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Abstract This paper presents a numerical investigation into the aerodynamic characteristics and fluid dynamics of a flying snake-like model employing vertical bending locomotion during aerial undulation in steady gliding. In addition to its typical horizontal undulation, the modeled kinematics incorporates vertical undulations and dorsal-to-ventral bending movements while in motion. Using a computational approach with an incompressible flow solver based on the immersed-boundary method, this study employs topological local mesh refinement mesh blocks to ensure the high resolution of the grid around the moving body. Initially, we applied a vertical wave undulation to a snake model undulating horizontally, investigating the effects of vertical wave amplitudes ( ${\psi }_m$). The vortex dynamics analysis unveiled alterations in leading-edge vortices formation within the midplane due to changes in the effective angle of attack resulting from vertical bending, directly influencing lift generation. Our findings highlighted peak lift production at ${\psi }_m={2.5}^{\circ }$and the highest lift-to-drag ratio (L/D) at ${\psi }_m=5^{\circ }$, with aerodynamic performance declining beyond this threshold. Subsequently, we studied the effects of the dorsal–ventral bending amplitude ( ${\psi }_{\text{DV}}$), showing that the tail-up/down body posture can result in different fore-aft body interactions. Compared to the baseline configuration, the lift generation is observed to increase by 17.3% at ${\psi }_{\text{DV}}$= 5°, while a preferable L/D is found at ${\psi }_{\text{DV}}$= −5°. This study explains the flow dynamics associated with vertical bending and uncovers fundamental mechanisms governing body–body interaction, contributing to the enhancement of lift production and efficiency of aerial undulation in snake-inspired gliding. 

Abstract This study investigates the interaction of a two-manta-ray school using computational fluid dynamics simulations. The baseline case consists of two in-phase undulating three-dimensional manta models arranged in a stacked configuration. Various vertical stacked and streamwise staggered configurations are studied by altering the locations of the top manta in the upstream and downstream directions. Additionally, phase differences between the two mantas are considered. Simulations are conducted using an in-house developed incompressible flow solver with an immersed boundary method. The results reveal that the follower will significantly benefit from the upstroke vortices (UVs) and downstroke vortices depending on its streamwise separation. We find that placing the top manta 0.5 body length (BL) downstream of the bottom manta optimizes its utilization of UVs from the bottom manta, facilitating the formation of leading-edge vortices (LEVs) on the top manta’s pectoral fins during the downstroke. This LEV strengthening mechanism, in turn, generates a forward suction force on the follower that results in a 72% higher cycle-averaged thrust than a solitary swimmer. This benefit harvested from UVs can be further improved by adjusting the phase of the top follower. By applying a phase difference of $\pi /3$to the top manta, the follower not only benefits from the UVs of the bottom manta but also leverages the auxiliary vortices during the upstroke, leading to stronger tip vortices and a more pronounced forward suction force. The newfound interaction observed in schooling studies offers significant insights that can aid in the development of robot formations inspired by manta rays.