Three-dimensional scaling laws of cetacean propulsion characterize the hydrodynamic interplay of flukes' shape and kinematics

Abstract

Cetaceans convert dorsoventral body oscillations into forward velocity with a complex interplay between their morphological and kinematic features and the fluid environment. However, it is unknown to what extent morpho-kinematic features of cetaceans are intertwined to maximize their efficiency. By interchanging the shape and kinematic variables of five cetacean species, the interplay of their flukes morpho-kinematic features is examined by characterizing their thrust, power and propulsive efficiency. It is determined that the shape and kinematics of the flukes have considerable influence on force production and power consumption. Three-dimensional heaving and pitching scaling laws are developed by considering both added mass and circulatory-based forces, which are shown to closely model the numerical data. Using the scaling relations as a guide, it is determined that the added mass forces are important in predicting the trend between the efficiency and aspect ratio, however, the thrust and power are driven predominately by the circulatory forces. The scaling laws also reveal that there is an optimal dimensionless heave-to-pitch ratio h* that maximizes the efficiency. Moreover, the optimal h* varies with the aspect ratio, the amplitude-to-chord ratio and the Lighthill number. This indicates that the shape and kinematics of propulsors are intertwined, that is, there are specific kinematics that are tailored to the shape of a propulsor in order to maximize its propulsive efficiency.

Form, function, and divergence of a generic fin shape in small cetaceans

“…accordance with known empirical and theoretical scaling relationships between the body mass, fluke's planform, swimming kinematic parameters and swimming speed [28,55,71].…”

“…Different combinations of low and high Λ and AR in tail flukes in cetaceans are related with lift production, reducing induced drag, and structural strength and stiffness of the fin tissues [ 14 , 55 , 68 – 70 ]. Dimensions of the tail flukes are related to the body length in accordance with known empirical and theoretical scaling relationships between the body mass, fluke’s planform, swimming kinematic parameters and swimming speed [ 28 , 55 , 71 ].…”

School cohesion, speed and efficiency are modulated by the swimmers flapping motion

“…These findings could be instrumental for understanding the role of the fluid medium as a mediator of the physical interactions between swimming fish, and to assess the hydrodynamic benefits to fish schooling. Fish have more complex flapping motions than simple heaving and pitching (Lin, Wu & Zhang 2019; Van Buren, Floryan & Smits 2019; Ayancik, Fish & Moored 2020), and the compliance of the fish body is believed to play an important role in the flapping efficiency and ability to extract energy from ambient flows (Beal et al. 2006; Lucas et al.…”

Flapping wing propulsion: Comparison between discrete vortex method and other models

“…The LDVM is able to run on a laptop. Reported efficiencies between 83% and 92% are found [1][2][3][4] for the propulsion by periodic flapping of a fluke used by the cetaceans. These wing flows are developing for much larger Reynolds numbers and amplitudes, combining heave and pitch, maintaining the wing section relative angle of attack to a small value, with an attached flow.…”

“…The periodic vertical plunging of a fluke is a means of propulsion for the cetaceans with efficiencies between 83% and 92% [1][2][3][4] . For a global overview, testifying of the continuous interest in the subject, see the recent review work by Wu et al 5 .…”