Wavenumber affects the lift of ray-inspired fins near a substrate

TL;DR

The paper investigates how wavenumber $k$—the ratio of chord to wavelength—affects lift on ray-like fins near a substrate. Using a controllable ray-inspired robotic fin, multi-modal measurements (force, SPIV), potential-flow analyses, and a simple dynamical model, it shows that near-ground the time-averaged lift becomes strongly negative and increases in magnitude as $k$ decreases, while thrust and power are less sensitive to ground proximity. Lift is negative because quasisteady lift dominates wake-induced lift; wake effects weaken with higher $k$, and the image-vortex configuration near the wall contributes to this behavior. A minimal dynamical-system model indicates oscillatory fins collide with the ground more quickly than undulatory fins, consistent with observations in benthic rays. These results offer a mechanistic interpretation of depth-dependent wavenumber in rays and provide design insights for ray-inspired robots operating near substrates.

Abstract

Rays and skates tend to have different fin kinematics depending on their proximity to a ground plane such as the seafloor. Near the ground, rays tend to be more undulatory (high wavenumber), while far from the ground, rays tend to be more oscillatory (low wavenumber). It is unknown whether these differences are driven by hydrodynamics or other biological pressures. Here we show that near the ground, the time-averaged lift on a ray-like fin is highly dependent on wavenumber. We support our claims using a ray-inspired robotic rig that can produce oscillatory and undulatory motions on the same fin. Potential flow simulations reveal that lift is always negative because quasisteady forces overcome wake-induced forces. Three-dimensional flow measurements demonstrate that oscillatory wakes are more disrupted by the ground than undulatory wakes. All these effects lead to a suction force toward the ground that is stronger and more destabilizing for oscillatory fins than undulatory fins. Our results suggest that wavenumber plays a role in the near-ground dynamics of ray-like fins, particularly in terms of dorsoventral accelerations. The fact that lower wavenumber is linked with stronger suction forces offers a new way to interpret the depth-dependent kinematics of rays and ray-inspired robots.

Figures (7)

• Figure 1: Oscillatory, semi-oscillatory, and undulatory motions. Left: Cownose Ray (Rhinoptera bonasus, $k=0.4$). Middle: Clearnose Skate (Rostroraja eglanteria, $k=0.9$). Right: Bluespotted Ribbontail Ray (Taeniura lymma, $k=1.4$). Wavenumbers extracted from rosenberger2001pectoral.
• Figure 2: (a) Side view and back view of the ray-inspired robotic platform. (b) Sketches of the wave shape at the fin tip. (c) A perspective view of the experimental setup. Test section: 0.38$\times$0.45$\times$1.52 m).
• Figure 3: A schematic of the potential flow simulation setup. An image system is used to enforce the no-flux boundary condition at the ground plane ($y=0$).
• Figure 4: (a-c) Time-averaged thrust $\overline{C}_T$, power $\overline{C}_P$, and lift $\overline{C}_L$ coefficients of oscillatory ($k=2/3$), semi-oscillatory ($k=3/3$), and undulatory ($k=4/3$) fins at a range of Strouhal number $St$ and ground proximity $d/c$ ($d$: ground distance; $c$: chord length). Shaded boxes are slices through the 3D plots at $St=0.57$. (d-f) Comparisons between experiments and simulations at $St=0.57$. Hollow circles in (d) correspond to locations of PIV cases. Note the difference in the scale of $\overline{C}_P$ in (e). The difference between experiment and simulation in $\overline{C}_P$ may be attributed to unmodeled physical effects in the 2D simulation.
• Figure 5: Force decomposition results obtained from potential flow simulations. (a) Total lift () and its decomposition for (b) oscillatory ($k=2/3$), (c) semi-oscillatory ($k=3/3$), and (d) undulatory ($k=4/3$) fins at $St=0.57$. The wake-induced lift $\overline{C}_{L,\mathrm{wake}}$ was represented by dotted lines ( ) and the quasisteady lift $\overline{C}_{L,\mathrm{quasisteady}}$ was represented by dashed lines ( ).