Zhihn Li ,Dn Xi ,*,Xufeng Zhou ,Jio Co ,Weishn Chen ,Xingsong Wng

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

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

Keywords:Self-rolling dolphin Parameter differential mode Numerical simulation Hydrodynamic performance Flow structures

ABSTRACT The novel autonomous rolling performance is realized by the pair of pectoral fins of a threedimensional(3-D) bionic dolphin in this paper numerically.3-D Navier-Stokes equations are employed to simulate the viscous fluid around the bionic dolphin.The effect of self-rolling manoeuvrability is explored using the dynamic mesh technology and user-defined function (UDF).By varying the parameter ratios,the interaction of flexible pectoral fins is divided into two motion modes,amplitude differential and frequency differential mode.As the primary driving source,the differential motion of a pair of pectoral fins can effectively provide the rolling torque,and the trajectory of the entire rolling process is approximately the clockwise spiral.The results demonstrate that the rolling angular velocity and driving torque in the steady state can be improved by increasing parameter ratios,and the rolling efficiency can reach the maximum under the optimal parameter ratio.Meanwhile,different parameter ratios do not affect the rolling radius of the self-rolling dolphin.The evolution process around the pair of pectoral fins is shown by the flow structures in self-rolling swimming,reasonably revealing that self-rolling locomotion is produced by the pressure and wake vortices surrounding the pair of pectoral fins,and the wake structures depend primarily on the variation of parameter ratio.It properly turns out that the application of the pair of pectoral fins can realize the self-rolling performance through parameter differential modes.

1.Introduction

As a vast treasure house of natural resources,the ocean with excellent development and utilization potential has recently attracted widespread attention in different fields,especially the research of automatic underwater vehicles (AUV) [1-4].The typical representative of AUV is the fish-like robot,which has flexible maneuverability,high-speed performance and low energy consumption.These remarkable abilities inspire the hydrodynamics study,and the numerous simulations of underwater dolphin are similar to the flow past bluff body reported by Arif [5-8].The accurate swimming mechanics and wake flow structure details have been researched in recent years [9-14].In addition to the self-rotation locomotion,the vast majority of existing research of fish-like locomotion currently focused on the self-propulsion and turning motion [15-20].The natural phenomenon that the dolphin rolls flexibly with physical coordination during the underwater process provides the inspiration for the study.Through the comparison with precedents of the normal swimming locomotion,the study of selfrolling motion for fish-like robots is still limited,and the hydrodynamics behind the self-rolling locomotion needs more systematic research.

Numerous control methods and experimental studies of fishlike robots were carried out and reported,showing that the caudal and pectoral fins are the most crucial maneuvering systems.Review papers [21]presented the first bionic robotic fish having the capacity to swim with caudal fin.Liao et al.[22]proposed that a dual caudal-fin system can help the miniature robotic fish achieve more stable and efficient propulsion and steering performance.Suebsaiprom and Lin [23]performed experiments using two Degrees of Freedom(DoFs) buoyancy mechanism and kinematic models to achieve complicated 3-D locomotion,including rolling stability.Morgansen et al.[24]designed 3-D robotic fish with equations of motion and briefly verified the correctness of the model through the turning motion.Yu et al.[25-27]provided an SMOcentered method for the multi-joint robotic dolphin to realize different bio-inspired actions,including gliding,pitching,even leaping out of water.These studies deepened the bionic fish’s understanding of the different motion laws from controller design and dynamic modelling.However,more flexible mechanisms still require further systematic investigations,such as the self-rolling method using pectoral or caudal fins.

In terms of numerical simulations of fish-like swimming,some innovative results about the hydrodynamics of fish-like motion have been recorded in preview studies.A bionic fish with carangiform mode was employed by Borazjani [28]to demonstrate the hydrodynamics of undulation motion with the variation of Strouhal and Reynolds numbers.Shao [29]applied an immersive boundary method to study the heave and pitch motion of different flapping wings numerically with different aspect ratios and the related wake structures.Xia [30]revealed that the propulsive efficiency of the fish-like robot could be optimized by head swing motion to some extent.Carling et al.[31]conducted the quantitative numerical simulations based on self-propulsion of the anguilliform swimming.For turning motion,Xu [18]proposed to use overlapping grids to realize the numerical simulation of the self-yaw movement by a pair of rigid pectoral fins.Yeo [32]used the hybrid meshfree-Cartesian grid to simulate a sharp clockwise turn through sweeping the tail of two-dimensional fish.Feng [33]concentrated on the hydrodynamics of C-turn maneuvering based on numerical simulation of a tuna-like body,which is in good agreement with the natural experimental results.These studies paid more attention to the different swimming patterns realized by different parts of the fish-like robot quantitatively and revealed the quantitative law of motion through the systematic parameter research.However,the specific investigation of self-rotation with pectoral fins is not reported in the literature.

These above experimental and numerical studies have generated significant results in the field of the biomimetic fish,and have provided a wealth of aquatic movement modes in the face of complex underwater environments.The asymmetrical movement of the pectoral fins on both sides and the flexible swing of the fish body prepare a reference for the kinematics of this study.Additionally,systematic governing parameters,such as hydrodynamic thrust and propulsive efficiency,bring inspiration for the quantitative numerical study of self-rotation.Compared with the caudal fin that can provide efficient propulsion to the dolphin,we make the most of the idle pectoral fins to adjust the dolphin’s posture flexibly.On one hand,the importance of this study lies in the fact that quantitative self-rotation plays an active role in accurate detection and obstacle avoidance.On the other hand,the underwater self-rotation provides the necessary theoretical basis for the future study of porpoising and rolling dolphins.

In this study,we extend linear forward motion to in-plane rotational motion and realize differential motion modes relying on symmetrical pectoral fins.With the objective of exploring the hydrodynamics and flow features of this little-known swimming mode,the numerical simulation of autonomous pectoral fins is conducted based on a fluid-structure interaction (FSI) method.Further research also will be studied: (1) The influence of varied parameters on the rotation of dolphin based on two motion modes,(2) the effects of the self-rolling locomotion on the driving torque,the rolling efficiency and the rolling radius,and (3) the impact of the self-rolling locomotion on evolving the pressure distribution and flow structure surrounding the dolphin.

2.Material and methods

2.1.Computational model and domain

Fig.1.Physical model and coordinate system of dolphin and pectoral fins.

Fig.2.physical model of the computational domain of dolphin.

In this study,a bionic prototype employed as the virtual swimmer is closely modelled with the reference of the shape and movement of an actual dolphin.The curve fitting method is used to sketch the smooth profile of the bionic dolphin,including caudal and pectoral fins.The study’s emphasis is on the coordination of body and pectoral fins,shown in Fig.1.The coordinate system of the dolphinoxdydzdis adopted as usual,while the coordinate systemsoxplyplzplandoxpryprzprdefined for the left and the right pectoral fins are applied to describe the configuration and motion.Especially,the position of the pectoral fins is defined on the planeoxdyd.Inoxdydzd,the shapes of two pectoral fins are symmetric separately regarding the planeoxplypland the planeoxprypr.The length of the dolphin robot isL,the length from the origin to the tip of pectoral fins is 0.25L,the angle betweenoxprandoxdisβ=30 °.

Commonly,the live dolphin’s complex kinematics is divided into two essential components: locomotion based on Media and/or Paired Fin (MPF) and locomotion based on Body and/or Caudal Fin(BCF).To investigate the feasibility of the self-rolling motion intently,the active locomotion of body and the caudal fin is not under consideration,and more attention is paid to the flexible undulation of pectoral fins.In terms of the shape parameters of the pectoral fins,the span lengthb=0.15L,the chord lengthc=0.125L,and the thickness of finsh=0.1care shown in Fig.1.Additionally,the cross-section of the pectoral fins is described as an oval.

Fig.2 depicts the physical details of the swimming domain,which is a 5L× 5L× 3Lcubic tank filled with water.The choice of the computing size is based on the computing cost and accuracy requirements.A suitable rolling space can effectively grasp the flow field structure and show the law of motion.The initial position of the bionic dolphin is defined according to the possible self-rolling trajectory.It is evident that the head of the dolphin is placed at 1.5Lfrom outlet plane in thez-axis direction.Along thex-axis,the center of the dolphin is placed at 1.5Lfrom the negative direction,while in they-axis direction,the distance between dolphin and positive y-direction is also 1.5L.

Fig.3.Different views of the virtual dolphin meshed with triangular elements.

The swing of the pectoral fins is a complicated deformation process with flexible boundaries,so we apply triangular elements to mesh the bionic dolphin in Fig.3.This meshing method has been widely used in previous studies [34-36],including the selfpropelled swimming of fish-like robots and the active deformation of various fins.

2.2.Kinematics

Based on biological observations and inspiration from the motion of caudal fin,the kinematical equations describing the lateral undulations of the pectoral fins are thus given separately as follows:

In the above equations,xplandxprare the axial directions measured alongoxplyplandoxpryprfrom the origin to the tip of the pectoral fins,ypl(xpl,t) andypr(xpr,t) represent the instantaneous lateral displacement of the left and right pectoral fins,A(xpl) andA(xpr)are the functions of amplitude,ωis the wave frequency,kis the wave number corresponding to a wave length,φis the phase difference.In this study,the basic swing of the pair of pectoral fins is the reverse differential method,that is,the phase differenceφof both pectoral fins is 180 °.In order to obtain better rolling performance,the direction of the rolling torque generated by two pectoral fins is always the same.The length of pectoral fins is usually assumed to be unchanged during the wavy motion.With the amplitudeA(x) defined by a quadratic curve of the form,it is not difficult to simulate the undulation of pectoral fins.

whereC0,C1andC2are the envelope amplitude coefficients.For example,a set of parameters to describe these motions can be listed asC0=0,C1=0.04 andC2=0.4/Lp,whereLp(0.25L) is the length from the origin to the tip of one pectoral fin as mentioned.

For self-rolling bionic dolphin,the further differential motion of both pectoral fins is mainly employed to realize better rolling performance by assuming that the effect of its center of mass on pitching and turning motion is weakened.The differential motion of undulating fins is divided into two modes: frequency differential and amplitude differential mode.In this study,we introduce the definition of frequency ratioαf=ωpl/ωprand amplitude ratioαA=A(xpl)/A(xpr)to explain the parameter differential motions,while the basic frequencyωpr=2 Hz and basic amplitudeA(xpr)=0.025Lare fixed for the right pectoral fin.The undulating movement of the basic(right) pectoral fin in one fin-swing cycle is shown in Fig.4.Additionally,iRbis expressed to indicate the rotation matrix,which is employed to convert from the initial coordinate frameCbto instant coordinate frameCireciprocally,θiis the instantaneous rolling angle during the rolling motion.

2.3.The governing equations and boundary conditions

Based on differential motion modes,the underwater self-rolling motion is studied as an incompressible flow problem,and there is a 3-D incompressible flow enclosing the bionic dolphin.To simulate the viscous fluid around the bionic dolphin,we employ the related governing equations,which are 3-D Navier-Stokes equations given by

where ?is the gradient operator,uis velocity component of the fluid,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,FandMZrepresent the fluid force and torque applied on the fish body separately,mis the robot mass,¨x is the swimming acceleration along the related axis,˙φcis the angular velocity of rolling axis and ¨φcis the angular acceleration,IZrepresents the inertial moment about the rolling axis.The fluid forceFand torqueMZare 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,ezis the unit vector along thez-direction.

For the numerical simulation of the virtual dolphin,the entire self-rolling motion contains two kinds of boundary conditions: the moving surface of the dolphin’s body and the boundary conditions of inflow and outflow in the computational domain.To simulate the kinematics of the underwater dolphin and the flow field surrounding the body,we adopt a no-slip boundary condition for the mobile interface with the equivalence of the dolphin velocityudand flow velocityu.In terms of the boundary conditions of the computational domain,the upstream boundary is zero velocity and zero pressure gradient,while the velocity and pressure imposed on the downstream boundary and other boundaries are zero gradient.

2.4.Numerical method

In our previous works [37-39],we focused on the numerical simulation of the self-propulsion of the fish-like robot and verified the 3-DoFs FSI method in detail.The simulation process is carried out on computational fluid dynamic (CFD) solver ANSYS Fluent.We employ the finite volume method to discretize the Navier-Stokes equations.In the spatial discretization,the setup of gradient interpolation is the Green-Gauss cell-based method,and the secondorder scheme is used for the pressure interpolation of each governing equations.Additionally,the second-order upwind scheme is adopted for convection terms,and the second-order implicit scheme is applied for the transient formation.We employ the SIMPLE algorithm to obtain the pressure-velocity coupling of the continuity equation.In the numerical calculation of fluid-structure coupling,the embedded DEFINE_GRID_MOTION macro is connected to the main code of the solver to realize the required flexible deformation motion.The user-defined function and dynamic mesh technology are applied for rigid movement.The mesh methods,including smoothing and remeshing method are used to capture the swimming performance and regenerate the mesh grids of the simulated body.Considering the accuracy of the results and computational cost,we control the number of grids at 5.3× 106with time stepdt=0.005T,Tis the cycle of pectoral fins beating water,which is related toω.It is important to note that depending on the frequency of motion of pectoral fins,we need to select appreciate values of time step.For each step,the numerical computation process of the instant flow-structure scheme is as follows:

Fig.4.Mid-line motions of basic pectoral fin in one beating period.

(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 grid 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.andrespectively represent the solution based on the front and back under-relaxation at the time iterationn+1.

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

Fig.5.Grid sensitivity test for virtual dolphin.

2.5.Grid sensitivity test and numerical validation

It is essential to select appropriate gird refinement mode based on model size and simulation environment to achieve accurate simulation results [40].Three successively grids,including 1.5(coarse),5.3 (standard) and 13.4 (fine) millions of grids,are used for a grid sensitivity study.Their corresponding uniform edge lengths are 0.016L,0.008Land 0.004Lrespectively.The grid sensitivity study is conducted with the frequency ratio of the paired pectoral fins unchanged,and the amplitude ratio controlled within the range of 2-3.In order to make the simulation results have reference value,the domain size,time step and boundary conditions are unified.Fig.5 shows the variation of the rotation center along thex-axis with the amplitude ratio for three different grid cells.It can be seen from the Fig.5 that both the nominal and fine grids give very close reasonable results,while the coarse grid cannot solve the flow problem well.We then conclude that the results obtained on the nominal grids are not sensitive to further mesh refinement.Therefore,around the bionic dolphin,refined uniform meshes with a grid size of 0.008Lare employed to discretize the working area.

To verify that the method can deal with the boundary problem of 3-D 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.Fig.6 shows the calculated time history curve of the force along the vibration direction of the cylinder,including the resultant force (solid line),the differential component (dotted line),and the viscous component (dashed line),as well as the results from the literature [41].It can be seen from Fig.6 that the calculated results (in red) are in good agreement with the results (in black) reported by Dutsch [41].Fig.7 shows the instantaneous pressure contours and vorticity contours of the surrounding fluid induced by the oscillatory cylinder locomotion with different phase angles,reflecting the formation and shedding of vortex streets in the process of cylindrical vibration.They are respectively dominated by two vortex streets with opposite spiral directions.The numerical methods presented in this part can deal with moving boundary problems like cylindrical harmonic vibration.

Fig.6.Time history of the net force and its pressure and viscous components from the present computations (red lines) compared with the results of Dutsch et al.[41].(black lines).

Fig.7.Pressure and vorticity contours at four different phase angles (γ=2 πft): (a)γ=0 °;(b) γ=96 °;(c) γ=192 °;(d) γ=288 ° using oscillatory cylinders case.

2.6.Calculation of performance parameters

With prescribed motions,the self-rolling bionic dolphin is undulated by the pectoral fins,and the dolphin body rolls from a still state.During the locomotion process,several computational parameters are applied to describe the self-rotation locomotion.The component of the instantaneous flow force along thex-axis andy-axis and instant fluid torque along thez-axis can be solved by pressure and viscous stress acting on the bionic dolphin as

whereXriis the position vector of the fish body surface in theoxdydzd,ejis the component of the normal vector on the dolphin body surfacedS,τijis viscous stress tensor.During the rotation of the fish body,the torque generated by the two pectoral fins needs to be considered quantitatively,depending on whetherMZ(t)is positive or negative,it can be separated as driving torqueMD(t)and braking torqueMB(t),the instant torque can be decomposed as follow:

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

The moment of inertia of the dolphin around the center of mass is calculated to realize the fish-shaped rotating motion.According to the situation that the fish body changes transiently in motion,we assume thatr(x,t) is the distance from a certain point of the rigid segment of the fish body to thez-axis,s(x,t) is the distance from a certain point of the flexible segment to thez-axis.The moment of inertia of the dolphin is a function of time expressed as follow:

wherexis the measure along thex-axis direction,lhis the distance from the root of one pectoral fin to thez-axis,lis the distance from the tip of one pectoral fin to thez-axis,ρis the density of dolphin,Ar(x) andAf(x) represent the cross-sectional function along thex-axis direction of the fish body.

For the convenience of research,the dolphin’s gravity and buoyancy that balance each other in they-direction are ignored.In addition to the consumed power in self-rolling motion,the power loss produced by the lateral motion along thexandy-axis needs to be considered,and the motion along thez-axis is too less to be ignored.The equation of power loss can be defined as:

In order to make different prototypes of robotic dolphin satisfy the conclusion of the numerical solution without consideration of external size separately,it is necessary to non-dimensionalize kinematic parameters of autonomous rolling.

whereCMDis the dimensionless driving torque,andCPLis the dimensionless lateral energetic consumption.

3.Results and discussion

This paper concentrates on a pair of pectoral fins acting as the rolling system to drive the bionic dolphin numerically,while the forward,turning and pitching motion are out of consideration.To evaluate the feasibility of this novel kinematics,we will analyze and discuss the simulation results based on two differential motion modes,frequency differential and amplitude differential mode.

3.1.Time history variations based on two motion modes

It is evident that the dolphin experiences a complicated kinematic process as the pectoral fins swing periodically and spontaneously.In this section,the amplitude differential mode is specialized with amplitude ratioαA=3,and frequency differential mode is specialized with frequency ratioαf=3.Both two differential motion modes can be divided into three states,starting at rest,accelerating gradually,and rolling at a steady state.During the process of starting,the pectoral fins undergo a row of waves along the base of the pectoral fins to the tip,however,due to the delayed action resulting from the limited effect of the initial swing,the movement of the dolphin is described as slightly shaking which is hard to observe.When the dolphin is in the free acceleration state,the rolling angular velocityωrgrows upward in steps from near zero,and the growth slope gradually becomes flat in the later period.After that,the dynamic processes converge to a steady state swimming.The time history of the rolling angular velocityωrand the rolling angleθrbased on the different modes is shown in Figs.8 and 9.

In the detailed view of movement law of the pair of pectoral fins in a cycle with amplitude ratioαA=3,which is shown in Fig.8 (a),the angular velocityωrshows a sinusoidal periodic change,while in Fig.8 (b) three peak amplitudes correspond to the frequency ratioαf=3.Closer inspection reveals that the first two peaks are almost the same size,while the third peak amplitude is smaller.This is due to the reason that the left pectoral fin is in the limit position three times,having more incredible water hitting efficiency,while the right pectoral fin is in the middle of the limit position and the equilibrium position for the first two peaks.The last peak is smaller because the right pectoral fin is in the process of returning,which contributes braking torque instead of driving torque.

Fig.8.Time history of rolling angular velocity (a) Amplitude differential mode(αA=3 &αf=1) (b) Frequency differential mode (αf=3 &αA=1).

Fig.9.Time history of the instantaneous rolling angle.

The instantaneous angle curve is displayed in Fig.9,and we can find that the rolling angleθrexperiences a swing back in the opposite direction during the start-up phase.After several cycles of fluctuations,the steady state is reached.Therefore,as time progresses,the rolling angle increases slowly at first and then gradually increases in a linear manner.The difference between two rolling modes is that the growth fluctuation of the angle under the frequency differential method is more stable than that of the amplitude differential method.By contrast,the angular acceleration of amplitude differential mode (αA=3) is larger than that of frequency differential mode (αf=3).

Fig.10.The trajectory of the self-rolling performance based on amplitude differential mode(αA=3).

Finally,these two motion modes can get a similar clockwise spiral path.Fig.10 visually shows the reference trajectory based on amplitude differential mode(αA=3) through combining the rolling angle along thez-axis and displacement in thex-yplane.The swimming pattern started from the origin with asymmetrical beating frequency of pectoral fins.Instead of rolling at origin,the bionic dolphin regulates its swimming position and rolls along the spiral track clockwise,lateral displacement along thex-axis andy-axis need to be considered.The values of the rolling angleθand current timetof the dolphin at five extreme positions are marked in Fig.10.We compare the trajectory with the law of the hour hand,whose turning angleδintroduced on a clock,and correspond the highest position of the dolphin(t=33T) to the zero on the clock.We can find that the rolling angle of dolphin can be described byθ=δ+β,βis defined in Fig.1,suggesting that the left pectoral fin swings almost along the lateral direction of the rolling dolphin.Clear observation reveals that the left pectoral fin acts as the main driver moving along the trajectory,and the differential law of the pair of pectoral fins is the cause of the self-rolling motion.In the following parts,the rotation trajectories of different cases based on some curve parameters,including rolling radiusRand rolling centerOR(xr,yr) are analyzed systematically.

To evidently study the lateral displacement accompanying the rotation showed in Fig.10,We depict the instantaneous velocity throughx-axis andy-axis respectively during the rolling motion with amplitude differential mode,shown in Fig.11.Both the instantaneous velocity curvesVxandVycan be described approximately as a sinusoidal change around zero,which can be divided into several sinusoidal periodic fluctuations for one cycle.After reaching the steady state,both instantaneous velocity curves have almost the same amplitude.The difference is that there is a 90-degree phase difference between Fig.10 (a) and (b).It is important to note that the amplitude of a single sinusoidal cycle is also changing regularly,confirmed in both directions.When the instantaneous speed reaches the maximum or minimum value,the velocity amplitude gradually converges,which can be understood that in the lateral movement,when the instantaneous speed is higher,the movement is more stable,otherwise the more violent.The frequency differential mode has similar curves in terms of the instantaneous lateral velocity along with two directions.Additionally,by comparing the Figs.8 (a) and 11,we find that the dolphin’s lateral movement process follows the self-rotation to stabilize at almost the same time.

Fig.11.Time history of instantaneous velocity (a) The instantaneous velocity along the x-direction (b) The instantaneous velocity along the y-direction.

3.2.Effect of different amplitude ratios

The magnitude of the amplitude is usually employed for the analysis of self-propelling dolphin.Similarly,we study the kinematics of the self-rolling mode of bionic dolphin with the different values of amplitude ratio.Firstly,we give the steady rolling angular velocityΩand the driving torque coefficientCMDcorresponding to the different amplitude ratios when controlling frequency ratioαf=1.From Fig.12 (a),the steady rolling angular velocityΩwitnesses a monotonous upward trend with the rise ofαA,andCMDalso rises,but the rate of rise grows faster.It can be observed that it is sufficient for the bionic dolphin to increase the rolling angular speed and the driving torque by appropriately increasing the amplitude ratio of pair of pectoral fins properly.

Fig.12 (b) illustrates the influence of the amplitude ratio in the view of energetic consumption.The lateral movement belongs to useless power loss during the rotation.The lateral energy consumption coefficientCPLC based on the variation of the amplitude ratio is similar to the growth curve of the driving torque coeffi-cient.In terms of rolling efficiency,the trend ofηis ascending slowly until a maximum value,and afterwards a dramatic decline appears.The comparison of cases with different amplitude ratios shows that when the amplitude ratio reaches 2.5,the efficiency of rotation can reach the maximum.

Fig.12.Variations of performance parameters as functions of αA (a)Relation between U,CMD and αA (b)Relation between CPL,η and αA (c)Relation between xr,yr,RandαA.

From Fig.10,during the self-rolling performance,the virtual dolphin will rotate around a new center of rotation when reaching the stable rolling maneuver,which is defined asOR(xr,yr).To investigate the spiral trajectory in detail,we fit thex-axis andy-axis coordinates of the motion extracted from the original simulation results based on Kalman filtering method to obtain a smoothing polynomialy=f(x),and then the radius of curvature of trajectory[16]can be computed by

The filtered trajectory data is subjected to circle fitting using the principle of least squares,and the coordinates of the center of the circle are obtained.The variation of the radiusRand the center of rotation coordinates due to different values of amplitude ratio is shown in Fig.12 (c).AsαAincreases,xrincreases until it gradually approaches a steady value,while the change ofyrandRfluctuates in a small range irregularly,especially the value of radiusR,which basically remains invariable.It turns out that different amplitude ratios have no effect on the rolling radius of the self-rolling dolphin,and the change ofxris mainly caused by the inertial motion of the bionic dolphin driven by the left pectoral fin during the start and acceleration phases.

3.3.Effect of different frequency ratios

To analyze the influence of the frequency differential mode on the self-rolling locomotion,we define that the amplitude ratioαAis invariable (αA=1).Fig.13 (a) and (b) show the variations of the steady rolling angular velocityΩ,the driving torque coefficientCMD,the power coefficientCPLand rolling efficiency as functions ofαf.Asαfrises,theΩincreases linearly,whileCMDandCPLrise with growth magnitude rising gradually.There is not a great deal of difference between the relation ofηwithαfand the relation ofηwithαA,except that after reaching the maximum efficiency,the reduction magnitude is declined slowly.It reasonably turns out that the bionic dolphin rolling with an appropriate frequency ratio is beneficial to improve the rolling efficiency relatively.

The trajectory parameters of rolling motion are plotted in Fig.13 (c).The variation ofxrexperiences a steady upward tendency with a gradually reduced growth,while the variation ofyrremains constant despite a little fluctuation.It is indicated that the rolling radiusRstabilize approximately 0.25 m,which is marginally larger than that of amplitude differential mode.

3.4.3-D flow structures

This work aims to study the motion mechanism of the swing of the left and right pectoral fins of the bionic dolphin.Compared with the traditional autonomous swimming achieved by the kinematics of caudal fin,this work emphasizes the feasibility of a pair of pectoral fins to coordinate movement with the combination of the pressure distribution and flow structure around the dolphin’s pectoral fins.Two different motion modes,frequency differential and amplitude differential mode are discussed separately.The pressure distribution contours generated by the rolling dolphin based on amplitude differential mode are shown in Fig.14.It is obvious that the pressure range of the left pectoral fin is more extensive than that of the right pectoral fin for amplitude differential mode.We take the left pectoral fin for example,when the left pectoral fin is in the left extreme position shown in Fig.14 (a),there is a high-pressure core and a low-pressure core separately generating on the two sides of the pectoral fin in the form of a semicircle,leading to high instantaneous rolling efficiency.As the fin moves to the right extreme position,the high-pressure core on both right sides continually travels to the tip of the fin,and sheds.After that,the new high-pressure core is formed on the root of the fin,while the low-pressure core is transferred to another side.The pressure cores are displayed on opposite sides in the right limit position,shown in Fig.14 (d).At this time,the bionic dolphin increases the braking torque gradually based on the imbalance of the opposite pressure difference around the pectoral fins,which hinders the rolling process.As the fin goes back to its initial extreme location,the pressure distribution mirrors the previous half circle.It can be concluded that the dolphin experiences a torque change in sinusoidal form,including the driving torque during the first half cycle,while the braking torque is created by return stroke during the second half cycle,which is visually displayed in Fig.14.

Fig.13.Variations of performance parameters as functions of αf (a) Relation between U,CMD and αf (b) Relation between CPL,η and αf (c) Relation between xr,yr,R and αf.

Fig.14.Variations of performance parameters as functions of αA (a) (n+0) T (b) (n+1/4) T (c) (n+2/5) T (d) (n+1/2) T (e) (n+3/4) T (f) (n+4/5) T.

Fig.15 plots the pressure distribution contours based on frequency differential mode.The left pectoral fin’s swing frequency is three times that of the right pectoral fin,so the pressure area on the left fin is larger than the right pectoral fin.Compared with amplitude differential mode,the left pectoral fin experiences six extreme positions,while the right one experiences twice,so the swing of the entire pair of pectoral fins has multiple combinations.Whent=(n+0)T,(n+4/12)Tand (n+8/12)T,the left pectoral fin is located in the left extreme position,leading the maximum driving torque in one swing.Whent=(n+2/12)T,(n+6/12)Tand (n+10/12)T,the left pectoral fin is located in the right extreme position,leading the maximum braking torque.Compared with the left pectoral fin,the right pectoral fin has less influence on the rolling motion of the bionic dolphin,and it has the same swing motion as the right one based on amplitude differential mode.It is important to note that the driving torque of the situation that both pectoral fins located in the left extreme position,shown in Fig.15 (a),is larger than that of the situation that one is in left extreme position while another is in the mid-point or right extreme position.

Fig.15.Variations of performance parameters as functions of αf (a) (n+0) T (n+1/12) T (b) (n+1/6) T (n+1/4) T (e) (n+1/3) T (f) (n+1/2) T (g) (n+2/3) T (h)(n+3/4) T (n+5/6) T (j) (n+1) T.

Fig.16.Vorticity contours based on different frequency ratios.(a) αf=2 (b) αf=3.

In order to illustrate the locomotion principle of the autonomous rotation of the bionic dolphin,it is crucial to explore the flow field in the dolphin’s self-rotation process from the vorticity contours formed by the swing of pectoral fins.Based on frequency differential mode,the vorticity contours after reaching the steady state are plotted in Fig.16,where Fig.16 (a) and (b) are severally the vorticity distribution in thex-yslice with frequency ratioαf=2 andαf=3 for pair of pectoral fins.In Fig.16 (a),the wake structure is indistinctly presented as the single row vortex structure,and each swing alternately falls off two vortex streets with opposite directions.The time of fall usually occurs when the fin swings to change the direction.As the frequency of the left pectoral fin increases,vortex shedding mode is changed from single to double-row,the path of vortex shedding follows the movement trajectory of the bionic dolphin shown in Fig.16 (b).For the right pectoral fins in both of Fig.16 (a) and (b),the shedding direction of the vortex is from the tip of fins to the root,because the shedding direction is affected by the lateral movement.We also find that as the swing frequency of the left pectoral fin increases,the flow field changes more drastically,resulting in greater vortex intensity,which positively affects the growth of driving torque.

Fig.17 depicts the vorticity contours in thex-yslice based on amplitude differential mode,where Fig.17 (a) and (b) are severally the vorticity distribution with amplitude ratioαA=2 andαA=3 for pair of pectoral fins.In Fig.17 (a),the wake pattern is featured as a single-row wake.Owing to the reason that one cycle of the swing of pectoral fins has relatively less influence on the movement of dolphins,the vortex street shedding in the previous cycle will be covered by the newly generated vortex street.With a relatively large amplitude ratio,the wake structure plotted in Fig.17 (b) is presented as a double-row wake.In the vortex structure arranged along the trajectory,the longitudinal distance between the two vortex streets with the same spiral direction on the inner side is smaller than that on the outer side,this is due to the particularity of the self-rolling movement.From the comparison of Fig.17 (a) and (b),we find that as the swing amplitude of the left pectoral fin increases,the rolling angular velocity increases,and the mutual compression between the shedding vortices in one cycle decreases,so the rise in the driving torque is more remarkable.Through the comparison of Figs.16 and 17,when the frequency ratio or amplitude ratio is high,the left pectoral fin will display a double-row wake structure,which can be explained by the equation ofStnumber [42].In the forward motion process with BCF,theStnumber is usually defined asSt=fA/U,wherefis swing frequency,Ais maximum amplitude andUis steady propulsion speed.Similarly,we also can employ this formula to explain the self-rotation motion withSt=αAαf/U,whereUis the steady lateral velocity along the trajectory.Typically depending on highSt,the vortex street splits transversely,and the double-row wake appears.

Fig.17.Vorticity contours based on different amplitude ratios.(a) αA=2 (b) αA=3.

4.Conclusion

In this paper,the bionic dolphin is employed to simulate a novel self-rolling locomotion numerically.We systematically study the hydrodynamic performance of the rolling motion by controlling the ups and downs of the left and right pectoral fins with different frequency ratios and amplitude ratios.The necessary kinematics and energetic characteristics are adopted to analyze the rolling law in different com putational conditions.The most remarkable findings of our study are as follows:

(1) On one hand,both parameter differential modes help the bionic dolphin achieve stable rotation after starting and accelerating,while the clockwise spiral trajectory is generated by the lateral force in thex-yplane.On the other hand,the frequency differential method has a better convergence effect than the amplitude difference,which is mainly reflected in the amplitude of steady-state rolling angular velocity and angle.

(2) Varying frequency ratioαfor amplitude ratioαAon the rotating performance is effective for rolling dolphin to increase steady rolling angular velocityΩlinearly,while the driving torque coefficientCMDand the power coefficientCPLrise with increasing growth magnitude.The relation betweenαforαAand rolling efficiencyηreveals that the bionic dolphin rolling with appropriate frequency ratio or amplitude ratio is beneficial to improve the rolling efficiency relatively.Our simulations also conclude that the rolling radiusRis not affected by the variation ofαforαA,on the contrary,the rolling center experiences lateral movement along thex-axis direction asαforαAincreases.

(3) The pressure distribution described as two pairs of pressure cores surrounding the bionic dolphin can explain the swing law and kinematics of the pair of pectoral fins vividly.The 3-D structure of the wake is shown to depend primarily on theSt.As the main driving pectoral fin,the left pectoral fin is shown with a more complicated and diverging wake structure based on the high-amplitude ratio or high-frequency ratio.This finding helps explain that the best way to improve the angular velocity of rotation is to increase the parameter ratio.

In conclusion,compared with the traditional bionic fish that can only achieve propulsion performance,the autonomous rolling performance provides more possibilities for the bionic dolphin.The rational use of the pectoral fins can help us better understand the dolphin’s multi-dimensional manipulation mechanism,and provide theoretical support for the design and production of the robotic dolphin.Based on this work,subsequent studies can concentrate on the interaction of the caudal and pectoral fins to further make the bionic dolphin flexible,and even prepare for the independent completion of the dolphin’s porpoising behavior.In the future,we will make contributions to improve the efficiency of rolling motion and realize the maneuverability of dolphins with multiple DoFs.

Declaration of Competing Interest

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.