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 Biomechanics – Hindawi Publishing Corporation
Published: Jun 24, 2020
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