Zhihn Li ,Dn Xi ,Jio Co ,Weishn Chen ,Xingsong Wng

a School of Mechanical Engineering,Southeast University,Nanjing,Jiangsu 211189,China

b State Key Laboratory of Robotics and System,Harbin Institute of Technology,Harbin 150080,China

Keywords:Self-yaw dolphin Spanwise flexibility Caudal fin Numerical simulation Hydrodynamic performance

ABSTRACT The hydrodynamic performance of the virtual underwater vehicle under self-yaw is investigated numerically in this paper,we aim to explore the fluid laws behind this plane motion achieved by the bionic flexibility,especially the spanwise flexibility of the caudal fin.The kinematics of the chordwise flexible body and the spanwise flexible caudal fin are explored through dynamic mesh technology and user-defined functions (UDF).The 3-D Navier-Stokes equations are applied to simulate the viscid fluid surrounding the bionic dolphin.The study focuses on quantitative problems about the fluid dynamics behind the specific motion law,including speed of movement,energy loss and working efficiency.The current results show that the self-yaw can be composed of two motions,autonomous propulsion and active steering.In addition,the degree of the flexible caudal fin can produce different yaw effects.The chordwise phase difference Ф is dominant in the propulsion function,while the spanwise phase difference δ has a more noticeable effect on the steering function.The pressure distribution on the surface of the dolphin and the wake vortex generated in the flow field reasonably reveal the evolution of self-yaw.It properly turns out that the dolphin can combine the spanwise flexible caudal fin and the chordwise flexible body to achieve self-yaw motion.

1.Introduction

The morphological structure of aquatic animals has excellent athletic ability within a long period of evolution.As a typical representative,flexible fish can complete many actions in complex underwater environments by relying on the Body and/or Caudal Fin (BCF) swimming pattern.Therefore,in recent years,automatic underwater vehicles (AUV) [1–4]used to imitate fish-shaped swimmers have been widely studied and applied.Suebsaiprom and Lin [5]applied a planar four-bar linkage model to derive a nonlinear,dynamic fish movement mode that allows the fish robot to move in a three-dimensional(3D) domain.Yu et al.[6–8]combined streamlined and high-thrust tail advance mechanism design and passive dorso-abdominal propulsion strategy to achieve robotic dolphin’s gliding and leaping.Furthermore,the interaction of caudal and pectoral fins is actuated as the salutary role to perform the multiple Degrees-of-Freedom(DOFs) actions [9].These bionic designs show the beneficial effect of the kinematics of these fins on the effective underwater thrust and driving torque.

As an emblematic aquatic mammal,the dolphin is considered a pliable and explosive swimmer.The abundant movement patterns of dolphins benefit from their fins,including pectoral fins,dorsal fins,and caudal fins.Significantly,the different motion laws of caudal fins not only play a role in high-speed propulsion,but also play a role in flexible manipulation [10].In the previous numerical investigations [11,12],the caudal fin was simplified as a rigid oscillating foil to depict a heave and pitch motion.This idealized scheme cannot indeed reveal the kinematics and energetics of the fish body,and the simulation results violate the principle of biological structure to some extent.With the continuous demand for biologically inspired engineering,the flexible caudal fin of the bionic dolphin has become a new research focus.

According to different flexibility directions,the caudal fin is mainly divided into chordal flexibility and spanwise flexibility.In terms of chordal flexibility,Yang et al.[13]found that adjusting the chordal bending amplitude of the tuna tail fin can produce higher propulsion efficiency through numerical simulation.Feng et al.[14]revealed that the chord-flexible caudal fin has faster starting performance and better heading stability than the rigid caudal fin under self-propulsion.Curatolo and Teresi [15]proposed the concept of active deformation to describe and model fish body muscle movements,emphasizing the role of string flexibility on propulsion.Compared with the systematic research system of chordal flexibility,the related research on spanwise flexibility is rarely reported.Zhu and Shoele [16,17]considered the law of spanwise flexibility of the caudal fin at a given advancing speed in the absence of the necessary swimming carrier.In our previous study[18],we properly adjusted the shape of the caudal fin’s spanwise water hitting surface with different trajectories through numerical simulation to obtain higher propulsion efficiency and increase the stability of self-propelled swimming.It can be noticed that the development of flexible caudal fins can greatly improve the maneuverability of underwater robots,but the previous research was limited to straight motion.Flexible caudal fins can also provide more possibilities for underwater turning,the study on self-yaw based on flexible caudal fin has never been explored.

The self-yaw motion mentioned above is a compound mechanism that requires to be weighed and considered,which is different from the symmetry of linear propulsion.Many observations and experiments [19–21]have shown that turning motion can be achieved by changing the frequency and amplitude of the fluctuations of the pectoral fins on both sides.In addition,extensive investigations focused on a fast and transient C-turning through the flexibility of the fish body.In terms of related experimental tests[22–24],researchers generally use particle image velocimetry(PIV)to study the response process of live fish and obtain corresponding dynamic laws.The wide application of computational fluid dynamics (CFD) methods has promoted in-depth research on numerical simulation of C-turning [25–28].More detailed hydrodynamic mechanisms and 3D wake vortex structures are obtained,providing theoretical support for the design of biomimetic robot fish.These studies pay more attention to the instantaneous turn of fish in the face of emergencies from a biological perspective.However,a continuous and periodic yawing method suitable for fish-like robots still lacks the systematic study.

In this paper,the effect of the body’s chord flexibility and the caudal fin’s spanwise flexibility on the self-yaw locomotion is studied under the premise of considering efficiency,stability and innovation.We break the limitations of previous studies [12,29]on selfswimming and extend linear forward motion to in-plane yaw motion,providing more possibilities for autonomous fish spatial motion.Inspired by this,the diversity of dolphins’ flexibility is generally motivated behind the development of bionic motion systems.While keeping the body kinematics unchanged,we propose an exact quantitative method to control the span and chord swing of the caudal fin.Then the self-yaw process of the bionic dolphin in 3D space is obtained through numerical simulation.Finally,professional parameters help explain the influence of spanwise flexibility on the improvement of yawing performance mathematically.

2.Material and methods

2.1.Computational model

In this work,by referring to the shape and movement of real dolphins,we apply a virtual swimmer to act as the computational model to realize the self-yaw mode.Combined with the body and caudal fin,the smooth outline of the entire virtual dolphin is sketched using a curve fitting method,shown in Fig.1.Some details about pectoral and dorsal fins are ignored.The coordinate systemoxbybzbandoxcfycfzcfare applied for the dolphin body and the caudal fin separately to describe the sophisticated configuration and motion.The length of the dolphin robot isL,and the span length of caudal fin isb.it is vividly exhibited that the chordwise shape of the caudal fin is NACA0018 wing section,while along the spanwise direction,it is depicted as the bifurcated sector.

Fig 1.Physical model and coordinate system of dolphin and caudal fin.

Fig 2.grid system for the computational domain and the dolphin.

The domain and grids used for numerical simulation are established based on computational cost and accuracy.The entire swimming area is a 15L× 3L× 12Lbox-shaped tank filled with water,shown in Fig.2 (A).The flexible swing of the bionic caudal fin is a complex deformation process.Therefore,the entire fluid area is discretized into tetrahedral meshes,while the volume surrounding the dolphin is gradually refined.The appropriate grid density can effectively grasp the flow field structure and show the law of motion.In Fig.2 (B),the surface of the dolphin is discretized into a uniform triangular mesh,and the mesh at the bend of the caudal fin is also refined to catch the geometric details.This meshing method has been widely used in previous studies [30–32],and is applied to the self-propelled swimming of fish-shaped robots and the active deformation of various fins.

Fig 3.Mid-line motions of chordwise flexible body in one beating period.

Fig 4.The prescribed kinematics of the bionic dolphin within half a cycle.

In the preparation stage of the numerical simulation,two boundary conditions are set in the entire self-yaw motion: the moving surface of the dolphin’s body and the boundary conditions of inflow and outflow in the computational domain.In order to simulate the kinematics of the underwater dolphin and the flow field around the body,a no-slip boundary condition is adopted for the moving interface,where the dolphin velocityudis equal to the flow velocityu.The upstream and downstream boundaries and the wall boundaries constitute the entire swimming area.All interfaces have zero pressure gradient.The difference is that the upstream boundary has zero velocity,while the velocity imposed on the downstream boundary and other boundaries has zero gradient.

2.2.Kinematics

The motion unit of the bionic dolphin employed for self-yaw includes a chordwise flexible body and a spanwise flexible caudal fin.The kinematics are described below separately.

Commonly,the flexible body of the dolphin can be abstracted as a spline curve,experiencing a series of travelling waves that gradually increase in amplitude and propagate from head to tail.The head and caudal fin have a certain lateral amplitude.We combine the travelling wave characteristics of body motion and apply a quadratic curve containing a constant term to fit the amplitude envelope to establish the kinematic description of the chordwise flexible body mathematically depicted as

whereyb(x,t) represents the instantaneous lateral moving distance of body,A(x) is the function of amplitude defined byA(x)=(C0+C1x+C2x2),C0,C1 andC2 are the envelope amplitude coefficients.fis the wave frequency,andkis the wave number corresponding to a wavelength.The length of the undulating body is usually assumed to be unchanged during the wavy motion.Fig.3 shows the motion curve of the midline of the dolphin during the swing cycle whenf=5Hz,displaying the chordal movement of the dolphin’s flexible body calculated based on Eq.(1).The overall motion amplitude completely follows the shape parameters of the dolphin,which is similar to the quadratic amplitude envelope fitted by the fish simulation curve of Borazjani and Sotiropoulos[33].

The travelling wave characteristics of the dolphin’s body can only produce linear propulsion,while the swing of the spanwise caudal fin provides the active turning function on the plane.The motion of the caudal fin can be regarded as a composite motion including translation and spanwise swing,in which the translation motion is generated by following the wave of the body,and the instant lateral translational displacement can be described as:

whereAcfis the elevation of the vertical joint of the tail,andLbis the length of the dolphin body.We also describe the spanwise flexible swing as a differential motion with a certain spanwise phase difference.Along the spanwise direction,the pitch of the caudal fin has different phase differences in the form of linear function at the same chordwise direction.the instant pitching angular displacementθ(z,t) is expressed as

whereθmaxrepresents the pitching amplitude of the caudal fin,Фrepresents the phrase between the lateral motion and pitching motion,bis the span length of the caudal fin,andδis the phrase between the left and right extreme position of the caudal fin along the spanwise direction.Fig 4.displays the prescribed kinematics of the bionic dolphin within half a cycle.Especially,δ=0 is the situation that the caudal fin pitches as the rigid flat plate without spanwise flexibility.All positions on the caudal fin have the same beating frequencyfand pitching amplitudeθmax,θmaxcan be expressed as

Fig 5.Grid sensitivity test for numerical simulation.

wherecis the body wave speed,αmaxis the maximum angle of attack of the caudal fin.

2.3.The governing equations

Realizing the self-yaw of the virtual dolphin and studying the influence of different parameters on steering efficiency are the emphasis of this research.The research method is based on the continuity and incompressible flow problem.To simulate the 3D viscous fluid surrounding the bionic dolphin,we define the relevant governing equations with the following Navier-Stokes equations:

where ?is the gradient operator,ρis the density,pis the pressure divided by the density andμis the dynamic viscosity.In this work,Newton’s equations of motion are adopted to describe the motion of the dolphin.

In these equations,(Fx,Fy,Fz) and (Mx,My,Mz) represent the fluid forces and moments working on the fish body and caudal fin in three directions,mis the mass of the robot,andJis the relative moment of inertia of the rotation axis,ris the position of the center of mass of the moving dolphin,andφis the deviation angle of the fish relative to the original position.From the perspective of hydrodynamics,fluid forces and moments can also be expressed as follows:

Whereσis the normal stress vector,nis the unit vector along the normal direction,dSis the differential unit area along the fish surface,and (ex,ey,ez) is the unit vector along the three directions.

2.4.Numerical method

Numerical simulation has been widely used in our previous investigations,including fish-shaped robots with different forms under self-propulsion,which indirectly verified the effectiveness of the 3-DOFs fluid-structure interaction (FSI) method.The solver ANSYS Fluent acts as the tool for computational fluid dynamics (CFD)in the simulation process [34].We use the finite volume method to discretize the Navier–Stokes equations: The Green-Gauss cellbased method is applied for the setup of gradient interpolation in the spatial discretization,and the second-order upwind scheme is adopted for turbulent kinetic energy and specific dissipation rate.We employ the second-order scheme for the pressure interpolation of each governing equations and the second-order implicit scheme for the transient formation.The SIMPLE algorithm is used to calculate the pressure-velocity coupling of the continuity equations.Based on Newton’s motion equations,the user-defined function and dynamic mesh technology [35]are applied for rigid movement,while the embedded DEFINE_GRID_MOTION macro is connected to the main code of the solver to realize the required deformation motion.To control the accuracy of the results and computational cost,we control the time-step sizedt=0.005T,T is the period of the dolphin swing.For each step,on the one hand,the mesh methods,including smoothing and remeshing method,are applied to regenerate and smooth the mesh grids of the simulated body,on the other hand,the numerical computation process of the instant flow-structure scheme is as follows:

(1) Obtain the deformation velocityudefof the fish surface through the specified dolphin kinematics,and calculate the movement velocityucand deformation velocityuc|defof the dolphin’s center of mass with Newton’s motion equations.

(2) Update the dynamic grids of the dolphin position and the surface-fluid interface using the following formula at timestepn.

(3) Solve the fluid Eqs.(5) and (6) by the boundary conditions defined on the fluid-solid interface,and advance the transient flow solution from timestepnton+1.

(4) Calculate the hydrodynamic forces and moments acting on the dolphin and smooth them with the subsequent underrelaxation program:

whereβrepresents under-relaxation factor,and the value ofβis determined by the cost and accuracy of the calculation.respectively represent the solution based on the front and back underrelaxation at the time iterationn+1.

(1) Advance the yaw angular velocity and rigid moving velocity of the dolphin with the computed fluid force and moment at timestepn+1.

2.5.Grid sensitivity test and numerical validation

Accurate simulation results depend on the appropriate mesh refinement mode.This section applies three sizes of grids to conduct a grid sensitivity test according to the model size and numerical environment.The total number of grids generated is 1.9 (coarse),6.2 (standard) and 14.8 (fine) million grids,corresponding to the uniform edge lengths of 0.016L,0.0 08Land 0.0 04L.To make the test content have a reference value,the domain size,time step and boundary conditions are unified.In the subsequent quantitative grid sensitivity test,we control the spanwise phase difference of the dolphin to 30 °,and the chordwise phase difference is controlled within the range of 60–90 °.Fig.5 shows the variation of the steady-state swimming velocityUwith the chordwise phase differenceФfor three different mesh cells.From the test results,we can see that the coarse grids cannot give reasonable results,while standard grids and fine grids can solve hydrodynamic problems well.We conclude that over-refined meshes cannot get more optimized results,and the use of a refined uniform mesh discretization work area with a size of 0.008Lis a suitable choice based on calculation efficiency and result accuracy.

To verify that the method can deal with the boundary problem of 3D locomotion and predict the fluid force and flow field structure,we simulate a classic example that the cylinder starts from a static state and performs a one-dimensional simple harmonic motion along the horizontal direction in previous studies [11,12].The calculated results are in good agreement with the results reported by Dutsch et al.[36],laterally verifying the feasibility of the numerical method applied to boundary motion.

2.6.Calculation of performance parameters

According to the previously prescribed kinematic equations,the self-yaw dolphin is driven by a combination of a chordwise flexible body and a spanwise flexible caudal fin.Two phrases of “autonomous propulsion” and “active steering” are responded to by this set of kinematics.The dolphin starts to yaw from a stationary state,and the swing of the caudal fin also contribute partly to the autonomous propulsion of the dolphin,shown in Fig.6.Since the roll and pitch fluctuations produced by the spanwise flexible swing are symmetrical,the moments in thexandzdirections are balanced with each other in one cycle.In this study,we put emphasis on three-dimensional straight and steering motion.

Fig 6.The composition of the yaw dolphin’s motion law.

Fig 7.The instantaneous time history of the turning parameters.

During the voluntary movement,the turning angle is constantly changing.We use the yaw matrixiRyto convert the dolphin’s initial coordinate systemCbto the instantaneous coordinate systemCiafter turning,whereθiis the instantaneous yaw angle during the steering motion.

Then in the entire movement process,several calculation parameters are applied to describe the self-yaw.The components of the instantaneous flow force along thexi,yi,andziaxes in the coordinate systemCican be solved by the compressive and shear stresses acting on the bionic dolphin

whereejis the component of the normal vector on the dolphin body surfacedS,τijis viscous stress tensor in the instantaneous yaw coordinate system.The macroscopic view of compressive stress is the differential pressure resistance of fish swimming,and the shear stress is the friction resistance.During the dolphins swimming cyclically,both the differential pressure resistance and friction resistance change periodically.It is judged that the change of the sign ofFxi(t) during the periodic swimming process can reflect its contribution to the thrustFT(t) and resistanceFR(t)

The power consumed by the vertical motion and spanwise motion of the dolphin body to overcome the action of the fluid during the autonomous yaw,can be expressed as

The instantaneous fluid torque component along theyaxis plays a key role in the autonomous yaw process and can be expressed as:

where Xriis the position vector of the fish body surface in theoxdydzd,During the yaw of the dolphin,the yaw moment generated by the soft swing of the caudal fin requires to be considered quantitatively.The principle is similar to the derivation in the Eq.(17),depending on whetherMy(t) is positive or negative,and it can be separated into driving torqueMD(t) and braking torqueMB(t),the instantaneous torque can be decomposed into:

After defining the driving and braking torque with above decomposition,we can reconstruct the equation of instantaneous net torque:

To analyze the superiority of the flexible caudal fin in self-yaw,the propulsion part of the stable turning process is analyzed according to the Froude propulsion efficiency equation [37],and the yaw efficiency can be similarly redefined:

To standardize the numerical solution of the virtual dolphin for different types and external dimensions,we define several important dimensionless energetic parameters based on the needs of quantitative research:

whereCFTandCMDare respectively dimensionless propulsion force and driving torque,andCPLis dimensionless energy loss.

3.Results and discusstio n

This section concentrates on the self-yaw law provided by the composite flexibility of the bionic dolphin.In addition to the travelling wave-characterized body acting as the propulsion system,the caudal fin with spanwise flexibility can provide effective driving force and torque in propulsion and steering.The feasibility of the bionic dolphin under self-yaw is evaluated by numerical methods,and the best simulation results are obtained by discussing and analyzing different performance situations.

3.1.Time history variations of performance parameters

The movement process of self-yaw can be decomposed into autonomous propulsion along the forward trajectory and active steering in thex-zplane around the center of mass.A virtual dolphin with the flexible caudal fin periodically swings through two steering states: explosive yaw and stable yaw.Fig.7 shows the instantaneous time history of the turning angular velocityωyand the turning angleβy.The kinematics performance parameters are specified by defining the swing frequencyω=5Hz,the maximum attack angleαmax=30 °,the chordwise phase differenceФ=90 ° and the spanwise phase differenceδ=20 °.In the process of starting,the dolphin’s turning angular velocityωyincreases explosively,reaching the maximum valueωmaxwithin a few swing cycles,andωygradually decreases in the following period,and the slowing slope becomes flat.Finally,the dynamic process of the dolphin converges to a stable yaw,and the turning angular velocityωyfluctuates at a fixed value.After comparing the two yaw states,we find that the amplitude ofωygradually converges from the explosive yaw in the early stage to the steady yaw.From the time history of the turning angleβy,the explosive yaw contributes most of the angular displacement from starting to stabilization,but the stable yaw can continuously output linearly varying angular displacements for the dolphin.

Along with the swing of the flexible body and the caudal fin,the dolphin inevitablyhas a propulsion effect during the yaw process,where the propulsion direction corresponds to the change of the yaw angle.Fig.8 shows the time history curve of the advancing velocityudand the lateral velocityulalong the forward trajectory.We can clearly see that the advancing velocityudkeeps increasing until it reaches an asymptotic average value,at which point the average thrust and the average drag balance.In this steady state,the steady-state swimming velocityUin one cycle remains constant.At the same time,even if the dolphin reaches a stable state,the velocityudfluctuates around the stable value.This conclusion is consistent with previous numerical tests of autonomous swimming [38,39],indicating that the flexible swing in the span direction does not affect the qualitative advancement of the dolphin.The changing law of the lateral velocity shows that the bionic dolphin exhibits periodic transitions,in which the lateral velocityulduring the period corresponding to the explosive yaw is slightly higher than the stable one,and the sideways translation is negligible.

Fig 8.The instantaneous time history of the propulsion parameters.

Fig 9.Time history of total propulsion force,lateral force,and turning moment.

For the yaw motion under the condition of autonomous three DoFs,the dynamic behavior of the dolphin is entirely dominated by complex hydrodynamics.The total propulsion force,lateral force,and turning moment are shown in Fig.9.In the steadystate cruise phase,the total propulsion force converges from the thrust-type force in the acceleration phase to a zero-average fluctuation pattern,while the lateral force has always maintained a zero-average fluctuation pattern.The yaw moment undergoes a process from the driving torque in the first angular acceleration stage to the braking torque in the angular deceleration stage and finally converges.The driving torque in the steady state is balanced with the braking torque,which is shown by Eq.(22).

Fig.10 shows that the entire motion trajectory of the virtual dolphin’s self-yaw based on thex-zplane.Since the lateral force of the dolphin has little effect on the overall motion in one cycle,the displacement in they-direction is negligible.To discuss the clockwise trajectory in detail,we first extract the instantaneousx-axis andz-axis coordinates of the motion from the original simulation results.The Kalman filter method is used as an effective fitting method to obtain the smoothing polynomialz=f(x),and then the radius [40]of the trajectory curvature can be obtained

Fig 10.The trajectory of the yaw performance.

Fig 11.Time history of the instantaneous yaw radius.

The time history curve of the instantaneous yaw radiusRis also calculated and shown in Fig.11.Due to the limited dynamic motion characteristics of the dolphin,the robotic dolphin is almost in the phase of marking time in the first few cycles.This part of the trajectory is an asymptotic short straight line affected by inertia,so the instantaneous yaw radiusRis almost infinite.With the dual acceleration of the bionic dolphin in propulsion and turning,the radius increases rapidly from the lowest value.When in the later stage of the burst of yaw,the instantaneous yaw radiusRslowly converges,but the amplitude ofRgradually increases in a single cycle.In the stable phase of propulsion and yaw,Ris maintained at a constant fluctuation,which conforms to the period of the flapping motion of the caudal fin.

3.2.Effect of chordwise and spanwise phrase on self-yaw performance

In this work,chordwise phase differenceФand spanwise phase differenceδwork as two critical parameters.We quantitatively study the self-yaw of the dolphin under different caudal fin shapes and reveal their effects on improving swimming performance and energy ratio for autonomous propulsion and active steering.

To explore the mapping relationship between the motion behavior of the caudal fin and the steady-state swimming velocityU,Fig.12 (a) shows the variation of steady-state swimming velocityUas functions ofФandδ.For the determined parameter combination ofФandδ,there is a unique convergence velocityUthat makes the fish body maintain steady-state swimming.On the contrary,to achieve a predetermined steady-state swimming velocity,there are multiple possible combinations of phase differences in two directions,that is,multiple motion behaviors.Specifically,we can see that as the spanwise phase differenceδgradually increases,the steady-state swimming velocityUalong the advancing direction linearly decreases,but the amplitude of the fluctuation is not apparent with the quantitative analysis using the chordwise phase differenceФas a parameter.When the value ofδis fixed,the maximum value of the steady-state swimming velocityUappears withФbetween 60 ° and 75 °,which is consistent with the regular test result of swing-wing from Anderson et al.[41]and Triantafyllou et al.[42],indirectly showing that the chordwise phase differenceФis dominant in the propulsion function.

To measure the difficulty of propulsion after the bionic prototype swims in a steady state,the average power consumption coefficientCPLand propulsion efficiencyηDof the dolphin in the steady state with respect toФandδare shown in Fig 12.It can be seen from Fig.12 (b) that as the chordwise phase differenceФincreases and the spanwise phase differenceδdecreases,theCPLmonotonically decreases and gradually stabilizes.This can be understood as a bionic prototype to achieve a predetermined steadystate travel speed.We can choose to increaseФand reduceδto reduce power consumption.The change law in propulsion efficiencyηDcalculated according to the Froude efficiency model is shown in Fig.12 (c).For a givenδ,ηDgradually increases with the rise ofФ,and the rate of rising grows slower.A better value ofηDcan be obtained whenФis close to 90 °.This optimal solution accords with our previous research conclusions [39]on self-propulsion,indicating that the propulsion law in yaw is not affected by spanwise flexibility.As the spanwise phase differenceδgradually increases,the propulsion efficiency witnesses a downward trend,which shows that the increase ofδof the caudal fin produces more energy loss,which is not conducive to the high propulsion efficiency of the bionic prototype.

Fig 12.Variations of propulsion performance parameters as functions of Ф and δ.

In addition to the research on propulsion in self-yaw,we also define the maximum instantaneous turning angular velocityωmax,steady-state turning angular velocityΩand steering efficiencyηTto explore the mapping relationship between motion behavior of caudal fin and yaw law.The maximum instantaneous turning angular velocityωmaxrepresents the explosive ability in the burst period during the self-yaw process,and the steady-state turning angular velocityΩrepresents the lasting ability to steer during the final steady-state yaw.Fig.13 (a) and (b) show the varying law ofωmax andΩas function ofФandδ.With the rise of theФand the decline ofδ,the curves ofωmaxandΩare similar,and both show a monotonous upward trend.Whenδis given,ωmaxis affected more sensitively by the change ofФthanΩ.The smaller the chord phase difference,the faster the rise speed.WhenФis 45 °,ωmaxhas a tendency to converge with the growth of the spanwise phase differenceδ.

Fig.13 (c) shows the varying curve of the steering efficiency with the phase differences in different directions.As the spanwise phase differenceδincreases and the chordwise phase differenceФdecreases,the steering efficiencyηTgradually decreases and tends to stabilize.A better value ofηTcan be obtained whenФis between 90 ° and 105 °.The yaw radiusRis an important indicator to measure the yaw capability,which depends on the balance between propulsion and steering performance.The smaller the yaw radius,the better the yaw effect.Fig.13 (d) shows the influence of the radius of steady-state yaw by varyingФandδ,and an intuitional inspection shows that asδincreases,the value of yaw radiusRgradually decreases,the declining yaw radius corresponds to the gradually increasing steering angular velocity in Fig.13 (a).WhenФis fixed to 45 °,the bionic dolphin can obtain the optimal yaw effect regardless of energy loss.

Fig 13.Variations of propulsion performance parameters as functions of Ф and δ.

In conclusion,the comparison of these cases shows that the degree of different flexibility of the caudal fin produces different self-yaw effects.Firstly,we combine Figs.13 (b) and 12 (a)(c) and reveal that the trend ofΩis ascending monotonously asδincreases.On the contrary,a dramatic decline appears in the curves of the steady-state swimming velocityUand propulsion efficiencyηD.The more flexible the caudal fin is in the span direction,the greater the proportion of the dolphin in the yaw compared to the propulsion effect.In terms of work efficiency,From Figs.12 (c) and 13 (c),whenФis between 90 ° and 105 °,the dolphin can obtain the best propulsion efficiency and steering efficiency.

3.3.Transient variation in pressure distribution and flow field

This work aims to analyze the impact of the pressure gradient covered by the surface of the dolphin on the hydrodynamic force in an all-around way,Fig.14 uses a mirroring method to display the pressure distribution profile within a period to avoid the emergence of dead ends.Firstly,there is always a high-pressure zone on the head of the dolphin.Whent=0T,the bionic dolphin is in the upper limit position,the lower edge of the body and the upper edge of the caudal fin intersect to form a high-pressure zone,while the low-pressure zone appears at the upper edge of the body and the lower edge of the caudal fin.From Fig.14,as the bionic dolphin moves from the upper limit position to the lower limit position,the high-pressure core on the upper edge of the body gradually moves to the tail due to the chordal travelling wave of the body,and a new high-pressure core is formed at the lower edge.The local pressure difference on both sides of the dolphin generates forward thrust.The pressure distribution law of the caudal fin is the same as that of the body in the chord direction.When the dolphin reaches the lower limit position,the resistance gradually decreases due to the wake formed by the low-pressure core falling off the rear edge of the caudal fin.The alternating law of the entire surface pressure within one cycle corresponds to the fluctuation law of the advancing velocityudwith an increasing sine wave in Fig.8,which is consistent with our previous conclusions about undulating fish bodies to achieve self-propelled swimming [11,12].

Fig 14.Evolution of pressure contours within one circle.

Fig 15.Vorticity contours within half one circle in three planes.

Along the spanwise direction,the high-pressure core of the caudal fin is formed at the tail stalk.We take the lower surface of the caudal fin displayed in the mirror for example.As time goes by,due to the phase difference between the left and right extreme positions of the caudal fin in the span direction,the edge of the left limit fin first forms a red high-pressure line att=1/8T,while the formation process at the right limit takes half a beat.Whent=2/8T,the high-pressure line at the left extreme edge gradually penetrates the midline of the caudal fin,and a gradual high-pressure area is formed on the entire caudal fin surface.The width and depth of the high-pressure distribution on the left side are obviously larger than that on the right side.Whent=3/8T,the highpressure distribution completely covers the lower surface of the entire caudal fin.On the contrary,the upper surface forms a lowpressure area.When the caudal fin returns to the lower limit position (t=4/8T),the upper surface of the tail fin repeats the work of the first half cycle.It can be concluded that,in general,the caudal fin with spanwise phase difference provides the dolphin with progressive pressure formation,distribution and shedding,and finally provides the driving torque for the dolphin’s yaw.

Fig.15 reveals the evolution of the instantaneous vorticity field of the dolphin during steady yaw.In order to facilitate the analysis of the details of spanwise flexibility,we intercept three planes along the spanwise direction of the caudal fin,namely the center line of the dolphin,the left and right limit of the caudal fin.It can be seen from Fig.15 that the dolphin in different planes produces a single-row vortex wake structure during self-yaw,and each swing alternately falls from two vortex streets in opposite directions.The red vortex corresponds to the counterclockwise direction,and the blue vortex corresponds to the clockwise direction.Because the caudal fin has a spanwise phase difference,the leftlimit caudal fin takes the lead to reach the upper limit position of swing,causing the vortex to fall off and become the first to generate thrust for the left part,thereby generating steering torque.The plane of the center line and the right limit at this moment shows that the vortex is still in the forming stage.Let’s taket=3/8Tas an example,the red vortex at the left limit has begun to dissipate,and the blue vortex is being formed.At this time,the red vortex at the center line has just fallen off,and the red vortex at the right limit is still in the formative stage.This process of “formation-growthshedding” of wake vortices with phase difference can be called“vortex dislocation” [43].Therefore,the progressive vortex shedding sequence along the span can produce a gradual and asymmetrical thrust effect on the trailing edge of the caudal fin,which is different from the traditional linear propulsion vortex law [38]and indirectly explains the movement mechanism of the dolphin’s yaw.

We compare the vortex fields of the three planes simultaneously and find that from the left to the right of the caudal fin,the vortices gradually accumulate and become clearer.Firstly,because the vortex of the right limit finally falls off and dissipates,there is a shedding time difference with the vortex structure of the left limit.Secondly,because the dolphin is in a yaw motion,the right extreme caudal fin is on the inside of the yaw motion,and the outer vortex can accumulate inward as the dolphin turns and dissipate together.The bionic dolphin makes full use of the wake area at the trailing edge of the caudal fin to absorb energy from the jet to the inside and rear of the orbit,and produces strong autonomous propulsion and stable active steering.The curved single-row Karman vortex street form gradually formed on the inner side of the trajectory verifies the motion mechanism of self-yaw.

4.Conclusion

In this article,we rely on the flexibility of a conventional dolphin to simulate a novel type of yaw motion numerically.We quantitatively study the hydrodynamic performance of self-yaw motion by controlling the dolphin’s chordwise flexible body and spanwise flexible caudal fin,and systematically analyze the yaw laws under different calculation conditions.The most notable findings of our research are as follows:

(1) Self-yaw motion can be defined as a combination of autonomous propulsion and active steering.The chordwise phase differenceФbetween the body and the caudal fin helps the bionic dolphin achieve a stable advance after starting and accelerating in the propulsion direction.The spanwise phase differenceδhas a more prominent effect on the steering function.In the early stage of yaw,explosive steering occurs due to a large steering torque,and then the steering angular velocity gradually decreases and tends to converge.

(2) The different flexibility of the caudal fin produces different self-yaw effects.The more flexible the caudal fin in the spanwise direction,the greater the proportion of the dolphin in yaw compared to the propulsion effect.When the spanwise phase differenceδis fixed and the chordwise phase differenceФis 45 °,the maximum turning angular velocity and the minimum yaw radiusRcan be obtained.In terms of work efficiency,whenФis between 90 ° and 105 °,the dolphin can obtain the best propulsion efficiency and steering efficiency.

(3) Describing the pressure distribution as the caudal fin surface’s spanwise direction of and the body’s chordwise direction in a cycle can vividly explain the dolphin’s swinging laws and kinematics.The 3D structure of the wake mainly depends on the position of the caudal fin.It is easier to accumulate vorticity if it is on the inner side of the trajectory(right limit position).The progressive vortex shedding sequence along the spanwise direction can produce a gradual and asymmetrical thrust effect on the trailing edge of the caudal fin.This finding helps to explain that the best way to increase the yaw rate is to increase the spanwise phase differenceδ.

In conclusion,different from the linear propulsion performance achieved by the conventional rigid caudal fin,the spanwise flexible caudal fin expands the linear forward motion into in-plane yaw motion.The rational use of flexible tail fins can help us better study the multi-dimensional manipulation mechanism of the dolphin,and provide theoretical support for the design and production of bionic dolphins with multiple DoFs.Based on this work,follow-up research can focus on the realization of precise yaw and the improvement of yaw efficiency.In the future,we will coordinate the work of the caudal and pectoral fins and try to realize the maneuverability of dolphins with multiple DoFs.

DeclarationofCompetingInterest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported by National Natural Science Foundation of China [grant number 51875101 ],and State Key Laboratory of Robotics and System (HIT) [grant number SKLRS-2018-KF-11 ].The authors are greatly grateful to the referees for their helpful comments and suggestions,which help improve this paper.