frequency and the natural frequency of the system and the damping coefficient. In particular, for a fixed damping coefficient, the maximum flow-induced heave amplitude is achieved at the resonant condition (frequency ratio of 1), which also gives rise to the highest energy harvesting performance. At this resonance condition, though a smaller damping produces a larger heave amplitude, the optimal energy harvesting performance is obtained consistently at an intermediate damping coefficient of 1.5. In addition, at the resonance condition, the heave amplitude, or Strouhal number, and the hydrodynamic forces on the foil are both found to collapse well for different reduced frequencies, suggesting a similarity in the vortex dynamics generated by the elastically mounted system. A low-order model based on classical vibration theory is formulated to reproduce the power coefficient using the damping coefficient and Strouhal number, and we find that the power coefficient predicted by the model agrees well with that measured in the experiment over the range of reduced frequency explored.

Figure 1

(a) Experimental system with the hydrofoil, force transducer, rotary and linear motors and encoders; (b) top view of one typical kinematics of the hydrofoil ($\pi /2$ phase difference between pitch and heave) with five different positions (only the downstroke is shown here due to its stroke symmetry); (c) the force-feedback control system diagram; (d) “ring-down” experiment comparing the measured heaving position and the theoretical position.

Figure 2

Nonsinusoidal pitch profiles: $-1<\beta <0$ for triangular profiles; $\beta =0$ for sinusoidal and $\beta >0$ for trapezoidal profiles.

Figure 3

Contour plots of (a) induced heave amplitude, $H_0$, (b) heave power coefficient, $C_h$, (c) pitch power coefficient, $C_p$, and (d) total power coefficient, $C_{\text{total}}$, (e) pitch efficiency, ${\eta }_p$, (f) heave efficiency, ${\eta }_h$, with respect to frequency ratio, $f/f_n$, and damping coefficient, $b^{*}$; parameters: prescribed sinusoidal pitch profile with pitch amplitude, ${\theta }_0={65}^{\circ }$.

Figure 4

Response of (a) heave power coefficient, (b) flow-induced heave amplitude, and (c) phase difference between the lift force and the flow-induced heave motion (circles $\circ$) and phase difference between the prescribed pitch motion and the flow-induced heave motion (cross $\times$) at a fixed damping coefficient of 1.5; (d) lift-heave portraits for three different frequency ratios: $f/f_n=0.6$ (red), $f/f_n=1.0$ (blue), $f/f_n=3.8$ (green). Foil kinematics in half a cycle: (e) $f/f_n=0.6$ (red), (f) $f/f_n=1.0$ (blue), (g) $f/f_n=3.8$ (green).

Figure 5

At a fixed frequency ratio ($f/f_n=1$), (a) the heave power coefficient, $C_h$, is plotted with respect to damping coefficient; (b) the lift-heave portrait demonstrates the work done by the lift force for different damping coefficient, $b^{*}$ = 0.3, 1.5, and 6.0, at a reduced frequency $f^{*}$ = 0.150; (c) the Strouhal number, $\text{St}=2f{H}_0/U_{\infty }$, is plotted with respect to damping coefficient, $b^{*}$, for different reduced frequency, $f^{*}=fc/U_{\infty }$.

Figure 6

Temporal evolution of (a) lift coefficient and (b) torque coefficient for different reduced frequencies, at a fixed frequency ratio of 1 and a fixed damping coefficient of 1.5. Data of half a cycle are shown for simplicity due to the dynamic symmetry.

Figure 7

Relationship between heave power coefficient, scaled by the Strouhal number squared, and the damping coefficient. Solid line: theoretical prediction [Eq. (16)]; symbols: experimental results at several frequencies, all at a fixed frequency ratio of 1.

Figure 8

(a) Heave power coefficient as a function of frequency ratio, $f/f_n$, at different pitch profiles ($\beta$); (b) heave power coefficient as a function of damping coefficient, $b^{*}$, at different pitch profiles ($\beta$).

Figure 9

(a) Heave power coefficient measured in the experiment and predicted by the theoretical model [Eq. (16)] and total power coefficient measured in the experiment, (b) pitch power coefficient, and (c) lift-heave phase portrait. Parameters: reduced frequency $f^{*}=0.125$, frequency ratio, $f/f_n=1$, damping coefficient, $b^{*}=1.5$.

Figure 10

Instantaneous (a) pitch acceleration (rad/${\mathrm{s}}^2$), (b) effective AOA, and (c) lift coefficient are shown for three pitch profiles ($\beta$ = $-0.975$ (triangular), 0 (sinusoidal), and 2.25 (trapezoidal), for half cycle). The instantaneous effective AOA is given by ${\theta }_e\left(t\right)=\theta \left(t\right)-arctan(V_h\left(t\right)/U_{\infty })$, where $V_h\left(t\right)$ is the instantaneous heave velocity. Parameters: reduced frequency, $f^{*}=0.125$, frequency ratio, $f/f_n=1$, damping coefficient, $b^{*}=1.5$.