Hydrodynamic modulation via cupping in a crustacean-inspired propulsor

Abstract

Shrimp, like many invertebrates swimming at intermediate Reynolds numbers ($Re$), rely on the interplay between morphology and kinematics to generate thrust while producing sufficient lift to overcome their negative buoyancy. Shrimp pleopods branch into an endopodite and an exopodite, whose relative motion varies the projected surface area during the swimming cycle. For this mechanism to function, the exopodite must be cambered relative to the endopodite at a set cupping angle $ζ$, which partially decouples the effective angle of attack of the exopodite from the overall leg kinematics. Here, we investigate the role of $ζ$ in modulating thrust$-$lift balance during steady forward locomotion. Using a dynamically scaled (40$\times$) robotic pleopod, we systematically varied $ζ$ from $0^\circ$ to $80^\circ$, measured hydrodynamic forces, and performed particle image velocimetry at $Re = 968$. Moderate cupping angles ($ζ= 20^\circ-40^\circ$), consistent with biological observations, provide optimal thrust$-$lift balance. At these angles, the exopodite abducts rapidly during the power stroke, maximizing projected area at peak flow velocity, and adducts early during the return stroke, minimizing resistive drag. A reduced-order force model reveals that the exopodite contributes 52$-$62\% of total lift, particularly at intermediate $ζ$, where a leading-edge vortex (LEV) forms and remains attached throughout the power stroke. At extreme cupping angles, LEV coherence degrades and force production weakens. These findings demonstrate that shrimp pleopods function as hybrid propulsors exploiting both drag- and lift-based forces, and that $ζ$ serves as a geometric control parameter capable of tuning thrust$-$lift balance independently of stroke kinematics.

Figures (9)

• Figure 1: Geometric parameters governing shrimp kinematics and morphology. Labeled parameters are seen in three distinct views. (A) The lateral view shows appendage length, $l$, the angle $\alpha$ between the protopodite (proximal part of pleopod) and the body axis (BA) passing through the roots of the five pleopods, the leeward angle $\beta$ between the protopodite and endopodite, and the $\psi$ angle between BA and the endopodite. (B) The zoomed lateral view of a pleopod shows the cupping angle, $\zeta$, which is the camber angle of the abducted exopodite relative to the endopodite. (C) The frontal view of a pleopod shows the out-of-plane abduction angle, $\gamma$.
• Figure 2: Schematics of the experimental setup for simultaneous force and PIV data collection. The horizontal laser sheet bisected the fully abducted exopodite. Two lasers mounted opposite each other produced two coplanar overlapping laser sheets to eliminate shadows (the figure shows only one laser, for clarity). The pleopod was mounted horizontally and oscillated along the horizontal plane (red arrow) by a servomotor. The force transducer was mounted above the waterline between the servo and the model using custom adapters, aligning the $x$-axis of the force transducer with the anterior face of the endopodite. Experiments were conducted in a 210-liter tank ($119\times43\times41$ cm$^3$), with an 11.5 cm long paddle, and a distance 1.2L to the nearest tank walls. Note that the experimental tank is not to scale.
• Figure 3: Force frame of reference. The forces are shown in three frames of reference: laboratory, shrimp, and global frames. Forces on the rotating paddle (blue) are measured in two directions, one that is perpendicular ($F_{\perp}$) and one that is parallel to the endopodite and exopodite ($F_{\parallel}$). In the shrimp frame of reference, $F_{T}$ is the force in the swimming direction, and $F_{L}$ is the force orthogonal to the swimming direction. In the global frame of reference, lift is the force pointing vertically, and thrust is the horizontal force. $\psi$ is the angle between the body axis and the endopodite, $\zeta$ is the angle between the endopodite and exopodite, and $\delta$ is the angle between the exopodite and the horizontal direction. The paddle illustration shows the lateral view in the laboratory frame of reference, with the axis of rotation, endopodite, and exopodite labeled.
• Figure 4: Variation of $\gamma$ throughout one beat cycle. a) Phased-average exopodite kinematics during a beat at varying cupping angles and $\Rey = 968$. The instantaneous time $t$ during the beat cycle is normalized to the beat period $T$. Shading indicates the standard deviation from the plotted mean values (solid lines) of three consecutive beats. The dashed black line represents biological observations obtained from the kinematics of tethered P. paludosus. Profiles are drawn for $\zeta = 35^o$, where the exopodite is shaded yellow. b) Normalized angular velocity ($\dot{\psi}(t)$) to maximum exopodite angle and abduction duration (proportion of the cycle during which the exopodite is abducted). Angular velocity is normalized to the shrimp value (*). Filled circles represent experimental data; asterisks represent biological shrimp data at $\zeta = 35^o$
• Figure 5: Dynamics of thrust and lift forces at variable cupping angles. Values for the top, middle, and bottom panels are: $\zeta = 0^\circ$, $\zeta = 35^\circ$, and $\zeta = 70^\circ$, respectively. The total measured hydrodynamic force is shown with a black line (with shaded standard deviation (STD)). The decomposed contributions from the exopodite and the endopodite are shown in blue and red lines, respectively. Time $t$ is normalised using the stroke period $T$.