Optimum Curvature Characteristics of Body/Caudal Fin Locomotion

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Journal of Marine Science and Engineering

Article Optimum Curvature Characteristics of Body/Caudal Fin Locomotion

Yanwen Liu and Hongzhou Jiang *

School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China; [email protected] * Correspondence: [email protected]; Tel.: +86-451-86413253

Abstract: Fish propelled by body and/or caudal fin (BCF) locomotion can achieve high-efficiency and high-speed swimming performance, by changing their body motion to interact with external fluids. This flexural body motion can be prescribed through its curvature profile. This work indicates that when the fish swims with high efficiency, the curvature amplitude reaches a maximum at the caudal peduncle. In the case of high-speed swimming, the curvature amplitude shows three maxima on the entire body length. It is also demonstrated that, when the Reynolds number is in the range of 104–106, the swimming speed, stride length, and Cost of Transport (COT) are all positively correlated with the tail-beat frequency. A sensitivity analysis of curvature amplitude explains which locations change the most when the fish switches from the high-efficiency swimming mode to the high-speed swimming mode. The comparison among three kinds of BCF fish shows that the optimal swimming performance of thunniform fish is almost the same as that of carangiform fish, while it is better not to neglect the reaction force acting on an anguilliform fish. This study provides a reference for curvature control of bionic fish in a future time.

Keywords: swimming performance; curvature proﬁle; sensitivity analysis; curvature distribution

Citation: Liu, Y.; Jiang, H. Optimum Curvature Characteristics of 1. Introduction Body/Caudal Fin Locomotion. J. Mar. Sci. Eng. 2021, 9, 537. https:// After hundreds of millions of years of evolution, fish show superiority in high- doi.org/10.3390/jmse9050537 efficiency and high-speed motion, which has attracted the attention of many researchers. The study on bionic fish can lead to a new design of Automatic Underwater Vehicles (AUV). Academic Editor: Alessandro Ridolfi In the nature, most fishes use the body and/or caudal fin (BCF) propulsion to generate thrust. These fishes are distinguished into three categories: anguilliform mode, carangiform J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 2 of 18 Received: 19 April 2021 mode, and thunniform mode (Figure1). The paramount differences between the three Accepted: 11 May 2021 modes are the amplitude envelope of the propulsive wave and the wavelength [1,2]. Published: 17 May 2021

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Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and (a) (b) (c) conditions of the Creative Commons Attribution (CC BY) license (https://Figure 1. FigureThree main 1. Three BCF main swimming BCF swimming modes. (a modes.) anguilliform (a) anguilliform mode; (b) mode; carangiform (b) carangiform mode; (c) mode; (c) creativecommons.org/licenses/by/thunniformthunniform mode (taken mode from (taken Lindsey from Lindsey [2]). [2]). 4.0/). A fish uses both the active and passive mechanisms while swimming [3]. Passive swimming reacts to external flow with no muscle activation [4]. In contrast, active swimming implies swimming generated through muscular activation [5]. BCF fish can autono- J. Mar. Sci. Eng. 2021, 9, 537. https://doi.org/10.3390/jmse9050537 https://www.mdpi.com/journal/jmse mously control their curvature profile to achieve the compliant body motion and realize active/passive swimming [6–11]. Hence, the research on curvature profile is important to understand the fish locomotion mechanism. The curvature distribution is closely related to the materials and shape of the fish body, including soft tissue (tendons, muscle, and ligaments) and hard parts (fin rays and vertebral column). From the biological perspec- tive, the curvature is controlled mostly by internal muscular activation. Specifically, fish can receive the stretch receptors that depend on body curvature to the central nervous system and then drive the calcium dynamics to affect the muscle activity, which can further modulate the swimming behavior [12,13]. Many in vivo experiments and electromyogram analyses have been conducted to discover the neuro–musculo–mechanical model, and much progress on the model has been made [14–18]. Williams and Bowtell developed a simple dynamic model to describe the interaction between body curvature and muscle activation [19,20]. McMillen et al. suggested that fish neuromuscular systems produce an intrinsic preferred curvature that can be derived from in vivo experiments [21,22]. Con- sidering the complexity and the difficulty of the comprehensive model, some researchers turn to study the curvature profile directly instead of the connection to muscular activation. Van Rees et al. and Eloy combined different hydrodynamic models and bi-objective optimization algorithms to obtain the optimal curvature profiles and body shapes for undulatory swimming [23,24]. Such numerical optimization can help overcome some limiting constraints in reality. Nevertheless, these optimized body shapes show a slightly unsmooth or strange geometry when compared to real fish. One of the possible reasons is that the fish in nature faces many constraints in the process of evolution. In this paper, empirical body shapes obtained by biological data of BCF fish are used to avoid this problem and the corresponding analysis of optimal curvature profiles can provide a more reasonable value of information. Certain techniques have been used to achieve variable curvature structures [3,25–29]. A common method is to tune the pressure of the soft pneumatic actuator. White et al. changed the local curvature by developing a Tunabot with different degrees of freedom [28,29]. In addition, some accurate curvature control methods have been proposed and applied in the field of soft robotics [30–32]. The developments in structure fabrication and the control methods, in turn, demand the further investigation of optimal curvature distribution of BCF fish. Therefore, swimming performance optimization through curvature control will become the focus of future related research on bionic robotic fish. The premise of curvature control is to understand the impact of varying the curvature distribution on swimming performance, which is discussed in this study.

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A fish uses both the active and passive mechanisms while swimming [3]. Passive swimming reacts to external flow with no muscle activation [4]. In contrast, active swimming implies swimming generated through muscular activation [5]. BCF fish can au- tonomously control their curvature profile to achieve the compliant body motion and realize active/passive swimming [6–11]. Hence, the research on curvature profile is important to understand the fish locomotion mechanism. The curvature distribution is closely related to the materials and shape of the fish body, including soft tissue (tendons, muscle, and ligaments) and hard parts (fin rays and vertebral column). From the biological per- spective, the curvature is controlled mostly by internal muscular activation. Specifically, fish can receive the stretch receptors that depend on body curvature to the central nervous system and then drive the calcium dynamics to affect the muscle activity, which can further modulate the swimming behavior [12,13]. Many in vivo experiments and electromyogram analyses have been conducted to discover the neuro–musculo–mechanical model, and much progress on the model has been made [14–18]. Williams and Bowtell developed a simple dynamic model to describe the interaction between body curvature and muscle activation [19,20]. McMillen et al. suggested that fish neuromuscular systems produce an intrinsic preferred curvature that can be derived from in vivo experiments [21,22]. Consid- ering the complexity and the difficulty of the comprehensive model, some researchers turn to study the curvature profile directly instead of the connection to muscular activation. Van Rees et al. and Eloy combined different hydrodynamic models and bi-objective optimization algorithms to obtain the optimal curvature profiles and body shapes for undulatory swimming [23,24]. Such numerical optimization can help overcome some limiting constraints in reality. Nevertheless, these optimized body shapes show a slightly unsmooth or strange geometry when compared to real fish. One of the possible reasons is that the fish in nature faces many constraints in the process of evolution. In this paper, empirical body shapes obtained by biological data of BCF fish are used to avoid this problem and the corresponding analysis of optimal curvature profiles can provide a more reasonable value of information. Certain techniques have been used to achieve variable curvature structures [3,25–29]. A common method is to tune the pressure of the soft pneumatic actuator. White et al. changed the local curvature by developing a Tunabot with different degrees of freedom [28,29]. In addition, some accurate curvature control methods have been proposed and applied in the field of soft robotics [30–32]. The developments in structure fabrication and the control methods, in turn, demand the further investigation of optimal curvature distribution of BCF fish. Therefore, swimming performance optimization through curvature control will become the focus of future related research on bionic robotic fish. The premise of curvature control is to understand the impact of varying the curvature distribution on swimming performance, which is discussed in this study. This paper is organized as follows: Section2 describes the optimization problem of swimming performance. Section3 presents the optimization results including the swimming motions, curvature profiles, the frequency effect, the change mechanism of curvature amplitude, and comparison among BCF fishes. Section4 discusses the similarities and differences between our optimization and other optimizations. Finally, conclusions are summarized in Section5.

2. Materials and Methods 2.1. Optimization Problem Definition Here we take the carangiform fish as an example for optimization. The optimization of the other two kinds of BCF fish will be conducted in Section 3.4. The shape of carangiform fish is determined by the empirical equations in [33] (see Figure2). The views are set out in the global coordinate system OXYZ where Y-axis is horizontal and Z-axis is vertical. The X-axis is determined by the right-handed rule. The swimming direction is along with the negative Y-axis. J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 3 of 18

This paper is organized as follows: Section 2 describes the optimization problem of swimming performance. Section 3 presents the optimization results including the swimming motions, curvature profiles, the frequency effect, the change mechanism of curvature amplitude, and comparison among BCF fishes. Section 4 discusses the similarities and differences between our optimization and other optimizations. Finally, conclusions are summarized in Section 5.

2. Materials and Methods 2.1. Optimization Problem Definition Here we take the carangiform fish as an example for optimization. The optimization of the other two kinds of BCF fish will be conducted in Section 3.4. The shape of carangiform fish is determined by the empirical equations in [33] (see Figure 2). The views are set out in the global coordinate system OXYZ where Y-axis is horizontal and Z-axis is vertical. J. Mar. Sci. Eng. 2021, 9, 537 3 of 17 The X-axis is determined by the right-handed rule. The swimming direction is along with the negative Y-axis.

FigureFigure 2. 2.Side Side view view (up (up) and) and top top view view (bottom (bottom) of carangiform) of carangiform fish. The fish. shape The of shape carangiform of carangiform fish is fish treatedis treated as numerous as numerous elliptical elliptical sections sections continuously continuously distributed distributed along the along spine. thea(s) spine. and b(s) a(s) mean and b(s) themean major the semi-axis major semi and-axis minor and semi-axis. minor semis is the-axis. arc lengths is the and arcL length= 0.3 m and is the L whole= 0.3 m body is the length. whole body length. The midline kinematics can be described by a curvature profile κ(s,t), defined as an amplitude function K(s) multiplied by a traveling wave [23], as: The midline kinematics can be described by a curvature profile κ(s,t), defined as an t s amplitude function K(s) κmultiplied(s, t) = K( bys) cos a traveling[2π( − τ wave(s) )] [23], as: (1) T L ts (s , t )=− K ( s )cos[2  (  ( s ) )] (1) where t is the time, T = 1 s is the tail-beat cycle, and τ(s) determinesTL the wavelength. Both K(s) and τ(s) are functions fitted at n = 11 interpolation points that are evenly distributed onwhere the midline. t is the Consideringtime, T = 1 s the is boundarythe tail-beat conditions, cycle, andK(0) τ =(sK) (determinesL) = τ(0) = 0, the the wavelength. kinematic Both optimizationK(s) and τ(s) problem are functions includes fitted 19 degrees at n = of 11 freedom interpolation in total. points According that to are the evenly biological distributed dataon the gathered midline. by ZuoConsidering Cui, the range the boundary of τ(s) is set conditions, as 0.5–2 [34 K].(0) = K(L) = τ(0) = 0, the kinematic optimizationIf a specific problem curvature includes profile is19 given, degrees the of swimming freedom kinematics in total. According can be determined to the biological accordingdata gathered to the by reduced-order Zuo Cui, the dynamic range ofmodel τ(s) is developedset as 0.5–2 by [34 Eloy]. [24]. The general introductionIf a specific of this curvature model is asprofile follows is given, (shown the in Figureswimming3). Once kinematics the initial can curvature be determined θ profileaccording is set, to the the angle reducedbetween-order the dynamic tangent to model the midline developed and horizontal by Eloy axis [24]. is obtained The general in- based on the mathematical definition of curvature. The position of any point r = (x, y) on troduction of this model is as follows (shown in Figure 3). Once the initial curvature pro- the midline is then derived by employing the integral to a trigonometric function of θ. As a result,file is theset, velocity the angle components θ between on the the tangentialtangent to and the normal midline direction and horizontal at any point axis of theis obtained J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 4 of 18 midline,based on namely the mathematicalu and v, can be definition calculated of through curvature. the differential The position to the of position any point function r = (x, y) on andthe midline the projection is then analysis. derived by employing the integral to a trigonometric function of θ. As a result, the velocity components on the tangential and normal direction at any point of the midline, namely u and v, can be calculatedThe position ofthrough the the Thedifferential velocity components to the position  d ddxx ddyy = (,)st The angle midline =+ functionPrescribed andd sthe projection analysis. s u s xxs=+cosd ddddtsts (,)st =+ (s , ts )d 0  0 0 0 ddyyddxx s =− yys=+sin d v 0 0 ddddtsts

 is solved xy,are solved 0 00 Equations of Swimming Kinematics conservation The forces on the body is obtained FF= u(,) v FigureFigure 3.3. TheThe general general introduction introduction of theof the reduced reduced dynamic dynamic model model proposed proposed by Eloy. by Eloy.

F The forcesforces acting acting on on the the ﬁsh fish body body that that include include reaction reaction force forcem, skin-friction Fm, skin-friction drag drag F , form drag F , and resistive force F⊥, can be expressed as the functions of the F//, form drag Fformform, and resistive force F⊥, can be expressed as the functions of the velocity velocity components. Their empirical formulas are shown in Equations (2)–(5), respectively. components. Their empirical formulas are shown in Equations (2)–(5), respectively. The The reaction forces are derived by the large-amplitude elongated-body theory [35]. The reactiondescription forces of skin-friction are derived drag by andthe resistivelarge-amplitude force is obtained elongated by- thebody hydrodynamics theory [35]. The de- scription of skin-friction drag and resistive force is obtained by the hydrodynamics analysis of Taylor [36]. The empirical formula of form drag caused by the non-streamlined body is concluded by Hoerner [37]. −(0.5muvmvmvntn 2 ） d()dF =−s (2) m st

(3) d2.9dFt// =− ρ νavus

max(b ) Fe=0.33ρUS2 (4) form 0 L Y (5) ddFn ⊥ =−CaD ρvvs where t and n are the unit vectors of tangential and normal directions of midlines, respectively. ρ = 1000 Kg/m3 is the density of both fish and fluid. m = a2 is the added mass of

each section. ν = 10−6 m2/s is the kinematic viscosity of the fluid. CD = 2-b/a is the approximation of the resistive force coefficient. U0 is the mean tangential velocity at the fish head. S = 0.0043 is a number that describes the streamlining of the fish body. The constants of integration that emerged during the process, such as θ0, x0, and y0, can be solved by the conservation of momentum and angular momentum (Equations (6) and (7)). Thus, the undulatory behavior is fully understood. 2 L d r FFFF+d + d +d −Ms d = 0 (6) form0 // m ⊥ dt2

22 L dd r IsMs eFFFrd(dd+d)d0++−= (7) 0 ddtt22z//m ⊥ where M = πab is the fish mass of each section. I = 0.25 πab3 is the moment of inertia of each section. The indices used in the simulation to measure the swimming performance are shown in Equations (8) and (9). Consequently, each curvature profile has its corresponding swimming performance. The bi-objective optimization was adopted for obtaining the maximum relative swimming speed U* and the minimum Cost of Transport (COT), deriving the optimal curvature profile. UUL* = / (8)

PPPMP++0.13270.8 5 COT==tm s musc = tot tot (9) Mtot gU M tot gU M tot gU

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analysis of Taylor [36]. The empirical formula of form drag caused by the non-streamlined body is concluded by Hoerner [37].

∂(muvn − 0.5mv2t) ∂mvn dF = ( − )ds (2) m ∂s ∂t q dF// = −2.9ρ |νav|uds t (3) max(b) F = 0.33ρU 2 × S e (4) form 0 L Y

dF⊥ = −CDaρv|v|ds n (5) where t and n are the unit vectors of tangential and normal directions of midlines, respectively. ρ = 1000 Kg/m3 is the density of both fish and fluid. m = ρ a2 is the added mass −6 2 of each section. ν = 10 m /s is the kinematic viscosity of the fluid. CD = 2 − b/a is the approximation of the resistive force coefficient. U0 is the mean tangential velocity at the fish head. S = 0.0043 is a number that describes the streamlining of the fish body. The constants of integration that emerged during the process, such as θ0, x0, and y0, can be solved by the conservation of momentum and angular momentum (Equations (6) and (7)). Thus, the undulatory behavior is fully understood.

Z L d2r Fform + dF// + dFm+dF⊥ − M ds = 0 (6) 0 dt2 ! Z L d2θ d2r I ezds + (dF// + dFm+dF⊥ − M × rds = 0 (7) 0 dt2 dt2

where M = ρπab is the fish mass of each section. I = 0.25ρπab3 is the moment of inertia of each section. The indices used in the simulation to measure the swimming performance are shown in Equations (8) and (9). Consequently, each curvature profile has its corresponding swimming performance. The bi-objective optimization was adopted for obtaining the maximum relative swimming speed U* and the minimum Cost of Transport (COT), deriving the optimal curvature profile. U∗ = U/L (8) P P + P 0.1327M 0.8 + 5P COT = tm = s musc = tot tot (9) MtotgU MtotgU MtotgU

where g is the acceleration of gravity and Mtot is the total fish mass. The total metabolic power Ptm is the sum of standard metabolic rate Ps and the metabolic power Pmusc consumed by the swimming muscles [38,39]. The unit of U* is Body Length/s, namely, BL/s. COT quantifies the total energy consumption per unit mass and distance [40]. The higher the value of COT, the lower the efficiency. Ptot serves as the total power required for swimming, which is described in Equation (10).

Ptot = Pm + P⊥ + P// + Pform + Pi (10)

where Pm, Pk, Pform, and P⊥ are the powers consumed by reaction force, skin-friction drag, form drag, and resistive force, respectively. These powers are easy to calculate by multiplying the forces and the corresponding velocities. Pi is the internal dissipation power (see Equation (11)). * !2 + Z L d2θ Pi = µI ( ds (11) 0 dsdt where µ = 1000 Pa s is the internal viscosity of ﬁsh and the brackets mean time averaging. J. Mar. Sci. Eng. 2021, 9, 537 5 of 17

The optimization problem is concluded here. Given the shape of carangiform ﬁsh, (K1, K2, ... , Kn−1, τ1, τ2, ... , τn) were chosen as variables and the two objective functions were the maximum U* and the minimum COT. In the process of parameter settings, if the Kmax value is too high, the search space will become wider and the simulation time will be longer, whereas, if the Kmax value is too low, true optimization results may not be obtained. Therefore, it is necessary to set the appropriate value of Kmax. Gazzola et al., Van Rees et al. and Eloy set this value to 2π/L, 3π/L, 10/L, respectively [23,24,41]. In this paper, the Kmax value was initially set to 10π/L (a number high enough to produce simulation results), while the maximum values of K corresponding to different lengths could be thus obtained once optimization was completed. The simulation showed that Kmax was approximately inversely proportional to L. The inverse proportional function was ﬁtted as Kmax = 8.4/L. This restriction can be relaxed in some cases, which are later discussed in Section 3.3.

2.2. Optimization Algorithm The NSGA-II algorithm for bi-objective optimization was used in this paper. NSGA-II is nowadays one of the most popular and efﬁcient multi-objective algorithms [42,43]. Before introducing NSGA-II, there are two important concepts to understand: dominance and Pareto front. For two individuals G1 and G2, if U*(G1) > U*(G2), COT(G1) ≤ COT(G2) or U*(G1) ≥ U*(G2), COT(G1) < COT(G2), G1 is said to dominate G2. Therefore, the set of all non-dominated points is called the Pareto front. Like genetic algorithms, the NSGA-II procedure is roughly as follows (Figure4). First, an offspring population was generated from the parent population through crossover and mutation. The probability for crossover was 0.8; next, offspring and parent population were combined into a mixed population and sorted according to non-dominated sorting (the conception of crowding distance was not used in our simulation); following, the set of all non-dominated points was selected for the next parent generation; ﬁnally, the above steps were repeated until the termination condition was met. The initial size and the maximum size of the population were set as J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 6 of 18 20 and 2000, respectively. The detailed algorithm implementation was coded using the open-source software Python.

Start

Initialize the population of Genetic operation: (K1, K2, …, Kn-1, τ1, τ2, …, τn) crossover and mutation

offspring population Are NO constraints met? Evaluating objective functions: U* and COT Evaluating objective functions: Non-dominated * U and COT sorting

Non-dominated Gen=Gen+1 sorting Max generations?

Stop

FigureFigure 4. 4.The The ﬂowchart flowchart of theof the NSGA-II NSGA algorithm.-II algorithm. 3. Results 3.1.3. Results Reference Case 3.1. TheReference case of Case the tail-beat frequency f = 1 Hz was taken as a reference case. Setting ϕ(s) =The−2π case·τ(s) ·ofs/ Lthe, ϕ (tails) was-beat the frequency phase of the f = curvature 1 Hz was proﬁle. taken Foras a the reference swimming case. forms Setting φ(s) of= minimum−2π·τ(s)·s/ COTL, φ and(s) was maximum the phase swimming of the speed curvature in the referenceprofile. case,For the the distributionsswimming forms of minimum COT and maximum swimming speed in the reference case, the distributions of τ and φ along the body length are both depicted in Figure 5. To validate the effectiveness of the simulation results, a single-objective optimization algorithm, covariance matrix ad- aptation evolution strategy (CMA-ES), was used to obtain the distributions of τ for the above-mentioned forms. It can be seen from Figure 5 that the numerical solutions based on CMA-ES were in good agreement with these based on NSGA-II. Furthermore, the distributions of τ and φ were similar for the two cases. τ increased sharply at first, followed by a decrease, but then again increased in the last fifth of body length. The phase distribution presented a step shape in general. As for the disagreement between algorithms around three-tenths of the body length for the minimum COT case (Figure 5a), it can be explained as follows. The curvature amplitude in this location was near zero for the minimum COT case (shown in Figure 6). Therefore, according to Equation (1), the change in τ value in this location had little effect on swimming performance. In spite of the disagreement of τ value in this location, the swimming performance obtained by the two algorithms was almost equal.

(a) (b) Figure 5. Distributions of τ and φ along the body length. (a) is for the case of minimum COT and (b) is for the case of maximum swimming speed. The average wavelengths of the former and latter cases were 1.17BL and 0.92BL, respectively. All the curves were fitted by piecewise cubic Hermite interpolation.

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Start

Initialize the population of Genetic operation: (K1, K2, …, Kn-1, τ1, τ2, …, τn) crossover and mutation

offspring population Are NO constraints met? Evaluating objective functions: U* and COT Evaluating objective functions: Non-dominated * U and COT sorting

Non-dominated Gen=Gen+1 sorting Max generations?

Stop

Figure 4. The flowchart of the NSGA-II algorithm.

3. Results 3.1. Reference Case The case of the tail-beat frequency f = 1 Hz was taken as a reference case. Setting φ(s) J. Mar. Sci. Eng. 2021, 9, 537 = −2π·τ(s)·s/L, φ(s) was the phase of the curvature profile. For the swimming6 offorms 17 of minimum COT and maximum swimming speed in the reference case, the distributions of τ and φ along the body length are both depicted in Figure 5. To validate the effectiveness of theof τ simulationand ϕ along re thesults, body a lengthsingle are-objective both depicted optimization in Figure 5algorithm,. To validate covariance the effectiveness matrix ad- aptationof the evolution simulation strategy results, a(CMA single-objective-ES), was optimizationused to obtain algorithm, the distributions covariance of matrix τ for the aboveadaptation-mentioned evolution forms. strategy It can (CMA-ES),be seen from was Figure used to5 obtainthat the the numerical distributions solutions of τ for based on theCMA above-mentioned-ES were in good forms. agreement It can be with seen these from based Figure on5 that NSGA the- numericalII. Furthermore solutions, the dis- tributionsbased on of CMA-ES τ and φwere were in similar good agreement for the two with cases. these τ basedincrease ond NSGA-II. sharply Furthermore,at first, followed by thea decrease, distributions but then of τ and againϕ were increase similard in for the the last two fifth cases. of bodyτ increased length. sharply The phase at ﬁrst, distri- butionfollowed present byed a decrease, a step shape but then in general. again increased As for inthe the disagreement last ﬁfth of body between length. algorithms The phase distribution presented a step shape in general. As for the disagreement between around three-tenths of the body length for the minimum COT case (Figure 5a), it can be algorithms around three-tenths of the body length for the minimum COT case (Figure5a) , explainedit can be as explained follows. as The follows. curvature The curvature amplitude amplitude in this location in this location was near was nearzero zerofor the for min- imumthe minimumCOT case COT(shown case in (shown Figure in 6). Figure Therefore,6). Therefore, according according to Equation to Equation (1), the (1), change the in τ valuechange in inthisτ value location in this ha locationd little hadeffect little on effectswimming on swimming performance. performance. In spite In spiteof the of disa- greementthe disagreement of τ value of inτ valuethis location, in this location, the swimming the swimming performance performance obtained obtained by bythe the two al- gorithmstwo algorithms was almost was equal. almost equal.

(a) (b)

FigureFigure 5. Distributions 5. Distributions of τ ofandτ and φ alongϕ along the the body body length. length. ( (aa)) is is for thethe casecase of of minimum minimum COT COT and and (b) ( isb for) is thefor casethe ofcase of J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 7 of 18 maximummaximum swimming swimming speed. speed. The The average average wavelengths wavelengths of of the the former former and latterlatter cases cases were were 1.17BL 1.17BL and and 0.92BL, 0.92BL, respectively. respectively. All theAll curves the curves were were fitted ﬁtted by piecewise by piecewise cubic cubic Hermite Hermite interpolation. interpolation.

(a) (b)

FigureFigure 6. The 6. amplitudeThe amplitude envelope envelope of ofcurvature curvature profiles proﬁles of of six six swimming forms. forms. The The circles circles of (ofa) ( representa) represent the discretethe discrete pointspoints of K(s). of K(b(s).) shows (b) shows the the swimming swimming speeds speeds and and the COTCOT values values of of the the six six swimming swimming forms forms A-F. All A- theF. All curves the ofcurves (a) are of (a) are fittedﬁtted by by piecewise piecewise cubic cubic Hermite interpolation.interpolation. For For the the different different swimming swimming forms forms A, B, A, C, D,B, E,C, andD, E, F, and the values F, the ofvaluesU* of U* werewere 0.837BL/s, 0.837BL/s, 1.022BL/s, 1.022BL/s, 1.144BL/s, 1.144BL/s, 1.236BL/s, 1.236BL/s, 1.340BL/s, 1.340BL/s, and and 1.423BL/s, respectively;respectively; the the values values of COTof COT were were 0.093, 0.093, 0.1, 0.112,0.1, 0.112, 0.129, 0.129, 0.165, 0.165, and and0.306, 0.306, respectively. respectively.

SixSix typical typical swimming swimming forms forms A A-F-F werewere extracted extracted from from the the reference reference case case (the speed(the speed rangerange was was divided divided into into fifths) ﬁfths).. FigureFigure6 presents6 presents the amplitudethe amplitude envelopes envelopes of these of curvature these curvature profiles. When U* was small (Case A in Figure 6), the curvature amplitude was near zero along with the first 2/3 of the body length, while its peak value appeared at the caudal peduncle. When U* reached the maximum (Case F in Figure 6), there were three local peaks of curvature amplitude in the whole body length, which were at the head, at the 2/3 of the body length, and at the caudal peduncle, respectively. It indicates that the undulatory motion mainly concentrates in the last third of the body length to achieve minimum COT, and the large undulation also exists in the head part to reach maximum speed. In nature, the fish head needs to keep rigid to protect the brain tissue, so the curvature of the head part cannot reach the simulation value of the swimming form F. It can therefore be assumed that the evolution of natural fish inclines towards the direction of maximizing swimming efficiency. However, this does not mean that the simulation result of F is meaningless. If different kinds of soft materials are applied in the fabrication of biomimetic fish to realize the curvature distribution of F, the simulated maximum speed can also be achieved in reality.

3.2. Frequency Effect Yue and Tokić showed that when the fish mass is about 0.5 kg (the fish mass in this paper was 0.48 kg), the range of tail-beat frequency is 1.25–4.5 Hz [44]. Therefore, optimization results of U* and COT at various tail-beat frequencies within the range of 1 Hz–4 Hz were obtained (Figure 7). Figures 7 and 8 demonstrate that the maximum U* and the maximum COT increased linearly with frequency while the minimum U* and the minimum COT remained constant. Real fish swimming data observed by Videler and Hess [7] indicate that the swimming speed is proportional to frequency when the frequency is in the range of 1 Hz–14 Hz, also supporting the above result. Another evaluation index often used in other references is introduced here: the stride length U’ = U*/f [7,24]. The stride length is a dimensionless number that is the number of fish lengths travelled during one tail-beat period. In our simulation, the maximum U’ remained stable at around 1.42, and the minimum U’ decreased from 0.84 to 0.245, with the increasing frequency.

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profiles. When U* was small (Case A in Figure6), the curvature amplitude was near zero along with the first 2/3 of the body length, while its peak value appeared at the caudal peduncle. When U* reached the maximum (Case F in Figure6), there were three local peaks of curvature amplitude in the whole body length, which were at the head, at the 2/3 of the body length, and at the caudal peduncle, respectively. It indicates that the undulatory motion mainly concentrates in the last third of the body length to achieve minimum COT, and the large undulation also exists in the head part to reach maximum speed. In nature, the fish head needs to keep rigid to protect the brain tissue, so the curvature of the head part cannot reach the simulation value of the swimming form F. It can therefore be assumed that the evolution of natural fish inclines towards the direction of maximizing swimming efficiency. However, this does not mean that the simulation result of F is meaningless. If different kinds of soft materials are applied in the fabrication of biomimetic fish to realize the curvature distribution of F, the simulated maximum speed can also be achieved in reality.

3.2. Frequency Effect Yue and Tokićshowed that when the fish mass is about 0.5 kg (the fish mass in this paper was 0.48 kg), the range of tail-beat frequency is 1.25–4.5 Hz [44]. Therefore, optimization results of U* and COT at various tail-beat frequencies within the range of 1 Hz–4 Hz were obtained (Figure7). Figures7 and8 demonstrate that the maximum U* and the maximum COT increased linearly with frequency while the minimum U* and the minimum COT remained constant. Real fish swimming data observed by Videler and Hess [7] indicate that the swimming speed is proportional to frequency when the frequency is in the range of 1 Hz–14 Hz, also supporting the above result. Another evaluation index often used in other references is introduced here: the stride length U’ = U*/f [7,24]. The J. Mar. Sci. Eng. 2021, 9, x FOR PEERstride REVIEW length is a dimensionless number that is the number of fish lengths travelled during 8 of 18 one tail-beat period. In our simulation, the maximum U’ remained stable at around 1.42, and the minimum U’ decreased from 0.84 to 0.245, with the increasing frequency.

FigureFigure 7. Optimization7. Optimization results results of U* and of COT U* atand different COT tail-beatat different frequencies. tail-beat The Paretofrequencies. front The Pareto front remainedremain unchangeded unchanged after 5000 after generations. 5000 generations. The corresponding The rangescorresponding of U* obtained ranges at different of U* obtained at different frequencies were: [1 Hz: 0.837 BL/s–1.423 BL/s], [1.5 Hz: 0.935 BL/s–2.166 BL/s], [2 Hz: 0.971 BL/s– frequencies were: [1 Hz: 0.837 BL/s–1.423 BL/s], [1.5 Hz: 0.935 BL/s–2.166 BL/s], [2 Hz: 0.971 BL/s– 2.903 BL/s], [2.5 Hz: 0.998 BL/s–3.663 BL/s], [3 Hz: 0.988 BL/s–4.416 BL/s], [3.5 Hz: 1.002 BL/s– 5.1492.90 BL/s],3 BL/s], [4 Hz: [2. 0.9805 Hz: BL/s–5.912 0.998 BL/s BL/s].–3.66 The3 corresponding BL/s], [3 Hz: ranges 0.98 of8 COTBL/s obtained–4.416 atBL/s], different [3.5 Hz: 1.002 BL/s–5.149 frequenciesBL/s], [4 were:Hz: [10.98 Hz:0 0.093–0.306],BL/s–5.91 [1.52 B Hz:L/s]. 0.088–0.492], The corresponding [2 Hz: 0.087–0.720], ranges [2.5 Hz:of COT 0.087–1.030], obtained at different fre- [3quencies Hz: 0.087–1.423], were: [3.5 [1 Hz:Hz: 0.087–1.738], 0.093–0.306], [4 Hz: [1. 0.087–2.260].5 Hz: 0.088–0.492], [2 Hz: 0.087–0.720], [2.5 Hz: 0.087–1.030], [3 Hz: 0.087–1.423], [3.5 Hz: 0.087–1.738], [4 Hz: 0.087–2.260].

Figure 8. The ranges of U* and COT at different tail-beat frequencies based on the optimization results shown in Figure 7.

Swimming kinematics with different frequencies are shown in Figure 9. In the case of minimum COT, the amplitude envelopes decreased with frequency. Especially, at 3 Hz– 4 Hz, fish relied solely on the tail to generate thrust. However, at the maximum U*, the amplitude envelopes remained the same.

(a) f = 1 Hz (b) f = 1.5 Hz

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* Figure 7. Optimization results of U* and COT at different tail--beat frequencies. The Pareto front * remainremained unchanged after 5000 generations. The corresponding ranges of U* obtained at different frequenciesfrequencies were:: [[1 Hz: 0.837 BL/s–1.423 BL/s],, [1.[1.5 Hz: 0.935 BL/s–2.166 BL/s], [[2 Hz: 0.971 BL/s– 2.903 BL/s], [2.[2.5 Hz: 0.998 BL/s–3.663 BL/s], [[3 Hz: 0.988 BL/s–4.416 BL/s], [3.[3.5 Hz: 1.002 BL/s–5.149 J. Mar. Sci. Eng. 2021, 9, 537 BL/s], [[4 Hz: 0.980 BL/s–5.912 BL/s]. The corresponding ranges of COT obtained at different fre-8 of 17 quencies were:: [[1 Hz: 0.093–0.306],.306], [1.[1.5 Hz: 0.088–0.492],.492], [[2 Hz: 0.087–0.720],.720], [2.[2.5 Hz: 0.087–1.030],.030], [[3 Hz: 0.087–1.423],.423], [3.[3.5 Hz: 0.087–1.738].738],, [[4 Hz: 0.087–2.260]..260].

* FigureFigure 8.8. TheThe rangesranges ofof UU** andand COTCOT atat differentdifferent tail-beattail--beat frequenciesfrequencies basedbased onon thethe optimizationoptimization resultsresultsresults shownshownshown ininin FigureFigureFigure7 7.7..

Swimming kinematics kinematics with with different different frequencies frequencies are are shown shown in in Figure Figure 9.9 .In In the the case case of ofminimum minimum COT, COT, the theamplitude amplitude envelopes envelopes decrease decreasedd with withfrequency. frequency. Especially, Especially, at 3 H atz– * 34 Hz–4Hz, fish Hz, relie ﬁshd relied solely solely on the on tail the to tail generate to generate thrust. thrust. However, However, at the at themaximum maximum U*,, Uthethe*, theamplitude amplitude envelopes envelopes remain remaineded thethe thesame.same. same.

((a)) ff = 1 Hz ((b)) ff = 1..5 Hz

((c)) ff = 2 Hz ((d)) ff = 2..5 Hz

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(e) f = 3 Hz (f) f = 3.5 Hz

(g) f = 4 Hz

Figure 9. Swimming kinematics at different tail-beat frequencies. (a), (b), (c), (d), (e), (f) and (g) show the swimming Figure 9. Swimming kinematics at different tail-beat frequencies. (a), (b), (c), (d), (e), (f) and (g) show the swimming kin- kinematicsematics with with 1 Hz 1 Hz,, 1.5 1.5Hz, Hz, 2 Hz, 2 Hz, 2.5 2.5Hz, Hz,3 Hz, 3 Hz,3.5 Hz 3.5 and Hz and4 Hz, 4 Hz,respectively. respectively. The left The side left siderepresents represents casescases with withthe mini- the minimummum COT COT and and the theright right side side represents represents cases cases with with the the maximum maximum UU*. *.The The lines lines with with green, green, blue, blue, cyan, cyan, magenta, magenta, yellow, royal,royal, orange, orange, violet, violet, pink, pink, and and dark-grey dark-grey color color refer refer to to the the swimming swimming kinematics kinematics at at the thet = t 0,= 0.10, 0.1T, T 0.2, 0.2T, 0.3T, 0.3T, 0.4T, 0.4T, 0.5T, 0.5T, 0.6T, 0.6T, 0.7 T, 0.7T, 0.8 T, 0.8T, and T, and 0.9 0.9T, respectively. T, respectively. The The black black line line represents represents the amplitudethe amplitude envelope. envelope.

There are other ways to study the frequency effect on swimming performance. For example, in the robotic fish swimming experiment presented in [45], the fishtail is driven by the steering gear. Liu et al. kept the swing angle amplitude of steering gear constant, meaning swimming kinematics do not change. Only the frequency is changed to investigate the swimming performance. Similar to this method, this paper studied the relationships of U*, U’, A*, and COT with frequency while keeping the swing angle of the fishtail (θtail) constant. θtail can be derived based on the different undulatory locomotion of the Pareto fronts. The premise is that these parameters should have a one-to-one mapping with θtail. The relationships of U*, U’, A*, and COT with θtail were checked, and the one-to- one mapping was generally satisfied. Figure 10 displays the trends of parameters U*, U’, A*, and COT vs. frequency when θtail was 25° and 35°, respectively. U* and COT values showed a sharp increase along with the rise in frequency while U’ and A* values exhibited a lighter increase. This shows that the swimming speed, stride length, and COT are all positively correlated with frequency, which agrees well with the trend of these metrics demonstrated in the experiments of White et al. [29].

(a) (b)

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(e) f = 3 Hz (f) f = 3.5 Hz

(g) f = 4 Hz

Figure 9. Swimming kinematics at different tail-beat frequencies. (a), (b), (c), (d), (e), (f) and (g) show the swimming kinematics with 1 Hz, 1.5 Hz, 2 Hz, 2.5 Hz, 3 Hz, 3.5 Hz and 4 Hz, respectively. The left side represents cases with the minimum COT and the right side represents cases with the maximum U*. The lines with green, blue, cyan, magenta, yellow, royal, orange, violet, pink, and dark-grey color refer to the swimming kinematics at the t = 0, 0.1 T, 0.2 T, 0.3 T, 0.4 T, 0.5 T, 0.6 T, 0.7 T, 0.8 T, and 0.9 T, respectively. The black line represents the amplitude envelope. J. Mar. Sci. Eng. 2021, 9, 537 9 of 17

There are other ways to study the frequency effect on swimming performance. For example, in the robotic fish swimming experiment presented in [45], the fishtail is driven by theThere steering are other gear. ways Liu toet studyal. kept the the frequency swing angle effect onamplitude swimming of performance.steering gear For constant, meaningexample, swimming in the robotic kinematics fish swimming do not experiment change. presentedOnly the infrequency [45], the fishtailis changed is driven to investi- by the steering gear. Liu et al. kept the swing angle amplitude of steering gear constant, gate the swimming performance. Similar to this method, this paper studied the relation- meaning swimming kinematics do not change. Only the frequency is changed to investigate * * shipsthe swimming of U , U’, performance. A , and COT Similar with frequency to this method, while this keeping paper studiedthe swing the relationshipsangle of the fishtail tail tail (θof U) *,constant.U’, A*, and θ COT can with be frequencyderived based while keepingon the different the swing undulatory angle of the fishtaillocomotion (θtail) of the Paretoconstant. fronts.θtail can The be premise derived basedis that on these the different parameters undulatory should locomotion have a one of- theto-one Pareto mapping withfronts. θtail The. The premise relationships is that theseof U*, parameters U’, A*, and should COT with have θ atail one-to-one were checked mapping, and with the one-to- oneθtail .mappin The relationshipsg was generally of U*, U satisfied.’, A*, and COT with θtail were checked, and the one-to-one mappingFigure was 10 generally displays satisfied. the trends of parameters U*, U’, A*, and COT vs. frequency when Figure 10 displays the trends of parameters U*, U’, A*, and COT vs. frequency when θtail was 25° and 35°, respectively. U* and COT values showed a sharp increase along with θ was 25◦ and 35◦, respectively. U* and COT values showed a sharp increase along with thetail rise in frequency while U’ and A* values exhibited a lighter increase. This shows that the rise in frequency while U’ and A* values exhibited a lighter increase. This shows that thethe swimming swimming speed,speed, stride stride length, length, and and COT COT are are all positivelyall positively correlated correlated with frequency, with frequency, whichwhich agrees wellwell withwith the the trend trend of theseof these metrics metrics demonstrated demonstrated in the in experiments the experiments of of WhiteWhite et al. [[29].29].

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(a) (b)

FigureFigure 10. 10. U*,*, U’, A** and COTCOT plotted plotted against against the the tail-beat tail-beat frequency, frequency, are shown are shown in (a), in (b ),(a (),c) ( andb), (c) and (d(d),), respectively. respectively. The The red red and and black black lines lines are are the casesthe cases of θ =of 35 θ◦ =and 35°θ and= 25 ◦θ, respectively.= 25°, respectively.

Nevertheless,Nevertheless, thethe above above simulation simulation result result does does not meannot mean that thethat swimming the swimming speed speed increasesincreases across thethe frequency frequency spectrum spectrum indeﬁnitely. indefinitely. It is worthIt is worth noting noting that the that proposed the proposed optimization model is suitable for Re within the range of 104 to 106. If Re lies beyond this optimization model is suitable for Re within the range of 104 to 106. If Re lies beyond this range, the calculation formula of forces will change, the model will no longer be valid and range,the relationship the calculation between formula swimming of forces speed will and change, frequency the will model also change. will no longer be valid and the relationship between swimming speed and frequency will also change.

3.3. Curvature Amplitude Figure 11 indicates that as the fish length increased from 0.3 m to 0.7 m, the minimum COT decreased, so the swimming efficiency increased, which is consistent with the optimization results presented in [44]. Two aspects give reason to this occurrence. First, as Kmax decreased with length, the lateral amplitude at the fish head decreased while that at the tail showed a small increase. Moreover, the slope of the amplitude envelope of swimming kinematics became smoother when the length increased (Figure 12). These all led to a reduction in both the recoil motion and the internal diffusion. Furthermore, though U* decreased with length, the actual swimming speed increased with length, leading to an increase in Re. The drag coefficient Cd~Re−0.5 also decreased with length [44].

Figure 11. Optimization results of U* and COT with different lengths. The dimensions of body geometry, such as a(s) and b(s), are scaled accordingly. When the length was 0.3 m, 0.4 m, 0.5 m, 0.6 m, and 0.7 m, the range of U* was 0.837 BL/s–1.423 BL/s, 0.738 BL/s–1.454 BL/s, 0.644 BL/s–1.462 BL/s, 0.576 BL/s–1.483 BL/s, and 0.518 BL/s–1.486 BL/s, respectively; that of COT was 0.093–0.306, 0.069– 0.291, 0.056–0.285, 0.047–0.280, and 0.041–0.290, respectively.

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(c) (d) Figure 10. U*, U’, A* and COT plotted against the tail-beat frequency, are shown in (a), (b), (c) and (d), respectively. The red and black lines are the cases of θ = 35° and θ = 25°, respectively.

Nevertheless, the above simulation result does not mean that the swimming speed increases across the frequency spectrum indefinitely. It is worth noting that the proposed optimization model is suitable for Re within the range of 104 to 106. If Re lies beyond this J. Mar. Sci. Eng. 2021, 9, 537 range, the calculation formula of forces will change, the model will no longer be10 valid of 17 and the relationship between swimming speed and frequency will also change.

3.3. Curvature Amplitude 3.3. Curvature Amplitude Figure 11 indicates that as the fish length increased from 0.3 m to 0.7 m, the minimum Figure 11 indicates that as the fish length increased from 0.3 m to 0.7 m, the minimum COT decreased, so the swimming efficiency increased, which is consistent with the opti- COT decreased, so the swimming efficiency increased, which is consistent with the optimization results presented in [44]. Two aspects give reason to this occurrence. First, as mization results presented in [44]. Two aspects give reason to this occurrence. First, as Kmax decreasedKmax decrease withd length,with length, the lateral the lateral amplitude amplitude at the fishat the head fish decreased head decrease whiled that while at the that at tailthe showedtail show a smalled a small increase. increase. Moreover, Moreover, the slope the of slope the amplitude of the amplitude envelope envelope of swimming of swim- kinematicsming kinematics became bec smootherame smoother when thewhen length the length increased increase (Figured (Figure 12). These 12). allThese led all to aled to reductiona reduction in in both both the the recoil recoil motion motion and and the the internal internal diffusion. diffusion. Furthermore, Furthermore, though thoughU* U* decreaseddecreased withwith length, length, the the actual actual swimming swimming speed speed increased increase withd with length, length, leading leading to an to an −0.5−0.5 increaseincrease in inRe Re.. The The drag drag coefficient coefficientCd C~dRe~Re also also decreased decreased with with length length [44 [].44].

FigureFigure 11.11. Optimization resultsresults ofofU U** andand COTCOT withwith differentdifferent lengths. lengths. TheThe dimensionsdimensions of of body body ge- geometry,ometry, such such as as aa((ss)) and and b(s),), areare scaledscaled accordingly. accordingly. When When the the length length was was 0.3 m,0.3 0.4 m, m,0.4 0.5 m, m, 0.5 0.6 m, m, 0.6 m, andand 0.70.7 m, m, the the range range of ofU* U was* was 0.837 0.83 BL/s–1.4237 BL/s–1.42 BL/s,3 BL/s, 0.738 0.73 BL/s–1.4548 BL/s–1.45 BL/s,4 B 0.644L/s, 0.64 BL/s–1.4624 BL/s–1.46 BL/s,2 BL/s, J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 11 of 18 0.5760.576 BL/s–1.483BL/s–1.483 B BL/s,L/s, and and 0.51 0.5188 B BL/s–1.486L/s–1.486 BL/s, BL/s, respectively; respectively; that that of ofCOT COT was was 0.09 0.093–0.306,3–0.306, 0.069– 0.069–0.291,0.291, 0.056– 0.056–0.285,0.285, 0.047 0.047–0.280,–0.280, and and0.04 0.041–0.290,1–0.290, respectively. respectively.

FigureFigure 12. 12.Amplitude Amplitude envelopes envelopes of of swimming swimming kinematics kinematics with with various various lengths lengths based based on theon the opti- opti- mizationmization results results shown shown in in Figure Figure 11 .11.

Next,Next, the the sensitivity sensitivity analysis analysis of ofK( sK)( wass) was conducted, conducted, namely, namely, the impactthe impact of varying of varying a a singlesingle feature feature of ofK (Ks()s on) on swimming swimming performance performance was was studied. studied. As displayedAs displayed in Figure in Figure6 , 6, fromfrom the the swimming swimming form form of of high-efficiency high-efficiency A toA theto the swimming swimming form form of high-speed of high-speed F, the F, the values of K at the second, eighth, and ninth interpolation point (represented as K2, K8, values of K at the second, eighth, and ninth interpolation point (represented as K2, K8, K9, K , respectively) increased at to different extent. On the other hand, the values of K at respectively)9 increased at to different extent. On the other hand, the values of K at five five interpolation points, from the third one to the seventh one, increased synchronously interpolation points, from the third one to the seventh one, increased synchronously roughly. Thus, the joint sensitivity analysis of these K values (represented as K3–7) was roughly. Thus, the joint sensitivity analysis of these K values (represented as K3–7) was conducted. To study the influence of Kmax on swimming performance, the K value at the conducted. To study the influence of Kmax on swimming performance, the K value at the tenth interpolation point (represented as K10) was also included in the discussion. The variationstenth interpolation of the aforementioned point (representedK values as onK10) the was swimming also included performance in the discussion. are shown The in var- Figureiations 13 of. Forthe convenience,aforementioned the curvature K values amplitudeson the swimming in the case performance of minimum are COT shown (Case in 0Figure) were13. For a reference convenience, for other the cases.curvature amplitudes in the case of minimum COT (Case 0) were a reference for other cases.

(a)

(b)

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Figure 12. Amplitude envelopes of swimming kinematics with various lengths based on the optimization results shown in Figure 11.

Next, the sensitivity analysis of K(s) was conducted, namely, the impact of varying a single feature of K(s) on swimming performance was studied. As displayed in Figure 6, from the swimming form of high-efficiency A to the swimming form of high-speed F, the values of K at the second, eighth, and ninth interpolation point (represented as K2, K8, K9, respectively) increased at to different extent. On the other hand, the values of K at five interpolation points, from the third one to the seventh one, increased synchronously roughly. Thus, the joint sensitivity analysis of these K values (represented as K3–7) was conducted. To study the influence of Kmax on swimming performance, the K value at the tenth interpolation point (represented as K10) was also included in the discussion. The var- J. Mar. Sci. Eng. 2021, 9, 537 11 of 17 iations of the aforementioned K values on the swimming performance are shown in Figure 13. For convenience, the curvature amplitudes in the case of minimum COT (Case 0) were a reference for other cases.

(a)

(b)

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(c)

(d)

(e)

Figure 13. 13. VariationsVariations of ofcurvature curvature amplitude amplitude on the on swimming the swimming performance performance U* andU COT.* and (a COT.), (b), (a), (b), (c), (c), (d) and (e) refer to the sensitivity analysis of K10, K9, K8, K3–7 and K2, respectively. The left col- (d) and (e) refer to the sensitivity analysis of K10, K9, K8, K3–7 and K2, respectively. The left column umn is the variation of K(s) at some discrete point and the right column is the effect of this varia- tionis the on variation swimming of performance.K(s) at some discrete point and the right column is the effect of this variation on swimming performance.

K10 was set to 12, 16, 21, 24, 28, and 32 from case −3 to case 2, respectively. K10 appeared to have little effect on the swimming performance, as illustrated in Figure 13a. As K10 increased, both speed and efficiency slightly increased at first and then decreased a little, implying that there must exist some K10 value that maximizes the speed and efficiency. Therefore, although the curvature profile changed from A to F in Figure 6, the value of K10 kept constant. Since the value of K10 affects the swimming performance a little, the restriction of Kmax can be relaxed appropriately. K9 was set to 5, 8, 10.7, 13, and 16 from case −2 to case 2, respectively (Figure 13b). K9 had a great influence on swimming speed. Higher K9 caused a dramatic improvement (up to 50%) in speed and a small change (up to 10%) in efficiency. In the bi-objective optimization process, case 0, case 1, and case 2 were all non-inferior solutions while both case -2 and case -1 were inferior solutions. Consequently, the minimum K9 illustrated in Figure 6 was the K9 value in case 0. K8 was set to 2, 4.1, 10, 15, and 20 from case −1 to case 3, respectively (Figure 13c). K8 had a great influence on both speed and efficiency. Higher K8 value resulted in a significant increase in speed and COT, by 62% and 68%, respectively. This change in COT meant

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K10 was set to 12, 16, 21, 24, 28, and 32 from case −3 to case 2, respectively. K10 appeared to have little effect on the swimming performance, as illustrated in Figure 13a. As K10 increased, both speed and efficiency slightly increased at first and then decreased a little, implying that there must exist some K10 value that maximizes the speed and efficiency. Therefore, although the curvature profile changed from A to F in Figure6, the value of K10 kept constant. Since the value of K10 affects the swimming performance a little, the restriction of Kmax can be relaxed appropriately. K9 was set to 5, 8, 10.7, 13, and 16 from case −2 to case 2, respectively (Figure 13b). K9 had a great influence on swimming speed. Higher K9 caused a dramatic improvement (up to 50%) in speed and a small change (up to 10%) in efficiency. In the bi-objective optimization process, case 0, case 1, and case 2 were all non-inferior solutions while both case -2 and case -1 were inferior solutions. Consequently, the minimum K9 illustrated in Figure6 was the K9 value in case 0. K8 was set to 2, 4.1, 10, 15, and 20 from case −1 to case 3, respectively (Figure 13c). K8 had a great influence on both speed and efficiency. Higher K8 value resulted in a significant increase in speed and COT, by 62% and 68%, respectively. This change in COT meant that the swimming efficiency was significantly reduced. Therefore, the value of K8 constantly increased from A to F in Figure6. The second-highest K value, K8, thus became the most important factor in the selection of Kmax. The minimum Kmax value must be higher than that of K8. Considering that the maximum K8 value in Figure6 was less than 20, Kmax could be relaxed to 6/L. K3–7 (the average value of the sequence from K3 to K7) was set to 0, 0.6, 1.4, 2.2, and 3 from case −1 to case 3, respectively (Figure 13d). The increase in K3–7 had a similar effect to the increase in K8, leading to a marked rise in speed and COT, by 30% and 26%, respectively. Since the increase in K3–7 was smaller than that of K8, the swimming performance was more sensitive to K3–7. K2 was set to 0, 0.85, 5, 10, 15, and 20 from case −1 to case 4, respectively (Figure 13e). K2 had little effect on speed. It meant that although K2 increased greatly from A to F in Figure6, it had little effect on improving the swimming speed. COT increased by 15% along with K2, indicating that the swimming efficiency decreased moderately.

3.4. Comparison among BCF Fish Two other kinds of fishes, anguilliform fish, and thunniform fish, are discussed in this part. The shapes of anguilliform fish and thunniform fish are described in [46] and [33], respectively. Based on the above two body shapes, optimization results of U* and COT are plotted in Figure 14. Interestingly, for the same U*, carangiform fish was more efficient, although thunniform fish is more streamlined in a general sense. This reveals that swimming performance optimization is a comprehensive balance of the body shape, kinematics, dynamics, not just some aspect among them. For the same COT, carangiform fish and thunniform fish were faster than anguilliform fish. The reason is that the caudal fin propulsion adopted by the former two can provide a higher reaction force than the body propulsion used by anguilliform fish. Portions of different powers included in the total power are plotted in Figure 15, where PD = P// + Pform is the power consumed by the total drag force. Not surprisingly, the change in power ratios of carangiform fish with the swimming speed was in agreement with that of thunniform fish. The greatest difference between anguilliform fish and the other two fishes was that it had a larger P⊥ ratio and a small Pm ratio. This indicates that anguilliform fish swimming is mainly a body propulsion mode while carangiform fish and thunniform fish combine body propulsion and caudal fin propulsion. As U* increased, all the elements of PD decreased, while all the elements of P⊥ increased. The main cause was that higher values of U* enhanced the lateral motion amplitude of the fish body, bringing about the increase in P⊥. Moreover, with the increase in U*, Pi ratios in thunniform fish and carangiform fish increase rapidly, reaching 66% and 50%, respectively. The Pm ratio in anguilliform fish was about 8%, half that of the ratio in the other two fish types. The J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 13 of 18

that the swimming efficiency was significantly reduced. Therefore, the value of K8 constantly increased from A to F in Figure 6. The second-highest K value, K8, thus became the most important factor in the selection of Kmax. The minimum Kmax value must be higher than that of K8. Considering that the maximum K8 value in Figure 6 was less than 20, Kmax could be relaxed to 6/L. K3–7 (the average value of the sequence from K3 to K7) was set to 0, 0.6, 1.4, 2.2, and 3 from case −1 to case 3, respectively (Figure 13d). The increase in K3–7 had a similar effect to the increase in K8, leading to a marked rise in speed and COT, by 30% and 26%, respectively. Since the increase in K3–7 was smaller than that of K8, the swimming performance was more sensitive to K3–7. K2 was set to 0, 0.85, 5, 10, 15, and 20 from case −1 to case 4, respectively (Figure 13e). K2 had little effect on speed. It meant that although K2 increased greatly from A to F in Figure 6, it had little effect on improving the swimming speed. COT increased by 15% along with K2, indicating that the swimming efficiency decreased moderately.

3.4. Comparison among BCF Fish Two other kinds of fishes, anguilliform fish, and thunniform fish, are discussed in J. Mar. Sci. Eng. 2021, 9, 537 this part. The shapes of anguilliform fish and thunniform fish are described in [46] and13 of 17 [33], respectively. Based on the above two body shapes, optimization results of U* and COT are plotted in Figure 14. Interestingly, for the same U*, carangiform fish was more efficient, although thunniform fish is more streamlined in a general sense. This reveals reaction force F at the tail is often neglected in some related studies of anguilliform that swimming performancem optimization is a comprehensive balance of the body shape, ﬁsh swimming [17,18,47,48]. Boyer et al. used Poincaré–Cosserat equations to solve the kinematics, dynamics, not just some aspect among them. For the same COT, carangiform problem of anguilliform ﬁsh swimming [48]. Compared to CFD simulation results, the fish and thunniform fish were faster than anguilliform fish. The reason is that the caudal forcefin propulsion in the swimming adopted direction by the former is quite two different. can provide The omittancea higher reaction of the reaction force than force the may bebody a factor propulsion attributing used by to thisanguilliform discrepancy. fish.

FigureFigure 14.14. Optimization results results of of UU* and* and COT COT of ofthree three kinds kinds of fish. of fish. In the In thecase case of the of thunniform the thunniform fish,fish, thethe U* range range was was 0.752 0.752 BL/s BL/s–1.460–1.460 BL/s BL/s,, and andthe COT the COTrange range was 0. was1–0.5, 0.1–0.5, which whichwas roughly was roughly consistentconsistent with the ranges ranges of of carangiform carangiform fish. fish. In In the the case case of ofthe the anguilliform anguilliform fish, fish, the theU* rangeU* range was was 0.5620.562 BL/s–0.965BL/s–0.965 BL/s BL/s,, and and the the COT COT range range was was 0.2– 0.2–0.5.0.5.

Portions of different powers included in the total power are plotted in Figure 15, where PD = P// + Pform is the power consumed by the total drag force. Not surprisingly, the change in power ratios of carangiform fish with the swimming speed was in agreement with that of thunniform fish. The greatest difference between anguilliform fish and the other two fishes was that it had a larger P⊥ ratio and a small Pm ratio. This indicates that anguilliform fish swimming is mainly a body propulsion mode while carangiform fish and thunniform fish combine body propulsion and caudal fin propulsion. As U* increased, all the elements of PD decreased, while all the elements of P⊥ increased. The main cause

Figure 15. The proportion of different powers in the total power as a function of U*. (a), (b) and (c) are the cases of carangiform fish, thunniform fish, and anguilliform fish, respectively. The discrete points in (a) refer to the six swimming forms in Figure6. The discrete points in ( b) and (c) represent the six swimming forms in thunniform fish and anguilliform fish, respectively. Both typical swimming forms are extracted from the Pareto fronts in Figure 14 (the black line and green line in Figure 14).

4. Discussion Triantafyllou et al. discovered that the St (St = fA/U, f is the tail-beat frequency, A is the peak-to-peak amplitude of the tail) range of swimming animals with foil-like tails is 0.2–0.4 [49]. In our simulation, the St range of carangiform fish was 0.19–0.30 and that of thunniform fish was 0.255–0.360, both of which can be identified for efficient propulsion. However, the St range of anguilliform fish was 0.42–0.5 and was beyond the St range proposed by Triantafyllou et al. On the other hand, Long et al. determined that St equals 0.56 based on the observation data of hagfish swimming and the range calculated by Kern et al. through computational fluid dynamics is 0.59–0.67 [9,50]. Thus, it is considered that the optimal St range, proposed by Triantafyllou et al., may not apply to anguilliform fish swimming. Moreover, Wiens and Hosoi found that optimally efficient swimming kinematics could be characterized by a non-dimensional variable Ψ (Equations (12) and (13)) [51]. When Ψ is

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between 0.3 and 1.0, the swimming efficiency is near-optimal. The Ψ range of carangiform fish in this simulation was 0.44–0.94. These analyses reflect the rationality of simulation.

sin(β) − πSt cos(β) Ψ = 1 − (12) β − πSt

U β = π St (13) V J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 15 of 18 where V is the wave propagation speed. U/V is the slip ratio that has been widely used and detailed calculation can be found in previous studies [51,52]. The sensitivity analysis of curvature amplitude showed that K9, K8, and K3–7 were the mostThe significantsensitivity analysis variables, of curvature which can amplitude instruct show the stiffnessed that K9 adjustment, K8, and K3–7 strategywere the for roboticmost significant fish. On thevariables, other hand,which itcan can instruct be seen the thatstiffnessK2 should adjustment always strategy be near for zerorobotic if the gentlefish. On effect the onother speed hand, is ignored.it can be Inseen other that words,K2 should there always is no be need near to zero improve if the thegentle speed effect on speed is ignored. In other words, there is no need to improve the speed by only by only one percent at the expense of keeping K2 so large. Following this way, the large curvatureone percent amplitudes at the expense mainly of keeping existed K in2 so the large. last thirdFollowing of the this fish way, body, the which large curvature conforms to theamplitudes biological mainly observation existed in in reality. the last Furthermore, third of the fish this body, discovery which alsoconforms reduced to the the biolog- difficulty ofical the observation stiffness distribution in reality. Furthermore implementation., this discovery also reduced the difficulty of the stiffnessThe distribution three differences implementation. betweenour model and Eloy’s model are as follows. First, the frequencyThe three wasdifferences set constant between in our model model, and which Eloy’s can model help are investigate as follows. the First, frequency the frequency was set constant in our model, which can help investigate the frequency effect; effect; second, the fish body shape was determined by the empirical equations. Strange second, the fish body shape was determined by the empirical equations. Strange shapes shapes (like Figure 5e,f in [24]) can be avoided through this way; third, COT, the commonly (like Figure 5e,f in [24]) can be avoided through this way; third, COT, the commonly used used measure in other references, was chosen as the efficiency measure instead of a novel measure in other references, was chosen as the efficiency measure instead of a novel meas- measure E* first adopted by Eloy. The discussion on the differences between the two ure E* first adopted by Eloy. The discussion on the differences between the two efficiency efficiency measures is not the scope of this paper. The phenomenon of bifurcation in the measures is not the scope of this paper. The phenomenon of bifurcation in the Pareto front Pareto front that ever emerged in Eloy’s model did not occur in our model. The main cause that ever emerged in Eloy’s model did not occur in our model. The main cause may be the maybiological be the fish biological shape adopted fish shape in our adopted model. in our model. InIn the the casecase ofof the highest efficiency, efficiency, the the swimming swimming speed speed and and the the midline midline kinematics kinematics obtainedobtained by by our our model model are are similar similar to to those those obtained obtained by by Eloy’s Eloy’s model model (the (the relative relative swimming swim- speedming inspeed Eloy’s in Eloy’s model mod is 0.727el is 0.727 and and the swimmingthe swimming kinematics kinematics are are illustrated illustrated in inFigure Figure 16 ). The16). similarityThe similarity shows shows that that the the fish fish body body shape shape usedused in our our model model is isnear near the the desired desired bodybody shapeshape forfor thethe mostmost efficientefficient swimming. swimming. The The midline midline kinematics kinematics of ofSaithe Saithe was was also also usedused toto makemake a comparison comparison with with the the results results of these of these two twomodels models (the relevant (the relevant data is data ex- is extractedtracted from from [53 [53]).]). The The corresponding corresponding amplitude amplitude envelopes envelopes are are depicted depicted in Figure in Figure 17. 17. ThereThere was was nono bigbig differencedifference between the the three three cases. cases. The The differences differences in inthe the head head and and tail tail amplitudesamplitudes maymay resultresult from the fish fish body body shape. shape. It Itcan can be be concluded concluded that that for forthe the most most efficientefficient swimmingswimming mode, mode, the the results results obtained obtained from from models models and andthe data the observed data observed in re- in realityality reach reacheded an an agreement, agreement, and and the the consistency consistency was was not not only only reflected reflected in body in body shape shape but but alsoalso in in swimming swimming kinematics.kinematics.

(a) (b) (c)

FigureFigure 16. 16The. The different different midline midline kinematics of of carangiform carangiform fish. fish. (a) (isa) for is forthe thecase case of the of minimum the minimum COT in COT our inmodel; our model; (b) is for the case of the highest efficiency in Eloy’s model; (c) is for the case of Saithe fish in nature through the biological (b) is for the case of the highest efficiency in Eloy’s model; (c) is for the case of Saithe fish in nature through the observation. biological observation.

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FigureFigure 17. 17.The The comparison comparison of amplitude of amplitude envelope envelope among three among cases. three cases. 5. Conclusions 5. ConclusionsOptimal curvature profiles were obtained by combining bi-objective optimization and theOptimal reduced dynamic curvature model profiles proposed were by Eloyobtained [24]. As by the combining swimming mode bi-objective of the optimization carangiformand the reduced fish transitioned dynamic from model high-efficiency proposed mode by to Eloy high-speed [24]. As mode, the the swimming number mode of the of local maxima of the curvature amplitude changed from one to three along with the bodycarangiform length. fish transitioned from high-efficiency mode to high-speed mode, the number of localThe frequency maxima effect of the on swimmingcurvature performanceamplitude waschange examined.d from The one maximum to threeU* along with the andbody the length. maximum COT increased linearly with the tail-beat frequency, while the minimum U* andThe the minimumfrequency COT effect remained on swimming constant. On performance the other hand, was when examined. swimming kine- The maximum U* maticsand the remained maximum unchanged, COT swimming increase speed,d linearly stride with length, the and tail COT-beat were frequency, all positively while the mini- correlated with the tail-beat frequency. * mumBased U and on the the sensitivity minimum analysis, COT the remain changinged consta mechanismnt. On of curvaturethe other amplitude hand, when swimming regardingkinematics swimming remain speeded unchanged, was revealed. swimming The curvature speed, amplitude stride at nine-tenths length, and of body COT were all posi- lengthtively ( Kcorrelated10) had little with effect the on both tail efficiency-beat frequency. and speed. The curvature amplitude at four- fifths ofBased body lengthon the (K sensitivity9) had a great analysis, influence onthe speed changing but had mechanism little effect on of efficiency. curvature amplitude The curvature amplitude at seven-tenths of body length (K ) and the curvature amplitude regarding swimming speed was revealed. The curvature8 amplitude at nine-tenths of body between fifths and three-fifths of body length (K3–7) both had a great impact on speed and 10 efficiency.length (K The) curvaturehad little amplitude effect on at both tenths efficiency of body length and (speed.K2) had The little curvature effect on speed, amplitude at four- althoughfifths of it body increased length from (K high-efficiency9) had a great mode influence to high-speed on speed mode. but In ha addition,d little it effect was on efficiency. foundThe curvature that the swimming amplitude efficiency at seven increased-tenths as the of fishbody body length was longer. (K8) and the curvature amplitude Optimization in the cases of the other two kinds of BCF fishes (anguilliform fish and between fifths and three-fifths of body length (K3–7) both had a great impact on speed and thunniform fish) indicated that anguilliform fish mainly adopts body propulsion, while carangiformefficiency. fish The and curvature thunniform amplitude fish combine at bodytenths propulsion of body and length caudal (K fin2) propulsion.had little effect on speed, Portionsalthough of differentit increase powersd from included high in-efficiency the total power mode were to analyzed.high-speed It was mode. indicated In addition, it was thatfound the lateralthat the motion swimming of anguilliform efficiency fish is increase large andd the as reactionthe fish force body of was anguilliform longer. fish is low.Optimization However, considering in the cases that of the the proportion other two of the kinds power of caused BCF fishes by the reaction(anguilliform fish and force (Pm) of anguilliform fish accounted for about 8%, it is better not to neglect it in the accuratethunniform swimming fish) analysis indicate ofd anguilliform that anguilliform fish. fish mainly adopts body propulsion, while carangiformThe above discussionfish and thunniform shows that if real-time fish combine and accurate body curvature propulsion control and can caudal be fin propul- achieved,sion. Portions bionic robot of different fish has the powers potential included to surpass in the the swimming total power performance were analyzed. of real It was indi- fishcated in terms that ofthe swimming lateral motion efficiency of and anguilliform speed. This studyfish is lays large a theoretical and the foundation reaction force of anguil- for the design and control of high-speed and efficient AUVs. liform fish is low. However, considering that the proportion of the power caused by the Authorreaction Contributions: force (PmConceptualization,) of anguilliform Y.L.; fish methodology, accounted Y.L. for and about H.J.; formal 8%, it analysis, is better Y.L. not to neglect it andin the H.J.; accurate resources, H.J.; swimming writing—original analysis draft of preparation, anguilliform Y.L.; writing—review fish. and editing, H.J.; supervision,The H.J.;above project discussion administration, shows H.J.; that funding if real acquisition,-time H.J.and All accurate authors have curvature read and control can be agreed to the published version of the manuscript. achieved, bionic robot fish has the potential to surpass the swimming performance of real Funding: This research was funded by the National Natural Science Foundation of China, grant numberfish in 51275127. terms of swimming efficiency and speed. This study lays a theoretical foundation for the design and control of high-speed and efficient AUVs.

Author Contributions: Conceptualization, Y.L.; methodology, Y.L. and H.J.; formal analysis, Y.L. and H.J.; resources, H.J.; writing—original draft preparation, Y.L.; writing—review and editing, H.J.; supervision, H.J.; project administration, H.J.; funding acquisition, H.J. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China, grant number 51275127. Institutional Review Board Statement: Not applicable.

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Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The numerical and experimental data used to support the findings of this study are included within the article. Acknowledgments: The authors would like to thank Christophe Eloy from Aix-Marseille University for his technical support. Conflicts of Interest: The authors declare no conflict of interest.

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