A Numerical Tool for the Analysis of Bioinspired Aquatic Locomotion 3031305477, 9783031305474

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SpringerBriefs in Applied Sciences and Technology PoliMI SpringerBriefs

Giovanni Bianchi

A Numerical Tool for the Analysis of Bioinspired Aquatic Locomotion

SpringerBriefs in Applied Sciences and Technology

PoliMI SpringerBriefs Series Editors Barbara Pernici, Politecnico di Milano, Milano, Italy Stefano Della Torre, Politecnico di Milano, Milano, Italy Bianca M. Colosimo, Politecnico di Milano, Milano, Italy Tiziano Faravelli, Politecnico di Milano, Milano, Italy Roberto Paolucci, Politecnico di Milano, Milano, Italy Silvia Piardi, Politecnico di Milano, Milano, Italy

Springer, in cooperation with Politecnico di Milano, publishes the PoliMI SpringerBriefs, concise summaries of cutting-edge research and practical applications across a wide spectrum of fields. Featuring compact volumes of 50 to 125 (150 as a maximum) pages, the series covers a range of contents from professional to academic in the following research areas carried out at Politecnico: • • • • • • • • • • • • • • • •

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Giovanni Bianchi

A Numerical Tool for the Analysis of Bioinspired Aquatic Locomotion

Giovanni Bianchi Dipartimento di Meccanica Politecnico di Milano Milan, Italy

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISSN 2282-2577 ISSN 2282-2585 (electronic) PoliMI SpringerBriefs ISBN 978-3-031-30547-4 ISBN 978-3-031-30548-1 (eBook) https://doi.org/10.1007/978-3-031-30548-1 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Nemo: “Mr. Ray! So how do the stingrays all know where to go?” Mr. Ray: “That’s what an instinct is, Nemo. Something deep inside you that feels so familiar that you have to listen to it. Like a song you’ve always known”. —Finding Dory

Preface

Autonomous Underwater Vehicles (AUVs) have numerous applications, such as scientific research, search and rescue operations, industrial monitoring, and military applications. However, AUVs still have worse swimming capabilities than fishes, which have achieved outstanding swimming performances through natural selection. Fishes have optimized their swimming strategies to adapt to their habitat and maximize their chance of survival, which is achieved thanks to their particular propulsion mechanism based on a different mechanical principle from man-made propellers. Therefore, investigating the locomotion strategy of fish is of great interest to exploit it for the propulsion of AUVs. The evaluation of swimming performance considers aspects such as velocity, maneuverability, energy efficiency, and stealth. Fishes have evolved locomotion strategies with high burst acceleration and speed, low drag coefficient, and high energy efficiency, enabling them to swim for long periods. Maneuvering performance is also essential, with fishes using their flexible bodies and fins to turn rapidly and with a small curvature radius. Finally, biological swimming strategies produce low-intensity noise, while AUVs generate harmonic noise and cavitation. The outstanding swimming abilities of fish provide inspiration for developing biomimetic autonomous underwater vehicles (AUVs) that are maneuverable, energy efficient, and stealthy. However, it is important to consider that fish evolved to maximize survival and reproduction, not for technical performance, and their organs often have multifunctional roles. Therefore, blindly copying every characteristic of fish propulsion may not be effective for designing AUVs, and it is necessary to analyze the underlying physical principles and remove constraints imposed by organs’ multifunctionality and evolutionary history. By understanding the hydrodynamics of fish swimming, it is possible to develop simpler and more optimized engineering adaptations of biological solutions for AUVs. This book presents a technique to analyze the mechanism of fish propulsion by solving the flow hydrodynamics around a flexible object moving in water, as well as its motion dynamics. The software OpenFOAM is utilized to solve the fluid dynamics, while an overset grid is used to manage the moving boundary. Additionally, a library

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Preface

has been developed to combine the built-in rigid body motion solver with a predetermined deformation of the object. This method has been used to analyze the hydrodynamics of a cownose ray, but it can be applied to any type of deformable underwater vehicle with known deformation kinematics. The intended audience for this book ranges from students to researchers and AUV designers who are interested in investigating biological swimming propulsion. The first two chapters introduce the general principles of swimming hydrodynamics, and the last two chapters describe the developed method and the results obtained with the application of this method to the swimming dynamics of the cownose ray. The genesis of this book can be attributed to the research undertaken throughout the duration of my doctoral program and I would like to express my sincere gratitude to Prof. Simone Cinquemani for his support and guidance throughout my doctoral studies. His expertise and insights were fundamental in shaping my research, and I am grateful for his mentoring and encouragement. Finally, I would like to extend my gratitude to my family for their unwavering support in enabling me to pursue my passions and achieve my academic goals. Milan, Italy March 2023

Giovanni Bianchi

Contents

1 Introduction and State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 7

2 Hydrodynamics of Swimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Classification of Swimming Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hydrodynamic Principles of Fishes and Cetacean Locomotion . . . . . 2.2.1 Swimming of a Waving Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Slender Body Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Vortices in the Wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Efficiency of Fish Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 14 15 21 27 36 38

3 Model of Cownose Ray Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Rajiform Swimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Bio-mechanical Model of Fin Kinematics . . . . . . . . . . . . . . . . . . . . . . . 3.3 CFD Model of Cownose Ray Forward Swimming . . . . . . . . . . . . . . . . 3.3.1 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 43 50 52 57

4 Wake Structure and Swimming Performance of the Cownose Ray . . . 4.1 Vortex Structures in the Wake and on the Leading Edge . . . . . . . . . . . 4.1.1 Leading-Edge Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Influence of Kinematic Parameters on Swimming Performances . . . . 4.2.1 Wavelength Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Frequency Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Angle of Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Strouhal Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 67 69 69 70 73 74 76

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Acronyms

AOA AUV BCF BL CFD COT DPIV FSI IBM LEV MPF NACA RANS SST URANS

Angle of Attack Autonomous Underwater Vehicle Body Caudal Fin Body Length Computational Fluid Dynamics Cost of Transport Digital Particle Image Velocimetry Fluid-Structure Interaction Immersed Boundary Method Leading-Edge Vortex Median Paired Fin National Advisory Committee for Aeronautics Reynolds-Averaged Navier-Stokes Shear Stress Transport Unsteady Reynolds-Averaged Navier-Stokes

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Chapter 1

Introduction and State of the Art

Abstract Autonomous Underwater Vehicles (AUVs) are increasing in popularity because they can explore the ocean’s depths and perform operations without risking human life, and their applications range from search and rescue to scientific research and military operations. However, despite recent progress, AUVs still have worse swimming capabilities than fishes. The locomotion strategies of fishes have achieved outstanding swimming performances because they have evolved through natural selection for millions of years, so investigating how fishes propel themselves is of great interest to exploit the same mechanism for AUV propulsion. An effective bioinspired design cannot be carried out by blindly copying all the characteristics of fish locomotion, but it should be based on a deep understanding of the underlying physical principles that make fish swim so efficiently. Fish swimming is a complex phenomenon involving the unsteady fluid dynamics of a deformable moving body immersed in water, and advanced numerical tools are needed to figure out the mechanism of fish propulsion. In this chapter, the most popular numerical methods used to analyze fish swimming are described, and the main novel aspects of the proposed technique based on an overset grid are introduced.

The interest in the development of Autonomous Underwater Vehicles (AUVs) has recently increased thanks to the technological progress in robotics [1]. Since the deep ocean is a hostile environment inaccessible to humans, which remains largely unexplored, AUVs open the possibility to explore the ocean seabed and perform operations without risks [1, 2]. Therefore, the applications for AUVs are manifold, ranging from search & rescue to scientific research or military operations. Equipping an AUV with a camera, it is possible to study underwater wildlife and map the seafloor for oceanographic, archaeological, or geological research; in addition, AUVs can be fitted with sensors for environmental monitoring. These vehicles are also employed for search & rescue operations to find survivors after natural disasters in flooded areas and to look for sunken ships or crashed airplanes. AUVs are also interesting for their industrial applications, such as the monitoring or even the repair of underwater infrastructures like oil & gas pipelines and telecommunication cables or the inspection of the seafloor to look for natural resources and to select suitable places to build new structures. Finally, AUVs are employed for military applications © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Bianchi, A Numerical Tool for the Analysis of Bioinspired Aquatic Locomotion, PoliMI SpringerBriefs, https://doi.org/10.1007/978-3-031-30548-1_1

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like surveillance, mine countermeasures, and anti-submarine warfare [2]. However, despite the recent progress in the development of AUVs, motivated by their numerous applications, they still have worse swimming capabilities than fishes [3]. The locomotion strategies of fishes have achieved outstanding swimming performances because they have been evolving for millions of years through natural selection as the first organisms that appeared on Earth were aquatic, and swimming is the first kind of locomotion they developed [4–6]. Seas and rivers are populated by a huge number of animals with different dimensions and behaviors, which have optimized their swimming strategies to adapt to their habitat and maximize their chance of survival. Hence, in nature, it is possible to find animals that swim at high speed with great energy efficiency or that move with extreme agility and excellent stealth capabilities [3, 6–8]. This expertise is achieved thanks to their particular propulsion mechanism, which is based on a different mechanical principle from the one used by man-made propellers and allows animals to achieve unparalleled performances [3]. Therefore, it is of great interest to investigate the locomotion strategy of fish in order to exploit it for the propulsion of AUVs. The main aspects relevant to evaluate swimming performances are swimming velocity, maneuverability, energy efficiency, and stealth [3]. The maximum speed is one of the most important performances for swimming animals since it is fundamental for escaping predators or chasing prey; thus, natural selection has led to the development of locomotion strategies featured by very strong burst acceleration and high speed, such as the one adopted by tunas and sharks, which can travel at 15 ∼ 20 ms−1 [9]. These fishes periodically oscillate their caudal fin, being propelled by the high lift force produced [6, 7, 10, 11], and they are much faster than the conventionally-propelled AUVs, that reach a maximum speed of 5.14 ms−1 [3]. Thrust is not the only factor affecting the maximum speed because also drag reduction mechanisms play a key role. The low drag coefficient is primarily due to the streamlined shape of fishes [12] whose fins have a rounded leading edge and a tapered trailing edge, which delay flow separation [3]. Moreover, fish skin is provided with other drag reduction mechanisms, such as microgrooves on the scales [13] and tubercles on the leading edge of the fins [14], affecting the flow in the boundary layer, or a hydrophobic substance on the skin surface reducing the drag coefficient [15]. Fishes swim at their maximum speed only for a short time since it requires high power consumption, so they usually swim at a lower cruising speed maximizing the energy efficiency of locomotion, which is another fundamental parameter of swimming locomotion associated with long endurance. Many fishes migrate through the oceans, relentlessly swimming for days without eating, and the manta ray never stops swimming for its entire lifespan. This would not be possible if the swimming strategy adopted were not featured by great energy efficiency, which is the most interesting characteristic of fish locomotion for the design of bioinspired AUVs. The most efficient swimmers are tunas, manta rays, and cetaceans, which reach an efficiency between 80 and 90% [3, 12, 16], meaning that the power lost in the wake is just a minimal fraction of the input power. Conversely, the maximum efficiency achieved by thrust propellers never exceeds 70%, and thrusters work with maximum energy

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efficiency only in a small velocity range, whereas fish swimming is characterized by very high efficiency in a wide speed range [3]. Another aspect that influences swimming is maneuvering performance, in particular, agility, i.e., the capability to turn with high angular speed without losing stability, and maneuverability, i.e., the ability to turn with a small curvature radius. Many fishes that use their caudal fin to propel, like tunas and pickerels, exploit the flexibility of their bodies to perform incredibly fast start maneuvers such as C-turns, where the animal bends its body in a “C” shape and then straightens, gaining high starting acceleration and turning very rapidly [17, 18]. The angular speed achieved with this maneuver is more than seven times as high as the maximum reached by AUVs [3]. Furthermore, a flexible body, combined with a high number of fins and appendages involved in the movement, represents a considerable advantage in terms of maneuverability for several animals, like squids, boxfishes, and rays, which can turn in place with a null curvature radius [19–21]. Conversely, a null curvature radius is impossible to achieve with torpedo-shaped AUVs, which usually have a long body and only one thruster on the posterior part [3]; nevertheless, some AUVs are equipped with several thrusters so they can perform null curvature radius turns as well [22]. Finally, another important advantage of biological swimming strategies is stealth since they only produce low-intensity noise due to the periodic displacement of the surrounding water, and some species, like the humpback whale, have fins with tubercles on their leading edge, which reduce vortex shedding, further decreasing the generated noise [3]. In contrast, thrusters that propel AUVs rotate at high speed generating harmonic noise and causing cavitation on the blades. The implosion of the bubbles generated by cavitation creates a shockwave, which is the prevalent source of noise for AUVs [23]. The outstanding swimming performances of fishes suggest that it is possible to solve the challenges AUVs face in terms of maneuverability, energy efficiency, and stealth just by replicating all the features of fish propulsion in a biomimetic robot. However, it must be considered that the locomotion strategies are the result of natural selection; thus, animals did not evolve to optimize a specific technical performance but to maximize their chances of surviving and reproducing [24]. Therefore, fishes, like any other living being, perform several functions other than swimming, and their organs are often multifunctional, like the dorsal fins of the sea robin, which do not generate any thrust but are only used for social signaling or the fins of the bluegill sunfish, which have a mechanosensory function [25]. Hence, during the design of a bioinspired autonomous vehicle, it is misleading to think that every characteristic of the animal’s body is fundamental for the function of interest, swimming in this case, and it must be recognized that organs’ multifunctionality introduces several constraints which can be relaxed for a biomimetic robot, simplifying the design process and potentially leading to better performances. Furthermore, the constraints imposed by the evolutionary path of the species under consideration must be taken into account to understand in which sense a biological solution is optimal. For example, for aquatic animals breathing with gills is much more energy efficient than with lungs because swimming near the water surface implies an additional drag. However, despite millions of years of evolution, cetaceans never developed gills because

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they evolved from other mammals, so they are constrained to use lungs for breathing [26]. Hence, blindly copying nature is not an effective solution to face engineering challenges, but it is necessary to analyze the biological solution to understand which is the main underlying physical principle and remove all the constraints imposed by organs’ multifunctionality and the evolutionary path. This way, it is possible to find an engineering adaptation of the biological solution that is less complex and more optimized for specific technical performance. Therefore, to develop highly performant and efficient bioinspired AUVs, it is useful to investigate the hydrodynamics of fish swimming to understand the geometric and kinematic characteristics that make them so efficient. The first step to fully understanding fish swimming dynamics is the investigation of how fishes are actuated by analyzing their neuro-muscular activity and how the kinematics of their movement affects the motion of the surrounding fluid and the generated forces. An accurate observation of animals’ internal structure is fundamental since it allows to figure out the mechanical properties of their tissues, like stiffness and mass distribution, and how they affect locomotion dynamics. The internal biomechanics of fishes are far from being completely understood; however, thanks to an accurate analysis of the skeletal structure and to experimental techniques like electromyography, biologists can reconstruct the kinematics of the movement of most fish species [27, 28]. Moreover, measuring the neuro-muscular activity while the fish is swimming provides information about how the external flow influences the swimming strategy of the animal [28]. This analysis highlights the biological aspects but cannot explain the underlying physical principle of fish locomotion because this is related to the interaction with the fluid. Hence, it is essential to figure out how the flow of the surrounding fluid is affected by the animal’s motion. The most widespread experimental technique to visualize flow is Digital Particle Image Velocimetry (DPIV), which consists of seeding the flow with particles and tracking their motion with a laser and a high-speed digital camera. This technique makes visible the velocity field around a fish putting in evidence the vortices shed in the wake and the water displaced by the fish movement [29], and it has been applied to study several fish species during steady-state swimming and maneuvering [29–33]. DPIV gives a very accurate description of the velocity field, but it does not allow direct quantification of the pressure field and of the forces exchanged by the fish and the surrounding water [27]. Furthermore, this technique requires using a real live fish, and it is extremely challenging to make it swim in the desired way and control its kinematic parameters [34]. Thus, numerical approaches are more common, simulating the dynamics of the fish’s body and the hydrodynamic of the surrounding fluid. Fish locomotion involves the dynamics of an actively deformable body immersed in a fluid, resulting in a complex problem of Fluid-Structure Interaction (FSI), where the body movement affects the flow of the surrounding fluid, generating forces that propel the fish and, at the same time, affect its body deformation. A numerical simulation of fish swimming includes a fluid-dynamics simulation solving the flow coupled with a multibody simulation solving the dynamics of the fish body. Generally, the fin and body deformation can be accurately reconstructed from the observation of swimming fishes, and it is used

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as input to the simulations to calculate pressure and velocity fields and compute the forces applied by the animal to move [27]. Hence, body deformation is imposed as a boundary condition to a Computational Fluid Dynamics (CFD) solver, which gives, as result, the characteristics of the surrounding flow and the forces exchanged with the body. Then, it is possible to provide as an input to a multibody dynamics solver the body deformation and the forces applied by the fluid previously computed to calculate the forces required for the fish to move. This work focuses on the solution of fluid dynamics, and the main element of innovation regards the way a moving boundary like the fish surface is treated. The simplest way to treat a moving boundary is by using a deformable mesh that follows the deformation of the fish. The grid-deforming algorithms move the mesh points on the boundaries according to the motion prescribed by the moving surface and impose a zero displacement condition on the points sufficiently far from the moving boundary. The displacement of all the points close to the moving surface is smoothed to avoid excessive element distortion and have a gradual transition from the moving boundary to the still mesh region [35, 36]. This method is widely employed for simulating animal swimming, Liu et al. employed this strategy to analyze the hydrodynamics of a tadpole [37], Chen et al. studied the swimming locomotion of batoids with a moving mesh [38], Safari et al. used a deformable grid to study the flapping motion of a manta ray [39], Fouladi et al. used this strategy to study the effect of vortices on the swimming strategy of trouts [40], and Doi et al. with this method investigated why salmons swim parallel to each other [41]. The main limitation of deformable meshes is that only small movements of the fish can be simulated; otherwise, the mesh elements distort excessively or even overlap. Thus, it is impossible to simulate the swimming locomotion in large of a fish, and usually, the fish’s center of mass is still, and only its fins are moving. To simulate a fish swimming forward, a relative velocity between the fish and the fluid is imposed by adding a constant velocity as a boundary condition at the inlet. This way, null acceleration is imposed on the fish’s body, so this method can give accurate results only when the fish swims at a constant speed. However, this condition is never met because of the intrinsic unsteadiness of the flow around a swimming body, which generates periodic forces and always has acceleration different from zero, even at the steady state. Therefore, the results are more accurate if the animal’s body is moved forward according to the acceleration resulting from the applied forces. To overcome the limitations of a deformable grid, adaptive remeshing can be used, deleting the elements not satisfying quality criteria and substituting them with new elements from an undistorted background grid, as Ramamurti et al. did to simulate the swimming locomotion of eels and carangiform fishes [35]. This method was also used by Li et al., who simulated the propulsion of a thunniform fish [42], and by Li et al., who investigated the hydrodynamics of a pufferfish [43]. Adaptive mesh smoothing and refinement ensures a mesh of good quality over the whole domain, but it implies topological changes to the grid at every time step, significantly increasing the computational effort. A different approach is represented by the Immersed Boundary Method (IBM), a non-boundary-conforming methodology in which the grid is fixed, and the body

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moves across the grid. The effect of the moving boundary is accounted for by adding an external forcing term in the equation of motion to ensure a no-slip velocity condition on the body surface. The boundary is discretized as a series of points, and this force is applied to the points of the grid overlapping the boundary [44]. Several variants of this method exist, and they differ according to how they treat the application of the force on the grid nodes and how they couple the fluid dynamics with the dynamics of the body representing the moving boundary. This method is suitable for simulating the forward swimming of an animal or when large deformations are involved because the grid does not distort following the body movement and because the mesh can always be structured with very high quality since the complex geometry of the moving boundary is not meshed. This feature makes IBM very suitable for the simulation of fish swimming, and it has been employed for several studies of fish swimming hydrodynamics. Borazjani et al. used the IBM method to compute the forces and analyze the vortex patterns of carangiform and thunniform swimming [45–47], Dong et al. employed it to find the forces generated by the fins of a bluegill sunfish [48]; Maertens et al. employed this methodology to run numerical simulations of fish swimming to compute the energy efficiency of their locomotion [49]. Bottom et al. used a curvilinear IBM to investigate the hydrodynamics of stingrays [50]; Cui et al. developed an IBM with a sharp interface to simulate BCF swimming and analyze the vortices in the wake [51]. Thekketil et al. proposed an improved version of IBM called Immersed Interface Method in which the structural dynamics of the moving body are solved too, and the velocity continuity condition is imposed directly on the boundary rather than interpolating it on the grid points and used to simulate batoid locomotion [52]. The same method was used by Gupta et al. to study the hydrodynamics of anguilliform and carangiform swimmers [53]. Not meshing the surface of the moving body is the main strength of this method because it avoids problems related to mesh distortion while the body moves, but at the same time, it does not allow to accurately solve the flow in the boundary layer due to the lack of a high-resolution, layered mesh in correspondence to the boundary. Thus, it is difficult to simulate phenomena like flow separation and reattachment with the IBM method [54]. The limitations of the immersed boundary method can be overcome by adopting an overset grid, which consists of two overlapping meshes, one still in the background and one attached to the moving body, which deforms and moves along with it. The solution is obtained by interpolating one grid on the other, obtaining more accurate results than IBM because the boundary layer can be solved with a high-resolution, layered mesh [55]. The elements on the background mesh cannot be distorted because they do not move, and the moving mesh quality is not affected by the forward motion of the body because it only undergoes small distortions due to the deformation of the body. Since the results obtained with this method are more reliable than those obtained with other methods, overset meshes are used to simulate very complex phenomena, like prey capturing by suction-feeding fishes or the swimming locomotion of fish larvae [56]. Li et al. used this method to simulate the collective behavior of a red nose tetra fish school [57], and Liu et al. studied the dependencies of the swimming performance of a BCF swimmer on kinematics parameters adopting an overset grid

References

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[58]. Xu et al. studied with this strategy the locomotion of a fish with rigid pectoral fins [59]; Yang et al. coupled the CFD solver based on an overset grid with a reinforcement learning tool to develop a control algorithm for the fish [60], and Sumikawa et al. analyzed the stability of batoid swimming during asymmetric fin flapping [61]. The method described in this book allows for solving the hydrodynamics of the flow around a deformable body moving in a fluid and the dynamics of its motion. The open-source software OpenFOAM was used to solve the fluid dynamics, an overset grid is used to deal with the moving boundary, and a library was implemented to combine the in-built rigid body motion solver with a prescribed deformation of the body. This tool has been applied to study the hydrodynamics of a cownose ray, and it can be applied to any kind of deformable underwater vehicle provided that its deformation kinematics is known in advance. The remaining part of this book is organized as follows. Chapter 2 presents the hydrodynamics of fish swimming, describing the different types of fish locomotion, illustrating simplified analytical models of fins that explain the underlying physical principle of fish propulsion, and relating the thrust generation mechanism to the vortex in the wake. Finally, some considerations about the energy efficiency and dimensionless parameters of swimming are made. Chapter 3 describes the kinematics of the cownose ray’s fin motion and illustrates how it was implemented in the CFD model to solve the hydrodynamics and the forward swimming dynamics of the fish. The results of several simulations are presented, relating the wake structure and the swimming performance to the kinematic parameters of fin movement.

References 1. Bogue R (2015) Underwater robots: a review of technologies and applications. Ind Robot: An Int J 42(3):186–191 2. Sahoo A, Dwivedy SK, Robi PS (2019) Advancements in the field of autonomous underwater vehicle. Ocean Eng 181:145–160 3. Fish FE (2020) Advantages of aquatic animals as models for bio-inspired drones over present AUV technology. Bioinspiration & Biom 15:025001 4. Gordon MS, Blickhan R, Dabiri JO, Videler JJ (2017) Animal locomotion: physical principles and adaptations. CRC Press—Taylor & Francis Group 5. Newman JN, Wu TY (1974) Swimming and flying in nature—volume 2. In: Wu TY, Brokaw CJ, Brennen C (eds), Proceedings of the second half of the symposium on swimming and flying in nature, vol 2 6. Sfakiotakis M, Lane DM, Davies J (1999) Review of fish swimming modes for aquatic locomotion. IEEE J Ocean Eng 24(2):237–252 7. Salazar R, Fuentes V, Abdelkefi A (2018) Classification of biological and bioinspired aquatic systems: a review. Ocean Eng 148:75–114 8. Buren T, Floryan D, Smits A (2020) Bioinspired underwater propulsors. Cambridge University Press, Cambridge 9. Videler JJ, Wardle CS (1991) Fish swimming stride by stride: speed limits and endurance. Rev Fish Biol Fish 1:23–40 10. Lighthill MJ (1969) Hydromechanics of aquatic animal propulsion. Ann Rev Fluid Mech 1:413–446 11. Lighthill MJ (1969) Note on the swimming of slender fish. J Fluid Mech 9:305–317

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12. Webb PW (1975) Hydrodynamics and energetics of fish propulsion. Department of the Environment Fisheries and Marine Services of Canada 13. Videler JJ (1995) Body surface adaptations to boundary-layer dynamics. Symposia Soc Exp Biol 49:1–20 14. Miklosovic D, Murray M, Howle L, Fish F (2004) Leading-edge tubercles delay stall on humpback whale (megaptera novaeangliae) flippers. Phys Fluids 16(39) 15. Daniel TL (1981) Fish mucus: in situ measurements of polymer drag reduction. Biol Bull 160(3) 16. Fish FE, Schreiber CM, Moored KW, Liu G, Dong H, Bart-Smith H (2016) Hydrodynamic performance of aquatic flapping: efficiency of underwater flight in the manta. Aerospace 3(20):3030020 17. Su Z, Yu J, Tan M, Zhang J (2014) Implementing flexible and fast turning maneuvers of a multijoint robotic fish. IEEE/ASME Trans Mechatron 19(1) 18. Weihs D (1973) The mechanism of rapid starting of slender fish. Biorheology 10:343–350 19. Parson JM, Fish FE, Nicastro AJ (2011) Turning performance of batoids: limitations of a rigid body. J Exp Marine Biol Ecol 402:12–18 20. Jastrebsky RA, Bartol IK, Krueger PS (2016) Turning performance in squid and cuttlefish: unique dual-mode, muscular hydrostatic systems. J Exp Biol 219:1317–1326 21. Walker JA (2000) Does a rigid body limit maneuverability? J Exp Biol 203:3391–3396 22. Wang X, Liang S (2019) Maneuverability analysis of a novel portable modular AUV. Math Probl Eng 23. Carlton J (2012) Marine propellers and propulsion. Butterworth-Heinemann 24. Bar-Cohen Y (2011) Biomimetics: nature based innovation. Taylor & Francis 25. Hale ME, Galdston S, Arnold BW, Song C (2022) The water to land transition submerged: multifunctional design of pectoral fins for use in swimming and in association with underwater substrate. Integr Comp Biol 62(4):908–921 26. Fish FE, Beneseki JT (2013) Evolution and bio-inspired design: natural limitations. Springer 27. Liu Y, Jiang H (2022) Research development on fish swimming. Chinese J Mech Eng 35(114) 28. Liao JC, Beal DN, Lauder GV, Triantafyllou MS (2003) Fish exploiting vortices decrease muscle activity. Science 302:1566–1568 29. Liao JC (2007) A review of fish swimming mechanics and behaviour in altered flows. Phylosophical Trans R Soc B 362:1973–1993 30. Lauder GV (2000) Function of the caudal fin during locomotion in fishes; kinematics, flow visualization, and evolutionary patterns. Am Zoolog 40:101–122 31. Tytell ED, Lauder GV (2004) The hydrodynamics of eel swimming: I. Wake structure. J Exp Biol 207(11):1825–1841 32. Clark RP, Smits AJ (2006) Thrust production and wake structure of a batoid-inspired oscillating fin. J Fluid Mech 562:415–429 33. Thandiackal R, Lauder GV (2020) How zebrafish turn: analysis of pressure force dynamics and mechiancal work. J Exp Biol 223 34. Flammang BE, Porter ME (2011) Bioinspiration: applying mechanical design to experimental biology. Integr Comp Biol 51(1):128–132 35. Ramamurti R, Lohner R, Sandberg WC (1999) Computation of the 3-d unsteady flow past deforming geometries. Int J Comput Fluid Dyn 13:83–99 36. Ebrahimi M, Abbaspour M (2015) Numerical investigation of the forward and backward travelling waves through an undulating propulsor: performance and wake pattern. Ships Offshore Struct 37. Liu H, Wasserug R, Kawachi K (1997) The three-dimensional hydrodynamics of tadpole locomotion. J Exp Biol 200:2807–2819 38. Chen W, Wu Z, Liu J, Shi S, Zhou Y (2011) Numerical simulation of batoid locomotion. J Hydrodyn 23(5):594–600 39. Safari H, Abbaspour M, Darbandi M (2021) Numerical study to evaluate the important parameters affecting the hydrodynamic performance of manta ray’s in flapping motion. Appl Ocean Res 109:102559

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40. Fouladi K, Coughlin DJ (2021) CFD investigation of trout-like configuration holding station near an obstruction. Fluids 6(204) 41. Doi K, Takagi T, Mitsunaga Y, Torisawa S (2020) Hydrodynamical effect of parallelly swimming fish using computational fluid dynamics method. PLoS One 16(5):e0250837 42. Li N, Liu H, Su Y (2017) Numerical study on the hydrodynamics of thunniform bio-inspired swimming under self-propulsion. PLoS One 12(3):e0174740 43. Li R, Xiao Q, Liu Y, Liu L, Liu H (2020) Computational investigation on a self-propelled pufferfish driven by multiple fins. Ocean Eng 197:106908 44. Gilmanov A, Sotiropoulos F (2005) A hybrid cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies. J Comput Phys 207:457–492 45. Borazjani I, Sotiropoulos F (2008) Numerical investigation of the hydrodynamics of carangiform swimming in the rtansitional and inertial flow regimes. J Exp Biol 211:1541–1558 46. Borazjani I, Sotiropoulos F (2010) On the role of form and kinematics on the hydrodynamics of self-propelled body/caudal fin swimming. J Exp Biol 23(89–107) 47. Borazjani I, Daghooghi M (2012) The fish tail motion forms an attached leading edge vortex. Proc R Soc B 280:20122071 48. Dong H, Bozkurttas M, Mittal R, Madden P, Lauder GV (2010) Computational modelling and analysis of the hydrodynamics of a highly deformable fish pectoral fin. J Fluid Mech 645:345–373 49. Maertens AP, Triantafyllou MS, Yue DKP (2015) Efficiency of fish propulsion. Bioinspiration & Biomim 10:046013 50. Bottom RG, Borazjani I, Blevins EL, Lauder GV (2016) Hydrodynamics of swimming in stingrays: numerical simulations and the role of the leading-edge vortex. J Fluid Mech 788:407– 443 51. Cui Z, Yang Z, Jiang H (2020) Sharp interface immersed boundary method for simulating three-dimensional swimming fish. Eng Appl Comput Fluid Mech 14(1):534–544 52. Thekkethil N, Sharma A, Agrawal A (2020) Three-dimensional biological hydrodynamics study on various types of batoid fishlike locomotion. Phys Rev Fluids 5:023101 53. Gupta S, Agrawal A, Hourigan K, Thompson MC, Sharma A (2022) Anguilliform and carangiform fish-inspired hydrodynamic study for an undulating hydrofoil: effect of shape and adaptive kinematics. Phys Rev Fluids 7(9):094102 54. van Noordt W, Ganju S, Brehm C (2022) An immersed boundary method for wall-modeled large-eddy simulation of turbulent high-mach-number flows. J Comput Phys 111583 55. Vreman AW (2020) Immersed boundary and overset grid methods assessed for stokes flow due to an oscillating sphere. J Comput Phys 423:109783 56. Li G, Muller UK, van Leeuwen JL, Liu H (2012) Body dynamics and hydrodynamics of swiming fish larvae: a computational study. J Exp Biol 215:4015–4033 57. Li G, Kolomenskiy D, Thiria B, Godoy-Diana R (2019) On the interference of vorticity and pressure fields of a minimal fish school. J Aero Aqua Bio-Mech 8(1):27–33 58. Liu J, Yu F, He B, Yan T (2022) Hydrodynamic numerical simulation and prediction of bionic fish based computational fluid dynamics and multilayered perceptron. Eng Appl Comput Fluid Mech 16(1):858–878 59. Xu Y, Wan D (2022) Numerical simulation of fish swimming with rigid pectoral fins. J Hydrodyn 24(2) 60. Yan L, Chang X, Tian R, Wang N, Zhang L, Liu W (2020) A numerical simulation method for bionic fish self-propelled swimming under control based on deep reinforcement learning. Fluid Mech 234(17):3397–3415 61. Sumikawa H, Naraoka Y, Fukue T, Miyoshi T (2022) Changes in rays’ swimming stability due to the phase difference between left and right pectoral fin movements. Nat Sci Rep 12:2362

Chapter 2

Hydrodynamics of Swimming

Abstract This chapter presents the physical principles that allow fishes to propel themselves and analyzes why they move with high energy efficiency while having outstanding swimming performances. Firstly, the different strategies of swimming locomotion of fishes and cetaceans are described emphasizing the differences in the characteristics of the surrounding flow. Then, two analytical models of fish swimming are introduced and compared: the Slender Body Theory and the Waving Plate Model. These models lie on several simplifying assumptions, which do not reflect the actual fish geometry and behavior; still, they are very useful to understand the general basic principles of fish propulsion from a mathematical point of view. The same simplified approach is then used to analyze the vortices in the wake and relate them to the produced thrust. Finally, it is described how the energy efficiency is measured for a self-propelling body, and it is related to the Strouhal number.

2.1 Classification of Swimming Modes Swimming propulsion is exploited by aquatic animals covering a wide range of body sizes and speeds, from blue whales, which are 30 m long and can reach a cruise speed of 12 m/s, to protists, characterized by a length of 50 µm, and by a swimming velocity of 80 µm/s. The physical principles used by live beings to swim differ according to their size since the characteristics of the flow in which swimming animals are immersed depend considerably on the animal dimensions. The dimensionless parameter that allows quantifying the contribution of the different forces to momentum transfer is the Reynolds number, defined as follows: Re =

Ul ν

(2.1)

where U is the velocity of the flow, l is the characteristic length of the animal, and ν is the kinematic viscosity of the fluid, which for water is equal to 1×10−6 m2 /s. The Reynolds number represents the ratio between inertia and viscous forces in a flow. The flow of large animals is mainly dominated by inertia forces, and its © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Bianchi, A Numerical Tool for the Analysis of Bioinspired Aquatic Locomotion, PoliMI SpringerBriefs, https://doi.org/10.1007/978-3-031-30548-1_2

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2 Hydrodynamics of Swimming

Reynolds number can even reach 108 for large cetaceans, whereas smaller animals swim with a much lower Reynolds number since, at that scale, viscous forces are predominant over inertia, and for the smallest live beings like microorganisms, the Reynolds number is 10−3 [1, 2]. While it is straightforward to identify the kinematic viscosity of the fluid, it is not as easy to define the characteristic length, and the velocity of the flow since the characteristic length is related both to the kinematics of fin movements and to the dimensions of the animal, and the velocity of the flow is the superimposition of forward locomotion and fin movement. Hence, it is necessary to introduce the Swimming number, a new dimensionless parameter capturing both length and motion scales [2]: ω AL (2.2) Sw = ν where ω is the circular frequency of fin movement, A is the maximum amplitude of fin displacement, L is the length of the animal, and ν is again the kinematic viscosity of water. In an incompressible and inviscid flow, a deforming body accelerates a mass of surrounding fluid that scales with ρ L 2 per unit span, and the fluid acceleration scales with Aω2 , so that the inertia force of the fluid scales with ρ L 2 Aω2 . Since the amplitude of fin movement is generally proportional to the animal length, the local angle formed by the fin scales with A/L, and the thrust force is ≈ ρ L A2 ω2 [2]. For a low Reynolds number (< 104 ), the boundary layer is laminar and the viscous drag resisting to motion scales with ρ(ν L)1/2 U 3/2 according to Blasius theory. Writing an equilibrium of forces in the longitudinal direction so that thrust is equal to drag, it is possible to obtain that the scaling of the forward velocity U : U ≈ L 1/3 ν −1/3 A4/3 ω4/3

(2.3)

Substituting the scaling of U into Eq. 2.1, it is possible to obtain: Re ≈

L 1/3 ν −1/3 A4/3 ω4/3 L = Sw 4/3 ν

(2.4)

On the other side, for a high Reynolds number (> 104 ), the boundary layer becomes turbulent and the pressure drag is predominant over the viscous drag, so that the drag force per unit span scales with ρU 2 L. Therefore, the scaling of the velocity U corresponds to [2]: U ≈ Aω, (2.5) and substituting it into Eq. 2.1, it results that: Re ≈

AωL = Sw ν

(2.6)

2.1 Classification of Swimming Modes

13

Fig. 2.1 Relative contribution of forces to propulsion as a function of the Reynolds number

This scaling law is confirmed by experimental observations, and it allows computing the Reynolds number of the flow combining the characteristic dimensions of the animal and of its swimming gait [2]. The momentum transfer mechanisms adopted by live beings can be based on four different forces: • viscous drag: friction between the animal skin and the fluid. It is the consequence of the viscosity of water and a velocity gradient in the boundary layer. It depends on the swimming speed and the nature of the boundary layer. • pressure/form drag: pressure caused by the flow distortion as the body pushes water aside to pass. It depends on the shape of the animal, and the majority of fast-swimming animals have a streamlined shape to reduce this force. • acceleration reaction: the inertia of water accelerated by the fins. It is strongly dependent on the size of the fish and it is fundamental in periodic movements. • lift: hydrodynamic force acting in a direction perpendicular to the fin. It contributes to propulsion when the fin rotates about the pitch or yaw axis. The relative contribution of these forces to propulsion is different according to the Reynolds number of the flow, as shown in Fig. 2.1 [3]. Swimming propulsion mechanisms can be classified into four categories, whose main characteristics are summarized in Table 2.1 and described in the following sections. Combining the information contained in Table 2.1 and in Fig. 2.1, it is possible to observe that the Reynolds number of flagellar propulsion is extremely low, so viscous forces govern their motion and that for jet propulsion, featured by a medium-low Reynolds number, there is a contribution of drag, as well as of inertia. Finally, bodycaudal fin (BCF) and median paired fin (MPF) locomotion strategies are featured by high Reynolds number, so inertia and lift forces are predominant. This means

Table 2.1 Classification of biological swimming modes and their main characteristics Flagellar Jet propulsion Median paired fin Body caudal fin propulsion Speed Reynolds number Efficiency Maneuverability

80 µm/s–1 mm/s 104 ∼ 100 10 ∼ 20% Very low

2 cm/s–1 m/s 100 ∼ 102 20 ∼ 50% Very low

0.1 m/s–5 m/s 103 ∼ 106 50 ∼ 90% Very high

0.2 m/s–30 m/s 103 ∼ 108 40 ∼ 90% High

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2 Hydrodynamics of Swimming

Fig. 2.2 Terminology used to identify the fins of fishes [3]

that the physical principles of flagellar and jet propulsion swimming are peculiar to these swimming strategies. In contrast, the body-caudal fin and median-paired fin swimming are based on similar principles, and they mainly differ by the parts of the body performing the movements responsible for thrust generation. Referring to the nomenclature presented in Fig. 2.2, the body-caudal fin swimming strategy involves propagating a wave along the body, culminating in a sizeable periodic displacement of the tail. Conversely, in the median-paired fin swimming mode, thrust is generated by the coupled movements of pectoral fins or dorsal and anal fins. Their movement often consists of a traveling wave similar to BCF swimming mode, but with the substantial difference that the body remains still, and the wave is present only on the surface of the fin. The BCF swimming mode is predominant among fishes, as only 15% use MPF swimming for propulsion. Nevertheless, BCF swimming animals can use MPF swimming strategies for maneuvering and stabilization.

2.2 Hydrodynamic Principles of Fishes and Cetacean Locomotion Fishes and cetaceans move with a high Reynolds number, and their swimming propulsion depends primarily on the inertia force. The fluid’s viscosity has an important role in creating the vorticity shed into the wake, whereas the viscous drag causing skin friction is of marginal importance, as it affects the flow only in a thin boundary layer surrounding the body surface. This means that thrust generation and drag resistance can be studied separately [1]. Both BCF and MPF swimming modes are characterized by a periodic motion of the fins pushing water backward, so a forward thrust is obtained thanks to momentum conservation. During constant velocity swimming, the momentum in the backward

2.2 Hydrodynamic Principles of Fishes and Cetacean Locomotion

15

jet created by the fin movement is counterbalanced by the momentum created by viscous friction as the fish drags forward some of the fluid surrounding its body. The mechanism of thrust generation is a very complex phenomenon, as it depends on the pressure distribution of a three-dimensional, unsteady flow, where the interaction between vortices is of fundamental importance. Moreover, each species is characterized by a peculiar fin geometry and kinematics, which significantly affect swimming performances. Nevertheless, the general basic principles behind fish propulsion, which are common to both BCF and MPF swimmers, can be understood by making some simplifications: • inviscid flow: the role of viscosity is marginal in thrust generation; thus, the propulsive forces can be analyzed considering only the variation of momentum in the flow caused by the movement of the fins. The drag resistance acting on the body surface can be studied separately. • amplitude of fin displacement much smaller than body length: the fin moves only in a direction perpendicular to the swimming direction of the fish, and the amplitude of motion is much smaller than the fish length [4–6]. • fin of infinitesimal thickness: this assumption is valid for the vast majority of MPF fishes and BCF swimmers having tails with a high aspect ratio, such as rays, tunas, and sharks [4–6]. Lying on these assumptions, two models that capture the mechanism of fish swimming propulsion have been developed, leading to similar results.

2.2.1 Swimming of a Waving Plate In the first model, developed by Wu, the fin can be regarded as a deformable flat plate, producing a waving motion, as shown in Fig. 2.3. The fin is considered to have an infinite span, so to neglect the side effects occurring at the fin extremities. These are different for every fish species and give rise to a spanwise flow of secondary importance. Thus, the flow can be considered only two-dimensional [6]. The flow generated by the plate satisfies the continuity equation and the NavierStokes equation for incompressible and inviscid fluids (Eq. 2.7), where v = (U + u)i + vj. ⎧ ⎨ ∂v + (v · ∇) v = − 1 ∇ p ∂t ρ (2.7) ⎩ ∇ ·v=0 Considering the swimming velocity of the fish U constant and much greater than the fin displacement velocity components in u and v, it is possible to simplify the Navier-Stokes equation, obtaining Eq. 2.8.

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2 Hydrodynamics of Swimming

Fig. 2.3 Plate of infinitesimal thickness producing a waving motion

∂v 1 ∂v +U = − ∇p ∂t ∂x ρ

(2.8)

The acceleration of the fluid is also equal to the gradient of a function φ(x, t), called acceleration potential, defined as follows: φ(x, t) =

p∞ − p , ρ

(2.9)

where p∞ is the pressure of the fluid at infinite distance from the body. Combining Eq. 2.9 and the continuity equation (Eq. 2.7) it is possible to obtain: ∇2φ = 0

(2.10)

This means that velocity and pressure fields can be found by solving the Laplace equation for the acceleration potential. Moreover, working with the acceleration potential is particularly convenient because it is a regular function in the whole domain, as pressure is continuous everywhere inside the flow, particularly in the wake, unlike velocity that, for an inviscid flow, may allow discontinuities in the wake [4]. The motion of the plate can be written in the most general form as: y = h(x, t)

−

L L 0.4, the spanwise flow is not strong enough to stabilize the leading-edge vortex; thus, it becomes unstable, and it is shed in the wake, interacting with the vortices in the Reverse Karman Street in a detrimental way and reducing energy efficiency [19, 20]. For the vast majority of BCF swimming animals, the caudal fin is the primary source of propulsion. However, the flow in its proximity is generally disturbed by other appendages, such as dorsal or anal fins. These fins also shed vortices in the wake, which interact with vortices formed by the caudal fin, playing an essential role in

2.2 Hydrodynamic Principles of Fishes and Cetacean Locomotion

35

(a) Fish exploiting the counterflow in a Karman Street of an obstacle

(b) Fish exploiting the counterflow between two Reverse Karman Streets of preceding fish in the school

Fig. 2.17 Mechanism of energy extraction from vortices present in the flow

propulsion efficiency. Although the main role of dorsal and anal fins is a stabilization function in turning maneuvers, they also increase the propulsive efficiency of steadystate swimming. The dorsal and the caudal fin can be regarded as two oscillating foils in tandem, and there is a phase difference in the motion of the two fins induced by the wave propagation on the animal body. The wake generated by the dorsal fin is a Reverse Karman Street too, with a smaller width and with vortices of smaller strength since its motion is characterized by a smaller amplitude. The caudal fin intercepts these vortices, and they have the same rotational sense as the developing circulation around them. Enhancing the circulation, they increase the force generated by the fin and the strength of the vortices in the Reverse Karman Street in the wake [22, 23]. Swimming animals not only take advantage of the interaction between vortices generated by their fins but also exploit the vorticity already present in the flow. For example, when a bluff obstacle is present in a river or inside a stream of water, it releases a Karman Street in its wake. Swimming behind the obstacles, fishes reduce the drag force, as they are in a region of reduced flow velocity. However, the most significant advantage can be achieved if they synchronize their motion with the periodic vortex shedding of the obstacle to interact constructively with such vortices and extract energy from the wake, as shown in Fig. 2.17a [24, 25]. Moreover, when fishes swim in schools, they encounter the wakes of the preceding members, and they can recapture some energy from the wake by arranging themselves in a diamond formation. When two fishes swim side by side, each of them releases a Reverse Karman Street, with a jet stream in the center of the individual wake and counterflow at the two sides, as shown in Fig. 2.17b. Thus, in the middle of the two wakes, a Karman Street analogous to the one generated by a bluff body is present, so the fish behind can exploit the strong counterflow in this region to interact with vortices and reduce the energy consumption of locomotion [25].

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2 Hydrodynamics of Swimming

2.3 Efficiency of Fish Propulsion The efficiency of propulsion is the ratio between the useful work and the consumed energy in the condition of constant speed motion. The useful work is the work of the thrust force, corresponding to the energy required to overcome the resisting drag force of the fluid. For most vehicles, the propeller is isolated from the rest of the body, and the thrust generated by the propeller is balanced by the drag acting on the hull of a ship or a rigid part of a different vehicle. Thus, it is straightforward to identify the useful work, as it is possible to separate the contributions of thrust and drag since they act on different parts of the vehicle, and the energy efficiency is defined as the Froude efficiency: TU (2.65) η= P¯in where the overbar means the average over one period. Conversely, in the case of fish swimming, or, more in general, of a self-propelled deformable body, the quantification of the energy efficiency is much more challenging since both thrust and drag are generated by the movements of the fins or of the body, and it is complicated to separate them. Moreover, in the case of steady-state swimming, the net force acting on the fish body is zero, and it does not perform any useful work, despite consuming some power, so that the energy efficiency would always be zero [26]. Nevertheless, this limitation can be overcome by introducing different measures of energy efficiency, one of which is the cost of transport (COT). The COT is defined as the total energy consumed per unit distance traveled [26, 27]: C OT =

E tot S

(2.66)

The metabolic COT is calculated considering all the energy consumed by the animal as input energy. Thus, it includes the inefficiencies in the conversion from metabolic power to muscle power and from muscle power to kinetic power, and it is usually estimated by measuring the oxygen flow rate in the breathing of the animal [27]. On the other side, the mechanical COT considers as input energy only the energy spent to deform the body or the fin, and it is a much more useful measure of the energy efficiency of propulsion. The main drawback of using COT is that it is a dimensional quantity [J/m], and there is no universal way to normalize it, so this coefficient depends on the dimensions of the animal and its mass. Hence, it could only be used to compare different gaits of the same animal, but it cannot provide any information about the absolute energy efficiency of locomotion [26]. Usually, this coefficient is normalized with respect to the mass of the animal, but COT does not scale linearly with mass, so it does not allow any fair comparison of the energy efficiency between different fish species [28].

2.3 Efficiency of Fish Propulsion

37

Some ways to normalize the COT and to obtain a dimensionless quantity have been developed, but with such normalizations, it may result that gaits of the same animal with the same COT have a different normalized COT if they are characterized by different wavelength or frequency which make this measure useless to investigate the best combination of kinematic parameters that allows obtaining high energy efficiency [26, 27]. Another measure of the energy efficiency of a self-propelled body is the quasipropulsive efficiency (η Q P ), defined as: ηQ P =

RU Pin

(2.67)

where R is the towed resistance of the body [26]. Since the efficiency has a meaning only in steady-state conditions, the towed resistance is equal to the thrust force, and the numerator of Eq. 2.67 corresponds to the useful power of swimming. Nevertheless, the resistance R is not equal to the drag force at every time instant of fish movement, but it is the drag force measured towing the fish at a constant speed U . The advantage of using η Q P is that it captures the hydrodynamic characteristics of the body, and it can be used to compare fishes or AUVs with different shapes or sizes. Nevertheless, the resistance R is not measured in natural swimming conditions, introducing a considerable limitation to measuring the efficiency of fish locomotion, as the drag force acting on the fish is variable during swimming. These variations occur not only because the fish changes its shape and, consequently, the drag coefficient changes but also because the flow surrounding a fish moving its fins is very different from the flow surrounding a towed dead fish, even if the velocity U is the same. Therefore, this coefficient may even be greater than 1 if, during swimming, the interaction between the body and the fluid makes the drag coefficient drop substantially [26]. In order to quantify the energy efficiency of fish propulsion, it is necessary to separate the contributions of thrust and drag that act together on the same body. This work provides a different definition of efficiency, which is suitable for calculating analytically or numerically the efficiency of self-propelled bodies like swimming fishes. Since data coming from numerical analyses allow distinguishing the contributions of pressure and viscous forces, it is possible to define efficiency properly. The input power Pin is the power spent by the fish to move its fins, so it is obtained as the scalar product between the forces acting on the fins and the relative velocity of the fins with respect to the fish body vrel . Since fins are moved only in a lateral or vertical direction, the power Pin is independent of the swimming velocity and the forces acting in a longitudinal direction. Hence, the input power is obtained as:  Pin = −



( pn + τ ) d

(2.68)

where the infinitesimal forces acting on each infinitesimal element of the fin are the pressure force pn, being n the normal versor of the fin surface, and the viscous

38

2 Hydrodynamics of Swimming

tangential stresses τ . The useful output power Pout is the result of the pressure acting on the fins projected on the swimming direction, whereas the pressure acting on the rigid part of the body and the tangential stresses acting on the whole surface contribute to the resistance.  Pout = − px n x d f ins (2.69)  f ins

Finally, the efficiency can be computed as follows: η=

Pout Pin

(2.70)

This definition of efficiency indicates how much power contributes to generating a force propelling the fish in the swimming direction. This is because the tangential forces acting on the surface result from dissipative viscous effects, and the pressure force acting on the rigid part of the body always resists forward motion.

References 1. Wu TY (1971) Hydromechanics of swimming of fishes and cetaceans. Adv Appl Mech 11:1–63 2. Gazzola M, Argentina M, Mahadevan L (2014) Scaling macroscopic aquatic locomotion. Nat Phys 10:758–761 3. Sfakiotakis M, Lane DM, Davies J (1999) Review of fish swimming modes for aquatic locomotion. IEEE J Ocean Eng 24(2):237–252 4. Wu TY (1960) Swimming of a waving plate. J Fluid Mech 10:321–344 5. Lighthill MJ (1969) Note on the swimming of slender fish. J Fluid Mech 9:305–317 6. Wu TY (1971) Hydromechanics of swimming propulsion. Part 1. Swimming of a twodimensional flexible plate at variable forward speeds in an inviscid fluid. J Fluid Mech 46:337– 355 (1971) 7. Ramesh K, Granlund K, Ol MV, Gopalarathnam A, Edwards JR (2018) Leading-edge flow criticality as a governing factor in leading-edge vortex initiation in unsteady airfoil flows. Theor Comput Fluid Dyn 32:109–136 8. Sparemberg JA (2002) Survey of the mathematical theory of fish locomotion. J Eng Math 44:395–448 9. Gordon MS, Blickhan R, Dabiri JO, Videler JJ (2017) Animal locomotion: physical principles and adaptations. CRC Press—Taylor & Francis Group 10. Wu TY (1971) Hydromechanics of swimming propulsion. Part 3. Swimming and optimum movements of slender fish with side fins. J Fluid Mech 46:545–568 11. Eloy C (2012) Optimal strouhal number for swimming animals. J Fluids Struct 30:205–218 12. Drucker EG, Lauder GV (1999) Locomotor forces on a swimming fish: three-dimensional vortex wake dynamics quantified using digital particle image velocimetry. J Exp Biol 202:2393– 2412 13. Wu JC (1981) Theory for aerodynamic force and moment in viscous flows. AIAA J 19(4):432– 441 14. Saffman PG, Schatzmann JC (1982) An inviscid model for the vortex-street wake. J Fluid Mech 122:467–486 15. Triantafyllou MS, Triantafyllou GS, Gopalkrishnan R (1991) Wake mechanics for thrust generation in oscillating foils. Phys Fluids A 3:2835–2837

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16. Saadat M, Fish FE, Domel AG, Di Santo V, Lauder GV, Haj-Hariri H (2017) On the rules for aquatic locomotion. Phys Rev Fluids 2:083102 17. Taylor GK, Nudds RL, Thomas ALR (2003) Flying and swimming animals cruise at a strouhal number tuned for high power efficiency. Lett Nat 425:707–710 18. Triantafyllou GS, Triantafyllou MS, Grosenbaugh MA (1993) Optimal thrust development in oscillating foils with application to fish propulsion. J Fluids Struct 7:205–224 19. Anderson JM, Streitlien K, Barrett DS, Triantafyllou MS (1998) Oscillating foils of high propulsive efficiency. J Fluid Mech 360:41–72 20. Borazjani I, Daghooghi M (2012) The fish tail motion forms an attached leading edge vortex. Proc R Soc B 280:20122071 21. Eldredge Jeff D, Jones Anya R (2019) Leading-edge vortices: mechanics and modeling. Ann Rev Fluid Mech 51:75–104 22. Drucker EG, Lauder GV (2001) Locomotor function of the dorsal fin in teleost fishes: experimental analysis of wake forces in sunfish. J Exp Biol 204:2943–2958 23. Han P, Lauder GV, Dong H (2020) Hydrodynamics of median-fin interactions in fish-like locomotion: effects of fin shape and movement. Phys Fluids 32:011902 24. Beal DN, Hover FS, Triantafyllou MS, Liao JC, Lauder GV (2006) Passive propulsion in vortex wakes. J Fluid Mech 549:385–402 25. Liao JC (2007) A review of fish swimming mechanics and behaviour in altered flows. Phylosophical Trans R Soc B 362:1973–1993 26. Maertens AP, Triantafyllou MS, Yue DKP (2015) Efficiency of fish propulsion. Bioinspiration Biomim 10:046013 27. Bale R, Hao M, Singh Bhalla AP, Patankar NA (2014) Energy efficiency and allometry of movement of swimming and flying animals. PNAS 111(21):7517–7521 28. Kendall JL, Lucey KS, Jones EA, Wang J, Ellerby DJ (2007) Mechanical and energetic factors underlying gait transitions in bluegill sunfish (lepomis macrochirus). J Exp Biol 210:4265–4271

Chapter 3

Model of Cownose Ray Locomotion

Abstract This chapter describes the kinematics of cownose ray swimming, relating it to fin geometry and skeletal structure. The equation of the deformed fin surface is presented, and the influence of different kinematic parameters on fin movement is analyzed. Previous numerical studies about batoid swimming are briefly described, and the numerical implementation of the CFD model of cownose ray swimming is presented.

The investigation into the swimming strategy of an animal begins with a morphological and kinematic analysis of a biological sample, specifically the cownose ray, scientifically known as Rhinoptera Bonasus (Mitchill 1815). This particular fish is renowned for being one of the most proficient swimmers in the ocean, capable of traveling long distances at a speed of 0.6 to 0.8 times its body length per second, which results in optimal swimming efficiency. Additionally, the ray’s flexible pectoral fins serve as a propulsive source, ensuring both stability and maneuverability. The kinematic model of the motion of its fins is used as input of the CFD model, which allows highlighting the flow generated by the movement of the fins so that the mechanism of thrust generation can be understood and an accurate calculation of the forces can be achieved.

3.1 Rajiform Swimming The cownose ray employs a swimming strategy categorized as MPF, where it primarily utilizes its large pectoral fins for propulsion. This locomotion strategy is typical of fish belonging to the Batoidea order, including mantas, skates, and rays, and is referred to as rajiform. Batoids are characterized by their unique morphology, with dorsoventrally flattened bodies and large pectoral fins fused to the head to form a wide, flat structure resembling a disc or diamond [1, 2]. These fins are incredibly flexible due to their cartilaginous ribs that extend laterally from the body into the fin. The single cartilage segments require only small movements to generate significant flapping of the pectoral fins [2, 3]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Bianchi, A Numerical Tool for the Analysis of Bioinspired Aquatic Locomotion, PoliMI SpringerBriefs, https://doi.org/10.1007/978-3-031-30548-1_3

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3 Model of Cownose Ray Locomotion

Individual ribs move vertically, creating a wave along the fin length that varies in kinematics across different species. This category of fish exhibits both undulatory and oscillatory behaviors, occupying the continuum between the two. The bluespotted ray (Taeniura lymma, Forsskal 1775) has the highest number of waves at 1.4, while the giant manta (Mobula birostris, Walbaum 1792) has the lowest at 0.15. Other species, such as the southern stingray (Hypanus americanus, Hildebrand and Schroeder 1928), the smooth butterfly ray (Gymnura micrura, Block and Schneider 1801), and the cownose ray, have wavenumbers 1.1, 0.6, and 0.4, respectively. Fin shape ranges from being circular for fully undulatory locomotion to triangular for fully oscillatory locomotion, with many species exhibiting a varying wavelength to optimize their swimming mode according to the situation. The fully oscillatory swimming mode of batoids can also be referred to as mobuliform [1]. Undulatory pectoral fins have evolved to optimize their flexibility, as this feature is not only fundamental for the generation of the traveling wave, which allows propulsion, but also for feeding. These fishes live near the seabed and hunt smaller prey with a tactic called tenting. They cover their prey with their body and trap it by pressing the edges of their fins against the seabed so that it cannot escape. Fins with extreme flexibility make these fishes extremely agile in turning maneuvers, giving them the ability to perform turns with a very small or even null curvature radius [4]. On the other side, oscillatory pectoral fins do not show this level of multifunctionality, and they have evolved only to optimize swimming performances. Although very flexible, they are slightly more rigid than undulatory batoid fins, particularly in the region surrounding the leading edge, where the stress concentration is the highest. This gives them the possibility of generating more thrust and achieving a higher speed, with the drawback of reduced maneuverability. These fishes live in pelagic environments and may exploit oceanic currents to glide, moving without consuming energy. Moreover, some species, like the giant manta, never stop swimming throughout their entire lifespan because they need to keep in motion to push oxygenated water into their gills and absorb plankton with their mouth, suggesting that their motion is characterized by a high energy efficiency [4]. Some batoids are negatively buoyant, and their fin movements are slightly asymmetric, with a downstroke more powerful than the upstroke, so that they can generate a lift force in a similar way to the flapping flight of birds [5]. The differences in fin movements not only concern the wavelength but also the amplitude and frequency of motion vary among the species. It has been observed that undulatory species, characterized by a shorter wavelength, exhibit a small amplitude of fin movement, about 0.2 ∼ 0.3 times the fin width, since their stretching capability is limited, despite the high flexibility of the fins. The frequency of the fin movement is relatively high, and it can 3 Hz for the Atlantic stingray (Dasyatis Sabinus, Lesueur, 1824). Conversely, oscillatory species display less than half of a wave along the fin length; thus, they can exploit all the flexibility of their fins to achieve a high amplitude of movement, up to about 0.4 ∼ 0.5 times the fin width. The frequency of fin movement decreases as the locomotion behavior becomes more oscillatory, with the cownose ray moving its fins at about 0.9 Hz and the giant manta at about 0.3 Hz.

3.2 Bio-mechanical Model of Fin Kinematics

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Being the frequency inversely proportional to the wavelength, the propagation velocity of the wave on the fin, normalized with respect to the fish size, is approximately the same for all batoids. Moreover, the ratio between the swimming speed and the wave propagation velocity is 0.8 ∼ 0.9 for all species [1]. This implies that the water is pushed backward by the traveling wave with a velocity just slightly greater than the relative velocity between the fish and the surrounding water, meaning that the energy lost in the wake is a small fraction of the consumed energy, and the efficiency of rajiform swimming mode is very high, reaching 90% for the largest species, and making them efficient swimmers on a par with thunniform fishes [6]. Different behavior between oscillatory and undulatory species is also observed during turning maneuvers. Fishes with a more undulatory behavior perform turning maneuvers propagating waves on the fins in opposite directions to generate two equal and opposite forces on the two fins, producing a moment about the yaw axis. This technique gives them high maneuverability since they can curve even with a null curvature radius. However, this maneuver is quite an energy-demanding since their wide flat shape is characterized by a significant moment of inertia about the yaw axis, so high forces must be generated on the fins. Instead, fishes with a more oscillatory swimming mode do not need such small curvature radii, as they tend to live in pelagic environments without obstacles; thus, they prefer using a more energyefficient turning maneuver called banking, which is similar to the one performed by airplanes. During turning, they stop fin flapping, and, assisted by the high swimming velocity, they glide by tilting their body about the rolling axis so that the generated lift force is directed laterally, and a moment about the yaw axis is produced. Fishes consume little energy to move their fins during this maneuver, but the curvature radius is large. Some species with behavior between fully undulatory and fully oscillatory can combine these two maneuvers, banking and generating small amplitude waves on their fins [7]. The high efficiency of rajiform locomotion, combined with their excellent turning performances, makes these fishes a valuable source of inspiration for many biomimetic robots, generating significant interest in the study of their propulsion mechanism.

3.2 Bio-mechanical Model of Fin Kinematics The fins of a batoid fish are composed of 100 ∼ 200 cartilage segments connected in series by joints forming structures called radial elements, which are the structural support of the fin, and extend radially from the fin root, attached to the main body, to the fin tip. In addition, the radial elements are connected with some “crossbracing” connective tissue. The disposition of these components varies significantly among species: undulatory species are featured by staggered joints positioned at the extremity of the fin, which guarantee very high flexibility, whereas oscillatory species present cross-braced joints in the center, which stiffen the fin, making it capable of bearing higher loads [8].

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Table 3.1 Average dimensions of the cownose ray and kinematic parameters of its swimming [9–11] Disc length Disc width Frequency Mid-disc amplitude (proportion of disc width) Wavespeed (proportion of disc length) Number of waves Phase velocity Stride length Mass

0.45 0.7 1.04 0.35

[m] [m] [Hz] [-]

2.62

[-]

0.4 0.76 0.6 12

[-] [-] [m] [kg]

The kinematic parameters characterizing the cownose ray swimming are listed below and summarized in Table 3.1: • fin-beat frequency f : number of fin-beats per second; • mid-disc amplitude A: half of the total dorsoventral displacement of the widest portion of the fin, usually non-dimensionalized with respect to the disc width; • number of waves N : number of waves present along the fin, it is obtained by dividing the length L of the body from the snout to the beginning of the tail for the wavelength λ; • propulsive wavespeed c: velocity of the traveling wave along the fin, it is obtained by multiplying wavelength λ and frequency f ; • phase velocity U/c: ratio between forward velocity U and velocity of the traveling wave; • stride length U/ f : the distance traveled in one fin-beat period. To analyze the fin kinematics, first, a reference frame is set, as can be seen in Fig. 3.1. The origin of the coordinate system is placed on the root point of the leading edge, i.e., the intersection point between the leading edge and the foil baseline, where deformation does not occur during linear cruising motion. The x-axis is parallel to the pitch axis, i.e., the lateral body axis, and points to the tail direction; the y-axis is parallel to the roll axis, i.e., the spanwise direction; the z-axis is parallel to the yaw axis, i.e., the dorso-ventral body axis. The contour of the fin is reconstructed from the interpolation of markers positioned on the edge of the fin, as shown in Fig. 3.1, and it is shown in Fig. 3.2. The values on the axes are dimensionless: the dimensions of the pectoral fin were normalized by the chordwise length of the fin base, which is the maximum length of the pectoral fin along the chordwise direction. The shape of the fin is triangular, with a rounded and swept back tip [10]. The kinematic analysis of fin movement is based on the observation of the skeletal structure of the fin, and the fin deformation is calculated from the position and the

3.2 Bio-mechanical Model of Fin Kinematics

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Fig. 3.1 Dorsal shape of a cownose ray sample. Representation of the coordinate system (blue), the fin bases, and three key sections (red): midbody, fin middle, and fin tip part

Fig. 3.2 Edge of the right pectoral fin of a cownose ray

1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

relative rotation of the joints composing the fin skeleton. The cartilage structure of the fin was described by Russo et al., and it is composed of hundreds of spanwise-oriented radial segments connected by flexible joints [11]. To provide a representation of the pectoral fin movement, the source of cownose ray propulsion, a set of geometrical and kinematic input parameters has been defined: • Non-dimensional wave number, i.e., the number of waves traveling along a defined direction:

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3 Model of Cownose Ray Locomotion

Fig. 3.3 Representation of the reference systems on the fish body and on the fin

– Nc : chordwise direction – Ns : spanwise direction • Deflection angle of the fin tip with respect to the horizontal plane θmax • Fin-beat frequency ω = 2π f • Asymmetric flapping coefficient δ, to achieve an asymmetric flapping movement The fin-beat frequency f and the number of the chordwise wave Nc allow the computation of the wavespeed c, i.e., the velocity of propagation of the traveling chordwise wave: Ly f ω (3.1) c= = γ Nc Ly 2π , and λ = is the wavelength of λ Nc the traveling chordwise wave [11]. It is possible to introduce an additional reference system, with the coordinates r and s, with s representing the curvilinear abscissa in a spanwise direction with origin on the fin root, and r representing the curvilinear abscissa in chordwise direction with origin on the leading edge, as shown in Fig. 3.3. The absolute rotation in each joint is expressed as: where γ is the wavenumber, which is equal to

θr s =

1 [θmax s cos (r + s − ωt) + δs] Lx

(3.2)

The variables  and in Eq. 3.2 represent the angular phases of the traveling wave in the chordwise and in the spanwise direction, and they can be calculated as: =

2π Nc Ly

=

2π Ns Lx

(3.3)

Once the rotation in each joint is computed, it is possible to obtain the position of the joints at every time instant of the periodic fin movement. The position of each

3.2 Bio-mechanical Model of Fin Kinematics

(a) Oscillatory motion inputs

, Φ, Ψ,

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(b) Spanwise curvature by radial segment rotation about joints

Fig. 3.4 Kinematic inputs and method to obtain spanwise curvature Table 3.2 Geometrical and kinematic parameters used to describe the movement of the pectoral fin Geometrical parameters Fin baseline length Spanwise elements Chordwise elements Transverse segment angle Kinematic parameters

Ly S R β(r, s)

0.5 404 501 π/2

[m] [-] [-] [rad]

Number of chordwise waves Number of spanwise waves Maximum element angular amplitude Asymmetric flapping coefficient Fin-beat frequency Scaled wavespeed

Nc Ns θmax δ f c/L y

0.4 0.2 65 0.05 1.04 2.60

[-] [-] [deg] [deg] [Hz] [-]

joint i is obtained from the previous joint i − 1 along the spanwise direction, as shown in Fig. 3.4 and in Eq. 3.4, where L is the length of each radial element, and β is the orientation of the radial element with respect to the longitudinal axis [11]. ⎧ ⎪ ⎨ xi (r, s, t) = L cos θ (r, s, t) sin β(r, s) + xi−1 (r, s, t) yi (r, s, t) = L cos θ (r, s, t) cos β(r, s) + yi−1 (r, s, t) ⎪ ⎩ z i (r, s, t) = L sin θ (r, s, t) + z i−1 (r, s, t)

(3.4)

As all the radial elements are almost parallel and aligned with the spanwise direcπ tion, it is possible to consider β = for all elements without introducing any sig2 nificant error in the model. The input geometrical and kinematic parameters given to the model are based on the observation of biologists [10, 11], and they are summarized in Table 3.2.

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3 Model of Cownose Ray Locomotion

(a) T/6

(b) T/3

(c) T/2

(d) 2/3 T

(e) 5/6 T

(f) T

Fig. 3.5 3D deformed pectoral fin during a period

The fin deformation resulting from this kinematic model is presented in Fig. 3.5, where a perspective view of the fin in different time instants of a period is shown, and in Fig. 3.6, where the contours of the vertical displacement of the fin surface are displayed. The red portions of the contour plot represent a motion with a positive heigth (z > 0), whereas the blue portions represent a negative one (z < 0). The contour plot reveals the oscillatory behavior: for t < T /2, where T is the oscillation period,

3.2 Bio-mechanical Model of Fin Kinematics

49

(a) T/6

(b) T/3

(c) T/2

(d) 2/3 T

(e) 5/6 T

(f) T

Fig. 3.6 Contour plot of the vertical displacement (z-direction) of the pectoral fin during a period

the pectoral fin moves downwards (downstroke phase), indeed the fin tip region is blue-colored; analogously, for t > T /2, the pectoral fin is moving upwards (upstroke phase). Furthermore, the contour plot highlights the longitudinal traveling of the chordwise wave, as can be seen by looking at the longitudinal transfer of color on the fin surface.

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3 Model of Cownose Ray Locomotion

Fig. 3.7 Flapping of the fin tip spanwise cross-section of the pectoral fin during a period

The fin tip spanwise cross-section is endowed with the most significant spanwise dimension. Thus, it is characterized by the largest amplitude of oscillation, and it is the most representative in the analysis of the pectoral fin motion. In Fig. 3.7, the position of the fin tip spanwise cross-section is displayed for several time instants during a period of fin oscillation. The position at the beginning of the downstroke is colored green, and the line gradually shifts to blue, becoming dark blue at the end of the upstroke. It is possible to note that the fin positions during upstroke and downstroke are very similar, but they do not precisely match. The reason for this difference is the spanwise wave propagation, which induces a slight delay on the fin tip. Due to the asymmetric flapping coefficient δ, the amplitude of the upstroke is greater than that of the downstroke, matching the observation of biologists [10, 11]. The presence of a traveling wave on the fin surface can also be appreciated by looking at Fig. 3.8, where the vertical positions during a period of a point at the middle of the leading edge of a point on the fin tip, and a point in the middle of the trailing edge are shown. The fin tip, being more distal from the fin root, is characterized by a movement of greater amplitude, and there is a phase delay between the three points.

3.3 CFD Model of Cownose Ray Forward Swimming The forces acting on the fins are calculated using a numerical approach, coupling a CFD analysis with the dynamics of the forward swimming of the fish. Prior research includes Fish’s investigation into the dynamics of a manta ray’s steady-state swimming using a boundary element method. The study involved a motionless fish in a flow with constant velocity in the opposite direction of swimming, with analysis of the wake vortices and propulsion efficiency linked to the Strouhal number [6]. Liu utilized numerical tools to analyze the wake structure of a

3.3 CFD Model of Cownose Ray Forward Swimming

51

Fig. 3.8 The harmonic altitude of three key points along chordwise direction: middle point on the leading edge (MPL), fin tip (FT), the middle point on the trailing edge (MPT)

robot inspired by a batoid fish swimming at constant velocity [12]. Zhan simulated the gliding of a manta beneath the sea surface and obtained drag and lift coefficients for stationary fins at different angles of attack [13]. Sharp’s numerical investigation focused on the forces acting on a fin that performs an undulatory motion and utilized Multiparticle Collision Dynamics [14]. Liu modeled the interaction of water with a deformable fin as a series of panels connected by springs [15], while Chen performed a dynamic analysis of the motion of a clearnose skate (Raja eglanteria, Bosc 1800) by coupling CFD with a dynamic solver [16]. Bottom conducted numerical simulations of stingray swimming locomotion, analyzing wake vortices [17]. Additionally, Thekkethil studied batoid fish hydrodynamics, describing the three-dimensional vortex structure and highlighting the cownose ray’s high thrust coefficient compared to other batoid fishes [18]. Huang conducted a hydrodynamic analysis with a complete six-degree-of-freedom (6-DOF) simulation of a bioinspired robot with undulating fins and two propellers. The study established a relationship between fin flapping frequency and robot speed and examined the robot’s stability during forward swimming and maneuvers [19]. The main focus of this book is to enhance the findings of the previous research by examining the acceleration of a cownose ray from a stationary position to a steadystate velocity during forward swimming. The deformation of the fin is predetermined, and the movement of the ray is computed along with the pressure and velocity of the surrounding fluid. The resulting wake structure and findings are then compared in terms of thrust, efficiency, and steady-state velocity, and correlated with the wavelength and frequency of fin deformation. The ultimate objective of this analysis is to provide a comprehensive explanation of the intricate mechanisms that facilitate batoid fishes to achieve high efficiency and to relate them to the frequency and wavelength of fin movement.

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3.3.1 Numerical Implementation The literature [9] provides information on the size and form of the fins, as demonstrated in Fig. 3.9. The fin’s profile is assumed to be a sequence of symmetrical airfoils that gradually narrow toward the tip. At the center of the body, the section is modeled as a NACA 0020, a NACA 0015 at the midspan of the fin, and a NACA 0012 near the end of the fin. The previous section presented the kinematics of fin motion, which is slightly simplified in order to obtain a continuous function and make the calculation of the displacement of the points on the fin surface independent from each other. Firstly, the asymmetry coefficient δ is neglected since its influence is deemed irrelevant, as shown in the previous section. Additionally, it has been observed that the tip deflection lag, which is determined by the spanwise traveling wave, provides a minor contribution to thrust generation compared to the chordwise wave [20]. Therefore, can also be ignored. Although it has been suggested that the tip lag contribution reduces losses due to the tip vortices [6], this effect will be analyzed in future work. Based on these assumptions, Eq. 3.2 is modified as follows. θ (s, y, t) =

1 θmax s sin (y − ωt) Ly

(3.5)

and x and z coordinates of each point of the fin can be calculated as: 

sin [θmax s sin (y − ωt)] L x θmax sin (y − ωt) 0 s 1 − cos [θmax s sin (y − ωt)] z(s, t) = sin [θ (σ, y, t)]dσ = L x θmax sin (y − ωt) 0

x(s, t) =

s

cos [θ (σ, y, t)]dσ =

(a) Airfoils approximating the fin cross-section

(3.6) (3.7)

(b) Dimensions of the fin for the numerical analysis

Fig. 3.9 Shape and dimensions of the fin for the numerical analysis

3.3 CFD Model of Cownose Ray Forward Swimming

53

The open-source software OpenFOAM is utilized to implement a CFD model of a cownose ray swimming forward. The simulation is conducted in three dimensions, and an overset mesh is used, consisting of two merged mesh grids. The first mesh remains still and encompasses the whole domain, while the second mesh, which includes a portion of fluid around the fish, moves jointly with it. The software interpolates between overlapping elements to find a unique solution. The motion of the internal mesh surrounding the fish is the result of two superimposed motions: forward swimming and fin deformation. The motion due to fin deformation is imposed on every point of the mesh surrounding the fish according to Eqs. 3.6 and 3.7, whereas the motion due to forward swimming is treated as a rigid body. The dynamics along the longitudinal axis of the fish are solved with the Newmark method, which is commonly used to integrate differential equations in rigid-body mechanics numerically, as shown in Eq. 3.8. ⎧   ⎨ x˙n+1 = x˙n + (1 − γ )x¨n + γ x¨n+1 t   ⎩ xn+1 = xn + x˙n t + 1 (1 − 2β)x¨n + 2β x¨n+1 t 2 2

(3.8)

By choosing β = 0.25 and γ = 0.5, the integration scheme assumes an average constant acceleration during the time step and the method is unconditionally stable. The forward displacement is the only degree of freedom of the fish, and all other degrees of freedom are constrained. Therefore, at each time step, the fins are moved, deforming the internal mesh, and the solver calculates the forces acting on the fins and the forward displacement of the fish. The equations solved for the flow are unsteady Reynolds-averaged Navier-Stokes (URANS), and the solver used in OpenFOAM is overPimpleDyMFoam, capable of solving unsteady, incompressible, turbulent flows with an overset mesh. The time derivatives are discretized with the backward Euler method, and gradients are discretized using a Gauss linear scheme. To improve accuracy, the divergence operator is discretized with a linear upwind scheme, and to improve stability, a limit on the non-orthogonal correction of the Laplacian is applied. The overset interpolation method adopted is the Cell Volume Weight method, the most accurate available, which creates a map between the overset and the target mesh, assigning weights to overlapping elements that correspond to the portion of the element that overlaps. The Navier-Stokes equations are solved using the PIMPLE algorithm, over-relaxing the pressure and velocity fields and applying non-orthogonal correctors. The simulation domain is depicted in Fig. 3.10, with the stationary mesh colored in purple and the moving mesh in yellow. The domain used in the simulation consists of 300,000 hexahedral cells, which gradually become finer near the region occupied by the moving mesh. The dimensions of the domain are 5 m × 3 m × 2 m and can be seen in Fig. 3.11a. The moving mesh is generated using snappyHexMesh on a 0.6 m × 0.2 m × 1.1 m block with three refinement levels on the surface of the fish, resulting in a mesh with 1,462,199 cells, as shown in Fig. 3.11b. To reduce interpolation errors, cells on the boundary of the moving mesh are of similar size to cells in the background.

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3 Model of Cownose Ray Locomotion

Fig. 3.10 Domain of the CFD simulations, with the background mesh colored in purple and the overset mesh colored in yellow

(a) Background mesh

(b) Overset mesh surrounding the fish

Fig. 3.11 Mesh used in the CFD simulations

Figure 3.12 presents the findings of the sensitivity analysis conducted on the grid, where velocities obtained from various mesh grids are compared. The coarsest mesh comprises two refinement levels on the fish surface, and the background cells are twice as large as those in the selected mesh, while the most refined grid has four refinement levels, and its cells are twice as small. The results show that the chosen element dimension has a negligible effect on the outcomes, as the average deviation between the normal and the most refined mesh is only 1.25%. The k − ω SST (Shear Stress Transport) turbulence model with standard wall functions is used to resolve turbulence. To assess the appropriateness of the chosen mesh size for this turbulence model, the dimensionless distance from a wall, known as the y + parameter, is calculated using the following equation: y+ =

yu T ν

(3.9)

3.3 CFD Model of Cownose Ray Forward Swimming

55

Fig. 3.12 Grid sensitivity analysis showing the velocity obtained with meshes of different sizes

Fig. 3.13 y + on the fish surface

where y is the distance from the wall, ν is the kinematic viscosity of the fluid, and u T is the friction velocity. The friction velocity is calculated as presented in Eq. 3.10 and is the velocity related to the shear stresses τw between layers of the flow. uT =

τw ρ

(3.10)

In Fig. 3.13, the surface of the fish is colored with y + for the standard mesh: the values are smaller than 300 everywhere, meaning that the results obtained with this turbulent model are reliable with the wall functions used. A comparable examination was conducted to determine if the outcomes were self-reliant on the extent of the domain. The outcomes are presented in Fig. 3.14,

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3 Model of Cownose Ray Locomotion

Fig. 3.14 Domain size sensitivity analysis showing the velocity obtained with domains of different sizes

where it can be observed that a minor difference in velocity is observed with an increase in domain size. In this investigation, the size of the overset domain was altered while maintaining the same proportions as the background domain. It should be noted that this analysis was carried out on a different simulation with different kinematic parameters than the mesh size analysis. In the initial time step, the fins are in their original shape, and the pressure is uniformly zero across the entire domain. A velocity of 1 x 10−3 m/s is applied to one end of the boundary, while zero-gradient velocity boundary conditions are imposed on the other sides of the domain. A moving wall velocity condition is imposed on the surface of the cownose ray. Pressure is set to zero on one end of the domain, while a zero-gradient condition is applied to all other boundaries. The mesh deformation caused by the fins’ motion is applied to the entire moving mesh using Eqs. 3.6 and 3.7 in the rigid body motion solver of OpenFOAM for all cells belonging to the moving domain. The Reynolds number of this problem is necessary to compute the parameters of the turbulence model, but it cannot be calculated beforehand because the steady-state velocity of the fish is unknown. However, the kinematic parameters of the fin motion ω are determined, and the speed of the traveling wave can be computed as c = ,  allowing for an estimation of the velocity. The generated thrust can only be obtained if the speed of the traveling wave c is greater than the forward swimming speed of the fish U . When equilibrium is reached, and the cownose ray is swimming at a constant speed, the generated thrust is small, just enough to overcome the drag, and the steadystate speed is slightly less than the traveling wave velocity. This prediction enables the estimation of the Reynolds number. For a swimming gait, it can be calculated as reported in Eq. 3.11 [21]: ω AL (3.11) Re ≈ Sw = ν where ω is the frequency of flapping, A is the amplitude of fin-tip displacement, and L is the average chord length of the fin, as shown in Fig. 3.15. The Reynolds number results to be ∼105 for all the analyses.

References

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Fig. 3.15 Characteristic dimensions of the cownose ray for Reynolds number calculation

References 1. Rosemberger LJ (2001) Pectoral fin locomotion in batoid fishes: undulation versus oscillation. J Exp Biol 204:379–394 2. Rosemberger LJ, Westneat MW (2009) Functional morphology of undulatory pectoral fin locomotion in the stingray taeniura lymma (chondrichthyes: Dasyatidae). J Exp Biol 202:3523– 3539 3. Salazar R, Fuentes V, Abdelkefi A (2018) Classification of biological and bioinspired aquatic systems: a review. Ocean Eng 148:75–114 4. Hall KC, Hundt PJ, Swenson JD, Summers AP, Crow KD (2018) The evolution of underwater flight: The redistribution of pectoral fin rays, in manta rays and their relatives (myliobatidae). J Morphol 279:1155–1170 5. Di Santo V, Kenaley CP (2016) Skating by: low energetic costs of swimming in a batoid fish. J Exp Biol 219:1804–1807 6. Fish FE, Schreiber CM, Moored KW, Liu G, Dong H, Bart-Smith H (2016) Hydrodynamic performance of aquatic flapping: efficiency of underwater flight in the manta. Aerospace 3(20):3030020 7. Parson JM, Fish FE, Nicastro AJ (2011) Turning performance of batoids: limitations of a rigid body. J Exp Marine Biol Ecol 402:12–18 8. Schaefer JT, Summers AP (2005) Batoid wing skeletal structure: novel morphologies, mechanical implications, and phylogenetic patterns. J Morphol 264:298–313 9. Cai Y, Bi S, Li G, Hildre HP, Zhang H (2018) From natural complexity to biomimetic simplification: realization of bionic fish inspired by the cownose ray. IEEE Robot Autom Mag 99:1–13 10. Cai Y, Bi S, Zheng L (2012) Design optimization of a bionic fish with multi-joint fin rays. Adv Robot 26:177–196 11. Russo RS, Blemker SS, Fish FE, Bart-Smith H (2015) Biomechanical model of batoid (skates and rays) pectoral fins predicts the influence of skeletal structure on fin kinematics: implications for bio-inspired design. Bioinspiration Biomim 10:046002 12. Liu G, Ren Y, Zhu J, Bart-Smith H, Dong H (2015) Thrust producing mechanisms in rayinspired underwater vehicle propulsion. Theor Appl Mech Lett 5:54–57 13. Zhan J, Gong Y, Li T (2014) Effect of angles of attack on the hydrodynamic forces of manta ray. In: Proceedings of the eleventh pacific/Asia offshore mechanics symposium

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14. Sharp N, Hagen-Gates V, Hemingway E, Syme M, Via J, Feaster J, Bayandor J, Jung S, Battaglia F, Kurdila A (2014) Computational analysis of undulatory batoid motion for underwater robotic propulsion. In: Proceedings of the ASME 2014 4th joint US-European fluids engineering division summer meeting 15. Liu X, Iwasaki T, Fish F (2013) Dynamic modeling and gait analysis of batoid swimming. In: 2013 American control conference (ACC) 16. Chen W, Wu Z, Liu J, Shi S, Zhou Y (2011) Numerical simulation of batoid locomotion. J Hydrodyn 23(5):594–600 17. Bottom RG, Borazjani I, Blevins EL, Lauder GV (2016) Hydrodynamics of swimming in stingrays: numerical simulations and the role of the leading-edge vortex. J Fluid Mech 788:407– 443 18. Thekkethil N, Sharma A, Agrawal A (2020) Three-dimensional biological hydrodynamics study on various types of batoid fishlike locomotion. Phys Rev Fluids 5:023101 19. Huang H, Sheng C, Wu J, Wu G, Zhou C, Wang H (2021) Hydrodynamic analysis and motion simulation of fin and propeller driven manta ray robot. Appl Ocean Res 108:102528 20. Moored KW, Dewey PA, Leftwich MC, Bart-Smith H, Smits AJ (2011) Bioinspired propulsion mechanisms based on manta ray locomotion. Marine Technol Soc J 45(4):110–118 21. Gazzola M, Argentina M, Mahadevan L (2014) Scaling macroscopic aquatic locomotion. Nat Phys 10:758–761

Chapter 4

Wake Structure and Swimming Performance of the Cownose Ray

Abstract This chapter describes the kinematics of cownose ray swimming, relating it to fin geometry and skeletal structure. The equation of the deformed fin surface is presented, and the influence of different kinematic parameters on fin movement is analyzed. The numerical implementation of the CFD model of cownose ray swimming is presented, and finally, the results are analyzed, highlighting how the swimming performances and the wake structure change according to different kinematic parameters. The main parameters that affect swimming performances are frequency and wavelength of fin motion and frequency resulted in being proportional to the swimming velocity, and it did not affect the dimensionless parameters like energy efficiency and the Strouhal number, whereas a variation in wavelength implies changing the angle of attack of the fin, resulting in a different flow and strongly affecting all swimming performances. The vortices in the wake form a Reverse Karman Street, and vortex rings are connected like in a chain, similarly to other swimming animals, and for some wavelengths, a leading-edge vortex can be observed too. The energy efficiency is one of the highest among fishes, reaching 89% for the best combination of parameters, and the Strouhal number of most analyzed swimming motions is comprised between 0.2 and 0.4.

4.1 Vortex Structures in the Wake and on the Leading Edge The simulation results discussed in this section focus on the case with ω = 4π rad/s and λ = 2.5 body length (BL). However, the qualitative interpretation of the results can be extended to all other cases analyzed in this study. The generation of thrust is mainly concentrated on the outer regions of the fins, where the amplitude of motion is higher. Conversely, the central part of the body generates most of the drag, as highlighted by other works [1]. The distribution of thrust generation over the fin surface is presented in Fig. 4.1, which shows the force acting in the direction of swimming per unit area of the element. A negative value (shown in blue) indicates a contribution to thrust, while a positive value (shown in red) represents a contribution to drag.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 G. Bianchi, A Numerical Tool for the Analysis of Bioinspired Aquatic Locomotion, PoliMI SpringerBriefs, https://doi.org/10.1007/978-3-031-30548-1_4

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Fig. 4.1 Force in swimming direction on the bottom side of the fish during upstroke

(a) Plane of the section shown in Figures 4.2b and 4.2c

(b) Pressure distribution around the fin during upstroke

(c) Pressure distribution around the fin during downstroke

Fig. 4.2 Pressure distribution around a section in the middle of the fin

The pressure distribution in the midspan plane, shown in Fig. 4.2a, of the fin during the upstroke and downstroke is presented in Fig. 4.2. This movement generates a region of low pressure in front of the fish, which propels it forward. By examining the pressure distribution on the fins and the time-dependent profile of thrust, it is apparent that an equal propulsive force is produced during both the upstroke and downstroke because the profile and fin motion are symmetric. Therefore, the generation of thrust during a flapping cycle exhibits a sinusoidal pattern with a frequency that is twice that of the flapping frequency. The magnitude of the propulsive force is directly proportional to the velocity of the fins, resulting in

4.1 Vortex Structures in the Wake and on the Leading Edge

(a) Force in the swimming direction during four periods

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(b) Velocity of the fish during four periods

Fig. 4.3 Force in the swimming direction and velocity of the fish

a maximum value at the midpoint of the flapping cycle and minimum values at the upper and lower limits of the stroke. At the top and bottom extremities of the stroke, the fins’ velocity is zero, and no momentum is imparted to the water, resulting in no thrust generation. This is illustrated in Fig. 4.3a, which displays the variation of thrust over four periods of fin flapping. As the fish accelerates, there is a net propulsive force over one period, whereas once it has attained a steady state, the average force over one period is zero. Figure 4.3b shows that the steady state is achieved after a few cycles, with a sinusoidal variation in velocity similar to that of thrust. When in steady state, the oscillations around the mean velocity are less than 5% of the mean velocity. The velocity of the fish is nondimensionalized relative to its body length, which is 0.58 m, and expressed in body length per second (BL/s), as seen in Fig. 4.2b. The sum of thrust and drag produces the force in the longitudinal direction, which oscillates around zero mean value at the steady state. As discussed in Chap. 2, only forces acting normally on the fin surface contribute to useful power, while tangential viscous forces and those on the rigid central body resist forward motion. Figure 4.4 illustrates this decomposition of the longitudinal force, where the thrust force is always positive and normal to the fin surface, with a null value at the top and bottom extremities of fin movement when velocity is zero, following the same trend as the simplified dynamical model. The drag force increases quadratically with velocity, and its oscillations during the period result from variations in fin shape and angle of attack, which affect the fin’s drag coefficient. By decomposing the longitudinal forces, it becomes possible to calculate the efficiency of swimming locomotion for various combinations of kinematic parameters, which will be discussed in the following paragraphs. To comprehend how the fins move the fluid and generate thrust, examining the wake’s vortex structure is crucial, and Fig. 4.5 displays the wake’s vortices from different viewpoints.

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Fig. 4.4 Forces acting on the fish along the swimming direction

The wake can be divided into three parts for each fin: • fin tip vortices • vortices behind the middle of the fin • vortices behind the center of the body The vortices in the wake are similar to those obtained by Fish [2], except for the absence of vortices near the body center in their study. This difference could be due to the simulation of a different batoid fish species with a larger body than the cownose ray, resulting in less impact on fluid motion around the body center from fin flapping. A fascinating outcome of the numerical analysis is the understanding of how these fishes achieve high efficiency through shedding vortices. Similar to many other fishes, cownose rays release a vortex in the wake during each half flapping period when the fin reaches the top or bottom of its motion. The direction of vortex rotation is clockwise or counterclockwise, depending on whether the fin performs an upstroke or downstroke [1]. When observing the wake structure in a plane perpendicular to the fin span, depicted in Fig. 4.2a, it becomes evident that the vortices are arranged similarly to a reversed Karman Street, as demonstrated in Fig. 4.6, with counterclockwise vorticity at the top and clockwise vorticity at the bottom, with the fish swimming leftward. This vortex arrangement produces a propulsive jet in the wake’s center while there is a counterflow in the surrounding volume. If the fish swims at a constant speed, the jet stream and counterflow balance, resulting in null net streamwise momentum in the wake [3, 4]. Figure 4.7 illustrates the process of vortex detachment through a perpendicular section of the fin’s span-axis at different time points, with absolute velocity streamlines. The fish moves from left to right, and the first time step is on the left: 1. As the fish swims to the right and performs an upstroke, a region of negative pressure forms on the lower side of the fin. Water from the upper side flows around the leading edge and rotates, while the rest of the water is pushed upward and backward by the traveling wave’s motion. 2. When the fin reaches the uppermost point in its motion, its velocity is zero, and there is no pressure difference between the two sides. As a result, the vor-

4.1 Vortex Structures in the Wake and on the Leading Edge

(a)

(b) Fig. 4.5 Contour of surfaces with λ2 = 0

Fig. 4.6 Vortices in the wake forming a Reverse Karman Street

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Fig. 4.7 Sequence of time steps during a quarter of a flapping cycle, showing the detachment of one of the vortices forming the reverse Karman street

tex around the leading edge disappears, and the water behind the trailing edge maintains its upward and backward velocity from the previous time step. 3. As the fin begins its downward motion, the flow around it becomes the opposite of the flow during the first time step, and the water behind the trailing edge is pushed downward. Behind it, the water still maintains its upward velocity from the upstroke and a vortex forms. This vortex formation mechanism is the same as that observed in heaving and pitching airfoils [5, 6]. Although vortices in the wake are associated with energy dissipation, it is inevitable that a periodically moving body in a viscous fluid generates some vortices, as previously discussed in Chap. 2. The vortices mentioned in the previous paragraph are a direct consequence of this type of motion, which is responsible for producing thrust [3]. Therefore, while it is desirable to minimize their size, they are a byproduct of a highly efficient movement that generates thrust. Figure 4.8 also shows that there is a surplus of momentum in the opposite direction to the swimming direction during acceleration, with the fish swimming from right to left. The velocity vectors in the wake at the top of the figure illustrate that water is pushed backward by the motion of the fin and alternates between upward and downward flow following the fin’s displacement. At the beginning of the acceleration, the amplitude of the velocity vectors is higher. The color scale at the bottom of the figure presents the velocity component in the swimming direction, revealing a

4.1 Vortex Structures in the Wake and on the Leading Edge

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Fig. 4.8 On the top velocity vectors; at the bottom velocity component in the swimming direction

propulsive jet stream in the center and water moving in the opposite direction at the top and bottom, corresponding to the vortices. Figure 4.9b displays another vortex that forms around each fin tip, as vectors are drawn on the plane presented in Fig. 4.9a. This vortex is generated by the pressure difference between the two sides of the fin. During the downstroke, the fin’s motion creates a negative pressure on the upward-facing side, causing the fluid on the downward-facing side to rotate around the fin tip and move to the opposite side, generating a vortex. The same mechanism occurs during the upstroke, generating a vortex that rotates in the opposite direction. The shape of the fish causes this vortex to form only behind the fin tip because the spanwise velocity of water that tends to move to the opposite side is smaller than the forward velocity of the fish divided by the tangent of the sweep angle of the fin. The fin’s curvature is optimized to reduce the strength of this vortex, as the high curvature of the fin makes its tip act as a winglet, hindering the passage of water from one side to the other. Lastly, Fig. 4.10 displays another vortex located at the center of the fish’s body. This vortex is created by the inward and outward motion of the fin tips during their flapping. These vortices have opposite rotations on the two sides of the fish and switch their direction between the upstroke and downstroke. The results obtained in this study are in agreement with the findings of Clark [7] and Dewey [8], who conducted experiments on a fin inspired by batoids and actuated by a traveling wave motion. Their results showed that vortices in the wake alternate in sign and are arranged similarly to a reverse Karman street, and a vortex is present around the fin tip. However, the results differ significantly from those obtained by

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(a) Plane just behind the fin tip

(b) Fin tip vortex

Fig. 4.9 Velocity vectors of the vortex around the fin tip during a downstroke

(a) Plane near the tail of the fish

(b) Central body vortex

Fig. 4.10 Velocity vectors of the vortex below the central part of the fish body

Bottom [9], who studied a stingray with a fin of a different shape than the cownose ray and undulated its fins with a much shorter wavelength. The vortices in the wake are depicted in Fig. 4.11 using vorticity vectors, and it can be observed that the vortices creating the reverse Karman street, with vorticity vectors laterally oriented, are linked to the vortices generated by the flow around the fin tip, which have vorticity vectors oriented longitudinally and vertically, and to the vortices produced behind the body center, with vorticity vectors directed longitudinally. These vortices are interlinked and form a cohesive ring-shaped structure that is typical of fish propulsion wake, as discussed in Chap. 2.

4.1 Vortex Structures in the Wake and on the Leading Edge

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Fig. 4.11 Surface of λ2 = 0 identifying vortices in the wake that form ring structures

4.1.1 Leading-Edge Vortex A vortex forming on the leading edge (LEV) of fins is common in animals that swim with lift-based propulsion, such as carangiform and thunniform fishes, as noted by Bottom and Borazjani in their research [9, 10]. In this type of locomotion, the fins act as heaving and pitching airfoils with a high angle of attack, producing a lift force that propels the fish forward. A stable leading-edge vortex is a widespread phenomenon in nature since it enhances lift generation, and it is also found on the wings of flying insects [9]. The motion of cownose rays’ fins can be described using Eqs. 3.6 and 3.7, regardless of the wavelength. However, the motion can be oscillatory or undulatory depending on the wavelength, and only oscillatory locomotion typically produces a leading-edge vortex [10]. In this study, swimming performance is compared for different wavelengths, and the effect of this parameter on the LEV is discussed. Figure 4.12a shows the streamlines of absolute velocity on the fin tip during a downstroke for a short wavelength (λ = 1.67 BL). A vortex around the fin tip is visible, with the fluid moving laterally from the bottom to the top side of the fin. However, there is no vortex on the leading edge, as the fluid streamlines passing around the leading edge remain parallel, and no flow separation is observed on the fin surface. Similarly, in Fig. 4.12b, where the relative velocity vectors are shown, the velocity vectors remain almost parallel to the swimming direction, indicating the absence of a leading-edge vortex. The relative velocity is calculated by subtracting the fluid velocity from the fish’s forward velocity, and it differs from the relative

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(a) Streamlines of absolute velocity around the fin tip

(b) Relative velocity vectors near the fin tip

Fig. 4.12 Velocity vectors and streamlines near the fin tip for λ = 1.67 BL

(a) Streamlines of absolute velocity around the fin tip

(b) Relative velocity vectors near the fin tip

Fig. 4.13 Velocity vectors and streamlines near the fin tip for λ = 20 BL

velocity between each section of the fin and the fluid, as the fin is also moving vertically. In contrast, a vortex close to the leading edge only forms when the wavelength is longer, and the motion is oscillatory. Figure 4.13a displays the streamlines of absolute velocity around the fin tip during a downstroke, revealing the presence of a vortex around the leading edge. Water flows from the top side to the bottom side of the leading edge, rotating and creating a vortex connected to the vortex around the fin tip. This vortex is also evident in Fig. 4.13b, which illustrates the relative velocity vectors. The vortex generates a low-pressure region at its center near the leading edge, which enhances thrust. The formation of an LEV during the oscillatory fin motion of batoid fishes has also been observed by Lu [11], who studied the vortex formed by the fin motion of a manta, which has a longer wavelength than the cownose ray. They have also noted that the strength of the LEV increases as the wavelength increases.

4.2 Influence of Kinematic Parameters on Swimming Performances

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4.2 Influence of Kinematic Parameters on Swimming Performances This study involved conducting various dynamic analyses to examine the efficiency of these fishes and uncover the mechanisms behind their performance by manipulating the frequency, amplitude, and wavelength of their fin motion.

4.2.1 Wavelength Effect The velocity of the cownose ray swimming with oscillatory fin motion and various wavelengths is illustrated in Fig. 4.14. According to Rosemberger and Russo’s studies, this species typically moves its fins at a frequency of 1 Hz, with a wave traveling at 2.5 body lengths per second. However, the wavelength and frequency of fin motion can be adjusted to optimize maneuverability, power consumption, and acceleration based on the fish’s requirements [12, 13]. As shown in Fig. 4.14, the steady-state velocity of the cownose ray increases as the wavelength increases. However, there is a maximum velocity reached, beyond which increasing the wavelength leads to a decrease in velocity. This is due to the fact that a longer wavelength results in a higher wave propagation velocity, thereby providing the fish with more momentum as water is pushed backward at a higher speed. However, a longer wavelength also means that the fin pushes water more vertically than backward, making it harder to generate thrust. This finding is in agreement with the observation that batoid fishes, which exhibit an oscillatory behavior with longer wavelengths, swim faster and are adapted to open ocean environments, while fishes with undulatory behavior, characterized by shorter wavelengths, swim at lower speeds and inhabit coral reefs or confined environments [12].

Fig. 4.14 Forward velocity of the swimming fish for different wavelengths 1 Hz

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(a) Force in the swimming direction acting on the fish for different wavelengths at 1 Hz

(b) Efficiency of the fish for different wavelengths at 1 Hz

Fig. 4.15 Steady-state force and efficiency at different wavelengths

During steady-state swimming, the force oscillates around an average value of zero, as shown in Fig. 4.3. Figure 4.15a illustrates the amplitude of these oscillations for different wavelengths λ. Increasing the wavelength enhances the generated force; however, for very long wavelengths, increasing the wavelength further does not improve the generated thrust. This is because, with a longer wavelength, the fin becomes flatter and pushes water more in a vertical direction than backward, making it more difficult to generate thrust. The fish’s behavior is similar even when the wavelength is significantly different because the number of waves on the fin, which is the reciprocal of the wavelength, is similar. Figure 4.15b plots the amplitude of the oscillations of power consumption at steady-state swimming and the efficiency of fish swimming for different wavelengths λ. Swimming with a higher wavelength results in flatter fins that push water more vertically, making a shorter wavelength a more efficient swimming strategy. The efficiency is calculated by averaging the power over a period and is meaningful only when the fish has reached its steady-state velocity.

4.2.2 Frequency Effect To study the effect of frequency on swimming performances, simulations were conducted at different frequencies while keeping the wavelength constant at 2.5 BL. This value was chosen because it balances the forward velocity and energy efficiency, as described in the previous section, and because it closely matches the behavior of cownose rays in their natural environment [12, 13]. Figure 4.16 illustrates the forward velocity of the fish at different frequencies, indicating that the steady-state velocity is directly proportional to the frequency, consistent with findings by Huang

4.2 Influence of Kinematic Parameters on Swimming Performances

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[14]. The varying lengths of the curves in Fig. 4.16 reflect the non-dimensionalized time unit of the graph, where each simulation has a different period duration. The relationship between frequency and force has been investigated, and the results are presented in Fig. 4.17a. The data shows that the force is directly proportional to the square of the frequency, which is expected since the primary force is due to the inertia of the water in motion, which is proportional to the square of the frequency. The dotted red line in the figure represents the parabolic curve that fits the data, and the yellow area is the 95% confidence interval of the fitting. This finding is significant because it suggests that the swimming of batoid fishes can be scaled

Fig. 4.16 Forward velocity of the swimming fish for different frequencies with λ = 2.5 BL

(a) Force in the swimming direction acting on the fish for different frequencies with = 2.5 BL

(b) Efficiency of the fish for different frequencies with = 2.5 BL

Fig. 4.17 Steady-state force and efficiency at different frequencies

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(a) Steady-state velocity for different wavelengths and frequencies

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(b) Efficiency for different wavelengths and frequencies

Fig. 4.18 Steady-state velocity and efficiency for different wavelengths and frequencies

Fig. 4.19 Energy efficiency as a function of the ratio between forward velocity and wave traveling speed

linearly with frequency, and the efficiency of swimming is not affected by changes in frequency, as demonstrated in Fig. 4.17b. Figure 4.18 provides a comprehensive overview of how wavelength and frequency affect swimming performance. The graph highlights that velocity rises with frequency and reaches a peak value for a specific wavelength. On the other hand, the efficiency of swimming relies solely on wavelength. Consequently, in terms of energy utilization, batoid fish swimming can be highly efficient, up to 89% for certain swimming conditions, and adjusting frequency is more efficient than changing wavelength to increase acceleration. Finally, in Fig. 4.19, the dependency of the energy efficiency on the ratio between the forward velocity and the wave traveling speed is shown. As predicted by the waving plate model [15] and the slender body theory [3] described in Chap. 2, energy

4.3 Angle of Attack

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efficiency is linearly dependent on this ratio, and when U → c, the efficiency tends to 1.

4.3 Angle of Attack As the fins of a cownose ray typically exhibit a small number of waves, approximately 0.4 [12], each section of the fin can be viewed as a heaving and pitching airfoil with a defined angle of attack. In Fig. Fig. 4.20, the angle of attack α is shown as the angle between the fin chord and the relative water velocity, with ψ being the pitch angle of the fin section. Although the horizontal velocity of the fins, vx , is constant across all sections, the vertical velocity v y and pitch angle ψ differ since the motion of the fin tip has a larger amplitude than that of the fin root. As a result, the angle of attack varies across the fin sections and can be calculated as the difference between the relative velocity angle and the pitch angle ψ. α = ψ − arctan

vy vx

(4.1)

Assuming negligible motion of the fin section along the lateral axis, the relative velocity vr el of a cownose ray fin section can be represented as the sum of its horizontal velocity vx , which equals the fish’s forward velocity, and its vertical velocity v y , represented by the derivative of Eq. 3.7. As demonstrated earlier, both vx and v y are linearly dependent on frequency. However, v y is also dependent on the wavelength of fin motion, whereas vx is significantly affected only by wavelength. Consequently, the angle of attack of a fin section is determined solely by the wavelength of fin motion, with pitch angle ψ also dependent on wavelength. The larger the wavelength, the flatter the fin, and the smaller the pitch angle. Figure 4.21 illustrates the angle of attack variation of a fin section near the fin tip over a period. A modification in wavelength has two opposing impacts on the angle of attack. The first effect is due to the variation in pitch angle (ψ), which causes an increase in α for shorter wavelengths. The second effect is a result of different forward velocities, which decrease α for shorter wavelengths. However, near the fin tip, where the thrust generation is concentrated, the second effect dominates, leading to a larger angle of attack for longer wavelengths. This implies that during the upstroke, the angle of attack is negative, whereas, during the downstroke, it is positive.

Fig. 4.20 Angle of attack of a fin section

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Fig. 4.21 Variation of the angle of attack of a fin section positioned at 90 of fin span over a period

The angle of attack plays a crucial role in governing the dynamics of the leadingedge vortex (LEV). It is observed that an LEV is formed only when the fish swims with a longer wavelength, i.e., at a higher angle of attack. This result is consistent with other studies, such as Eldredge et al. [16], which suggest that a high angle of attack is essential for LEV formation.

4.4 Strouhal Number The Strouhal number, which characterizes the swimming performance of a fish, is defined in Chap. 2 using Eq. 4.2, where f is the flapping frequency, A is the peakto-peak amplitude of the trailing edge of the fin, and U is the swimming velocity [4]. fA (4.2) St = U However, for batoid fish, the trailing edge amplitude varies along the fin span, and measuring the vertical distance between two vortex cores in the wake provides a more accurate estimation of the peak-to-peak amplitude. This distance corresponds to the farthest span of the fin before the onset of the fin tip vortex. Figure 4.22 illustrates the Strouhal number for different frequencies and wavelengths. The Strouhal number varies with wavelength but not with frequency since the steady-state velocity is proportional to frequency. For most simulated swimming modes, the Strouhal number of the cownose ray is between 0.2 and 0.4, consistent with previous findings by researchers [2, 4, 17, 18]. This confirms the model’s ability to accurately reproduce the cownose ray’s behavior. The simulation with f = 1 Hz and λ = 2.5 BL, typical swimming parameters for the cownose ray, yields a Strouhal number perfectly in the middle of this optimal range. Decreasing the wavelength results in a lower Strouhal number, reducing propulsive efficiency. On the other hand, increasing the frequency to achieve a higher Strouhal number is not practical

4.4 Strouhal Number

(a) Strouhal number for different wavelengths

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(b) Strouhal number for different frequencies

Fig. 4.22 Strouhal number of cownose ray locomotion

due to the deformation it would cause on the fin surface. The Strouhal number also affects the formation of the leading-edge vortex, as observed by Bottom [9] in their investigation of stingray vortices. They found that a Strouhal number of about 0.3 favored LEV formation, consistent with the same Strouhal numbers for the cownose ray swimming with a large wavelength. However, the geometry and motion of the stingray fin are significantly different from those of the cownose ray’s fins. This study has developed a CFD model with a dynamic solver, which utilizes a deformable overset grid to simulate the swimming of a cownose ray. This method can be applied to other fish species as long as their movements are known. The overset mesh movements were superimposed using custom libraries specifically designed for this purpose. The analysis of the wake structure revealed the presence of vortices forming a reverse Karman street, which is typical of thrust generation in most fishes. Additionally, other vortices were observed around the fin tips and near the body center. The simulations were conducted using different wavelengths and frequencies of fin motion, and the swimming performances were compared. For fin motion with a long wavelength, the presence of a leading-edge vortex was highlighted, but this is not typical of a cownose ray’s motion, and the energy efficiency is low. The primary contribution to thrust was found to be the traveling wave pushing water in the opposite direction. The study showed that the energy efficiency of swimming locomotion is highly dependent on the wavelength, and it does not vary with frequency. The steady-state velocity is proportional to frequency, and it is highest at a specific wavelength. The calculated Strouhal number for the cownose ray falls within the same optimal range as other swimming animals. The developed model is a valuable tool that can be used to investigate the dynamics of maneuvering of batoid fishes in future research and to understand the impact of spanwise wave propagation on the fin.

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