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Zhang, Z.; Wang, Q.; Zhang, S. Applications of Biomimetic Propulsion for Underwater Vehicles. Encyclopedia. Available online: https://encyclopedia.pub/entry/54712 (accessed on 18 May 2024).
Zhang Z, Wang Q, Zhang S. Applications of Biomimetic Propulsion for Underwater Vehicles. Encyclopedia. Available at: https://encyclopedia.pub/entry/54712. Accessed May 18, 2024.
Zhang, Zhijun, Qigan Wang, Shujun Zhang. "Applications of Biomimetic Propulsion for Underwater Vehicles" Encyclopedia, https://encyclopedia.pub/entry/54712 (accessed May 18, 2024).
Zhang, Z., Wang, Q., & Zhang, S. (2024, February 02). Applications of Biomimetic Propulsion for Underwater Vehicles. In Encyclopedia. https://encyclopedia.pub/entry/54712
Zhang, Zhijun, et al. "Applications of Biomimetic Propulsion for Underwater Vehicles." Encyclopedia. Web. 02 February, 2024.
Applications of Biomimetic Propulsion for Underwater Vehicles
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This entry is adapted from the peer-reviewed paper 10.3390/biomimetics9020079

Biomimetics and bio-inspiration, distinct yet complementary, both derive insights from nature’s ingenuity in science and engineering. Biomimetics, specifically for underwater vehicles, studies and emulates the efficient shapes and behaviors of aquatic creatures like fishes, dolphins, and whales, leading to innovative underwater vehicle designs with enhanced speed, thrust, maneuverability, and reduced water drag.

underwater vehicles bioinspired design biomimetic propulsion drag reduction noise reduction computational fluid dynamics (CFD)

1. Introduction

Biomimetics and bio-inspiration, distinct yet complementary, both derive insights from nature’s ingenuity in science and engineering. Biomimetics, specifically for underwater vehicles, studies and emulates the efficient shapes and behaviors of aquatic creatures like fishes [1][2][3], dolphins [4][5][6], and whales [7][8][9], leading to innovative underwater vehicle designs with enhanced speed, thrust, maneuverability, and reduced water drag. Meanwhile, bio-inspiration, adopting a broader perspective, seeks abstract inspiration from nature, influencing diverse fields. It applies nature’s principles to foster innovation, extending beyond the direct replication of natural systems.
Table 1 presents the Reynolds number ranges for various aquatic organisms, alongside those of typical Autonomous Underwater Vehicle (AUV) models. This comparison sets the stage for the in-depth discussion on biomimetics. The diversity in Reynolds number ranges shown in the table reflects the broad spectrum of adaptations and efficiencies found in aquatic life. This diversity acts as a rich source of inspiration for the design and development of AUVs, embodying the core principles of biomimetic and bio-inspired design.
CFD is pivotal in the development of underwater vehicles, offering predictions and simulations of fluid behavior. This technology aids designers in optimizing underwater vehicle structures at early design stages, enhancing efficiency and resource conservation. Moreover, CFD has demonstrated close alignment with experimental outcomes, positioning it as an efficient alternative for underwater vehicle experiments. A prominent example of this is the SUBOFF model [10]. Developed by the United States Naval Surface Warfare Center Carderock Division (NSWCCD), the SUBOFF model encompasses both fully appended and bare hull forms. It was specifically designed for experimental studies and CFD research. As a benchmark model, it facilitates the study of hydrodynamic characteristics in submarine-like bodies, focusing particularly on aspects like drag, flow separation, boundary layer transitions, and vortex generation [11].
Table 1. The Reynolds number ranges of various aquatic organisms and typical AUVs.
Biological Species Reynolds Number Reference AUV Models Reynolds Number Reference
American eel 1006 [12] Pirajuba 3.5 × 106 [13]
Bluegill sunfish 1440 [14] Spray 4 × 105–6 × 105 [15]
Yellowfin tuna 1.9 × 104 [12] Slocum 5.2 × 105–7.5 × 105 [16]
Cownose ray 105 [17] Seaglider 4.9 × 105 [18]
Scomber scombrus 3.2 × 105 [19] SeaExplorer 4.9 × 105–9.9 × 105 [16]
Manta ray 5.1 × 105 [20] Petrel-II 7.2 × 105–1.8 × 106 [21]
Greenland shark 106 [22] Petrel-L 7.7 × 105 [23]
longfin inshore squid 4.4 × 105–1.5 × 106 [24] Petrel-4000 1.2 × 106 [25]
Basking shark 6.9 × 106 [26] ALBAC 7 × 105–1.4 × 106 [27]
Humpback whale 8.8 × 106 [7] Miniature 5.5 × 104 [28]
Dolphin 2.3 × 106–1.6 × 107 [29] ALEX 9.9 × 104–4.9 × 105 [30]
Researchers and engineers utilize the model to validate CFD codes, reinforcing the reliability of CFD [31][32][33]. Additionally, the application of CFD in validating factors like propeller thrust [34][35][36] and hydrodynamic noise [37][38][39] further affirms its dependability and efficiency.

2. Applications of Biomimetic Propulsion

AUVs have garnered significant interest due to their extensive applicability and multifunctional utility. These applications include but are not limited to, deep-sea exploration [40][41], seabed geological research [42][43], marine resource extraction [44], and underwater infrastructure maintenance [45][46]. However, the conventional propulsion method of AUVs, relying on propellers, poses several significant challenges. These challenges encompass excessive energy consumption, increased resistance, and heightened noise pollution, especially in complex marine environments.
Consequently, scientists have been observing and analyzing the swimming and maneuvering techniques of aquatic creatures, dedicating themselves to the development of innovative, biomimetic AUVs. These organisms use their physical structures and aquatic maneuverability for propulsion, producing undulating movements through the coordination of body, pectoral fins [47], and tail movements [48][49][50]. They also manipulate the surrounding water flow. This innovative approach offers a propulsion system that not only surpasses the speed and noise reduction capabilities of traditional AUVs but also draws attention to its groundbreaking energy efficiency and maneuverability.

2.1. Biomimetic Hydrofoil-like Tail Fin Propulsion

Inspired by aquatic organisms, the propulsion technique utilizing tail-fin flapping offers significant benefits, including enhanced propulsion efficiency, increased maneuverability and flexibility, and the ability to maintain stability in complex environments. These advantages unlock vast potential for the design of highly efficient and adaptable underwater vehicles.
Numerous researchers have performed numerical simulations on biomimetic propulsion using caudal fin-like hydrofoils [51][52][53]. They have endeavored to uncover the fundamental relationship between design parameters such as frequency, amplitude, aspect ratio, and others, and their effects on propulsion efficiency and other dynamic performances.
In the study of flapping-hydrofoil propulsion mechanisms at low Reynolds numbers, a significant number of research efforts have been directed toward understanding the impact of the hydrofoil’s shape and motion patterns on propulsion efficiency.
For instance, Karbasian et al. [54] and Gupta et al. [55] explored the influence of hydrofoil shape on propulsive performance. Karbasian et al. [54], drawing inspiration from fish fin morphology, introduced a fish-like flapping hydrofoil motion pattern. In contrast, Gupta et al. [55] focused on how different hydrofoil shapes affect the strength of wake vortices.
Additionally, Abbaspour and Ebrahimi [56], as well as Han et al. [57], compared the propulsive characteristics of hydrofoils under flapping and oscillating mechanisms and examined the impact of viscosity on flapping hydrofoil performance.
Abbaspour and Ebrahimi [56] observed pronounced leading-edge vortices in the wakes of flapping hydrofoils, while Han et al. [57] utilized the LBM-IBM method to investigate the flow field characteristics of 3D flapping hydrofoils across various Reynolds numbers. Moreover, certain studies have concentrated on hydrofoil flexibility and motion modes. You et al. [58] created deformable hydrofoils that mimic fish or cetacean fin propulsion, while Martin et al. [59] and Wei et al. [60] investigated the effects of varying Strouhal numbers and wavelengths on the propulsive performance of a NACA0012 hydrofoil.
Furthermore, Vijayakumaran and Krishnankutty [61], along with Alberti et al. [62], scrutinized the effects of diverse motion parameters, including Strouhal number, angle of attack, pitch amplitude, and phase angle, on hydrofoil propulsion.
Vijayakumaran and Krishnankutty [61] explored hydrofoils that combine swinging and yawing movements, whereas Alberti et al. [62] focused on NACA0015 hydrofoils performing combined sinusoidal rise and pitch motions.
To advance this field further, Liu et al. [63] and Zhou et al. [64] utilized CFD methods to study self-propelled NACA0012 hydrofoil models and a biomimetic NACA0013 hydrofoil, while analyzing various factors affecting their propulsive performance.
Finally, Zhang et al. [65] and Khalid et al. [66] introduced innovative hydrofoil design and motion paths. Zhang et al. [65] proposed a flapping hydrofoil with a three-degree-of-freedom motion path, and Khalid et al. [66] studied the fluid dynamics performance of NACA0012-like hydrofoils at different Reynolds numbers using an IBM-based computational solver, investigating the effects of wavelength and Strouhal number as control parameters.
These studies collectively indicate that the efficiency of hydrofoil propulsion and its fluid dynamics performance are multifaceted issues, involving factors such as hydrofoil shape, motion patterns, and the fluid environment. For example, comparisons between flapping and oscillating mechanisms reveal differences in vortex structures and propulsion efficiency across varying motion patterns.
Meanwhile, studying the effects of different motion parameters, such as Strouhal number, angle of attack, and pitch amplitude, unveils the complexity involved in designing more efficient hydrofoil systems.
Additionally, by mimicking biological motion characteristics and introducing new paths of motion, such as multi-degree-of-freedom paths, researchers are exploring novel ways to enhance hydrofoil propulsion efficiency.
These studies not only complement each other, providing a more comprehensive understanding of hydrofoil propulsion mechanisms but also lay the theoretical and experimental groundwork for designing future efficient propulsion systems.

2.2. Biomimetic Robotic Fish Propulsion

The exceptional propulsion performance and agile maneuverability of fish undoubtedly arouse interest in biomimetic robotic fish research. These biomimetic robotic fish play a crucial role in the design and analysis of underwater propulsion devices, prompting academia to intensify research and discussion on this subject [2][67][68][69][70]. Particularly for the application in small AUVs, biomimetic underwater robotic fish propulsion devices hold immense potential and could be extensively exploited and utilized in future scientific research.
It is widely recognized that the swimming method of fish will significantly influence the design of future robotic fish. The oscillatory motion of the fish’s tail and abdomen significantly affects the surrounding fluid flow. However, due to the instability in these effects, a comprehensive understanding and analysis of vortex dynamics and FSI are required.
In this process, CFD numerical simulations have made significant contributions to the research of many scholars. Numerous researchers have investigated the hydrodynamic performance of robotic fish using this method [3][71][72][73], advancing future related research. Lamas and Rodriguez [74] conducted a comprehensive review of numerical simulations in hydrodynamics and biomimetic propulsion, highlighting the importance of numerical simulations in studying fish swimming patterns.
To address the scarcity of reference geometric models for freshwater fish and the inadequacy of applicable numerical methods, Khan et al. [75] developed a numerical model utilizing OpenFOAM. This model used a realistic fish-shaped geometric model and was calibrated with laboratory-measured values. Similarly, Düzbastilar and Şentürk [19] developed Computer-Aided Design (CAD) models for three fish species (Scomber scombrus, Sarda sarda, and Thunnus thynnus) and conducted numerical simulations to assess their drag and propulsion performance.
Regarding the application of FSI models, Fouladi and Coughlin [76] proposed an FSI model to simulate the swimming behavior of fish in water. This model, utilizing commercial CFD software and user-defined functions, facilitates establishing numerical simulations of oscillatory fish swimming behaviors and serves as a reference for developing hydrodynamic numerical models for biomimetic underwater vehicles. Chung et al. [77] used an FSI computational framework using accurate Riemann solvers and the FVM to simulate the flapping behavior of fish fins and joint systems. The coupling of CFD and Computational Structural Dynamics (CSD) solvers enabled them to examine the impact of bidirectional FSI on fluid flow, and they corroborated their findings with experimental data.
Furthermore, Wright et al. [78] utilized FSI analysis to investigate how the material properties of robotic fish caudal fins affect hydrodynamic performance and efficiency. In a distinctive approach, Li et al. [79] studied live pufferfish and developed a numerical model, integrating CFD with multibody dynamics. Their study focused on fluid–fish interactions and highlighted the influence of flexible fins on the propulsion performance of fish.
Following this, by combining biomimetic design with fluid dynamics, Zangeneh and Musa [80] simulated the swimming of fish in water using OpenFOAM’s dynamic mesh technology to investigate their hydrodynamic characteristics. This research not only enhanced the understanding of underwater robotic fish’s dynamic behaviors but also provided valuable insights for designing, remotely controlling, and optimizing their flexibility. Collectively, these studies underscore the importance of using diverse methods and perspectives in understanding and optimizing the hydrodynamic characteristics of underwater vehicles.
Also focusing on caudal fin propulsion, Palit et al. [81] conducted a detailed CFD analysis on tilapia, focusing their research on how tail and abdominal vibrations influence tilapia’s hydrodynamic characteristics. They particularly emphasized variations in drag and lift coefficients, providing crucial insights into the dynamics of fish swimming. Their study complements the work of Chowdhury et al. [82], who constructed a robotic fish model that imitates the tail-fin propulsion mechanism of fish in order to assess its hydrodynamic properties during linear motion. Their findings provide a deep understanding of the role of caudal fin-driven mechanisms in enhancing the performance of biomimetic robotic fish, thus furthering advancements in robotic fish design.
Vignesh et al. [83] used CFD for both steady and unsteady simulations of bio-inspired AUVs, aiming to accurately calculate their hydrodynamic derivatives. Their research endeavors to provide key data for the design of more efficient AUVs. Meanwhile, Li et al. [84] used numerical simulations, focusing on the hydrodynamic performance of autonomously propelled tuna, including aspects like velocity, power requirements, and wake vortices. Their work offers valuable insights into the performance and driving mechanisms behind autonomous swimming, especially in terms of efficient propulsion and fluid dynamic optimization.
In their study of fish schooling behavior, Li et al. [85] investigated both the hydrodynamic characteristics and flow field structures of fish schools across various vertical modes, aiming to enhance the swimming efficiency of robotic fish schools. Pan and Dong [86], along with Ren et al. [73], conducted numerical simulations on fish in high-density, diamond-shaped schools to analyze the hydrodynamic interactions within the school. They discovered that fish in dense schools exhibit both higher thrust and improved propulsive efficiency compared with those in sparse schools, primarily due to the pronounced wall effect.
In the realm of motion control and design optimization of robotic fish, Tian et al. [87] developed a CFD simulation platform that focuses on adjusting the motion control parameters of robotic fish, thus offering new perspectives for design and optimization. Furthermore, the research by Ji et al. [88] and Zou et al. [89] focused on the practical functionalities of robotic fish, including object detection, tracking, and collision avoidance, thereby enhancing the performance and safety of robotic fish with CFD analysis. At the same time, Zhang et al. [90] and Chen et al. [91] used a comprehensive approach that combines data-driven methods and CFD technology. They developed multi-objective, multidisciplinary design optimization strategies for the motion control of biomimetic robotic fish, demonstrating the potential of technological integration in advancing robotic fish design.
Overall, CFD numerical simulations have played a crucial role in exploring the hydrodynamic performance of robotic fish and biomimetic underwater vehicles.
Moreover, these studies also emphasize the immense potential for technological integration. Examples including the CFD simulation platform by Tian et al. [87] and the multi-objective design optimization strategy by Chen et al. [91] highlight the importance of interdisciplinary collaboration in solving complex engineering challenges. With these advanced research efforts, researchers are not only able to design more efficient and flexible biomimetic robotic fishes but also gain a deeper understanding of the locomotion mechanisms of underwater organisms, thus profoundly impacting fields like marine engineering, environmental conservation, and biological studies.

2.3. Biomimetic Batoid-like Propulsion

Among aquatic organisms, batoids and rays use a distinctive swimming technique by flapping their pectoral fins, thereby exhibiting efficiency levels comparable to other fish species. Despite this, their superior agility and precision in executing turns distinguish them. Their streamlined bodies and low-drag skin contribute partially to this advantage. Their unique fin movement, facilitating greater propulsion with less energy, is equally critical. To replicate this biological characteristic, wherein batoids achieve maneuverability through pectoral fin flapping, researchers utilize CFD to perform detailed numerical simulations [92][93].
In the realm of simulating and understanding the dynamics of batoids, the work of Huang et al. [94][95] revealed how motion frequency, amplitude, and thrust interrelate in deformable airfoils inspired by batoids, delving into the hydrodynamic performance and wake structure of both the airfoil and Rhinoptera javanica. Similarly, studies by Bao et al. [96] and Luo et al. [97] uncovered dynamic pressure and velocity variations in the flapping fin motion of batoids, as well as the impact of pectoral fin movements on torque generation.
Regarding group swimming behaviors and their hydrodynamic effects, Gao et al. [98] conducted an in-depth investigation into the collective swimming behaviors of batoids and tuna, yielding new insights into the hydrodynamic effects of individual and coordinated swimming behaviors.
In the context of biomimetic batoid modeling and mechanisms, Lee and Kwon [99] utilized the commercial software package ADINA to simulate the journey distance and speed of a ray, while Rayapureddi and Mitra [100] developed an IBM-FSI algorithm using OpenFOAM to address challenges associated with biologically inspired self-propelled batoid robotic devices in 3D hydrodynamic flow fields. Furthermore, Liu et al. [20] proposed a novel design for a remotely controlled soft material robotic batoid. Separately, Huang et al. [101] conducted a hydrodynamic analysis using a six-degree-of-freedom motion equation, resulting in a design featuring dual pectoral fins and an auxiliary power vertical thruster. Abbaspour et al. [102] designed wave gliders of two different geometric shapes, showcasing the advantages of manta ray gliders in stable energy absorption.
Lastly, research by Bianchi et al. [103][104] focused on efficient locomotion mechanisms in underwater manta ray designs. They replicated the movement of the cownose ray for dynamic numerical analysis and utilized CFD models to study the hydrodynamic characteristics of ray swimming, thereby investigating efficient propulsion mechanisms.
Collectively, these studies lay a crucial foundation for a deeper understanding and simulation of batoid dynamics. Through in-depth exploration of the dynamics and motion mechanisms of batoids, these research efforts yield invaluable insights for the design of more efficient and agile underwater robotic batoids.

2.4. Biomimetic Dolphin Propulsion

Dolphins, recognized as some of the most remarkable swimmers among aquatic mammals, have always been admired for their efficient cruising capabilities. This efficiency primarily relies on their unique dorsoventral propulsion mechanism, which is widely used by other aquatic mammals. To a large extent, this influences the methods researchers use when seeking to understand and mimic the swimming mechanism of dolphins, particularly the underlying mechanisms of dorsoventral propulsion. Based on this, researchers often attempt to imitate dolphins from a kinematic perspective [105][106], as it is believed to be the quickest path to designing high-performance underwater vehicles. Many modern underwater vehicles’ design philosophies and technical inspirations stem from research on the propulsion movements of dolphins. Researchers further found that the interaction and coordination between a dolphin’s body and its pectoral, dorsal, and caudal fins substantially affect their swimming efficiency, offering the dolphin extraordinary agility and highly efficient propulsion power. Therefore, further deepening the understanding and interpretation of the interaction mechanism between dolphins’ bodies and their various fins, and how to mimic this mechanism, will be crucial in enhancing underwater vehicle design and exploring the propulsion mechanism of dolphin swimming.
Recent studies in the design of dolphin-inspired robotic systems have made significant strides, especially in simulating the locomotion mechanisms of dolphins and enhancing the hydrodynamic performance of these robots. Xue et al. [107] investigated the C-turning, pitching, and flapping propulsion mechanisms of a dolphin robot in their work, successfully proposing an accurate and stable maneuverability model. This model is vital for understanding and simulating the complex motion characteristics of dolphins.
Furthermore, Cao et al. [108] enhanced the pitching performance of dolphin robots by developing an elliptical-trajectory pectoral fin oscillation model. This innovation not only improved the control precision of the robot but also opened up new possibilities for its application in varying aquatic environments.
Wu and his team [109][110][111] utilized a comprehensive approach in the design of bionic dolphin robots. By integrating the advantages of mechanical dolphins and underwater gliders, they achieved significant improvements in maneuverability, speed, and endurance. In particular, the innovative biomimetic dolphin-like underwater glider described in reference [112] combines the agility of mechanical dolphins with the long-range stability of underwater gliders, demonstrating the efficient design of underwater robots achievable by simulating dolphin motion characteristics.
In the biomimetic study of dolphin hydrodynamic characteristics, Wang et al. [113] combined experimental and numerical methods to examine thrust generation, wake structure, and surface pressure of dolphins at different swimming speeds. Their research revealed that the dolphin’s caudal fin maintains a highly effective attack angle throughout most of each stroke and observed a significant difference in flow and surface pressure between low-speed and high-speed swimming.
Han et al. [114] delved into the dynamic characteristics of the dorsoventral propulsion mechanism of dolphins. They developed a 3D model of dolphin swimming and used an incompressible CFD solver based on the IBM approach to investigate the hydrodynamics and wake structure of dolphin swimming. Wang et al. [115] used theoretical analysis and numerical methods to calculate the swimming speeds of dolphins, highlighting the powerful thrust and efficient propulsion resulting from fin movement in high-speed swimming. Xia et al. [116] improved the understanding of dolphin motion mechanisms with a comparative analysis of different swimming modes.
Tanaka et al. [117] recorded the swimming process of dolphins using high-speed camera technology and quantified the dynamics of dolphins during acceleration using CFD technology. Meanwhile, Feng et al. [29] divided dolphin motion into three phases: oscillation of the caudal fin, deformation of the caudal fin, and oscillation of the posterior third of the body, discussing the mechanisms for achieving fast and efficient propulsion.
Lastly, Guo et al. [118] developed a realistic 3D model of a dolphin and used CFD technology to investigate the oscillatory hydrodynamics of dolphins.
Collectively, these studies form an essential foundation for a deep understanding of dolphin hydrodynamic characteristics and CFD simulations, offering valuable references for designing efficient and agile underwater robots. The efficient swimming mechanisms of dolphins not only inspire the design of novel underwater robots but also pave the way for new directions and possibilities in the future development of underwater robotic technologies.

2.5. Biomimetic Squid Propulsion

The efficiency of marine organisms’ underwater movements is significantly enhanced by their streamlined external structures, which helps to reduce potential hydrodynamic drag. Furthermore, the propulsion methodology significantly influences acceleration duration and cruising speed during aquatic life’s movements. It is noteworthy that squid species exhibit a unique propulsion style, markedly distinct from that of other aquatic beings and the conventional propeller propulsion used by most underwater vessels.
Squids possess an extraordinarily effective jet propulsion mechanism [119][120][121]. This specialized evasion system is instantly activated in response to threats, particularly those posed by predators. This fast propulsion process provides sufficient thrust for squids to swiftly elude dangers, making their underwater evasion performance an effectively strategized survival mechanism.
Squids also exhibit superior underwater structural features, including streamlined body shapes that are highly hydrodynamic, designed to minimize the drag experienced during swimming. This combination of shape and hydrodynamics significantly reduces resistance during swimming, substantially enhancing the efficiency of their underwater movements.
The application of jet evasion provides squids with a significant advantage in acceleration. This unique acceleration capability, coupled with their remarkable hydrodynamic characteristics, enables squids to adeptly navigate a variety of complex oceanic conditions. This proficiency in underwater swimming not only highlights their exceptional skills but also attracts considerable scholarly interest for research in this area.
In the field of underwater soft robotics, the utilization of jet propulsion mechanisms of squids and other cephalopods represents a significant innovation. The study by Zhu and Xiao [122] offers a comprehensive overview of current research, highlighting the potential of jet propulsion technology in the development of soft underwater robots.
Olcay et al. [24] constructed a 3D squid model using tomography and computed the resistance, drag coefficient, swimming speed, and propulsion efficiency at varying nozzle diameters based on this model. Their findings highlight that the viscous resistance in squids with different Reynolds numbers is twice that of the pressure resistance and that expanding the nozzle diameter from 1 cm to 2 cm can lead to a 20% increase in propulsion efficiency. Their study elucidates the performance of jet propulsion mechanisms under varying physical parameters, which is essential for understanding and optimizing propulsion systems in underwater robots.
Additionally, they developed an improved squid model. The study revealed that the improved squid model requires minimal thrust during the acceleration phase of the time-dependent velocity profile [123]. This finding is informative for designing more efficient underwater propulsion systems. In another study, they discovered that using a larger nozzle diameter, smaller angles of attack, and eliminating fins could increase the propulsion efficiency of the squid to approximately 80%, thereby significantly enhancing propulsion efficiency [124].
Luo et al. [125] designed a 2D propulsion system to simulate squid swimming, inspired by the jet propulsion mechanism of squids and other cephalopods. Simulation analysis results indicate that higher Reynolds numbers result in a larger driving force and higher efficiency, which can be attributed to the strong jet-induced vortex and effective reduction in the external body vortex. In a turbulent environment, either increasing the Reynolds number or reducing the nozzle size will accelerate the formation of symmetry-breaking instability. These findings provide significant insights into simulating the performance of underwater robots in real marine environments.
In another study, they constructed a 3D pulsed jet propulsion model, composed of a flexible body and a controllable bending nozzle. The results validated the efficacy of curved nozzles for thrust vectoring and determined that the external surface viscous friction is predominantly influenced by variations in Reynolds numbers [126].
Subsequently, they conducted a numerical study on a squid-inspired jet propulsion system by regulating body deflection during a single-emission process, aimed at investigating the impact of jet speed on the formation of a vortex ring and the system’s propulsion performance. The researchers concluded that at a specified maximum stroke ratio, the inverse-cosine jet speed can create a second vortex ring.
Hou et al. [127] simulated squids jumping out of the water using CFD technology. They analyzed the flow characteristics of squids in relation to the launch angle and carried out a quantitative analysis of the motion parameters of flying squids. The results demonstrated that jet propulsion tends to generate significant average thrust rather than high propulsion efficiency and revealed that the speed of flight is inversely related to the launch angle. These discoveries have contributed to performance enhancements in water-to-air transport tools.
Li and his team added tentacles to a robot squid in a simulation, analyzing the influence of the number, frequency, and maximum extension of these tentacles on propulsion capacity. Their findings suggested that a robot squid equipped with three tentacles achieves the best propulsion performance and that increasing the tentacle frequency can effectively enhance the steady-state velocity coefficient and propulsion efficiency. However, it is imperative to control the maximum bending range of the tentacles within a certain limit, as exceeding it may result in adverse effects [128].
In another study, they designed an underwater robot equipped with dual-driven composite tentacles, using overlapping grid technology to simulate incompressible viscous flow. After comparing three driving modes (reverse mode, homologous mode, and interlace mode), they discovered that the reverse mode demonstrated the best energy savings and propulsion efficiency. Compared with traditional fish-shaped robots, this underwater robot exhibited enhanced self-driving capabilities [129].
In summary, these studies provide rich insights into the understanding and optimization of underwater jet propulsion mechanisms, significantly contributing to the design and fabrication of more efficient, complex, and marine environment-adapted underwater robots and biomimetic propulsors.

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