Large-amplitude oscillations of foils for efficient propulsion

Large-amplitude oscillations of foils have been observed to yield greater propulsive efficiency than small-amplitude oscillations. Using scaling relations and experiments on foils with peak-to-peak trailing edge amplitudes of up to two chord lengths, we explain why this is so. In the process, we reveal the importance of drag, specifically how it can significantly reduce the efficiency, and how this effect depends on amplitude. The scaling relations and experimental data also reveal a fundamental tradeoff between high thrust and high efficiency, where the drag also plays a crucial role.

Figure 1

Experimental setup and sketch of motions.

Figure 2

An illustration of two motions that produce the same thrust. Velocity profiles are shown in the frame of reference where the fluid at infinity is at rest.

Figure 3

Efficiency as a function of maximum angle of attack at the leading edge for our data. Color indicates pitch angle (in degrees).

Figure 4

Coefficient of thrust as a function of (a) St for raw thrust and (b) ${\text{St}}^2$ for drag-corrected thrust. Color indicates pitch amplitude ${\theta }_0$.

Figure 5

Drag curve at $\text{Re}=8000$ for the foil used in all experiments. Values are reported relative to the drag at $\theta =0$. Color indicates each of the three trials.

Figure 6

Coefficient of power as a function of (a) St and (b) the scaling variable $f^{*}{\text{St}}^2(1-H^{*}{\mathrm{\Theta }}^{*})$. Color indicates ${\theta }_0$.

Figure 7

Coefficient of power as a function of $f^{*}{\text{St}}^2$ and $1-H^{*}{\mathrm{\Theta }}^{*}$. Color indicates $C_P$. The two plots are the same, but viewed from different angles.

Figure 8

Efficiency as a function of St. Color indicates (a) dimensionless amplitude $A^{*}$, (b) pitch amplitude ${\theta }_0$.

Figure 9

(a) Raw efficiency and (b) reduced-drag efficiency as functions of St. Color indicates $A^{*}$.

Figure 10

Efficiency as a function of thrust coefficient, with the drag-free scaling shown. Color indicates $C_T/1.2A_{\theta }^{*}$.