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Hydrodynamics of an actively heaving flexible foil under an incident surface wave

Published online by Cambridge University Press:  14 October 2025

Sang Jin Ji ,
Sung Goon Park Open the ORCID record for Sung Goon Park [Opens in a new window] ,
Ming Li  and
Lian Shen Open the ORCID record for Lian Shen [Opens in a new window]
Sang Jin Ji
Affiliation:
Department of Mechanical Engineering, Seoul National University of Science and Technology, 232 Gongneung-ro, Nowon-gu, Seoul 01811, Korea
Sung Goon Park*
Affiliation:
Department of Mechanical Engineering, Seoul National University of Science and Technology, 232 Gongneung-ro, Nowon-gu, Seoul 01811, Korea
Ming Li
Affiliation:
Department of Mechanical Engineering and St Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN 55455, USA
Lian Shen*
Affiliation:
Department of Mechanical Engineering and St Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN 55455, USA
*
Corresponding authors: Sung Goon Park, psg@seoultech.ac.kr; Lian Shen, shen@umn.edu
Corresponding authors: Sung Goon Park, psg@seoultech.ac.kr; Lian Shen, shen@umn.edu
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Abstract

We numerically investigate the hydrodynamics of an actively heaving flexible foil flapping under a wave surface. The coupled level set and volume-of-fluid method is used to capture the air–water interface, and the immersed-boundary method is used to capture the fluid–structure interaction. A sinusoidal heaving motion is imposed at the foil’s leading edge, and its posterior parts oscillate passively according to its flexible characteristics, allowing dynamic interactions with the wave-induced flow. The propulsive performance of the foil is examined for the influence of three main factors: the ratio of the heaving frequency ($f_{\!f}$) to the wave frequency ($f_w$), the phase difference between the heaving motion and the incident wave ($\mathit \varPhi$) and the submergence depth of the foil ($D$). At $\mathit \varPhi = 0$, the results reveal that the propulsion of the flexible foil benefits from flapping near the wave surface when $f_{\!f}/f_w = 0.5$, and the propulsive efficiency is optimised at $D/L = 1$, where $L$ is the foil’s length. However, when $f_{\!f}/f_w$ = 1.0 and 2.0, the propulsion of the flexible foil is hindered near the wave surface. This hydrodynamic hindrance is closely related to vortex splitting and roll-up phenomena, which induce the formation of a drag wake. By adjusting the phase difference $\mathit \varPhi$, the hindrance in the flexible foil propulsion can be mitigated to enhance propulsive performance. To further understand the relationship between the flapping kinematics and propulsive dynamics, we perform a scaling analysis based on lift force and added mass force, offering good quantification of propulsive performance.

JFM classification

Waves/Free-surface Flows: Wave-structure interactions Vortex Flows: Vortex interactions Biological Fluid Dynamics: Swimming/flying

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1021 , 25 October 2025 , A5
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© The Author(s), 2025. Published by Cambridge University Press

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