Flow speed has little impact on propulsive characteristics of oscillating foils

Experiments are reported on the performance of a pitching and heaving two-dimensional foil in a water channel in either continuous or intermittent motion. We find that the thrust and power are independent of the mean free-stream velocity for twofold changes in the mean velocity (fourfold in the dynamic pressure) and for oscillations in the velocity up to 38% of the mean, where the oscillations are intended to mimic those of freely swimming motions where the thrust varies during the flapping cycle. We demonstrate that the correct velocity scale is not the flow velocity but the mean velocity of the trailing edge. We also find little or no impact of streamwise velocity change on the wake characteristics such as vortex organization, vortex strength, and time-averaged velocity profile development—the wake is both qualitatively and quantitatively unchanged. Our results suggest that constant velocity studies can be used to make robust conclusions about swimming performance without a need to explore the free-swimming condition.

Figure 1

Experimental setup.

Figure 2

Propulsor mean thrust force for continuous (a) pitching and (b) heaving motions for ${\overline{U}}_{\infty }=60\mathrm{mm}/\mathrm{s}$ (white circles); 80 mm/s (light gray circles); 100 mm/s (dark gray circles); 120 mm/s (black circles). One pitch and three heave amplitudes are shown. Adapted from Floryan et al. [18] with permission.

Figure 3

Propulsor mean (a) thrust and (b) power undergoing streamwise velocity oscillations $0\le u_a/{\overline{U}}_{\infty }\le 0.38$ (dark to light symbols), for ${\overline{U}}_{\infty }=80$ mm/s. Continuously heaving (circular symbols), pitching (square symbols), and pitching combined with heaving (triangular symbols) foil. Inlaid graphic represents the relative leading edge lateral position—or pitch angle for pitch only cases (red solid line), and streamwise position (dashed blue line).

Figure 4

Propulsor mean (a) thrust and (b) power undergoing streamwise velocity oscillations $0\le u_a/{\overline{U}}_{\infty }\le 0.38$ (dark to light symbols). Intermittently heaving (circular symbols), pitching (square symbols), and pitching combined with heaving (triangular symbols) foil (duty cycle of 0.5). Inlaid graphic represents the relative leading-edge lateral position—or pitch angle for pitch-only cases (red solid line) and streamwise position (dashed blue line).

Figure 5

Propulsor mean (a) thrust and (b) power nondimensionalized by the trailing-edge velocity. Symbols and colors as in Fig. 4.

Figure 6

Phase-averaged vorticity for (i) continuous and (ii) intermittent heaving motions at streamwise velocity amplitudes (a) $u_a/{\overline{U}}_{\infty }=0$ and (b) 0.38. Phase angle of heaving motion $\psi =0^{\circ }$ (passing through the neutral position moving into the positive $y$ domain).

Figure 7

Time-averaged streamwise velocity profiles at (a) $x/c=0.25$, (b) 1, and (c) 1.75 for continuous heaving (circular symbols) and pitching (square symbols) motions. Streamwise velocity amplitudes $u_a/{\overline{U}}_{\infty }=0$ (dark symbols) and 0.38 (light symbols).

Figure 8

(a) Fluke-beat frequency and (b) nondimensional fluke-beat amplitude as functions of swimming speed per body length for several odontocete cetaceans. Adapted from Rohr and Fish [27], and reproduced from Floryan et al. [18] with permission.