Dan Xia (夏丹), Wei-shan Chen (陳維山), Jun-kao Liu (劉軍考), Xiang Luo (羅翔)

1. School of Mechanical Engineering, Southeast University, Nanjing 211189, China

2. State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China

Abstract: This paper examines the beneficial effects of the spanwise flexibility of the caudal fin for the improvement of the swimming performance for small fishlike robots.A virtual swimmer is adopted for controlled numerical experiments by varying the spanwise flexible trajectories and the spanwise flexible size of the caudal fin while keeping the body kinematics fixed.3-D Navier-Stokes equations are used to compute the viscous flow over the robot.Elliptical, parabolic and hyperbola trajectories are chosen to describe the spanwise flexible profile of the caudal fin.According to the sign (positive or negative) of the phase difference of the swinging motion,the spanwise flexibility can be divided into the fin surface of “bow” and the fin surface of “scoop”.It is observed that for both the fin surface of “bow” and the fin surface of “scoop”, the spanwise elliptical trajectory has the optimal swimming velocity, thrust, lateral force, and efficiency.With comparisons, using the flexible caudal fin with the fin surface of “bow”, the lateral force and the power consumption can be reduced effectively and the swimming stability can be increased while reducing little the swimming velocity and thrust.Meanwhile, using the flexible caudal fin with the fin surface of “scoop” can greatly improve the swimming velocity, thrust, and efficiency while increasing part of the lateral force and the power consumption.Three-dimensional flow structures clearly indicate the evolution process around the swimming robot.It is suggested that the fish, the dolphin, and other aquatic animals may benefit their hydrodynamic performance by the spanwise flexibility of the caudal fin.

Key words: Spanwise flexibility, caudal fin, fishlike robot, swimming performance, flow structures

Introduction

Fishes often enjoy a superior swimming performance in water.In the past decade, the bio-inspired propeller imitating the manner of the fish swimming is widely applied in the autonomous underwater vehicle(AUV)[1].In turns, the rapid development of the AUV promotes the study of the swimming mechanisms of the fish.Recently, the structural flexibility of the fish has attracted much attention owing to their possible applications in the bio-inspired engineering[2-4].A flexible fin is known to contribute to the highly efficient generation of thrust, lift, and other forces[3-4].The numerical and experimental studies[3,5]show the beneficial effects of the flexibility of the fish fins in the efficient generation of thrust and lift forces.

The fish fins, including the intermediate fin, the pair fin, and the caudal fin, play a key role in the selfswimming.The most effective among thesefins is the caudal fin, which plays not only a promoting role similar to the screw propeller, but also a steering role similar to the rudder[6-7].The caudal fin, made of several elastic fin rays and a flexible webbed, can have a three-dimensional deformation motion[1,6].It can be inferred that the differences between the rigid caudal fin and the flexible one make a significant change of the kinematics and the energetics of the fish in the swimming process.So far, the studies of the flexibility focused mainly on the flapping foils with two-dimensional kinematics (heaving and pitching)and unidirectional deformability (chordwise or spanwise)[3,6-8].The effect of the flexibility of the caudal fin was rarely reported.

The flexibility of the caudal fin can be divided into the chordwise flexibility and the spanwise flexibility.The chordwise flexibility mainly concerns the flapping foils[9-11], and the spanwise flexibility remains not well explored.The only report about the spanwise flexibility of the caudal fin can be found in Ref.[3], where the swinging motion of the caudal fin is taken into account in the spanwise flexibility at a given swimming speed, but the impact on the whole fish self-swimming is not considered.The beneficial effect of the spanwise flexibility of the caudal fin on the whole fish self-swimming perfor- mance remains to be further explored.

It is well known that the structural flexibility can improve the performance of the flapping foils[4-5,10-11].Heathcote[11]performed water tunnel experiments to explain the thrust generation and the propulsive efficiency of a chordwise flexible airfoil in the plunging motion.Most studies of the dynamics of the ray fins focused on the experimental observation of the kinematics, the structural deformation, and the flow field around the fins of live fish[10,12].Furthermore, mechanical devices imitating the fish fins were developed and tested[13].Despite the above mentioned studies to characterize the dynamics of the flexible foils, the hydrodynamic effects of the spanwise flexibility of the caudal fin on the entire fish swimming are still not made very clear.In particular,it is desirable to explain the performance differences of different flexible trajectories of the fin surfaces and the mechanisms of the spanwise flexibility of the caudal fin in improving the whole fish swimming performance.

In this paper, a numerical investigation is presented to examine the beneficial effects of the spanwise flexibility of the caudal fin on the improvement of the swimming performance for the fishlike robot, in various cases where the swinging motions of the caudal fin are controlled by imposing different phases at the points along the spanwise direction with the same chordwise position, and with the body kinematics unchanged.We focus on the comparisons of two aspects: the self-swimming performance of the spanwise flexibility trajectories of three kinds, in shapes of the ellipse, the parabola and the hyperbola,to see the performance differences yielded by different spanwise trajectories with the same flexible size, and the self-swimming performance of the fishlike robot generated by the caudal fin with spanwise flexibility and with spanwise rigidity, to explain the effects of the spanwise flexible size on the improvement of the swimming performance.

1.Physical model and kinematics

1.1 Physical model

A small tuna model is adopted as the fishlike robot, composed of a main body with a smooth profile and a high-aspect-ratio caudal fin, without considering all other minor fins.The robot sizex×y×zis 0.2 m×0.03 m×0.06 m, as shown in Fig.1(a).The caudal fin is like a crescent-shaped hydrofoil in its vertical cross-section and a low resistance airfoil in its horizontal cross-section[14], as shown in Fig.1(b).A vortex ring may be generated during the fish swimming and shed from the rear edge, to reduce the consumed energy and improve the propulsive efficiency[14].The coordinate system of the fish bodyObxbybzband that of the caudal finOcfxcfycfzcfare adopted.The shape parameters of the caudal fin include the span lengthb, the chord lengthc, the fin areaSand the aspect ratioR=b2|S.InOcfxcfycfzcf,the shape of the caudal fin is symmetric with respect to the planeOcfxcfycfand the planeOcfxcfzcf.

Fig.1 (Color online) Physical modelsof fishlikerobotand caudal fin

As is shown in Fig.1(b), three points 1, 2, 3 are marked on the leading edge of the caudal fin.Points 1,3 are, respectively, located on the upper and lower edges of the front end, and point 2 is located at the front end of the central axis.Points 7, 9 are marked in the junction of the upper edge, the lower edge and the trailing edge, point 8 is marked at the intersection of the connection between points 7, 9 and the axis, point 5 is marked at the intersection of the trailing edge of the caudal fin and the axis and points 4 and 6 are marked at the intersection of the straight line along the spanwise and the upper edge and the lower edge,respectively.The marked points (1, 2, 3), (4, 5, 6) and(7, 8, 9) are in the same tangential position and the connections of points 1, 4, 7, 2, 5, 8 and 3, 6, 9 are defined as the upper edge, the lower edge and the central axis of the caudal fin.The introduction of the marker definition helps to describe the kinematics of the spanwise flexibility of the caudal fin.

1.2 Kinematics

The kinematics of the fishlike robot is supposed to resemble the tuna motion observed in a live tuna,but variations are also introduced to investigate the effect of the spanwise flexibility of the caudal fin.In general, the kinematics of the robot has two basic components: as represented by a two-dimensional flexible spline curve and the caudal fin described by an oscillating foil[14].The origin of the spline curve coincides with the head of the robot.In this sense, the spline body is responsible for the caudal fin’s heave and the caudal fin’s own rotation is responsible for its pitch.The spline body is treated as a traveling wave expressed by

The body kinematics is always assumed to be unchanged during the swimming movement.The caudal fin can be simplified as a flexible foil moving at a specific combination of the pitch and heave motions[14].The instant transverse displacementycf(t) is expressed as

whereAbis the heave amplitude of the caudal perpendicular joint,Lbis the length of the body segments.

The spanwise flexibility of the caudal fin is mainly reflected in the movements of different points along the spanwise direction at the same chordwise position, as the marked points (1, 2, 3), (4, 5, 6) and (7,8, 9) shown in Fig.1(b), with different phase differences of the pitch motion.In order to easily describe the spanwise flexibility kinematics, the marked points are chosen to define, respectively, the pitching angular displacements of the upper edgeθ1,4,7(t), the central axisθ2,5,8(t) and the lower edgeθ3,6,9(t), as described as

whereθmaxis the pitch amplitude of the caudal fin,φis the phase angle of the heave leading the pitch,γis the phase angle of the central axis leading the upper and lower edges.All points on the caudal fin have the same frequencyωand the same pitch amplitudeθmax.θmaxcan be described as

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

It should be noted that whenγtakes a positive value, the pitch phase of the central axis is ahead of that of the upper and lower edges, and the larger the value ofγis, the larger the phase ahead is and the larger the spanwise flexibility is.Whenγtakes a negative value, the pitch phase of the central axis lags that of the upper and lower edges, and the smaller the value ofγis, the larger the lagging phase is and the larger the spanwise flexibility is.The schematic diagram for the size of theγvalue and the spanwise flexibility is shown in Fig.2.

Fig.2 Schematic diagram of spanwise flexibility and γ value

In the caudal fin movement according to Eqs.(2),(3) and (4), we connect the points along the spanwise direction at the same chordwise position into a curve.The marked points (1, 2, 3), (4, 5, 6) and (7, 8, 9) in Fig.1(b) form different profiles of the spanwise flexibility.In order to easily discuss the effect of the spanwise flexibility on the swimming performance,we choose the spanwise elliptical trajectory, parabolic trajectory and hyperbolic trajectory in the simulations.These three trajectories will be discussed in what follows.

(1) Spanwise elliptical trajectory

Several points along the spanwise direction at the same chordwise position are chosen to represent the spanwise profile of the caudal fin within a cycle.Taking the marked points (7, 8, 9) as an example, the span lengthbof the caudal fin is defined as the elliptic semi-major axis and the difference of the lateral displacements of points 7, 8 as semi-minor axis of the ellipse to establish the elliptic equations of the spanwise flexibility.We produce the motion sequence in time within a cycle according to the elliptic nature.Whenγis 40° or-40°, the motion sequences in time within a cycle at the connection of the marked points 7, 8, 9 at the trailing edge are shown in Fig.3.It can be seen that whenγis positive, the pitch phase of the central axis is ahead of that on the upper and lower edges.And the track of the spanwise flexibility is rendered as an outwardly convex “bow”as the caudal fin swings.Whenγis negative, the pitch phase of the central axis lags that of the upper and lower edges.And the track of the spanwise flexibility is rendered as an inwardly concave “scoop”as the caudal fin swings.

Fig.3 Spanwise elliptic profiles

In the caudal fin motion of the spanwise elliptical trajectory, it is assumed that the translational amplitudes of all points on the caudal fin are identical.Figure 4 shows the swimming posture of the fishlike robot of the spanwise elliptical trajectory within half a cycle whenγis 40°.Comparing Fig.4 with Fig.3(a), it is observed that at the time 0, the marked point 8 of the central axis reaches the left limited position and the marked points of the upper and lower ends of the trailing edge lag behind the marked point 8 in motion.At this time, the trajectory in the spanwise direction of the caudal fin is in an obviously bent state and the connection of the marked points 7, 8, 9 forms an inwardly concave elliptical profile.At the timeT/ 8, the trajectory in the spanwise direction of the caudal fin is a straight line.During the period from 0 toT/8, the central axis and the upper and lower edges of the caudal fin complete the profile change from an inwardly concave ellipse to a straight line.At the timeT/ 4, the trajectory in the spanwise direction of the caudal fin returns to an obviously bent state and the connection of the marked points 7, 8, 9 forms an outwardly convex elliptical profile, producing a trailing edge of the “bow” shape mentioned above.At the time 3T/8, the trajectory of the caudal fin appears still as a “bow”, until the central axis reaches the right limited position at the timeT/2.At this time, the caudal fin finishes the swing motion of half a cycle.Its motion from the timeT/2 to the timeTis completely symmetrical to that from 0 toT/2.

Fig.4 Swimming postures of the fishlike robot

(2) Spanwise parabolic trajectory

As a contrast to the elliptical trajectory, the spanwise parabolic profile can be constructed with the marked points 7, 8, 9 on the caudal fin, with the transverse motion trajectory of the marked point 8 as the symmetry axis of the parabola to establish the parabolic equation with the marked points 7, 9, the two endpoints of the parabola.The motion sequence of the parabolic trajectory is similar to that of the elliptical trajectory for the fishlike robot and thus is omitted here.

(3) Spanwise hyperbolic trajectory

As a contrast to the elliptical trajectory and the parabolic trajectory, the spanwise hyperbolic profile can be constructed with the marked points 7, 8 and 9 on the caudal fin, with the transverse motion trajectory of the marked point 8 as the real axis of the hyperbola and the vertical axis parallel to the spanwise direction to establish the spanwise hyperbolic equation with the marked points 7, 9, the two endpoints of the hyperbola.The motion sequence of the hyperbolic trajectory is similar to that of the above two trajectories and can be described in Fig.3.

It is important to note that the body kinematics of the fishlike robot is always unchanged during the swimming movement.In this work, we focus on the comparison on two aspects: on the one hand, we compare the swimming performance of the spanwise flexibility trajectories of three kinds, the ellipse, the parabola and the hyperbola, to explain the differences of different spanwise trajectories with the same flexibility, on the other hand, we compare the swimming performance of the caudal fin with a spanwise flexibility and with a spanwise rigidity, to explain the effect of the spanwise flexible size of the caudal fin on the improvement of the swimming performance.

2.Numerical method

2.1 Governing equations

With the objective of investigating the effect of the spanwise flexibility of the caudal fin on the swimming performance for the fishlike robot swimming, we consider a 3-D incompressible flow over the robot in self swimming.The equations governing the motion of a viscous fluid are the 3-D Navier-Stokes equations given by:

whereuis the fluid velocity vector,ρis the density,pis the pressure,μis the dynamic viscosity and ?is the gradient operator.To solve the equations in a domain containing the robot, a no-slip condition is imposed on the moving interface with the fluid velocityand the fish velocityas

In this work, the motion of the fishlike robot is in turn described by the Newton’s equations of motion as

whereFandMZare the fluid force and the torque acting on the fish,mis the fish mass,is the swimming acceleration,andare the angular velocity and the angular acceleration, andIZis the inertial moment about the yaw axis.The feedback of the torque is limited in the yaw direction to simplify the computations.The fluid forceFand the torqueMZare expressed as follows:

whereis the normal stress vector,nis the unit vector along the normal direction, dSis the differential unit area along the fish surface, ande3is the unit vector along thezdirection.

2.2 Numerical method

The Navier-Stokes equations are discretized using a finite volume method: with a second-order Crank-Nicolson scheme for the unsteady term, a second-order upwind scheme for the convective term and a second-order central difference scheme for the diffusion term.The pressure velocity coupling in the continuity equation is dealt with by using the SIMPLE algorithm.The Newton’s motion equation for the fishlike robot is solved by using the user-defined function.The coupling procedure is implemented by using an improved staggered integration algorithm[10,14].The mesh grids are locally refined near the fish and the wake region.To capture the movement of the fishlike robot in the 3-D domain, a dynamic mesh technique is used.At each updated time instant, the grids around the robot are regenerated and smoothed using the regridding and smoothing methods.The tail beat periodTis divided into 200 time steps, i.e.,Δt=T/200.To ensure the grid quality updated at each time step, a small time step size is selected depending on the tail beat frequency.

2.3 Swimming performance parameters

Several parameters are used to quantify the swimming performance.The component of the instant fluid force along thexdirection (which for simplicity will be denoted asF(t)) can be computed by integrating the pressure and the viscous forces on the robot as

whereejis thejth component of the unit normal vector andijτis the viscous stress tensor.Depending on whetherF(t) is negative or positive, it contributes to either the dragFD(t) or the thrustFT(t).To separate them, we decompose the instant force as follows:

By constructing the thrustFT(t) and the dragFD(t) given by Eqs.(13), (14), we have the unified fluid forceF(t), i.e.,F(t) =FT(t) +FD(t).The power loss due to the lateral motion of the fishlike robot is expressed as

The Froude efficiencyηdefined based on the axial fluid forceF(t) is zero for the steady swimming.Thus, it is helpful to define a Froude efficiency based on the thrust force and the power loss as follows[15]

whereis the average thrust force over a cycle,Uis the steady swimming velocity, andis the average power loss over the cycle due to the lateral motions.

For the swimming velocity, the components of the instant swimming velocity along thexandydirections (which are denoted asuc(t) andvc(t),respectively) can be non dimensionalized as

For the acting force, the components of the instant thrust force and the lateral force (which are denoted asFT(t) andFL(t), respectively) can be non dimensionalized.Furthermore, the power loss due to the lateral motion can also be non dimensionalized as:

whereCT,CLare the dimensionless thrust and lateral forces, andCPLis the dimensionless power.

Fig.5 (Color online) Time history of the hydrodynamic force and its pressure and viscous components from the present work compared with the results of Dutsch et al.[16]

Fig.6 Contours of pressures and vorticities for four different phase angles.(Fig.6 in Dutsch et al.[16])

2.4 Numerical validation

To validate the ability of our numerical method to predict the forces and the flow structures, we consider the case of a cylinder starting to oscillate in the horizontal direction in a fluid initially at rest.The resulting flow in this case provides a stable vortex shedding and two fixed stagnation points at the front and the back of the cylinder.The detailed studies were reported in Dutsch et al.[16].The cylinder motion is given by a harmonic oscillation.For validation, we choose the case with the same parameters as used in Dutsch et al.[16], where both the experimental and numerical results are reported.The computations are found to yield sufficiently accurate forces.

Figure 5 compares the calculated hydrodynamic force (solid lines) and its pressure (dotted lines) and viscous (broken lines) components in comparison with reported results[16].It is clear that the calculated forces are in good agreement with the reported results.Figure 6 shows the instant contours of the pressure and vorticity fields calculated for four different phase angles of the oscillatory cylinder motion.The flow structures reflect the vortex formation during the forward and backward motions, dominated by two counter-rotating vortices.The computed results here are in good agreement with Dutsch et al.’s results[16],which are not shown here but are reported clearly in their paper.

Fig.7 Time history of swimming velocities

3.Results and discussions

3.1 Time history variations of swimming velocity and flow structure due to spanwise flexibility

In the self-swimming of the caudal fin with the spanwise flexibility, the fishlike robot undergoes dynamic processes from starting at rest to gradually accelerating and converging to a steady state swimming.This process can be divided into two stages, the velocity convergence and the steady swimming.Throughout the entire self-swimming process, the velocity and the force see unsteady changes.Figure 7(a) shows the time history of the swimming velocity of different spanwise flexibility profiles.There are three convergence curves of the swimming velocityuc, and they are the elliptical, parabolic and hyperbolic trajectories for a givenγvalue of -40°.The distributions of the three trajectories are similar to those for otherγvalues.Figure 7(b) shows the convergence curves of the swimming velocityucof the elliptical trajectory for differentγvalues.It can be seen from Fig.7(a) that the fishlike robot with the spanwise elliptical trajectory achieves the highest steady swimming velocity, the next is that with the parabolic trajectory and the lowest is that with the hyperbola trajectory.For the elliptical trajectory in differentγvalues, it is found that whenγincreases from 0° to 40°, the fin surface of “bow”attacks the water,ucgradually decreases with a comparatively small decreasing extent.However,whenγdecreases from 0° to -40°, the fin surface of“scoop” attacks the water,ucgradually increases with a comparatively large increasing extent.

Fig.8 (Color online) Vorticity contours and velocity vectors

For the instant flow structure, we use the vorticity contours of the wake region formed by the fishlike robot in swimming to illustrate the similarities and differences of the caudal fin with different spanwise flexibilities attacking the water.Figure 8 shows the vorticity contours and the velocity vectors of the spanwise elliptical trajectory within a cycle whenγis equal to 40° and -40°, after the fishlike robot has reached to a steady state.Comparing the elliptical trajectories forγtaking values of 40° and-40°, the wake distributions formed by the fishlike robot in self-swimming are both the single row inverse Karman vortexes, along the longitudinal direction.Two opposite rotary vortexes are alternately shed off in each cycle, at the time when the swing direction changes, that is, at the time (15+0/10)Tor(15+5/10)T.Two opposite rotary vortexes, shed off in a cycle, are a pair, inducing a jet in a sinusoidal manner between them.And the jet plays an important role in advancing.

We analyze the fin surface of “bow” withγtaking the value of 40° and the fin surface of “scoop”withγtaking the value of -40°, with the same body kinematics, and it is found that the longitudinal spacing of the vortex fin surface of “scoop” is larger than that of the fin surface of “bow”.Because at this moment, the swing frequenciesfand the swing amplitudesAmaxin both case are identical, the steady swimming velocity of the fin surface of “scoop” is larger than that of the fin surface of “bow”.From the velocity vector distributions, we can see that compared with the fin surface of “bow”, the head of the robot with the fin surface of “scoop” has a larger velocity inxdirection.It means that under the same conditions, the robot with the fin surface of “scoop”has a larger swimming velocity.

Fig.9 Relation between swimming velocity and flexible size

3.2 Effect of spanwise flexibility on velocity

Fig.10 The vorticity iso-surfaces of different flexible sizes

To further study the effect of the caudal fin with the spanwise flexibility on the improvement of the robot self-swimming performance, we make comparisons in the following two aspects: on the one hand,we compare the self-swimming performance of three kinds of the spanwise flexibility trajectories to explain the performance differences of different spanwise trajectories in the cases of the same flexibility, on the other hand, we compare the self-swimming performance of the caudal fins with the spanwise flexibility and with the spanwise rigidity, to explain the effect of the flexible size of the caudal fin on the improvement of the swimming performance.

By changingγfrom -40°, the effect of the flexibility size of the caudal fin on the steady velocityUcan be seen.Whenγis less than 0, the swing phase of the central axis lags that of the upper and lower edges.And the smaller the value ofγis, the larger the flexibility size is.Whenγis more than 0,the swing phase of the central axis is ahead of that of the upper and lower edges.And the larger the value ofγis, the larger the flexibility size is.Whenγis 0,the swing phase of the central axis is the same as that of the upper and lower edges, and the flexibility size is 0.The influences ofγof three kinds of spanwise trajectories onUare shown in Fig.9.For the ellipse trajectory, asγincreases,Ugradually decreases and its reduction magnitude is gradually decreased.For the parabolic trajectory, asγincreases,Ulinearly decreases.For the hyperbolic trajectory, asγincreases,Ugradually decreases and its reduction magnitude gradually increases.When analyzing the trajectories of ellipse, parabola and hyperbola for the sameγ, the robot with the elliptical trajectory can obtain the highest swimming velocity, followed by the robot with the parabolic trajectory and the lowest is that with the hyperbola trajectory.

3.3 Effect of spanwise flexibility on swimming forces and efficiency

To illustrate the effect of the spanwise flexibility on the fishlike swimming performance, the thrust coefficientCTfor three kinds of spanwise trajectories against the spanwise flexibility sizeγis shown in Fig.10(a).It can be observed that, for the elliptical trajectory, asγincreases,CTgradually decreases and its reduction speed is small.For the parabolic trajectory, asγincreases,CTlinearly decreases.For the hyperbolic trajectory, asγincreases,CTdecreases and its reduction speed gradually increases.For the spanwise trajectories of ellipse, parabola and hyperbola with the same value ofγ, the robot with the elliptical trajectory can achieve the highest thrust, the next is that with the parabolic trajectory and the lowest is that with the hyperbolic trajectory.

The lateral force in the fish swimming directly drives its lateral motion, with a lateral acceleration.In the same motion cycle, the excessive lateral force will produce a large lateral velocity and lateral displacement, and thus increase its own power consumption.Meanwhile, if the difference of the lateral forces of the head and the tail is great, it is easy to generate a large yawing moments, for the robot to turn the fleet.In this sense, the lateral force of the fish swimming can not be ignored.An important advantage of a robot with the caudal spanwise flexibility is that the lateral force can be significantly reduced in the swimming process.The relation between the lateral force coefficientCLand the spanwise flexible sizeγis shown in Fig.10(b).Asγincreases, in all cases,CLfirstly increases and then decreases when the maximum is reached withγbeing nearly -30°.If the fin surface of “bow” is adopted to attack the water, the lateral force can be significantly reduced in the swimming process and the larger the flexibility is, the more significant the effect of reducing the lateral force is.In contrast, if the fin surface of “scoop’ is adopted to attack the water, it will increase the lateral force in the swimming process.

To further illustrate the effect of the caudal fin spanwise flexibility on the swimming power consumption, the average power coefficientCPLwithin a cycle againstγis shown in Fig.10(c).Asγincreases,CPLfirstly increases and then decreases.If the fin surface of “bow” is adopted to attack the water,the power consumption can significantly be reduced in the swimming process and the larger the flexibility is,the more significant the effect of reducing the power consumption is.In contrast, if the fin surface of“scoop” is adopted to attack the water, the power consumption will be increased to some extent.Among the spanwise trajectories of ellipse, parabola and hyperbola,the power of the elliptical trajectory is the smallest, the next is that of the parabolic trajectory and the largest is that of the hyperbolic trajectory.

The variation of the propulsive efficiencyηdue toγcalculated according to the Froude efficiency model is shown in Fig.10(d).For the three kinds of spanwise flexibility trajectories, asγincreases,ηdecreases until it gradually reaches a stable value.Because with the fin surface of “scoop”to attack the water can help to increase the steady swimming speed and the thrust force, its contribution to the propulsive power is very large.Though the fin surface of “scoop” to some extent increases its power consumption, the increase of the propulsive power is much larger than that of the power consumption and thus the efficiency gradually increases.However,attacking the water with the fin surface of “bow” can help to reduce the power consumption, but both the steady swimming speed and the thrust force will be reduced, so the efficiency decreases gradually until it reaches a stable value.For the three kinds of spanwise trajectories, the relations ofηwithγare almost the same, except that the efficiency of the elliptical trajectory is the highest, the next is that of the parabolic trajectory and the lowest is that of the hyperbolic trajectory.

Fig.11 (Color online)The vorticity iso-surfacesofelliptic profile

3.4 Effect of spanwise flexibility on flow structures

To reveal the self-swimming performance of the fishlike robot with the spanwise flexibility caudal fin,we study the effects of the trajectories of ellipse,parabola and hyperbola on the yielding flow structure and extract the vorticity iso-surfaces of the three kinds of trajectories by theq-criterion[14].A comparison shows that the vorticity iso-surfaces of the three kinds of trajectories have similar distribution patterns due to the similar spanwise trajectory shape.So, only the vorticity iso-surface of the elliptical trajectory is shown here, as well as the effect of the spanwise flexibility size on the flow structure distribution.Three views of the vorticity iso-surface of the elliptical trajectory are shown in Fig.11.It can be seen that the tail wake is composed of a series of vertically symmetrical vortex rings, as large as the span length of the caudal fin.A careful observation shows that the flexible caudal fin between two adjacent vortex rings has an extended fork structure,in a “hairpin” shape proposed in the PIV data of the bluegill swimming trials by Tytell and Lauder[17].Interestingly, the “hairpin” vortex ring is generated due to the sinusoidal swing of the caudal fin.

Fig.12 (Color online) The vorticity iso-surfaces for different flexible sizes

The flow structure yielded by the elliptical trajectory caudal fin of different flexible sizes is shown in Fig.12, including the vorticity iso-surfaces in the planesOxyandOxzwhenγvaries from 40°--40°.For different flexible sizes, all flow structures are in the alternately arranged “hairpin”shape.Within a cycle, two opposite rotary vortexes are shed and the spatial distribution of the flow is in a single column structure.This flow structure is identical to the flow distribution obtained in the PIV experimental test by Tytell and Lauder[17].With the decrease ofγ, the caudal fin is transformed from the fin surface of “bow” to that of “scoop”, and both the lateral width and the longitudinal length of the vortex streets increase.And in the same distance, the number of vortexes gradually decreases and the wake strength increases, corresponding to increased energy loss in the wake and power consumption.This finding is identical to the above energetics result.For different values ofγ, the spanwise height of the vortex streets is almost the same as the span of the caudal fin.

It is important to note that, whenγis 0°, the length and the width of the vortex generated by the rigid caudal fin have a moderate size.Whenγgradually increases, the rigid caudal fin gradually becomes flexible one with the fin surface of “bow”,and the length and the width of the vortex are reduced and the vortex is shed without changing its own kinematics.And therefore, the energy consumption is reduced in the wake, and then the swimming velocity decreases.Whenγgradually decreases, the rigid caudal fin gradually becomes flexible with the fin surface of “scoop”.And it can acquire a higher swimming velocity without changing its own kinematics and the energy consumption in the wake also gradually increases.

Fig.13 (Color online) The pressure distributions for different flexible sizes

The pressure distributions in the planesOxyandOxzof the ellipse trajectory for different flexible sizes are shown in Fig.13.It can be seen that the surrounding pressure distributions of the fishlike robot of different flexible sizes are similar.They have a negative pressure gradient from the high-pressure center in the head, and positive pressure gradients from two pairs of high and low pressure cores in the body and the caudal fin.From the pressure distributions in the planeOxz, it is seen that whenγis 40°,because the surface attacking the water is the fin surface of “bow”, there are two high-pressure centers on the upper and lower edges of the caudal fin and then they are shed off.Whenγdecreases to 0°, the surface attacking the water is a plane, and the pressure of the high-pressure center gradually decreases and will be shed off from the trailing edge.Whenγis-40°, the surface attacking the water is the fin surface of “scoop”.Because the flow velocity is large, the high-pressure center is in the central axis and is shed off from the trailing edge.From the pressure value in the figure, it is seen that the caudal fin with the fin surface of “scoop” can acquire the highest pressure to attack the water, the next is that with the rigid caudal fin, the smallest is that with the fin surface of “bow”.This is consistent with the above variations of the thrust coefficient against the flexible sizeγ.The pressure contours in this figure are very similar to those obtained experimentally[17-18]using the DPIV near the caudal fin of a swimmer.

4.Discussions

Fig.14 (Color online) Comparisons of the kinematics of flexible and rigid caudal fins

Table1 Comparisons of the energetics performance of flexible and rigid caudal fins

It is confirmed that the fishlike robot with the spanwise ellipse trajectory has the best self-swimming performance.To better understand the effect of the spanwise flexibility of the caudal fin on the fishlike robot self-swimming performance, the differences of the self swimming performance of the flexible and rigid caudal fins of the elliptical trajectory are studied.Comparisons of the kinematics of the spanwise flexible caudal fin of the ellipse trajectory withγtaking values of 40° and -40° and the rigid caudal fin are shown in Fig.14.The swimming posture of the robot is compared within half a cycle.It is clear that the body kinematics under these three conditions are identical, except the difference of the caudal kinematics due to the spanwise flexibility.It can be inferred that it is the small difference of the caudal fin flexible size that leads to substantial changes of the swimming performance of the fishlike robot.Specifically, whenγis 40°, the caudal fin is in the shape of“bow” in the direction attacking the water, whenγis 0°, the caudal fin is a rigid flat plate, whenγis 40°, the caudal fin is in a shape of “scoop”.

Comparisons of the swimming performance of the flexible and rigid caudal fins are shown in Table1.The caudal fin withγtaking the values of 40° and-40° are, respectively, chosen to compare with the rigid one.By comparing the fin surfaces of “bow” of the flexible fin and the rigid caudal fin, it is shown that for the same body kinematics, adopting the flexible caudal fin with the fin surface of “bow” can greatly reduce the lateral force in the swimming process and thereby reduce the power consumption and increase the swimming stability.At the same time,the steady swimming velocity and thrust force are smaller than those of the rigid caudal fin.In view of the integrated propulsive power and lateral power, the propulsive efficiency of the flexible caudal fin with the fin surface of “bow” is lower than that of the rigid one.By comparison of the fin surfaces of “scoop” of the flexible fin and the rigid caudal fin, it is shown that for the same body kinematics, adopting the flexible caudal fin with the fin surface of “scoop” can greatly increase the lateral force in the swimming process and thereby can achieve a higher propulsive efficiency.At the same time, its lateral force and power consumption are larger than those of the rigid caudal fin, but its increase extent of the power consumption is far less than that of the swimming velocity and the thrust of the caudal fin with the fin surface of “scoop”.Therefore, its efficiency is higher than that of the rigid caudal fin.The findings in this table are very similar to those obtained experimentally[12,17]using the DPIV for the swimming fish and numerically[19-20]with the mathematical model and the MATLAB code for fish swimming simulations.

It can be inferred that in the self-swimming process, the fishlike robot with the flexible caudal fin can generate a larger thrust force or a higher propulsive efficiency by appropriate adjustment of the fin surface attacking the water.Furthermore, it can reduce the lateral force and the power consumption and increase the swimming stability.Comparisons of kinematics, energetics and flow structures of the flexible and rigid caudal fins, reveal the effect of small differences of the flexible and rigid caudal fins on the yielding self-swimming performance of the fishlike robot and also the considerable role of the caudal fin attacking the water.This inference is supported by early experimental results[13]for a robotic model.

Specifically, using the flexible caudal fin with the fin surface of “bow” can effectively reduce the lateral force and power consumption and increase the swimming stability under the premise of reducing little the swimming speed and thrust.Therefore, from a biological perspective, it is more suitable for long time parade of the fishlike robot.Using the flexible caudal fin with the fin surface of “scoop” can greatly enhance the swimming velocity and thrust, improve the efficiency under the condition of increasing part of the lateral force and its own power consumption.Therefore, it is more suitable for the escape mode of the fishlike robot.

5.Conclusions

In the present study, a virtual tuna is constructed with the flexible caudal fin to examine the beneficial effects of the spanwise flexibility of the caudal fin on the improvement of the swimming performance of the fishlike robot.The virtual swimmer allows us to perform controlled numerical experiments by varying the spanwise flexible trajectories and the spanwise flexible sizes of the caudal fin while keeping the body kinematics fixed, which cannot be realized by purely experiments.The most important findings of our work are summarized as follows.

(1) The elliptic, parabolic and hyperbola trajectories are used to describe the spanwise flexible profile in the study.According to the phase difference of the swinging motion of the central axis and the upper and lower edges (positive or negative), the spanwise flexibility is divided into the fin surface of “bow” and the fin surface of “scoop”.By comparing the kinematics and the swimming performance of three kinds of trajectories, it is observed that for both the fin surfaces of “bow” or “scoop”, the spanwise elliptical trajectory has the optimal swimming speed, thrust,lateral force, power consumption and propulsive efficiency in performance.

(2) On this basis, by comparing the flexible caudal fin of the spanwise elliptical trajectory with the rigid one, it is seen that by using the spanwise flexible caudal fin with the fin surface of “bow”, the lateral force and the power consumption can be reduced effectively and the swimming stability is increased with reducing little the swimming speed and thrust.Therefore, from a biological perspective, it is more suitable for a long time parade of the fishlike robot.By using the flexible caudal fin with the fin surface of“scoop”, the swimming velocity and thrust can be greatly increased, the efficiency is improved under the condition of increasing part of the lateral force and its own power consumption.Therefore, it is more suitable for the escape mode of the fishlike robot.

(3) The comparative study of two aspects shows that by using the spanwise flexible caudal fin with the elliptical trajectory, the self-swimming performance is greatly improved with respect to the rigid caudal fin.By appropriately adjusting the shape of the fin surface of the caudal fin attacking the water, a larger thrust force or a higher propulsive efficiency can be achieved.On the other hand, the lateral force and the power consumption are reduced and the swimming stability is increased.These findings suggest that the fish, the dolphin, and other aquatic animals may benefit in their hydrodynamic performance from the spanwise flexibility of the caudal fin.

Acknowledgement

This work was supported by the State Key Laboratory of Robotics and System, Harbin Institute of Technology (Grant No.SKLRS-2018-KF-11).