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The impact of wavelength and phase difference on the collective performance of two parallel fish

Published online by Cambridge University Press:  06 April 2026

Xingjian Lin Open the ORCID record for Xingjian Lin [Opens in a new window] ,
Haoyuan Liao ,
Hao Xie  and
Jie Wu Open the ORCID record for Jie Wu [Opens in a new window]
Xingjian Lin*
Affiliation:
School of Mechanical Engineering, Nanjing Institute of Technology, Nanjing, Jiangsu 211167, PR China
Haoyuan Liao
Affiliation:
School of Mechanical Engineering, Nanjing Institute of Technology, Nanjing, Jiangsu 211167, PR China
Hao Xie
Affiliation:
School of Mechanical Engineering, Nanjing Institute of Technology, Nanjing, Jiangsu 211167, PR China
Jie Wu
Affiliation:
Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, PR China
*
Corresponding author: Xingjian Lin, xjlinf@njit.edu.cn
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Abstract

The influence of hydrodynamic interactions on the schooling behaviour of fish is still poorly understood. This paper numerically investigates the collective motion of two parallel fish that move freely in both the longitudinal and lateral directions, focusing on the effects of wavelength and phase difference on their stable formations, swimming speed and energy efficiency. It is found that two parallel fish can achieve stable formations in both longitudinal and lateral directions, only via the hydrodynamic interactions. Three distinct modes are classified based on the cycle-averaged longitudinal speed, i.e. the steady slow mode, the steady fast mode and the fluctuating fast mode; which mode occurs depends on the wavelength and phase difference. Compared to a single fish, two fish in the steady slow mode swim slower, whereas they swim faster in both the steady fast and fluctuating fast modes, with a maximum speed increase of 12 % observed in the latter mode. Moreover, the fish school exhibits higher propulsive efficiency than a single fish in most cases. Furthermore, the power consumption and propulsive efficiency of each fish in different modes are discussed in detail. Finally, the mechanism behind the stable formations has been analysed. These results may shed some light on understanding the underlying mechanisms of fish schooling behaviour.

JFM classification

Biological Fluid Dynamics: Collective behaviour Biological Fluid Dynamics: Propulsion Biological Fluid Dynamics: Swimming/flying

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1032 , 10 April 2026 , A47
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Copyright
© The Author(s), 2026. Published by Cambridge University Press

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