Results of dynamic testing, heaving motions, of the rectangular foil.

<p>Generally, the pressure-based calculations were able to accurately reproduce both the shape and the magnitude of the measured time traces, but less so than during 0° angle of attack motions. Agreement declined slightly as actuation frequency increased. (A) Comparisons of phase-averaged (n = 3) measured and calculated time traces. (B) Midline kinematics over one motion cycle, corresponding to the time traces on the left. <i>F</i>_<i>x</i> − streamwise forces. <i>F</i>_<i>y</i> − lateral forces. <i>T</i>_<i>z</i> − torques about the vertical axis. St—Strouhal number. Silhouettes represent standard deviations.</p

Dynamic tests—Quantitative comparison^a of forces and torques at increasing actuation frequencies.

<p>Dynamic tests—Quantitative comparison<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0189225#t003fn002" target="_blank">^a</a> of forces and torques at increasing actuation frequencies.</p

Pressure fields in the horizontal plane, at the foil’s midline.

<p>Pressure fields at the foils’ midlines around the rectangular (Rect) and tail-shaped (Tail) foils during 0° angle of attack and pure heaving motions at three points in a stroke cycle, corresponding to the snapshots in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0189225#pone.0189225.g013" target="_blank">Fig 13</a>. The foils were moved at 1.5 Hz actuation frequency and 1.5 cm heave amplitude, in an oncoming flow of 30 cm/s. Before direction reversal, the foils move downward. Color bar indicates the coefficient of pressure (C_P). During 0° angle of attack motions, pressure gradients peak near the trailing edge. In the heaving program, pressure peaks in the leading edge vortex, and deteriorates into complex patterns as the vortex is impacted by the foil.</p

Static tests—Quantitative comparison of forces and torques at increasing oncoming flow speeds.

<p>Static tests—Quantitative comparison of forces and torques at increasing oncoming flow speeds.</p

Image processing steps in making foil masks and force calculation boundaries.

<p>Boundary coordinates for foil masks and force calculation were generated using binary image processing. (A)-(D) illustrate mask generation, and the same process was used to produce force calculation boundaries. (A) A frame extracted from video of foil motion. Fluorescent paint at the foil’s midline appears as a bright line, and the portion of the foil below the light sheet is visible due to parallax of 3D structures. (B) The automatically-detected midline of the foil. (C) Binary image dilation widened the detected midline. (D) 200 equally-spaced points on the black-white boundary in (C) were extracted to use as a mask enclosing both the foil’s midline and the portion of the foil visible below the light sheet. The points depicted here were smoothed to remove jagged edges. (E) Smoothed foil mask plotted as a silhouette, and the 200-point force calculation boundary produced by the same process. (F) Pressure contour for the video frame, with the foil mask and force calculation boundary drawn in black.</p

Velocity vector fields in the transverse plane.

<p>Velocity vector fields in the transverse plane as the foils approach direction reversal, for 1.5 Hz actuation frequency, 1.5 cm heave amplitude, and 30 cm/s oncoming flow (the conditions used for 3D testing). The foils move toward the left. Vertical velocities (<i>V</i>_<i>z</i>) were taken along a vertical line immediately to the right of the foil. Bright spots represent either the edges of the foil or the strips of fluorescent paint. Colors in velocity traces represent different trials. (A) Rectangular foil, 0° angle of attack program. (B) Rectangular foil, heaving program. (C) Tail-shaped foil, 0° angle of attack program. (D) Tail-shaped foil, heaving program. <i>V</i>_<i>z</i>*—<i>V</i>_<i>z</i> normalized by the total velocity at the measurement location, plus or minus standard deviation. Rect—rectangular foil. Tail—tail-shaped foil.</p

Time step selection for pressure field calculation.

<p>A comparison of the measured and calculated lateral force (<i>F</i>_<i>y</i>) values when the rectangular foil was operating in 0° angle of attack motions at 2.0 Hz actuation frequency under different time-steps. (A) The noise in the force trace from pressure-based force calculations decreased as time step (dT) increased. (B) When a low-pass filter was applied to the noisy time traces, nearly identical traces resulted, and these traces resembled the trace produced the time step was 0.01s. Streamwise forces (<i>F</i>_<i>x</i>) and vertical torques (<i>T</i>_<i>z</i>) followed similar trends to those displayed here.</p

Tail-shaped foil—Quantitative comparisons^a of forces and torques from three-dimensional tests.

<p>Tail-shaped foil—Quantitative comparisons<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0189225#t002fn002" target="_blank">^a</a> of forces and torques from three-dimensional tests.</p

Processing of highly repeatable measured force and torque data.

<p>Measured force data were highly repeatable. (A) Example raw and filtered lateral force (<i>F</i>_<i>y</i>) traces, taken during dynamic testing. Three motion cycles during 1.0 Hz (top) and 2.0 Hz (bottom) 0° angle of attack motions are shown. (B) Filtered, phase-averaged traces of the data from (A). Silhouettes represent standard deviations. Streamwise forces (<i>F</i>_<i>x</i>) and vertical torques (<i>T</i>_<i>z</i>) followed similar trends to those displayed here.</p

Forces calculated at different boundary positions.

<p>The magnitudes of the forces calculated using the pressure-based technique did not decline substantially when the calculation loop was within ~2.5 boundary-layer-widths (<i>δ</i>_<i>99</i>) from the foil’s midline. To determine how close to the foil the force calculation loop needed to be for high accuracy results, the pressure-based calculation was conducted on multiple loops around the foil. Tests were conducted using pressure data from the dynamic, 2.0 Hz, 0° angle of attack trial. Loop position was measured in <i>δ</i>_<i>99</i>-widths from the foil’s midline. (A) All of the calculation loops examined, drawn on the original image of the foil. (B) Non-dimensional streamwise forces (<i>F</i>_<i>x</i>*) for the different loops, over three periods of foil motion. (C) Non-dimensional lateral forces (<i>F</i>_<i>y</i>*) for the different loops, over three periods of foil motion.</p