Tapered foils favor traveling-wave kinematics to enhance the performance of flapping propulsion

We report results from an experimental investigation on the fluid-structure interactions of flapping foils with tapered thickness profiles actuated in a quiescent viscous fluid. We seek to assess the propulsive performance of two sets of flapping foils; one with a fixed average bending stiffness, the other one with a fixed mass ratio. We find that foils that are stiffer towards the root than at their tip produce higher values of thrust and efficiency simultaneously, over a wide range of driving frequencies. Our kinematic analysis reveals that more tapered foils naturally develop a traveling-wave-dominated motion. We perform particle image velocimetry to relate the dynamics and kinematics of the flapping foils to the dynamics of the surrounding fluid. For more tapered foils, we observe a stronger vorticity production and a wake pattern with enhanced downstream speed of the fluid. Our paper provides experimental evidence that tapered stiffness distributions robustly enhances propulsive performance.

Figure 1

(a) Photograph and (b) schematic of a flexible tapered flapping foil, heaving with amplitude $A_0$ along the $y$ axis. (c) Schematic of the flapping foil and reaction forces $F_x\mathrm{and}{F}_y$ measured at the clamp. (d) Center-line deformation for a flat specimen ($\alpha =0$) (d1) and a highly tapered one ($\alpha =0.90$) (d2), actuated at three times their respective resonant frequency, $f=3f_0$. (e) Period-averaged velocity field of the surrounding fluid measured with particle image velocimetry (PIV). The bending profiles measurements and the PIV were performed separately (flapping foil $\alpha =0.90$ at $f=2f_0$).

Figure 2

Fabrication, experimental setup, and force measurements. (a1) and (a2) Photograph and schematic of the mold used to fabricate the flapping foils. (b) Experimental apparatus. (c) Time series of propulsive force $F_x$ (top) and the driving force $F_y$ (bottom) as raw (dots) and low-passed (line) signals for $\alpha =0.54$, $\tilde{f}=1$.

Figure 3

Performance of the (a)–(c) iso-$\langle B\rangle$, and (d) iso-$M$ foils as a function of the driving frequency. (a) Thrust coefficient $C_T$ and (b) power coefficient $C_P$ versus the driving frequency ratio $\tilde{f}$. (c) and (d) Efficiency $\varepsilon$ as a function of $\tilde{f}$ for the (c) iso-$\langle B\rangle$ and (d) iso-$M$ foils. The error bars in (a)–(c) correspond to the standard deviation of five runs. No error bars are included in (d) since only one run was performed for this set.

Figure 4

Kinematics of the flapping foils. (a) Normalized tip deflection $\tilde{a}$ and (b) phase $\tilde{\phi }$ of the tip relative to the root as functions of the frequency ratio $\tilde{f}$. (c) Snapshots of a deformed profile of the flapping foil with $\alpha =0.54$ at $\tilde{f}=2$. (c1) Measured center line and (c2) reconstruction from the first mode of complex orthogonal decomposition (COD) with its (c3) traveling and (c4) standing-wave components. (d) Traveling index $\mathcal{I}$ as a function of the frequency ratio $\tilde{f}$.

Figure 5

PIV results and derived vorticity and circulation measures. (a) Snapshots of the nondimensional vorticity $\tilde{\omega }$, phase averaged over one stroke ($\alpha =0.54,\tilde{f}=1.55$). The boundary $S$ used to compute the circulation $\mathrm{\Gamma }$ is shown as a dashed line. (b) Mean circulation $\tilde{\mathrm{\Gamma }}$ versus $\tilde{f}$. The error bars correspond to a threshold value $\pm 20\%$ above/below ${\tilde{\omega }}_{\mathrm{t}}=10$, demonstrating that the observed trends are weakly dependent on this choice. (c) Mean velocity flow at $\tilde{f}=3$ for (c1) the flat flapping foil ($\alpha =0$) and (c2) the most tapered one ($\alpha =0.90$) [see Fig. 1 for the corresponding kinematics]. The scale of the arrows size is different in the two plots, whereas their actual magnitude $\left|\right(\tilde{u},\tilde{v}\left)\right|$ is encoded in the adjacent color map.