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The Analysis of Biomimetic Caudal Fin Propulsion Mechanism with CFD

The Analysis of Biomimetic Caudal Fin Propulsion Mechanism with CFD Hindawi Applied Bionics and Biomechanics Volume 2020, Article ID 7839049, 11 pages https://doi.org/10.1155/2020/7839049 Research Article The Analysis of Biomimetic Caudal Fin Propulsion Mechanism with CFD 1,2 1,2 1,2 1,2 1,2 Guijie Liu , Shuikuan Liu, Yingchun Xie , Dingxin Leng, and Guanghao Li Department of Mechanical and Electrical Engineering, College of Engineering, Ocean University of China, Qingdao 266100, China Key Laboratory of Ocean Engineering of Shandong Province, Ocean University of China, Qingdao 266100, China Correspondence should be addressed to Yingchun Xie; xieyc@ouc.edu.cn Received 27 July 2019; Revised 22 January 2020; Accepted 25 February 2020; Published 24 June 2020 Academic Editor: Mohammad Rahimi-Gorji Copyright © 2020 Guijie Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In nature, fish not only have extraordinary ability of underwater movement but also have high mobility and flexibility. The low energy consumption and high efficiency of fish propulsive method provide a new idea for the research of bionic underwater robot and bionic propulsive technology. In this paper, the swordfish was taken as the research object, and the mechanism of the caudal fin propulsion was preliminarily explored by analyzing the flow field structure generated by the swing of caudal fin. Subsequently, the influence of the phase difference of the heaving and pitching movement, the swing amplitude of caudal fin, and Strouhal number (St number) on the propulsion performance of fish was discussed. The results demonstrated that the fish can obtain a greater propulsion force by optimizing the motion parameters of the caudal fin in a certain range. Lastly, through the mathematical model analysis of the tail of the swordfish, the producing propulsive force principle of the caudal fin and the caudal peduncle was obtained. Hence, the proposed method provided a theoretical basis for the design of a high-efficiency bionic propulsion system. 1. Introduction The most effective movements of swimming aquatic ani- mals of almost all sizes appear to have the form of a trans- verse wave progressing along the body from fish head to Autonomous underwater vehicles (AUVs) are a type of marine equipment that play a significant role in improving fish tail [3], and the fishes that have faster speed are using the daily life of human beings, such as to monitor the marine the biomimetic caudal fin propulsion way. They have a high environment or safeguard modern military operations. Thus, hydrodynamic efficiency and are applicable to long-time, it is gradually becoming an extensive research topic both at long-distance swimming in this way [4]. However, the study of its hydrodynamic characteristics has not come to a unified home and abroad [1]. However, AUVs have some shortages, which greatly limit the application in the narrow, complex, conclusion. and dynamic environment. For instance, the propulsive effi- Early in 1970s, Lighthill [5, 6] employed the influence of ciency is low. Moreover, maneuvering performance and the swing of the caudal fin on the flow field according to the concealment are poor and they have a negative influence “slender body theory” and then came up with “large-scale slender body theory” that is more suitable for analyzing fish on environment. In nature, fish evolved into the swimming mechanism propulsion patterns. Until 2011, Candelier et al. [7] extended that has an outstanding capability to produce high thrust effi- the “slender body theory” to a three-dimensional case to ciently and gains high performance in maneuvering flexibil- obtain the pressure expression and momentum expression ity and controllability. Recently, interest in the motion of of the slender fish body. The above researchers made large contributions to estab- fish has increased. A lot of attempts have been made to mimic the motion of fish and apply it to underwater vehicles and lish and develop the “slender body theory.” There is no deny- robots in the field of oceanography [2]. ing that the theory laid solid foundation for exploring the 2 Applied Bionics and Biomechanics propulsion mechanism of fish. Caudal fin is one of the most Then, a three-dimensional model was imported into Solid- important parts in fish body to generate a propulsive force. works software. In the software, the transverse symmetry sur- As the simplest propulsive mode of fish, caudal fin swing pro- face of the caudal fin was sliced to obtain the transverse pulsive has been concerned by extensive researchers since the interface of the caudal fin. The two-dimensional calculation beginning of the last century. The first to study the relation- model of the caudal fin is shown in Figure 1. ship between the parameters of the caudal fin swing and the The CFD calculation domain setting model is displayed propulsive force was the “resistance hydrodynamic model” in Figure 2. The flow field was established to be 1800 mm × established by Taylor in 1952 [8], which was applicable to 600 mm. The caudal fin has a length of 150 mm and a maxi- low Reynolds number. mum thickness of 15 mm. The outer rectangular border is the As our country keeps a watchful eye to the marine flow field boundary. In order to ensure accuracy and control resources, more and more scientific institutions have begun the number of meshes, a sufficient number of nodes was to do research work in the field of caudal fin propulsion arranged on the profile of caudal fin to encrypt the meshes and have made certain achievements. The influence of cau- near the caudal fin, and the meshes for the flow boundary dal fin stiffness [9, 10], caudal fin area [11], fin strip move- were less relative. ment [12], and swing phase [13] on caudal fin propulsion, The boundary conditions are exhibited in Table 1, and velocity, and efficiency has been preliminarily studied by the movement of the caudal fin model was controlled by a researchers. Besides, Liu et al. [14] considered different UDF function. The mesh model is shown in Figure 3, and thrusts at different frequencies and found that they had a the number of grids was about 31652. specific optimum frequency under a specific flexible con- 2.2. The Motion of Fish Body and Function of Fish Body nection. Xin and Wu [15] studied the effect of the shape Wave. In the process of propulsion, the fish mainly relies of caudal fin on swimming speed and efficiency in fish free on the fluctuations of the spine curve to generate a propulsive propulsion and found that the shape of the optimal caudal force. Through extensive biological observations and experi- fin varies with different swimming modes. Tomita et al. mental studies of fish behavior, researchers have found that [16] clarify developmental processes of the white shark cau- an implicit traveling wave is in the propulsive motion gener- dal fin, based on morphological observations of the caudal ated by the swinging caudal fin and the flexible body, which fin over several developmental stages. travels from the posterior neck to the tail. The bending of The above researchers mainly explored the relationship the spine and muscle tissue makes the fish appear wavy mor- between various influencing factors in theory, and some phology, and the amplitude is gradually bigger from the fish researchers have studied the advancement of fish by estab- head to the fish tail. The wave velocity of the traveling wave lishing a physical model and combining theory with exper- also known as “fish body wave” is greater than the forward iment [17, 18]. In 2016, Yin et al. [19] took into account speed of the fish body. The corresponding mathematical the thrust and resistance acting on the robot, and the thrust function expression is called the fish body wave function. characteristic is an effective factor for calculating the thrust. To some extent, the fish body wave function can be seen as In 2018, Zhong et al. [20] considered the interaction synthesized by the fish wave envelope and sinusoidal curve, between the pectoral fin and the caudal fin, founding that as shown in Figure 4. the dynamics of the pectoral fin and the caudal fin can be The wave function of the fish body begins from the center used to estimate the overall swimming speed of the biomi- of the inertia force of the fish body and gradually extends to metic fish. the caudal peduncle, and its curve equation [21] can be At present, few studies have been carried out on the shape expressed as of caudal fin and its propulsion principle of swordfish. For the sake of bridging this research gap, in this paper, the key y x, t = c x + c x sin kx + ωt , ð1Þ ðÞ ðÞ parameters of caudal fin were firstly calculated by using body 1 2 dynamic mesh technology in Fluent software. The generation of eddy currents and the generation of anti-Karman vortex where y is the lateral displacement of the fish, x is the body streets were analyzed. Secondly, by changing the major axial displacement of the fish, k is the multiple of wavelength parameters of the caudal fin, the propulsion mechanism (k =2π/λ), λ is the wavelength of the fish body wave, c x + was explored. Finally, through the study of the movement 2 c x is the fish wave amplitude envelope function, c is the 2 1 law of the fish tail, the biomimetic mechanism is fitted to primary coefficient of the fish body wave amplitude enve- the movement, which is verified to be correct and reasonable. lope, c is the quadratic coefficient of the fish body wave Meanwhile, the mechanism of the propeller force during the amplitude envelope, and ω is the fish body wave frequency propulsion process is further explored. By using the linkage, (ω =2πf ). the caudal peduncle was fitted by a bionic method, which laid The swing amplitude of the caudal fin and the distribu- the foundation for the underwater robots to realize high- tion of the body wave amplitude can be adjusted by adjusting efficiency propulsion. the value of c and c . 1 2 2.3. Main Parameters of Hydrodynamic Performance of 2. Kinematic Modelling Based on CFD Caudal Fin. The St number is a parameter that expresses 2.1. Establishment of a Finite Element Model. The data of cau- the characteristics of the wake structure [22]. It indicates dal fin was collected by 3D scanning and reverse engineering. the frequency of the swirl and the distance between them. Applied Bionics and Biomechanics 3 Figure 1: Data model of caudal fin. Caudal fin model For the fluctuating caudal fin, the St number is calculated by Wall the following formula: Inlet Outlet fA St = , ð2Þ Wall Flow field where f represents the swing frequency of caudal fin (Hz), A represents the caudal fin heaving motion amplitude, and V is Figure 2: Calculation domain setting. the average swimming velocity. The angle of attack δ is defined as when the fins pass max Table 1: Boundary condition setting. the equilibrium position, the angle moves between the tan- gential direction of the propulsive wave and the axis of sym- Type Value metry of the caudal fin, which can be expressed as Inlet Velocity-inlet 0 m/s Outlet Pressure-outlet 0 Pa δ = ϕ − θ , ð3Þ max 0 Upper and lower boundary Wall — Caudal fin model Wall — where ϕ indicates the angle between the X-axis and the tan- gential direction of the propulsive wave and θ indicates the angle between the geometric axis of symmetry and the X -axis when the tail fin passes the equilibrium position. 2.4. Basic Equation Based on CFD Numerical Calculation. CFD is a numerical calculation method for solving flow con- trol equations [23]. Considering viscous and incompressible flow, the following continuity equation and motion equation are established. ∂ρ ∂ + ρu =0, ð4Þ ðÞ ∂t ∂x Figure 3: Finite element model. ∂ ρu u ∂ ∂p ∂p ∂u i j i ′ ′ ðÞ ρu + = − + μ − ρu u + S , i i j i ∂t ∂x ∂x ∂x ∂z j i j j Fish wave envelope ð5Þ where ρ represents the density of fluid, t represents the time, u represents the velocity of fluid, x is the space coor- dinates, p represents the fluid pressure, μ represents the kinematic viscosity coefficient, and S represents the user- Fish wave defined source term. Caudal fin In order to solve Equations (4) and (5), it is also necessary to add a turbulent transport equation. It has been calculated Figure 4: Fish body wave and fish amplitude envelope. that the Reynolds number of all the working conditions is A 4 Applied Bionics and Biomechanics 4 5 between 4×10 and 1:4×10 , so the standard k‐ε [24, 25] As depicted in Figure 6, it can be seen that when the fre- model is used for calculation. It has been verified that the quency is constant, as the St coefficient increases, the flow velocity decreases, and the magnitude of the thrust coefficient standard k‐ε model is suitable for practical engineering flow calculations because of its high robustness and reasonable gradually becomes smaller. The lower limit of the thrust coef- ficient is substantially the same under any working condition accuracy. For further solving the equations above, the because the upper limit of the thrust coefficient decreases as coupled implicit algorithm is utilized; hence, variables such as pressure, velocity, and stress can be obtained simulta- the St coefficient increases. When the frequency is 1 Hz, it can be seen that although the magnitude of the thrust coeffi- neously [26]. cient changes with the change of the St number, the ampli- tude of the upper limit changes significantly. At the same 3. Hydrodynamic Calculation time, as the frequency increases, the lower limit of the thrust 3.1. Analysis of the Mechanism of Caudal Fin Propulsion. The coefficient also increases. Since the caudal fin moves in the negative direction of the fluctuation frequency f =0:8 Hz, the swing amplitude A = 120 mm, the fluctuation period T =1:25 s, and the phase dif- X-axis, the negative value in Figure 6 indicates the same pro- ference of the heaving and pitching movement 60 are pulsive force as the caudal fin move direction, and the posi- tive value indicates the resistance. selected for calculation [5, 13, 27]. As shown in Figure 5, when t =0:01 s, the caudal fin It can be seen from Figure 7 that the variation law of the average force curve is gradually increasing with the increase begins to move forward. At this time, caudal fin is around in anticlockwise rotation, and the upper side of the caudal of the St number, but the range of the increasing amplitude fin fluid pressure gradually increased. Meanwhile, the lower is gradually smaller. When St = 0:25, the force generated by the fins at f =0:5 Hz is not conducive to the advancement side of the pressure becomes smaller due to the formation of a low-pressure area. of the fish body. It can be obtained that the swing frequency has a huge influence on the thrust coefficient. The fish body When t = 1/4 T (0.31 s), the caudal fin reaches to the highest position of the swing. The lower side of the fish tail can overcome the flow resistance by adjusting its own tail- forms a low-pressure zone, and the swirl current is generated end frequency in time according to water velocity to avoid the force of the caudal fin to hinder the movement. by the front end of the caudal fin, which is the beginning of the second swirl. In t =0:5 s, the rotation of the swirl direc- tion is counterclockwise rotation, which is opposite to the 3.3. Effects of Phase Difference and the Angle of Attack. first swirl direction. During the process of fluctuation of the caudal fin, there is When t = 1/2 T (0.63 s), the second swirl completely falls a phase difference between the heaving and pitching move- off and the high- and low-pressure zones on both sides of the ment. The different motions can be obtained through chang- caudal fin appear mutative. The lower side is the high- ing the phase differences and then the hydrodynamic pressure area, and the upper side is the low-pressure area. numerical simulation can be analyzed, respectively. t =1:74 s and t =1:77 s are the fourth swirl belonging to The thrust coefficient changing with times is shown in the second cycle, and mechanism is similar with the second Figure 8. It can be obtained that the force of the caudal fin swirl. t =2:35 s is the fifth swirl, and it also belongs to the is the same as the direction of advancement at the beginning. second cycle, the mechanism of which is similar with the The direction of force changes with the heaving and pitching third swirl. movement, which becomes the opposite direction with the After the analysis of the caudal fin swimming process, we fish swimming, and is not conducive to fish for forward. In can acquire that the caudal fin swims in a wave manner. On Figure 8, the shaded portion below the line of y =0 indicates the upper and lower sides of the caudal fin, the high-pressure that it is conducive to fish for moving forward while the and low-pressure regions are formed according to the pitch- shaded portion above the straight line of y =0 indicates that ing direction. The forward swirl is gradually formed at the it is not conducive to fish for moving forward. The longer front end of the caudal fin. The swirl of the body becomes time it takes to promote the advancing force of the caudal larger as the caudal fin swings. Meanwhile, the swirl moves fin, the more favorable to forward. Comparing toward the end of the caudal fin, and it finally falls off.By Figures 8(a)–8(d), it can be seen that as the increase of the observing the direction of rotation of the five shedding swirls, phase difference, the larger the shadow area below the y =0 it can be found that the first, third, and fifth swirls are below line, which means the longer time to propulsion in a cycle. the X-axis and the direction is clockwise. The second and As displayed in Figure 9, when the phase difference is 50- fourth swirls are above the X-axis, and the direction of rota- ° ° 60 (the angle of attack is 21.6-25.6 ), the caudal fin is pro- tion is clockwise. It can be found that these five shedding pelled by a large force, so it is a relatively optimized mode swirls are in the tangential direction of X-axis and opposite of motion in this condition. to the swimming direction of the caudal fin. And then the Karman vortex shedding is formed, forming a backward jet to result in forward thrust. 3.4. Effects of Swing Amplitude. Based on above analysis, the following parameters are selected for calculation: the phase 3.2. Effects of St Number. The operating condition is selected difference is 60 , and the angle of attack is 23.3. The swing with the swing amplitude A = 120 mm, and the phase differ- amplitude A is selected as the following values: A =90 mm, ence is 60 . 120 mm, 150 mm, 180 mm, and 210 mm. In order to obtain Applied Bionics and Biomechanics 5 t = 0.47s t = 0.01s t = 1/4T (0.31s) (a) (b) (c) t = 0.50s t = 1/2T (0.63s) t = 1.12s (d) (e) (f) t = 1.77s t = 1.74s t = T (1.25s) (g) (h) (i) t = 2.35s (j) Figure 5: The fluid pressure distribution of the fluctuation caudal finatdifferent times. 40 100 –50 –20 –100 –40 –150 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Times (s) Times (s) St = 0.40 St = 0.40 St = 0.25 St = 0.25 St = 0.30 St = 0.45 St = 0.30 St = 0.45 St = 0.35 St = 0.35 (a) (b) Figure 6: Forced condition of different values of St: (a) f =0:5 Hz; (b) f =1 Hz. the relationship between the thrust coefficient and the swing swing amplitude is 180 mm and 210 mm, a slightly higher amplitude intuitively, Figure 10 was drawn. frequency can provide a more effective propulsion force for the caudal fin. As shown in Figure 10, it can be seen that as the swing amplitude increases, the propulsion force of the caudal fin in the swimming direction increases. When the swing fre- 3.5. Study on Propulsion Performance of Double Caudal Fins quency is f =0:8 Hz and the swing amplitude is 90 mm in Flow Field. In many cases, the fish do not swim alone. The and 120 mm, the swing of the caudal fin will generate a caudal fin swing of the former fish will cause the flow field to produce a certain regular wake vortex. The analysis of the force that hinders the advancement of the fish body. When the swing amplitude is larger than 150 mm, the propulsive way of the caudal fin using the wake vortex energy will be force for promoting motion will be generated. In addition, conducive to provide the development of basic theory for comparing the two curves, it can be seen that when the underwater biomimetic propulsion. rust coefficient rust coefficient 6 Applied Bionics and Biomechanics 15 formed at the trailing edge position of the double caudal fins. Between t =0:28 s and t =0:65 s, the rear caudal fin swayed with the front one and destroyed the wake vortex formed during the swinging of the front caudal fin. During the latter half of the caudal fin motion (between t =1:03 s and t =1:35 s), the wake vortex caused by the swing of the rear caudal fin failed to overlap with the eddy current of the front caudal fin. Due to the difference of angle of attack between double caudal fins, the rear caudal fin will destroy the vortex pro- duced by the front caudal fin, resulting in vortex dissipation –5 that is not conducive to the effective advancement of the rear 0.2 0.25 0.3 0.35 0.4 0.45 caudal fin. Value of St f = 0.5 Hz 4. The Control Method of Tail Movement f = 1 Hz 4.1. The Analysis of Tail Movement. The biomimetic tail pro- Figure 7: The mean value of the force of the caudal fin changes with pulsion mechanism mainly includes the caudal peduncle and St number. the caudal fins. The movement of the caudal fin is driven by the caudal peduncle. Swing frequency, swing amplitude, the phase difference, 4.1.1. Caudal Fin Simplified Model. When biomimetic under- and the distance between two caudal fins will affect the water vehicle is in the process of swimming, the process is mutual vortices in the flow field. Among these parameters, shown in Figure 14. Firstly, the static state is shown in the effect of the two caudal fins’ angle of attack on the wake Figure 14(a). The caudal fin in the quiescent state does not vortex is mainly studied. In the process of group swimming, occur angular swing. Then, as shown in Figure 14(b), the cau- the swimming gait of fish is basically similar. So, we mainly dal peduncle does not occur swing and the caudal fin begins changed the angle of attack during the hydrodynamic to swing upward. Subsequently, the caudal peduncle and the numerical simulation analysis, and other parameters are set caudal fin swing together and the caudal peduncle swings in a to the same. large angle as shown in Figure 14(c). Lastly, as shown in Figure 14(d), the caudal peduncle and the caudal fin swing 3.5.1. Simulation Model Establishment. The flow field is to the initial position from the maximum swing angle. After established with a length of 2300 mm and a width of that, they swing from the equilibrium position to the oppo- 1000 mm. The distance between the two caudal fins is site direction. 250 mm. The caudal fin model and boundary conditions are set to the same as before. The calculation domain creation 4.1.2. Tail Movement Model Establishment. The main part of and meshing are shown in Figure 11. the tail movement part includes the caudal peduncle move- ment and the caudal fin movement. 3.5.2. Wake Vortex of Double Caudal Fins. In Figure 12, the In the study of fish tail movement, the caudal fin move- double swinging caudal fins have the same swing amplitude ment is simplified as a rigid hydrofoil moving in the uniform in A = 150 mm. The swing frequency is f =0:8 Hz. And the flow field. The way of movement is around itself doing pitch- heaving and pitching movement phase difference is 60 .At ing and heaving swing compound movement. The equation t =0:04 s, the double caudal fins started to move, and the of motion is expressed as follows: trailing edges of the double caudal fins began to form eddy currents. From t =0:18sto t =0:29 s, the double caudal fins y = A × sinðÞ 2πft , ð6Þ swing simultaneously, and the two vortices formed at the θ = θ × sinðÞ 2πft‐φ : right rear of the double caudal fins gradually converge into 1 0 a large vortex. Double swinging caudal fins simultaneously sway at t =0:65sto t =0:90 s and form the upper and lower We can get the rising-sinking speed along with the Y-axis vortices at the trailing edge of caudal fin. At t =1:28 s, two of caudal fin and the pitching angular velocity around the Z -axis through the derivation of formula (6): rows of eddy currents are formed on the lower two sides of the swimming track. Due to the proper spacing and the same swing parameters during the entire swing, the rear ð7Þ V =2πfA × cosðÞ 2πft , caudal fin can add its own vortices to others without destroying the wake vortex of front caudal fin. The superim- ω =2πf θ cos 2πft − φ : ð8Þ ðÞ posed vortex will be beneficial to the rear caudal fins to pro- duce more efficient propulsion. The half of caudal fin expansion is r, so the caudal fin swing speed is 3.5.3. Double Caudal Fin Motion Tail Vortex Dissipation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mode. As shown in Figure 13, when t =0:01 s, the double V = V + ωr +2V ωr cos θ : ð9Þ ðÞ 1 1 1 caudal fins just started to swing and a wake vortex was rust coefficient Applied Bionics and Biomechanics 7 200 150 0 0 –50 –100 –100 –200 –150 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Time (s) Time (s) (a) (b) 100 100 50 50 0 0 –50 –50 –100 –100 –150 –150 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Time (s) Time (s) (c) (d) ° ° Figure 8: Resistance coefficient versus time in a cycle under different phase difference. (a) Phase difference is 30 , angle of attack is 31.7 ; (b) ° ° ° ° ° phase difference is 40 , angle of attack is 28.4 ; (c) phase difference is 50 , angle of attack is 25.6 ; (d) phase difference is 60 , angle of attack is 21.6 . –15 –20 –25 –30 –20 –35 –40 –40 30 40 50 60 70 80 90 –60 Phase difference (°) –80 Figure 9: Mean thrust coefficient diagram of caudal fin under 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 different phase difference. Swing amplitude (m) f = 0.5 Hz Then, we can get the speed of the caudal fin relative to f = 0.8 Hz the fluid: Figure 10: Comparison of average thrust coefficient in different swing amplitude. * * * V = V + V, ð10Þ 1t 0 The swing angular velocity of the caudal peduncle can be where V indicates the flow velocity and V indicates the expressed by the equation after the derivation of the above moving speed of caudal fin along the Y-axis. formula: As there is a phase difference ϕ between the caudal peduncle and the caudal fin when they swing, the swing ðÞ t =2πfθ × cosðÞ 2πft‐ϕ : ð12Þ 2 0 amplitude of caudal peduncle is set as A , so the swing law of caudal peduncle can be expressed as According to the theory of the wave plate [28], the caudal peduncle movement model can be simplified as a rigid plate. θ ðÞ t = θ × sinðÞ 2πft‐ϕ : ð11Þ 2 0 The swing speed of the caudal peduncle can be approximated Mean thrust coefficient rust coefficient rust coefficient rust coefficient rust coefficient Average thrust coefficient 8 Applied Bionics and Biomechanics 1t α = arctg + θ 2 2 2 2 2 2 4π f A cosðÞ 2πft + ω r +2V ωr cos θ 1 1 = arctg + θ , ð18Þ where θ is the instantaneous swing angle of the caudal fin, L is the length of slender, and C is the chord length. The driving force produced by the lifting of the caudal Figure 11: Calculation domain creation and meshing. finis by the linear velocity of the center of gravity of the caudal F =2πρLCV sin α cos α sin θ peduncle. As shown in Figure 15, the distance of caudal s1 1 1t 2 2 2 2 2 2 2 peduncle to the fish swing joints is set as r . 2 =2πρLC V +4π f A cos 2πft + ω r ðÞ When the caudal peduncle is moving in accordance with +2V ωr cos θ Þ sin α cos α sinðÞ θ × sinðÞ 2πft‐φ : 1 1 0 sinusoidal law, the displacement of the center of gravity can be approximated represented by the following formula: ð19Þ From the above analysis, we can obtain the total pro- x = r × sin θ = r × θ × sin 2πft‐ϕ : ð13Þ ðÞ ðÞ 2 2 2 2 0 pulsion produced by the moving caudal fin: The speed of the caudal peduncle center of gravity can be 2 2 2 2 2 2 2 F = F + F = ρS V +4π f A cos 2πft + ω r ðÞ expressed as 1 t1 s1 1 0 +2V ωr cos θ Þ sin θ +2πρLC V 1 1 1 V ðÞ t =2πfr × θ × cosðÞ 2πft‐ϕ : ð14Þ 2 2 0 2 2 2 2 2 2 +4π f A cos 2πft + ω r ðÞ +2V ωr cos θ Þ sin α cos α sin θ × sin 2πft‐φ : ðÞ ðÞ 1 1 0 Flow velocity is set as V , so that the relative velocity of ð20Þ the center point of the caudal peduncle is The analysis and calculation of the propulsion is simi- * * * V ðÞ t = V + V ðÞ t : ð15Þ lar to that of the caudal fin. The total propulsion calcula- 3 0 2 tion method of the caudal peduncle can be expressed as follows: 4.2. Establishment of a Kinematic Model of Tail Motion. When the fish is moving in the flow field, the tail will be sub- 2 2 2 2 2 2 jected to the pressure of the fluid from all directions. The F = ρ V +4π f r × θ × cos 2πft‐ϕ ðÞ 2 0 2 0 fluid pressure on the surface of the caudal fin is set as F . 2 2 2 2 2 × S × sin θ +2πρLC V +4π f r × θ According to the Bernoulli principle, the analysis of force is 2 2 0 2 0 as shown in Figure 16. 2 × cos 2πft‐ϕ Þ sin α cos α sin θ × sin 2πft‐ϕ : ðÞ ðÞ ðÞ 2 2 0 The propulsive force generated by fluid pressure: ð21Þ 1 1 2 2 2 2 2 2 F = ρV S sin θ = ρS V +4π f A cosðÞ 2πft 4.3. Gait Fitting of Fish Body Tail Biomimetic Mechanism. t1 1 1 1t 1 0 2 2 In Section 4, we used the link mechanism to simulate the 2 2 + ω r +2V ωr cos θ sin θ : 1 1 1 fish body wave, so it is necessary to control the swing angle of each joint in order that each connection endpoint ð16Þ can approximately fit the fish wave curve. The λ is defined as the ratio of the length of the tail swing to the whole According to the wing theory [29, 30], the lift effect on wavelength. It is essential to ensure that each linkage is caudal fin is set as F; the stress analysis is shown in Figure 16. continuous and the end point is at the end of the last link- The lift force of the fluid on the caudal fin in the vertical age on the fish body wave curve. The end position of link- direction of the caudal finis age satisfies the following equations: F =2πρLCV sin α cos α, ð17Þ s 1t x − x + y − y = L , i,j i,j−1 i,j i,j−1 j ð22Þ 2π where α is the instantaneous relative angle of attack of the : y ðÞ x, t = c x + c x sin kx − i , 1 i 2 i i i,j caudal fin; Applied Bionics and Biomechanics 9 t = 0.04s t = 0.18s t = 0.29s t = 0.65s (a) (b) (c) (d) t = 0.74s t = 0.90s t = 1.28s t = 1.49s (e) (f) (g) (h) Figure 12: Wake vortex of double caudal fin pressure cloud. t = 0.35s t = 0.65s t = 0.01s t = 0.28s (a) (b) (c) (d) t = 1.03s t = 1.15s t = 1.28s t = 1.35s (e) (f) (g) (h) Figure 13: Vortex’s dissipation model pressure cloud. (a) (b) (c) (d) Figure 14: Tail swing diagram. Y Y Caudal fin Caudal peduncle O x 𝛼 x O 𝜃 → → r v v 2 1t 1t F m Figure 15: Schematic diagram of caudal peduncle swing. Figure 16: Schematic diagram of the caudal fin’s fluid pressure decomposition and lift force analysis. 10 Applied Bionics and Biomechanics (2) When St number is in the range of 0.25 to 0.45, the where ðx , y Þ is the angular coordinate of the “j” linkage i,j i,j propulsive force generated by the caudal fin in one at the moment i in the swing period, x =0, x = λ ·2π, i,o i,5 swing period gradually increases as the St number 1 ≤ j ≤ 5, 0 ≤ i ≤ M. increases, but the range of the increasing amplitude Bring L into the formula (23), we can solve the coordi- gradually becomes smaller. In addition, when the nates of each endpoint at i =0, i =1 until i = M. After that, ° ° phase difference is in the range of 50 ~60 , the pro- we can get the angle θ between each linkage and fish body, i,j pulsion of the caudal fin is relatively large and this which can be expressed as is a more optimized motion mode "# (3) For hydrodynamic studies of double caudal fins, x − x i,j i,j−1 changing the angle of attack of the double caudal fins θ = artan : ð23Þ i,j y − y will produce different wake flow field structures. A i,j i,j−1 reasonable use of the wake vortex generated by the front caudal fin will help the rear caudal fin to reduce The swing angles of the five steering gears are calculated, the resistance and generate a propulsive force more respectively, effectively 8  (4) By studying the motion law of the tail of swordfish, y − y 1 0 > the motion fitting of the tail swing was carried out φ = arctan , > x − x 1 0 by using the link mechanism which was widely used > in machinery. By calculating the thrust of the simpli- > y − y > 2 1 φ = arctan − φ , fied tail swing model, the principle of the thrust gen- 2 1 x − x 2 1 erated by the caudal fin and caudal peduncle in the y − y process of propulsion was analyzed. By controlling 3 2 φ = arctan − φ , ð24Þ 3 2 the angle of the steering gear, the fish body wave x − x > 3 2 was fitted to the fish tail motion. In this way, the bio- > y − y 4 3 > mimetic motion mechanism of the caudal fin was > φ = arctan − φ , > 4 3 x − x preliminarily studied 4 3 y − y 4 3 φ = arctan − φ : : 6. Future Work 5 4 x − x 4 3 Based on the present study on theoretical calculation analysis of the caudal fin, the authors will build an experimental plat- Through the analysis of the fish body wave function, we form for the experimental analysis of the swing angle of the can obtain that as the number of joints increases, it is easier caudal fin by using the linkage mechanism, which further to fit the fish body wave curve, but it is more difficult to coor- verifies the correctness of our simulation results in the future dinate control between the steering gear (the motion of the work. tail swing is controlled by the steering gear). Thus, the end- points of the five joints should be fitted to the corresponding Data Availability fish body wave curve as much as possible. More importantly, the working angle of each steering gear and their mutual The data used to support the findings of this study are avail- position relationship should be well controlled. In this way, able from the corresponding author upon request. the bionic underwater vehicle can be moved like a fish, thus improving the propulsion efficiency and saving energy. Conflicts of Interest 5. Conclusion The authors declare that there is no conflict of interest regarding the publication of this paper. In this paper, the fast-moving swordfish was taken as the research object to explore the flow field structure of the Acknowledgments swordfish caudal fin swinging. The mechanism of the caudal fin propulsion was preliminarily investigated, and the bionic This research was supported by the National Science mechanism motion fitting was carried out. The main conclu- Foundation of China (No. 61540010 and No. 51979259) sions were obtained as follows: and the Shandong Natural Science Foundation (No. ZR201709240210). (1) During the swinging process, the fish-tail forms a wake vortex due to the transformation of the high- References pressure and low-pressure zones. 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The Analysis of Biomimetic Caudal Fin Propulsion Mechanism with CFD

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Hindawi Applied Bionics and Biomechanics Volume 2020, Article ID 7839049, 11 pages https://doi.org/10.1155/2020/7839049 Research Article The Analysis of Biomimetic Caudal Fin Propulsion Mechanism with CFD 1,2 1,2 1,2 1,2 1,2 Guijie Liu , Shuikuan Liu, Yingchun Xie , Dingxin Leng, and Guanghao Li Department of Mechanical and Electrical Engineering, College of Engineering, Ocean University of China, Qingdao 266100, China Key Laboratory of Ocean Engineering of Shandong Province, Ocean University of China, Qingdao 266100, China Correspondence should be addressed to Yingchun Xie; xieyc@ouc.edu.cn Received 27 July 2019; Revised 22 January 2020; Accepted 25 February 2020; Published 24 June 2020 Academic Editor: Mohammad Rahimi-Gorji Copyright © 2020 Guijie Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In nature, fish not only have extraordinary ability of underwater movement but also have high mobility and flexibility. The low energy consumption and high efficiency of fish propulsive method provide a new idea for the research of bionic underwater robot and bionic propulsive technology. In this paper, the swordfish was taken as the research object, and the mechanism of the caudal fin propulsion was preliminarily explored by analyzing the flow field structure generated by the swing of caudal fin. Subsequently, the influence of the phase difference of the heaving and pitching movement, the swing amplitude of caudal fin, and Strouhal number (St number) on the propulsion performance of fish was discussed. The results demonstrated that the fish can obtain a greater propulsion force by optimizing the motion parameters of the caudal fin in a certain range. Lastly, through the mathematical model analysis of the tail of the swordfish, the producing propulsive force principle of the caudal fin and the caudal peduncle was obtained. Hence, the proposed method provided a theoretical basis for the design of a high-efficiency bionic propulsion system. 1. Introduction The most effective movements of swimming aquatic ani- mals of almost all sizes appear to have the form of a trans- verse wave progressing along the body from fish head to Autonomous underwater vehicles (AUVs) are a type of marine equipment that play a significant role in improving fish tail [3], and the fishes that have faster speed are using the daily life of human beings, such as to monitor the marine the biomimetic caudal fin propulsion way. They have a high environment or safeguard modern military operations. Thus, hydrodynamic efficiency and are applicable to long-time, it is gradually becoming an extensive research topic both at long-distance swimming in this way [4]. However, the study of its hydrodynamic characteristics has not come to a unified home and abroad [1]. However, AUVs have some shortages, which greatly limit the application in the narrow, complex, conclusion. and dynamic environment. For instance, the propulsive effi- Early in 1970s, Lighthill [5, 6] employed the influence of ciency is low. Moreover, maneuvering performance and the swing of the caudal fin on the flow field according to the concealment are poor and they have a negative influence “slender body theory” and then came up with “large-scale slender body theory” that is more suitable for analyzing fish on environment. In nature, fish evolved into the swimming mechanism propulsion patterns. Until 2011, Candelier et al. [7] extended that has an outstanding capability to produce high thrust effi- the “slender body theory” to a three-dimensional case to ciently and gains high performance in maneuvering flexibil- obtain the pressure expression and momentum expression ity and controllability. Recently, interest in the motion of of the slender fish body. The above researchers made large contributions to estab- fish has increased. A lot of attempts have been made to mimic the motion of fish and apply it to underwater vehicles and lish and develop the “slender body theory.” There is no deny- robots in the field of oceanography [2]. ing that the theory laid solid foundation for exploring the 2 Applied Bionics and Biomechanics propulsion mechanism of fish. Caudal fin is one of the most Then, a three-dimensional model was imported into Solid- important parts in fish body to generate a propulsive force. works software. In the software, the transverse symmetry sur- As the simplest propulsive mode of fish, caudal fin swing pro- face of the caudal fin was sliced to obtain the transverse pulsive has been concerned by extensive researchers since the interface of the caudal fin. The two-dimensional calculation beginning of the last century. The first to study the relation- model of the caudal fin is shown in Figure 1. ship between the parameters of the caudal fin swing and the The CFD calculation domain setting model is displayed propulsive force was the “resistance hydrodynamic model” in Figure 2. The flow field was established to be 1800 mm × established by Taylor in 1952 [8], which was applicable to 600 mm. The caudal fin has a length of 150 mm and a maxi- low Reynolds number. mum thickness of 15 mm. The outer rectangular border is the As our country keeps a watchful eye to the marine flow field boundary. In order to ensure accuracy and control resources, more and more scientific institutions have begun the number of meshes, a sufficient number of nodes was to do research work in the field of caudal fin propulsion arranged on the profile of caudal fin to encrypt the meshes and have made certain achievements. The influence of cau- near the caudal fin, and the meshes for the flow boundary dal fin stiffness [9, 10], caudal fin area [11], fin strip move- were less relative. ment [12], and swing phase [13] on caudal fin propulsion, The boundary conditions are exhibited in Table 1, and velocity, and efficiency has been preliminarily studied by the movement of the caudal fin model was controlled by a researchers. Besides, Liu et al. [14] considered different UDF function. The mesh model is shown in Figure 3, and thrusts at different frequencies and found that they had a the number of grids was about 31652. specific optimum frequency under a specific flexible con- 2.2. The Motion of Fish Body and Function of Fish Body nection. Xin and Wu [15] studied the effect of the shape Wave. In the process of propulsion, the fish mainly relies of caudal fin on swimming speed and efficiency in fish free on the fluctuations of the spine curve to generate a propulsive propulsion and found that the shape of the optimal caudal force. Through extensive biological observations and experi- fin varies with different swimming modes. Tomita et al. mental studies of fish behavior, researchers have found that [16] clarify developmental processes of the white shark cau- an implicit traveling wave is in the propulsive motion gener- dal fin, based on morphological observations of the caudal ated by the swinging caudal fin and the flexible body, which fin over several developmental stages. travels from the posterior neck to the tail. The bending of The above researchers mainly explored the relationship the spine and muscle tissue makes the fish appear wavy mor- between various influencing factors in theory, and some phology, and the amplitude is gradually bigger from the fish researchers have studied the advancement of fish by estab- head to the fish tail. The wave velocity of the traveling wave lishing a physical model and combining theory with exper- also known as “fish body wave” is greater than the forward iment [17, 18]. In 2016, Yin et al. [19] took into account speed of the fish body. The corresponding mathematical the thrust and resistance acting on the robot, and the thrust function expression is called the fish body wave function. characteristic is an effective factor for calculating the thrust. To some extent, the fish body wave function can be seen as In 2018, Zhong et al. [20] considered the interaction synthesized by the fish wave envelope and sinusoidal curve, between the pectoral fin and the caudal fin, founding that as shown in Figure 4. the dynamics of the pectoral fin and the caudal fin can be The wave function of the fish body begins from the center used to estimate the overall swimming speed of the biomi- of the inertia force of the fish body and gradually extends to metic fish. the caudal peduncle, and its curve equation [21] can be At present, few studies have been carried out on the shape expressed as of caudal fin and its propulsion principle of swordfish. For the sake of bridging this research gap, in this paper, the key y x, t = c x + c x sin kx + ωt , ð1Þ ðÞ ðÞ parameters of caudal fin were firstly calculated by using body 1 2 dynamic mesh technology in Fluent software. The generation of eddy currents and the generation of anti-Karman vortex where y is the lateral displacement of the fish, x is the body streets were analyzed. Secondly, by changing the major axial displacement of the fish, k is the multiple of wavelength parameters of the caudal fin, the propulsion mechanism (k =2π/λ), λ is the wavelength of the fish body wave, c x + was explored. Finally, through the study of the movement 2 c x is the fish wave amplitude envelope function, c is the 2 1 law of the fish tail, the biomimetic mechanism is fitted to primary coefficient of the fish body wave amplitude enve- the movement, which is verified to be correct and reasonable. lope, c is the quadratic coefficient of the fish body wave Meanwhile, the mechanism of the propeller force during the amplitude envelope, and ω is the fish body wave frequency propulsion process is further explored. By using the linkage, (ω =2πf ). the caudal peduncle was fitted by a bionic method, which laid The swing amplitude of the caudal fin and the distribu- the foundation for the underwater robots to realize high- tion of the body wave amplitude can be adjusted by adjusting efficiency propulsion. the value of c and c . 1 2 2.3. Main Parameters of Hydrodynamic Performance of 2. Kinematic Modelling Based on CFD Caudal Fin. The St number is a parameter that expresses 2.1. Establishment of a Finite Element Model. The data of cau- the characteristics of the wake structure [22]. It indicates dal fin was collected by 3D scanning and reverse engineering. the frequency of the swirl and the distance between them. Applied Bionics and Biomechanics 3 Figure 1: Data model of caudal fin. Caudal fin model For the fluctuating caudal fin, the St number is calculated by Wall the following formula: Inlet Outlet fA St = , ð2Þ Wall Flow field where f represents the swing frequency of caudal fin (Hz), A represents the caudal fin heaving motion amplitude, and V is Figure 2: Calculation domain setting. the average swimming velocity. The angle of attack δ is defined as when the fins pass max Table 1: Boundary condition setting. the equilibrium position, the angle moves between the tan- gential direction of the propulsive wave and the axis of sym- Type Value metry of the caudal fin, which can be expressed as Inlet Velocity-inlet 0 m/s Outlet Pressure-outlet 0 Pa δ = ϕ − θ , ð3Þ max 0 Upper and lower boundary Wall — Caudal fin model Wall — where ϕ indicates the angle between the X-axis and the tan- gential direction of the propulsive wave and θ indicates the angle between the geometric axis of symmetry and the X -axis when the tail fin passes the equilibrium position. 2.4. Basic Equation Based on CFD Numerical Calculation. CFD is a numerical calculation method for solving flow con- trol equations [23]. Considering viscous and incompressible flow, the following continuity equation and motion equation are established. ∂ρ ∂ + ρu =0, ð4Þ ðÞ ∂t ∂x Figure 3: Finite element model. ∂ ρu u ∂ ∂p ∂p ∂u i j i ′ ′ ðÞ ρu + = − + μ − ρu u + S , i i j i ∂t ∂x ∂x ∂x ∂z j i j j Fish wave envelope ð5Þ where ρ represents the density of fluid, t represents the time, u represents the velocity of fluid, x is the space coor- dinates, p represents the fluid pressure, μ represents the kinematic viscosity coefficient, and S represents the user- Fish wave defined source term. Caudal fin In order to solve Equations (4) and (5), it is also necessary to add a turbulent transport equation. It has been calculated Figure 4: Fish body wave and fish amplitude envelope. that the Reynolds number of all the working conditions is A 4 Applied Bionics and Biomechanics 4 5 between 4×10 and 1:4×10 , so the standard k‐ε [24, 25] As depicted in Figure 6, it can be seen that when the fre- model is used for calculation. It has been verified that the quency is constant, as the St coefficient increases, the flow velocity decreases, and the magnitude of the thrust coefficient standard k‐ε model is suitable for practical engineering flow calculations because of its high robustness and reasonable gradually becomes smaller. The lower limit of the thrust coef- ficient is substantially the same under any working condition accuracy. For further solving the equations above, the because the upper limit of the thrust coefficient decreases as coupled implicit algorithm is utilized; hence, variables such as pressure, velocity, and stress can be obtained simulta- the St coefficient increases. When the frequency is 1 Hz, it can be seen that although the magnitude of the thrust coeffi- neously [26]. cient changes with the change of the St number, the ampli- tude of the upper limit changes significantly. At the same 3. Hydrodynamic Calculation time, as the frequency increases, the lower limit of the thrust 3.1. Analysis of the Mechanism of Caudal Fin Propulsion. The coefficient also increases. Since the caudal fin moves in the negative direction of the fluctuation frequency f =0:8 Hz, the swing amplitude A = 120 mm, the fluctuation period T =1:25 s, and the phase dif- X-axis, the negative value in Figure 6 indicates the same pro- ference of the heaving and pitching movement 60 are pulsive force as the caudal fin move direction, and the posi- tive value indicates the resistance. selected for calculation [5, 13, 27]. As shown in Figure 5, when t =0:01 s, the caudal fin It can be seen from Figure 7 that the variation law of the average force curve is gradually increasing with the increase begins to move forward. At this time, caudal fin is around in anticlockwise rotation, and the upper side of the caudal of the St number, but the range of the increasing amplitude fin fluid pressure gradually increased. Meanwhile, the lower is gradually smaller. When St = 0:25, the force generated by the fins at f =0:5 Hz is not conducive to the advancement side of the pressure becomes smaller due to the formation of a low-pressure area. of the fish body. It can be obtained that the swing frequency has a huge influence on the thrust coefficient. The fish body When t = 1/4 T (0.31 s), the caudal fin reaches to the highest position of the swing. The lower side of the fish tail can overcome the flow resistance by adjusting its own tail- forms a low-pressure zone, and the swirl current is generated end frequency in time according to water velocity to avoid the force of the caudal fin to hinder the movement. by the front end of the caudal fin, which is the beginning of the second swirl. In t =0:5 s, the rotation of the swirl direc- tion is counterclockwise rotation, which is opposite to the 3.3. Effects of Phase Difference and the Angle of Attack. first swirl direction. During the process of fluctuation of the caudal fin, there is When t = 1/2 T (0.63 s), the second swirl completely falls a phase difference between the heaving and pitching move- off and the high- and low-pressure zones on both sides of the ment. The different motions can be obtained through chang- caudal fin appear mutative. The lower side is the high- ing the phase differences and then the hydrodynamic pressure area, and the upper side is the low-pressure area. numerical simulation can be analyzed, respectively. t =1:74 s and t =1:77 s are the fourth swirl belonging to The thrust coefficient changing with times is shown in the second cycle, and mechanism is similar with the second Figure 8. It can be obtained that the force of the caudal fin swirl. t =2:35 s is the fifth swirl, and it also belongs to the is the same as the direction of advancement at the beginning. second cycle, the mechanism of which is similar with the The direction of force changes with the heaving and pitching third swirl. movement, which becomes the opposite direction with the After the analysis of the caudal fin swimming process, we fish swimming, and is not conducive to fish for forward. In can acquire that the caudal fin swims in a wave manner. On Figure 8, the shaded portion below the line of y =0 indicates the upper and lower sides of the caudal fin, the high-pressure that it is conducive to fish for moving forward while the and low-pressure regions are formed according to the pitch- shaded portion above the straight line of y =0 indicates that ing direction. The forward swirl is gradually formed at the it is not conducive to fish for moving forward. The longer front end of the caudal fin. The swirl of the body becomes time it takes to promote the advancing force of the caudal larger as the caudal fin swings. Meanwhile, the swirl moves fin, the more favorable to forward. Comparing toward the end of the caudal fin, and it finally falls off.By Figures 8(a)–8(d), it can be seen that as the increase of the observing the direction of rotation of the five shedding swirls, phase difference, the larger the shadow area below the y =0 it can be found that the first, third, and fifth swirls are below line, which means the longer time to propulsion in a cycle. the X-axis and the direction is clockwise. The second and As displayed in Figure 9, when the phase difference is 50- fourth swirls are above the X-axis, and the direction of rota- ° ° 60 (the angle of attack is 21.6-25.6 ), the caudal fin is pro- tion is clockwise. It can be found that these five shedding pelled by a large force, so it is a relatively optimized mode swirls are in the tangential direction of X-axis and opposite of motion in this condition. to the swimming direction of the caudal fin. And then the Karman vortex shedding is formed, forming a backward jet to result in forward thrust. 3.4. Effects of Swing Amplitude. Based on above analysis, the following parameters are selected for calculation: the phase 3.2. Effects of St Number. The operating condition is selected difference is 60 , and the angle of attack is 23.3. The swing with the swing amplitude A = 120 mm, and the phase differ- amplitude A is selected as the following values: A =90 mm, ence is 60 . 120 mm, 150 mm, 180 mm, and 210 mm. In order to obtain Applied Bionics and Biomechanics 5 t = 0.47s t = 0.01s t = 1/4T (0.31s) (a) (b) (c) t = 0.50s t = 1/2T (0.63s) t = 1.12s (d) (e) (f) t = 1.77s t = 1.74s t = T (1.25s) (g) (h) (i) t = 2.35s (j) Figure 5: The fluid pressure distribution of the fluctuation caudal finatdifferent times. 40 100 –50 –20 –100 –40 –150 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Times (s) Times (s) St = 0.40 St = 0.40 St = 0.25 St = 0.25 St = 0.30 St = 0.45 St = 0.30 St = 0.45 St = 0.35 St = 0.35 (a) (b) Figure 6: Forced condition of different values of St: (a) f =0:5 Hz; (b) f =1 Hz. the relationship between the thrust coefficient and the swing swing amplitude is 180 mm and 210 mm, a slightly higher amplitude intuitively, Figure 10 was drawn. frequency can provide a more effective propulsion force for the caudal fin. As shown in Figure 10, it can be seen that as the swing amplitude increases, the propulsion force of the caudal fin in the swimming direction increases. When the swing fre- 3.5. Study on Propulsion Performance of Double Caudal Fins quency is f =0:8 Hz and the swing amplitude is 90 mm in Flow Field. In many cases, the fish do not swim alone. The and 120 mm, the swing of the caudal fin will generate a caudal fin swing of the former fish will cause the flow field to produce a certain regular wake vortex. The analysis of the force that hinders the advancement of the fish body. When the swing amplitude is larger than 150 mm, the propulsive way of the caudal fin using the wake vortex energy will be force for promoting motion will be generated. In addition, conducive to provide the development of basic theory for comparing the two curves, it can be seen that when the underwater biomimetic propulsion. rust coefficient rust coefficient 6 Applied Bionics and Biomechanics 15 formed at the trailing edge position of the double caudal fins. Between t =0:28 s and t =0:65 s, the rear caudal fin swayed with the front one and destroyed the wake vortex formed during the swinging of the front caudal fin. During the latter half of the caudal fin motion (between t =1:03 s and t =1:35 s), the wake vortex caused by the swing of the rear caudal fin failed to overlap with the eddy current of the front caudal fin. Due to the difference of angle of attack between double caudal fins, the rear caudal fin will destroy the vortex pro- duced by the front caudal fin, resulting in vortex dissipation –5 that is not conducive to the effective advancement of the rear 0.2 0.25 0.3 0.35 0.4 0.45 caudal fin. Value of St f = 0.5 Hz 4. The Control Method of Tail Movement f = 1 Hz 4.1. The Analysis of Tail Movement. The biomimetic tail pro- Figure 7: The mean value of the force of the caudal fin changes with pulsion mechanism mainly includes the caudal peduncle and St number. the caudal fins. The movement of the caudal fin is driven by the caudal peduncle. Swing frequency, swing amplitude, the phase difference, 4.1.1. Caudal Fin Simplified Model. When biomimetic under- and the distance between two caudal fins will affect the water vehicle is in the process of swimming, the process is mutual vortices in the flow field. Among these parameters, shown in Figure 14. Firstly, the static state is shown in the effect of the two caudal fins’ angle of attack on the wake Figure 14(a). The caudal fin in the quiescent state does not vortex is mainly studied. In the process of group swimming, occur angular swing. Then, as shown in Figure 14(b), the cau- the swimming gait of fish is basically similar. So, we mainly dal peduncle does not occur swing and the caudal fin begins changed the angle of attack during the hydrodynamic to swing upward. Subsequently, the caudal peduncle and the numerical simulation analysis, and other parameters are set caudal fin swing together and the caudal peduncle swings in a to the same. large angle as shown in Figure 14(c). Lastly, as shown in Figure 14(d), the caudal peduncle and the caudal fin swing 3.5.1. Simulation Model Establishment. The flow field is to the initial position from the maximum swing angle. After established with a length of 2300 mm and a width of that, they swing from the equilibrium position to the oppo- 1000 mm. The distance between the two caudal fins is site direction. 250 mm. The caudal fin model and boundary conditions are set to the same as before. The calculation domain creation 4.1.2. Tail Movement Model Establishment. The main part of and meshing are shown in Figure 11. the tail movement part includes the caudal peduncle move- ment and the caudal fin movement. 3.5.2. Wake Vortex of Double Caudal Fins. In Figure 12, the In the study of fish tail movement, the caudal fin move- double swinging caudal fins have the same swing amplitude ment is simplified as a rigid hydrofoil moving in the uniform in A = 150 mm. The swing frequency is f =0:8 Hz. And the flow field. The way of movement is around itself doing pitch- heaving and pitching movement phase difference is 60 .At ing and heaving swing compound movement. The equation t =0:04 s, the double caudal fins started to move, and the of motion is expressed as follows: trailing edges of the double caudal fins began to form eddy currents. From t =0:18sto t =0:29 s, the double caudal fins y = A × sinðÞ 2πft , ð6Þ swing simultaneously, and the two vortices formed at the θ = θ × sinðÞ 2πft‐φ : right rear of the double caudal fins gradually converge into 1 0 a large vortex. Double swinging caudal fins simultaneously sway at t =0:65sto t =0:90 s and form the upper and lower We can get the rising-sinking speed along with the Y-axis vortices at the trailing edge of caudal fin. At t =1:28 s, two of caudal fin and the pitching angular velocity around the Z -axis through the derivation of formula (6): rows of eddy currents are formed on the lower two sides of the swimming track. Due to the proper spacing and the same swing parameters during the entire swing, the rear ð7Þ V =2πfA × cosðÞ 2πft , caudal fin can add its own vortices to others without destroying the wake vortex of front caudal fin. The superim- ω =2πf θ cos 2πft − φ : ð8Þ ðÞ posed vortex will be beneficial to the rear caudal fins to pro- duce more efficient propulsion. The half of caudal fin expansion is r, so the caudal fin swing speed is 3.5.3. Double Caudal Fin Motion Tail Vortex Dissipation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mode. As shown in Figure 13, when t =0:01 s, the double V = V + ωr +2V ωr cos θ : ð9Þ ðÞ 1 1 1 caudal fins just started to swing and a wake vortex was rust coefficient Applied Bionics and Biomechanics 7 200 150 0 0 –50 –100 –100 –200 –150 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Time (s) Time (s) (a) (b) 100 100 50 50 0 0 –50 –50 –100 –100 –150 –150 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Time (s) Time (s) (c) (d) ° ° Figure 8: Resistance coefficient versus time in a cycle under different phase difference. (a) Phase difference is 30 , angle of attack is 31.7 ; (b) ° ° ° ° ° phase difference is 40 , angle of attack is 28.4 ; (c) phase difference is 50 , angle of attack is 25.6 ; (d) phase difference is 60 , angle of attack is 21.6 . –15 –20 –25 –30 –20 –35 –40 –40 30 40 50 60 70 80 90 –60 Phase difference (°) –80 Figure 9: Mean thrust coefficient diagram of caudal fin under 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 different phase difference. Swing amplitude (m) f = 0.5 Hz Then, we can get the speed of the caudal fin relative to f = 0.8 Hz the fluid: Figure 10: Comparison of average thrust coefficient in different swing amplitude. * * * V = V + V, ð10Þ 1t 0 The swing angular velocity of the caudal peduncle can be where V indicates the flow velocity and V indicates the expressed by the equation after the derivation of the above moving speed of caudal fin along the Y-axis. formula: As there is a phase difference ϕ between the caudal peduncle and the caudal fin when they swing, the swing ðÞ t =2πfθ × cosðÞ 2πft‐ϕ : ð12Þ 2 0 amplitude of caudal peduncle is set as A , so the swing law of caudal peduncle can be expressed as According to the theory of the wave plate [28], the caudal peduncle movement model can be simplified as a rigid plate. θ ðÞ t = θ × sinðÞ 2πft‐ϕ : ð11Þ 2 0 The swing speed of the caudal peduncle can be approximated Mean thrust coefficient rust coefficient rust coefficient rust coefficient rust coefficient Average thrust coefficient 8 Applied Bionics and Biomechanics 1t α = arctg + θ 2 2 2 2 2 2 4π f A cosðÞ 2πft + ω r +2V ωr cos θ 1 1 = arctg + θ , ð18Þ where θ is the instantaneous swing angle of the caudal fin, L is the length of slender, and C is the chord length. The driving force produced by the lifting of the caudal Figure 11: Calculation domain creation and meshing. finis by the linear velocity of the center of gravity of the caudal F =2πρLCV sin α cos α sin θ peduncle. As shown in Figure 15, the distance of caudal s1 1 1t 2 2 2 2 2 2 2 peduncle to the fish swing joints is set as r . 2 =2πρLC V +4π f A cos 2πft + ω r ðÞ When the caudal peduncle is moving in accordance with +2V ωr cos θ Þ sin α cos α sinðÞ θ × sinðÞ 2πft‐φ : 1 1 0 sinusoidal law, the displacement of the center of gravity can be approximated represented by the following formula: ð19Þ From the above analysis, we can obtain the total pro- x = r × sin θ = r × θ × sin 2πft‐ϕ : ð13Þ ðÞ ðÞ 2 2 2 2 0 pulsion produced by the moving caudal fin: The speed of the caudal peduncle center of gravity can be 2 2 2 2 2 2 2 F = F + F = ρS V +4π f A cos 2πft + ω r ðÞ expressed as 1 t1 s1 1 0 +2V ωr cos θ Þ sin θ +2πρLC V 1 1 1 V ðÞ t =2πfr × θ × cosðÞ 2πft‐ϕ : ð14Þ 2 2 0 2 2 2 2 2 2 +4π f A cos 2πft + ω r ðÞ +2V ωr cos θ Þ sin α cos α sin θ × sin 2πft‐φ : ðÞ ðÞ 1 1 0 Flow velocity is set as V , so that the relative velocity of ð20Þ the center point of the caudal peduncle is The analysis and calculation of the propulsion is simi- * * * V ðÞ t = V + V ðÞ t : ð15Þ lar to that of the caudal fin. The total propulsion calcula- 3 0 2 tion method of the caudal peduncle can be expressed as follows: 4.2. Establishment of a Kinematic Model of Tail Motion. When the fish is moving in the flow field, the tail will be sub- 2 2 2 2 2 2 jected to the pressure of the fluid from all directions. The F = ρ V +4π f r × θ × cos 2πft‐ϕ ðÞ 2 0 2 0 fluid pressure on the surface of the caudal fin is set as F . 2 2 2 2 2 × S × sin θ +2πρLC V +4π f r × θ According to the Bernoulli principle, the analysis of force is 2 2 0 2 0 as shown in Figure 16. 2 × cos 2πft‐ϕ Þ sin α cos α sin θ × sin 2πft‐ϕ : ðÞ ðÞ ðÞ 2 2 0 The propulsive force generated by fluid pressure: ð21Þ 1 1 2 2 2 2 2 2 F = ρV S sin θ = ρS V +4π f A cosðÞ 2πft 4.3. Gait Fitting of Fish Body Tail Biomimetic Mechanism. t1 1 1 1t 1 0 2 2 In Section 4, we used the link mechanism to simulate the 2 2 + ω r +2V ωr cos θ sin θ : 1 1 1 fish body wave, so it is necessary to control the swing angle of each joint in order that each connection endpoint ð16Þ can approximately fit the fish wave curve. The λ is defined as the ratio of the length of the tail swing to the whole According to the wing theory [29, 30], the lift effect on wavelength. It is essential to ensure that each linkage is caudal fin is set as F; the stress analysis is shown in Figure 16. continuous and the end point is at the end of the last link- The lift force of the fluid on the caudal fin in the vertical age on the fish body wave curve. The end position of link- direction of the caudal finis age satisfies the following equations: F =2πρLCV sin α cos α, ð17Þ s 1t x − x + y − y = L , i,j i,j−1 i,j i,j−1 j ð22Þ 2π where α is the instantaneous relative angle of attack of the : y ðÞ x, t = c x + c x sin kx − i , 1 i 2 i i i,j caudal fin; Applied Bionics and Biomechanics 9 t = 0.04s t = 0.18s t = 0.29s t = 0.65s (a) (b) (c) (d) t = 0.74s t = 0.90s t = 1.28s t = 1.49s (e) (f) (g) (h) Figure 12: Wake vortex of double caudal fin pressure cloud. t = 0.35s t = 0.65s t = 0.01s t = 0.28s (a) (b) (c) (d) t = 1.03s t = 1.15s t = 1.28s t = 1.35s (e) (f) (g) (h) Figure 13: Vortex’s dissipation model pressure cloud. (a) (b) (c) (d) Figure 14: Tail swing diagram. Y Y Caudal fin Caudal peduncle O x 𝛼 x O 𝜃 → → r v v 2 1t 1t F m Figure 15: Schematic diagram of caudal peduncle swing. Figure 16: Schematic diagram of the caudal fin’s fluid pressure decomposition and lift force analysis. 10 Applied Bionics and Biomechanics (2) When St number is in the range of 0.25 to 0.45, the where ðx , y Þ is the angular coordinate of the “j” linkage i,j i,j propulsive force generated by the caudal fin in one at the moment i in the swing period, x =0, x = λ ·2π, i,o i,5 swing period gradually increases as the St number 1 ≤ j ≤ 5, 0 ≤ i ≤ M. increases, but the range of the increasing amplitude Bring L into the formula (23), we can solve the coordi- gradually becomes smaller. In addition, when the nates of each endpoint at i =0, i =1 until i = M. After that, ° ° phase difference is in the range of 50 ~60 , the pro- we can get the angle θ between each linkage and fish body, i,j pulsion of the caudal fin is relatively large and this which can be expressed as is a more optimized motion mode "# (3) For hydrodynamic studies of double caudal fins, x − x i,j i,j−1 changing the angle of attack of the double caudal fins θ = artan : ð23Þ i,j y − y will produce different wake flow field structures. A i,j i,j−1 reasonable use of the wake vortex generated by the front caudal fin will help the rear caudal fin to reduce The swing angles of the five steering gears are calculated, the resistance and generate a propulsive force more respectively, effectively 8  (4) By studying the motion law of the tail of swordfish, y − y 1 0 > the motion fitting of the tail swing was carried out φ = arctan , > x − x 1 0 by using the link mechanism which was widely used > in machinery. By calculating the thrust of the simpli- > y − y > 2 1 φ = arctan − φ , fied tail swing model, the principle of the thrust gen- 2 1 x − x 2 1 erated by the caudal fin and caudal peduncle in the y − y process of propulsion was analyzed. By controlling 3 2 φ = arctan − φ , ð24Þ 3 2 the angle of the steering gear, the fish body wave x − x > 3 2 was fitted to the fish tail motion. In this way, the bio- > y − y 4 3 > mimetic motion mechanism of the caudal fin was > φ = arctan − φ , > 4 3 x − x preliminarily studied 4 3 y − y 4 3 φ = arctan − φ : : 6. Future Work 5 4 x − x 4 3 Based on the present study on theoretical calculation analysis of the caudal fin, the authors will build an experimental plat- Through the analysis of the fish body wave function, we form for the experimental analysis of the swing angle of the can obtain that as the number of joints increases, it is easier caudal fin by using the linkage mechanism, which further to fit the fish body wave curve, but it is more difficult to coor- verifies the correctness of our simulation results in the future dinate control between the steering gear (the motion of the work. tail swing is controlled by the steering gear). Thus, the end- points of the five joints should be fitted to the corresponding Data Availability fish body wave curve as much as possible. More importantly, the working angle of each steering gear and their mutual The data used to support the findings of this study are avail- position relationship should be well controlled. In this way, able from the corresponding author upon request. the bionic underwater vehicle can be moved like a fish, thus improving the propulsion efficiency and saving energy. Conflicts of Interest 5. Conclusion The authors declare that there is no conflict of interest regarding the publication of this paper. In this paper, the fast-moving swordfish was taken as the research object to explore the flow field structure of the Acknowledgments swordfish caudal fin swinging. The mechanism of the caudal fin propulsion was preliminarily investigated, and the bionic This research was supported by the National Science mechanism motion fitting was carried out. The main conclu- Foundation of China (No. 61540010 and No. 51979259) sions were obtained as follows: and the Shandong Natural Science Foundation (No. ZR201709240210). (1) During the swinging process, the fish-tail forms a wake vortex due to the transformation of the high- References pressure and low-pressure zones. 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Applied Bionics and BiomechanicsHindawi Publishing Corporation

Published: Jun 24, 2020

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