Separability of drag and thrust in undulatory animals and machines

Complex Modal Characteristic Analysis of a Tensegrity Robotic Fish's Body Waves

A bionic robotic fish based on compliant structure can excite the natural modes of vibration, thereby mimicking the body waves of real fish to generate thrust and realize undulate propulsion. The fish body wave is a result of the fish body's mechanical characteristics interacting with the surrounding fluid. Thoroughly analyzing the complex modal characteristics in such robotic fish contributes to a better understanding of the locomotion behavior, consequently enhancing the swimming performance. Therefore, the complex orthogonal decomposition (COD) method is used in this article. The traveling index is used to quantitatively describe the difference between the real and imaginary modes of the fish body wave. It is defined as the reciprocal of the condition number between the real and imaginary components. After introducing the BCF (body and/or caudal fin) the fish's body wave curves and the COD method, the structural design and parameter configuration of the tensegrity robotic fish are introduced. The complex modal characteristics of the tensegrity robotic fish and real fish are analyzed. The results show that their traveling indexes are close, with two similar complex mode shapes. Subsequently, the relationship between the traveling index and swimming performance is expressed using indicators reflecting linear correlation (correlation coefficient (Rc) and p value). Based on this correlation, a preliminary optimization strategy for the traveling index is proposed, with the potential to improve the swimming performance of the robotic fish.

Energetic benefits in coordinated circular swimming motion of two swimmers

The coordinated movement of multiple swimmers is a crucial component of fish schools. Fish swimming in different formations, such as tandem, side-by-side, diamond, and phalanx, can achieve significant energetic advantages. However, the energetic benefits of nonstraight swimming behaviors, such as the collective motion of a milling pattern, are not well understood. To fill in this gap, we consider two swimmers in circular tracks, controlled by a PID approach to reach stable configurations. Our study finds that the optimal phase is affected by circumferential effects, and that substantial energy savings can result from both propulsion and turning. We also explore the radial effect in terms of energetic benefits. In a milling pattern, the inner swimmers can easily gain a certain energetic benefit (-8%), while their peers on the outside must be close enough to the inner swimmer with a proper phase to gain the energetic benefit (-14%). When the radial spacing becomes larger or is in an unmatched phase, the swimming of the outer swimmers becomes more laborious (+16%). Our results indicate that swimmers who maintain a matched phase and minimum radial effect obtain the highest energetic benefits (-26%). These findings highlight the energetic benefits of swimmers, even in a milling pattern, where the position difference dominates the extent of benefit.

Identification of the trade-off between speed and efficiency in undulatory swimming using a bio-inspired robot

Anguilliform swimmers, like eels or lampreys, are highly efficient swimmers. Key to understanding their performances is the relationship between the body's kinematics and resulting swimming speed and efficiency. But, we cannot prescribe kinematics to living fish, and it is challenging to measure their power consumption. Here, we characterise the swimming speed and cost of transport of a free-swimming undulatory bio-inspired robot as we vary its kinematic parameters, including joint amplitude, body wavelength, and frequency. We identify a trade-off between speed and efficiency. Speed, in terms of stride length, increases for increasing maximum tail angle, described by the newly proposed specific tail amplitude and reaches a maximum value around the specific tail amplitude of unity. Efficiency, in terms of the cost of transport, is affected by the whole-body motion. Cost of transport decreases for increasing travelling wave-like kinematics, and lower specific tail amplitudes. Our results suggest that live eels tend to choose efficiency over speed and provide insights into the key characteristics affecting undulatory swimming performance.

Force scaling and efficiency of elongated median fin propulsion

Several fishes swim by undulating a thin and elongated median fin while the body is mostly kept straight, allowing them to perform forward and directional maneuvers. We used a robotic vessel with similar fin propulsion to determine the thrust scaling and efficiency. Using precise force and swimming kinematics measurements with the robotic vessel, the thrust generated by the undulating fin was found to scale with the square of the relative velocity between the free streaming flow and the wave speed. A hydrodynamic efficiency is presented based on propulsive force measurements and modelling of the power required to oscillate the fin laterally. It was found that the propulsive efficiency has a broadly high performance versus swimming speed, with a maximum efficiency of 75%. An expression to calculate the swimming speed over wave speed was found to depend on two parameters:A_p/A_e(ratio between body frontal area to fin swept area) andC_D/C_x(ratio of body drag to fin thrust coefficient). The models used to calculate propulsive force and free-swimming speed were compared with experimental results. The broader impacts of these results are discussed in relation to morphology and the function of undulating fin swimmers. In particular, we suggest that the ratio of fin and body height found in natural swimmers could be due to a trade-off between swimming efficiency and swimming speed.

A numerical study of fish adaption behaviors in complex environments with a deep reinforcement learning and immersed boundary-lattice Boltzmann method

Fish adaption behaviors in complex environments are of great importance in improving the performance of underwater vehicles. This work presents a numerical study of the adaption behaviors of self-propelled fish in complex environments by developing a numerical framework of deep learning and immersed boundary-lattice Boltzmann method (IB-LBM). In this framework, the fish swimming in a viscous incompressible flow is simulated with an IB-LBM which is validated by conducting two benchmark problems including a uniform flow over a stationary cylinder and a self-propelled anguilliform swimming in a quiescent flow. Furthermore, a deep recurrent Q-network (DRQN) is incorporated with the IB-LBM to train the fish model to adapt its motion to optimally achieve a specific task, such as prey capture, rheotaxis and Kármán gaiting. Compared to existing learning models for fish, this work incorporates the fish position, velocity and acceleration into the state space in the DRQN; and it considers the amplitude and frequency action spaces as well as the historical effects. This framework makes use of the high computational efficiency of the IB-LBM which is of crucial importance for the effective coupling with learning algorithms. Applications of the proposed numerical framework in point-to-point swimming in quiescent flow and position holding both in a uniform stream and a Kármán vortex street demonstrate the strategies used to adapt to different situations.

Thrust generation during steady swimming and acceleration from rest in anguilliform swimmers

Escape swimming is a crucial behavior by which undulatory swimmers evade potential threats. The hydrodynamics of escape swimming have not been well studied, particularly for anguilliform swimmers, such as the sea lamprey Petromyzon marinus. For this study, we compared the kinematics and hydrodynamics of larval sea lampreys with those of lampreys accelerating from rest during escape swimming. We used experimentally derived velocity fields to calculate pressure fields and distributions of thrust and drag along the body. Lampreys initiated acceleration from rest with the formation of a high-amplitude body bend at approximately one-quarter body length posterior to the head. This deep body bend produced two high-pressure regions from which the majority of thrust for acceleration was derived. In contrast, steady swimming was characterized by shallower body bends and negative-pressure-derived thrust, which was strongest near the tail. The distinct mechanisms used for steady swimming and acceleration from rest may reflect the differing demands of the two behaviors. High-pressure-based mechanisms, such as the one used for acceleration from rest, could also be important for low-speed maneuvering during which drag-based turning mechanisms are less effective. The design of swimming robots may benefit from the incorporation of such insights from unsteady swimming.

Immersed Methods for Fluid-Structure Interaction

Fluid-structure interaction is ubiquitous in nature and occurs at all biological scales. Immersed methods provide mathematical and computational frameworks for modeling fluid-structure systems. These methods, which typically use an Eulerian description of the fluid and a Lagrangian description of the structure, can treat thin immersed boundaries and volumetric bodies, and they can model structures that are flexible or rigid or that move with prescribed deformational kinematics. Immersed formulations do not require body-fitted discretizations and thereby avoid the frequent grid regeneration that can otherwise be required for models involving large deformations and displacements. This article reviews immersed methods for both elastic structures and structures with prescribed kinematics. It considers formulations using integral operators to connect the Eulerian and Lagrangian frames and methods that directly apply jump conditions along fluid-structure interfaces. Benchmark problems demonstrate the effectiveness of these methods, and selected applications at Reynolds numbers up to approximately 20,000 highlight their impact in biological and biomedical modeling and simulation.

Robotic device shows lack of momentum enhancement for gymnotiform swimmers

Many fish generate thrust by undulating one or multiple elongated fins while keeping their body straight. This propulsion mechanism has stimulated interest in both biology and bio-inspired marine propulsion because its maneuverability and efficiency at low speed. Analytical studies have found that a fin attached to a rigid flat body can produce substantially higher thrust compared to a fin without a body, three- to four-fold for natural swimmers. However, this momentum enhancement has not been confirmed experimentally. In this work, a robotic ribbon fin model with an adjustable-height body was used to test the momentum enhancement for gymontiform swimmers where the undulating fin runs along the ventral side of the body. In a series of experiments, the force generated by the robotic device was measured as the body height of the robot, the undulating fin frequency and the flow speed were changed. It was found that the thrust generated by the ribbon fin is not affected by the presence of a body, thereby resulting in no momentum enhancement due to the fin-body interaction. These results suggest that if there is a benefit at a specific fin-body height ratio of the fishes, the momentum enhancement is not the reason. This result has broader implications in understanding the evolutionary adaption of undulatory fin propulsion and underwater vehicles designs.

Hydrodynamics of linear acceleration in bluegill sunfish, Lepomis macrochirus

In their natural habitat, fish rarely swim steadily. Instead they frequently accelerate and decelerate. Relatively little is known about how fish produce extra force for acceleration in routine swimming behavior. In this study, we examined the flow around bluegill sunfish Lepomis macrochirus during steady swimming and during forward acceleration, starting at a range of initial swimming speeds. We found that bluegill produce vortices with higher circulation during acceleration, indicating a higher force per tail beat, but they do not substantially redirect the force. We quantified the flow patterns using high speed video and particle image velocimetry and measured acceleration with small inertial measurement units attached to each fish. Even in steady tail beats, the fish accelerates slightly during each tail beat, and the magnitude of the acceleration varies. In steady tail beats, however, a high acceleration is followed by a lower acceleration or a deceleration, so that the swimming speed is maintained; in unsteady tail beats, the fish maintains the acceleration over several tail beats, so that the swimming speed increases. We can thus compare the wake and kinematics during single steady and unsteady tail beats that have the same peak acceleration. During unsteady tail beats when the fish accelerates forward for several tail beats, the wake vortex forces are much higher than those at the same acceleration during single tail beats in steady swimming. The fish also undulates its body at higher amplitude and frequency during unsteady tail beats. These kinematic changes likely increase the fluid dynamic added mass of the body, increasing the forces required to sustain acceleration over several tail beats. The high amplitude and high frequency movements are also likely required to generate the higher forces needed for acceleration. Thus, it appears that bluegill sunfish face a trade-off during acceleration: the body movements required for acceleration also make it harder to accelerate.

A pressure-based force and torque prediction technique for the study of fish-like swimming

Many outstanding questions about the evolution and function of fish morphology are linked to swimming dynamics, and a detailed knowledge of time-varying forces and torques along the animal’s body is a key component in answering many of these questions. Yet, quantifying these forces and torques experimentally represents a major challenge that to date prevents a full understanding of fish-like swimming. Here, we develop a method for obtaining these force and torque data non-invasively using standard 2D digital particle image velocimetry in conjunction with a pressure field algorithm. We use a mechanical flapping foil apparatus to model fish-like swimming and measure forces and torques directly with a load cell, and compare these measured values to those estimated simultaneously using our pressure-based approach. We demonstrate that, when out-of-plane flows are relatively small compared to the planar flow, and when pressure effects sufficiently dominate shear effects, this technique is able to accurately reproduce the shape, magnitude, and timing of locomotor forces and torques experienced by a fish-like swimmer. We conclude by exploring of the limits of this approach and its feasibility in the study of freely-swimming fishes.

How the bending kinematics of swimming lampreys build negative pressure fields for suction thrust

Swimming animals commonly bend their bodies to generate thrust. For undulating animals such as eels and lampreys, their bodies bend in the form of waves that travel from head to tail. These kinematics accelerate the flow of adjacent fluids, which alters the pressure field in a manner that generates thrust. We used a comparative approach to evaluate the cause-and-effect relationships in this process by quantifying the hydrodynamic effects of body kinematics at the body-fluid interface of the lamprey, Petromyzon marinus, during steady-state swimming. We compared the kinematics and hydrodynamics of healthy control lampreys to lampreys whose spinal cord had been transected mid-body, resulting in passive kinematics along the posterior half of their body. Using high-speed particle image velocimetry (PIV) and a method for quantifying pressure fields, we detail how the active bending kinematics of the control lampreys were crucial for setting up strong negative pressure fields (relative to ambient fields) that generated high-thrust regions at the bends as they traveled all along the body. The passive kinematics of the transected lamprey were only able to generate significant thrust at the tail, relying on positive pressure fields. These different pressure and thrust scenarios are due to differences in how active versus passive body waves generated and controlled vorticity. This demonstrates why it is more effective for undulating lampreys to pull, rather than push, themselves through the fluid.

Propulsive performance of an under-actuated robotic ribbon fin

Many aquatic animals propelled by elongated undulatory fins can perform complex maneuvers and swim with high efficiency at low speeds. In this propulsion, one or multiple waves travel along an elastic fin composed of flexible rays. In this study, we explore the potential benefits or disadvantages of passive fin motion based on the coupling of fluid-structure interactions and elasto-mechanical responses of the undulatory fin. The motivation is to understand how an under-actuated undulating fin can modify its active and passive fin motion to effectively control the hydrodynamic force and propulsive efficiency. We study the kinematics and propulsive performance of an under-actuated ribbon fin using a robotic device. During two experimental sets for fully-actuated fin and under-actuated fin respectively, we measured fin kinematics, surge forces and power consumption. Our results show that under-actuated fin can generate smaller thrust but consume less power comparing to a fully-actuated counterpart. The thrust generated by an under-actuated fin scales similarly to a fully-actuated fin-linear with the enclosed area and quadratic with the relative velocity. Power consumption scales with cube of lateral tangential velocity. Furthermore, we find that the under-actuated fin can keep the same propulsive efficiency as the fully-actuated fin at low relative velocities. This finding has profound implications to both natural swimmers and underwater vehicles using undulating fin-based propulsion, as it suggests that they can potentially exploit passive fin motion without decrementing propulsive efficiency. For underwater vehicles with undulatory fins, an under-actuated design can greatly simplify the mechanical design and control complexity of a versatile propulsion system.

Suction-based propulsion as a basis for efficient animal swimming

A central and long-standing tenet in the conceptualization of animal swimming is the idea that propulsive thrust is generated by pushing the surrounding water rearward. Inherent in this perspective is the assumption that locomotion involves the generation of locally elevated pressures in the fluid to achieve the expected downstream push of the surrounding water mass. Here we show that rather than pushing against the surrounding fluid, efficient swimming animals primarily pull themselves through the water via suction. This distinction is manifested in dominant low-pressure regions generated in the fluid surrounding the animal body, which are observed by using particle image velocimetry and a pressure calculation algorithm applied to freely swimming lampreys and jellyfish. These results suggest a rethinking of the evolutionary adaptations observed in swimming animals as well as the mechanistic basis for bio-inspired and biomimetic engineered vehicles.

Swimming animals are generally assumed to generate forward thrust by pushing surrounding water rearwards. Here, Gemmell et al. show that efficient swimming in lampreys and jellyfish is achieved primarily through suction, as vortex-associated low pressure regions are synchronized by undulations of the body.

Gray's paradox: a fluid mechanical perspective

Nearly eighty years ago, Gray reported that the drag power experienced by a dolphin was larger than the estimated muscle power – this is termed as Gray's paradox. We provide a fluid mechanical perspective of this paradox. The viewpoint that swimmers necessarily spend muscle energy to overcome drag in the direction of swimming needs revision. For example, in undulatory swimming most of the muscle energy is directly expended to generate lateral undulations of the body, and the drag power is balanced not by the muscle power but by the thrust power. Depending on drag model utilized, the drag power may be greater than muscle power without being paradoxical.