Ting-ting Li , Wen-rong Hu , Xu-yang Chen

1. Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

2.Shanghai Jiao Tong University and Chiba University International Cooperative Research Center (SJTU-CUICRC), Shanghai Jiao Tong University, Shanghai 200240, China

3. MOE Key Laboratory of Hydrodynamics, Shanghai Jiao Tong University, Shanghai 200240, China

4.Department of Mechanical Engineering, Tokyo Institute of Technology, Tokyo, Japan

Abstract: Tadpole swimming, including a solitary tadpole swimming and schooling side-by-side in an in-phase mode, is investigated numerically in the present paper.The three-dimensional Navier-Stokes equations for the unsteady incompressible viscous flow are solved.A dynamic mesh fitting the tadpole’s deforming body surface is adopted.The results showed that for a solitary tadpole swimming, two vortex rings are shed in each undulating period.However, as the resultant force on the tadpole is drag, the vortex rings are obviously asymmetric, shaped like “C ”.When the resultant force in the swimming direction approaches zero, the axes of the vortex rings are nearly vertical to the swimming direction.Distorted vortex rings are found when the resultant force on the tadpole is thrust.When the tadpole model obtains the optimum propulsive efficiency, its swimming speed and undulating frequency are close to the values observed in nature.For tadpoles swimming side-by-side in an in-phase mode, the vortex structures in the wake may merge, split and recombine.Compared with a solitary tadpole swimming, only a small hydrodynamic advantage occurs with schooling in parallel, which may be one of the reasons why tadpoles rarely, if ever swim, side by side for any amount of time or distance in nature.The effect of the undulating frequency on the tadpoles schooling is similar to that on a solitary tadpole.In addition,with an increase in the Reynolds number, the thrust force and the propulsive efficiency both increase, while the power consumption decreases.We also found that the tadpole benefits from the vortex pair shedding from its blunt snout, which can strengthen the vortex intensity in the wake and improve the pressure distribution.

Key words: 3-D, tadpole, parallel schooling, numerical simulation

Introduction

With millions years of evolutionary history,aquatic animals have acquired various morphological features that enable many to swim with high speed,efficiency, or agility.The relationship between the morphology of aquatic animals and swimming performance has long interested biologists.It is well known that the morphology of tadpoles and fish is quite different.As the larva of frogs and toads, the abrupt transition from tadpoles’ globose bodies to the laterally compressed tails makes them seem less“streamlined”[1].Apart from the interest of biologists,the study on the swimming of tadpoles that have blunt snouts could provide new ideas for the design of biomimetic underwater vehicles as well as drag reduction of underwater structures.

Tadpoles swim in a different way from fish because of their different morphology.When tadpoles swim, not only do their tail tips make large lateral excursions during each tail beat, but also their snouts oscillate[2].It seems that tadpoles are characterized by awkward shapes and awkward kinematics as well.It is not surprising then that tadpoles have long been presumed to be inefficient swimmers, at least compared with most fishes.In fact, tadpoles are good at swimming.Wassersug and Hoff[2]measured the propulsive efficiency of a tadpole and found that the efficiency of a tadpole is as high as that of a fish.In addition, the terminal portion of the tadpole’s tail does not serve a primary role in thrust generation.Liu et al.(1996, 1997) found that the tadpoles’ ability for efficient swimming is attributed to their highfrequency and large-amplitude undulation.Azizi et al.[3]confirmed that the lateral deflections at the snout in fact help generate thrust.

Besides, group living is a widespread behavior,observed in many aquatic animals, including some tadpoles and fishes.In recent years, much progress has been made in understanding fish schooling.Swimming in schools provides fish with a number of behavioral and ecological advantages, including reproduction, feeding, reduced predation risk[4], and migration[5].In addition, it is also found that fish in school can take advantage of the adjacent vortices shed from neighbors to save energy in swimming and gain hydrodynamic benefits[6].Inspired by schooling from hydrodynamic perspective, a group of fish can be modelled as a collection of individuals arranged in diamond, rectangular, triangular, tandem and side-byside configurations.The hydrodynamic advantages are attainable in a diagonal schooling configuration[7]and the efficiency of the downstream fish is increased by up to 80% by reducing the flapping amplitude[8].Mohsen and Iman[9]first carried out the 3-D simulations to investigate self-propelled synchronized swimmers in various rectangular patterns.The fish in the school with the smallest lateral distance swims 20% faster than a solitary swimmer while consuming similar amount of energy.Both 2-D and 3-D studies of fish schooling found that fish can gain energetic benefits from the hydrodynamic interactions when they swim in a school.For three fishes in a school, i.e.,one upstream and two downstream, the upstream fish enhances the thrust while the power consumption of the downstream fish decreases.For two traveling wavy foils in a tandem configuration, the downstream fish saves energy at any position by appropriately modulating its body motions according to the oncoming vortices shed from the upstream fish[8].The upstream fish benefits only for a small streamwise gap distance[8,10].Dong and Lu[11]numerically investigated flow over undulating foils arranged in side-by-side arrangement with anti-phase and in-phase traveling wavy movements.They demonstrated the saving of swimming power for the in-phase traveling and the increasing of thrust generation for the anti-phase traveling.Wang et al.[12]also found that the power consumption of the anti-phase case increases as the lateral distance decreases, which is opposite to that of the in-phase case.The propulsive efficiency in both cases increases slightly compared with the performance of a solitary self-propelled fish swimming.

As far as tadpole schooling is concerned, there are only a few experimental observations.Caldwell[13]found that individual tadpoles in schooling continually swim toward the middle of the school, presumably where predation intensity was the lowest.In the experiments of Heursel and Haddad[14], the schooling behavior of these tadpoles may enhance the effect of their warning coloration.Based on a computer image analysis system, Breden et al.[15]observed that aggregating tadpoles of different sizes orient in different directions.According to the laboratory data of Wassersug et al.[16], tadpoles ofBufo woodhouseiandXenopus laevisavoid positions directly in front of or behind neighbors.Tadpoles exhibit parallel orientation, but rarely show a preferred distance to neighbors.They also found that tadpoles move little while schooling.Hence, they speculated that it seems unlikely there could be a hydrodynamic advantage in maintaining precise distances between neighbors.Actually, tadpoles rarely, if ever swim, side by side for any amount of time or distance in nature.However,in our previous two-dimensional numerical study of both in-phase and anti-phase modes in tadpoles’parallel schooling[17], we found that tadpoles can save energy in in-phase mode while the total power consumption increases in anti-phase mode.In order to explain the paradox between the observation in nature and the results in 2-D study, it is necessary to know whether tadpoles’ parallel schooling in three dimensions provides significant hydrodynamic advantages.The details of flow characteristics of tadpoles’ parallel schooling in three dimensions are still not clear, too.

In this paper, we carry out a 3-D numerical study on the tadpole swimming including a solitary tadpole swimming and schooling side-by-side.The hydrodynamics and vortex structures in the flow field are analyzed.The influence of various controllable parameters is studied such as lateral distance,undulating frequency and Reynolds number.Swimming in the turbulent flow is studied as well.In addition, the effect of the tadpole’s blunt snout is discussed in detail.

1.Physical model and numerical method

1.1 Physical model

It’s noted that morphology varies less for tadpoles than for fishes[18].Tadpoles ofAnaxyrus americanus(American toad) are relatively small and have a proportionately short tail (tail length equals 60% of total body length).According to the observation and measurement, K?hler[19]gave the side view and top view of the tadpole of American toad.The snout of the tadpole is similar to an ellipse which the widest part equals 24% of body length and the highest part accounts for 17% of body length.In the present study,the tadpole model was simplified and established as Figs.1 (a), 1(b).

Fig.1(a) Vertical view

Fig.1(b) side view of the tadpole model of American toad

In our model, the tadpole undulates actively in a free stream.The transverse motion of the tadpole can be described by the following function

Hereλis the wavelength which is set to be 0.87L(Lrepresenting the body length of a tadpole) in the present study according to Wassersug[2].fis the frequency of the tadpole undulation.The amplitude of the tadpole along the body length,A(x), is calculated by spline interpolation from original data[2,20].The kinematic parameters are chosen based on the experimental observation[21].

1.2 Numerical method

In this paper, open source software OpenFOAM is used to simulate hydrodynamic performance and vortex structures of tadpole swimming.

The finite volume method is used to solve the 3-D Navier-Stokes (N-S) equations for the unsteady incompressible viscous flow.The PIMPLE method,which is the combination of the semi-implicit method for pressure-linked equations (SIMPLE) method and the pressure implicit with splitting of operators (PISO)method, is specially introduced to solve the unsteady flow.The SIMPLE method is used to solve the N-S equations within each iteration step, while the PISO method is used in the time march.

In most cases in this study, the Reynolds number based on the tadpole body length is 800, which matches the swimming environment of the American toad tadpole[2].The flow is laminar.It is also assumed that the tadpole model swims in turbulent flow at relative high Reynolds number.Currently the most popular approach for tackling industrial turbulent flow problem is the one based on solving the Reynolds averaged Navier-Stokes (RANS) equations.The shear stress transport (SST)k-ωmodel is used here.In the areas close to the boundary, the technique of wall function is adopted to solve the effect of the wall surface.In this study, a special dynamic mesh, which is calculated by transfinite interpolation method (TFI),is employed.In the near-wall region, the mesh deforming is adaptive to the deforming body surface at each time step, matching tadpole’s high-amplitude periodic motion.As a result, the special dynamic mesh can guarantee well the mesh quality when applied to simulate wavy bodies at high Reynolds number.

Here we carried out a numerical study of the tadpole swimming including a solitary tadpole swimming and tadpoles schooling side-by-side.For a solitary tadpole swimming, the computation domain size is from -4Lto 10Lin thexdirection, -5Lto 5Lin theydirection, and -5Lto 5Lin thezdirection.The no-slip boundary condition is specified on the surface of the tadpole.Uniform flow is set on the inflow boundary, and the normal and shear stress are specified to be zero at the downstream outflow boundary.As shown in Fig.2, we assumed that the number of the tadpole models is large enough in parallel schooling.Periodic boundary condition is applied to both lateral sides of a tadpole, just as Dong and Lu adopted[11].Hence, for tadpoles schooling in side-by-side arrangement, the lateral boundaries are locatedS/2 away from the center line.

Fig.2 Schematic diagram of an in-phase traveling wavy system consisted of tadpoles in side-by-side configuration

1.3 Hydrodynamic parameters

In this study, the hydrodynamic performance of the undulating tadpole in a free stream is modeled based on the numerical results.The resultant hydrodynamic force component in the incoming direction is the so called total drag force (FD) which consists of a friction drag (Ff) and a pressure difference drag (Fp).The corresponding drag coefficients are defined as the following, respectively

whereUis the speed of the oncoming flow.If the drag force is negative, the resultant force in the swimming direction is called thrust forceFT=-FD.Thus the thrust coefficient isCT=FT/(0.5ρU2A).Under these definitions, when the tadpole swims at a constant speed, both of the time-average drag and time-average thrust force are zero.

The total powerPrequired for the propulsive motion of the tadpole consists of two parts.One is the swimming powerPS, required to produce the lateral oscillation of traveling wave motion, and is defined as

wherefydenotes the lateral force along the tadpole surface,uyis the sideways velocity of each edge.The other is the power needed to overcome the drag force, and is represented asPD=FD U.Thus, the total power is obtained byP=PS+PD.The coefficient of the total power consumption is defined as

Furthermore, the mean thrust coefficientand the mean total power coefficientin one period are calculated.

In order to define the propulsive efficiency, the actual thrustis introduced, which is the sum of the force component in the swimming direction on each grid cell on the tadpole surface.The corresponding coefficient is defined as

The propulsive efficiency is defined as the ratio of useful power to input power, i.e.,.Note that, if we useFTinstead ofhere, the propulsive efficiency would be zero when a tadpole swims at constant velocity.

The second order invariant of the velocity tensorQcan well analyze the vortex structures of the 3-D model and is defined as

whereΩi jdenotes the anti-symmetric part of velocity tensor andSijrepresents the symmetric part of velocity tensor.

2.Numerical validation

In order to validate our numerical method, an elliptic foil undergoing a plunging motion is simulated at the Reynolds number of 1 000.As plotted in Fig.3,the comparison of the lift force coefficient and the drag force coefficient exhibits good agreement with the numerical results of Lu et al.[22](Fig.3).In addition, we also simulated a NACA0012 airfoil undergoing a fish-like undulating motion in both free stream and vortex wakes to validate our method[20].

Fig.3 (Color Online) Comparison of time history

Apart from that, the feasibility ofk-ωSST turbulence model also needs to be verified in the present study.A pitching NACA0020 foil fixed in a free stream is simulated atRe=1× 106as shown in Fig.4.Compared with the turbulence model in the literature[23], the hydrodynamic performance in the present study is closer to the experimental results.

Hence, the numerical methods employed in this study are reliable for the simulation of undulating objects either in the laminar flow or in the turbulent flow.

Here, the convergence test with different time steps and grid sizes has also been conducted.Timedependent lift coefficient of a solitary tadpole undulating at the frequencyf*=4.0 is shown in Fig.5.It can be seen that the computed results are independent of the time steps and the grid sizes in the present calculation.The time step used in this study is 0.3 and the entire mesh contains 2 268 587 cell numbers.

Fig.4 Time-dependent lift and drag coefficients for a pitching NACA0012 foil at Re=1× 106

Fig.5 (Color Online) Time-dependent lift coefficient for the wavy motion at f*=4.0.Condition-1: grid number 2 268 587, time step 0.30.Conditon-2: grid number 3 311 753, time step 0.15

3.Results and discussion

3.1 A solitary tadpole swimming

A solitary tadpole swimming at different undulating frequencies is modeled in this subsection.As presented in Fig.6, the time averaged resultant force on the tadpole is drag force at the nondimensional undulating frequenciesf*=fL/U=1.0,1.8 while at the frequenciesf*=3.0, 4.0, the time averaged resultant force on the tadpole is thrust force.It’s also found that the mean thrust coefficient and the mean total power coefficient increase with the increasing undulating frequency, which is consistent with the general swimming law of aquatic animals.However, the propulsive efficiency is maximized at the frequencyf*=2.4 when the average thrust is nearly zero.It is noted that the tadpole also swims at this velocity and undulating frequency in nature[2].

Fig.6 Performance of a solitary tadpole swimming at different non-dimensional frequencies

To investigate the possible role of vortex dynamics on the hydrodynamic performance of tadpoles, vortex structures are visualized and compared.Figure 7 shows the 3-D frequencies.It can be seen that two vortex rings are shed in each undulating period.However, when the resultant force in the swimming direction on the tadpole is drag, the vortex rings are obviously asymmetric, shaped like “C” (Fig.7(a)).At the frequency when the resultant force in the swimming direction approaches zero, the axes of the vortex rings are nearly vertical to the swimming direction (Fig.7(b)).

Fig.7 (Color Online) 3-D wake structures visualized by the iso-surfaces of Q-criterion for a solitary tadpole swimming at different frequencies

This can be demonstrated by the in-line counterrotating vortices that will be shown in Fig.8(b).It is consistent with the previous study[24]that vortex centers are aligned approximately when the resultant force in the swimming direction is almost equal to zero.In addition, distorted vortex rings are found when the resultant force in the swimming direction on the tadpole is thrust.

The contours of vorticity in thezdirection（Ωz), velocity in thexdirection （Ux) at the typical moment 0.325Tof a solitary tadpole swimming at different undulating frequencies on the mid-planez=0 are presented in Figs.8, 9.As shown in Fig.8, four vortices are formed in one undulating period.The higher undulating frequency results in the stronger vortices in the wake.For the lower undulating frequencyf*=1.0, each vortex shed from the tail in an undulating period splits into a strong and a weak vortex.The weaker ones are so weak that they dissipate soon.Thus the vortex rings are asymmetric and the wake flow is similar to the Kármán vortex street.The resultant force on the tadpole body in the swimming direction is drag force because of the obvious low velocity region.It is also found that two reverse Kármán vortex streets are formed and deflect away from the symmetric line at the high undulating frequencyf*=4.0 (Fig.8(c)).Hence, two strong backward jets are formed which contribute to the thrust generation (Fig.9(c)).When the thrust is little at the undulating frequencyf*=2.4, the counter-rotating vortices are arranged nearly a line in each vortex street.So the backward jet is very weak.In additional to the vortices shed from the tail of the tadpole, a pair of positive and negative vortices is formed behind the blunt snout.

Fig.8 (Color Online) Contours of vorticity zΩ at the midplane z=0 of a solitary tadpole swimming at different non-dimensional frequencies

3.2 Paralleled tadpoles swimming in an in-phase pattern

3.2.1 Effect of the lateral distance

It is generally believed that the main hydrodynamic reason for fish schooling is that fish can consume less energy by swimming in schools.It was found in previous studies that in the parallel swimming, fish or tadpoles save energy in the inphase pattern while they even consume more energy in the anti-phase pattern[11-12,17].From the point of view of saving energy, the in-phase pattern is more favorable for schooling.Therefore, in this paper, we only study the in-phase pattern.Here the kinematic parameters are the same as those in solitary swimming and the non-dimensional frequency is 4.0.

Fig.9 (Color Online) Contours of velocity Ux at the midplane z=0 of a solitary tadpole swimming at different non-dimensional frequencies

Figure 10 shows the variation of hydrodynamic performance against the lateral space of parallel swimming.It shows that the smaller lateral distance between tadpoles always leads to the lower thrust and power consumption but higher propulsive efficiency.However, compared with a solitary tadpole swimming,schooling tadpoles gain only a small hydrodynamic advantages in terms of energy-saving benefits and higher propulsive efficiency.

The reason for the hydrodynamics variation from a solitary tadpole swimming to tadpoles schooling lies in the channel effect.With the lateral distance decrease, the flow velocity near the tadpole surface increases because of the channel effect.According to the wave plate theory[25], the thrust is proportional to the velocity ratioc/U(crepresenting the propagating wave velocity), thus smaller lateral distance between adjacent tadpoles results in lower thrust force of tadpoles schooling.On the other hand, owing to the increasing flow velocity near the tadpole surface, the drag of the snout increases with the decreasing lateral distance.Besides, the main reason resulting in the decrease of total power consumption is that the transverse force decreases because of the synchronizing motion of the tadpoles.The improvement of hydrodynamic performance in the 3-D study is less than 3%, which is far lower than that in our previous 2-D study[24].The effect of the schooling on the hydrodynamic performance in three dimensions is much weaker because the fluid flow around 3-D bodies, especially the flat snout, can weaken the channel effect.This may prove the guess proposed by Wassersug et al.[16]that it seems unlikely there could be hydrodynamic advantages while paralleled tadpoles maintain a precise distances between neighbors.

Fig.10 (Color Online) Performance of tadpole school swimming at different lateral distance at f*=4.0

Figures 11-13 exhibit the instantaneous wake flow field at the typical moment 0.325Tfor different lateral distances.As shown in Fig.11, A pair of vortices is formed behind the blunt snout of the tadpole.However, in fish schooling, there are no vortices shed behind the snout of fish[11].Furthermore,there are also four vortices shed from the tail tip in each undulating period when the tadpoles lead a parallel schooling.However, when the vortices with the same direction shed by the adjacent tadpoles are close enough, they will merge together, forming a reverse Kármán vortex street.As shown in Fig.11(a)whereS=0.875L, there are two reverse Kármán vortex streets merged by the three adjacent tadpoles.As the lateral distance decreases (Figs.11(b), 11(c)),the merged reverse Kármán vortex street then splits into two reverse Kármán vortex streets.Each new split reverse Kármán vortex street gradually becomes close to the center line of each tadpole.When the lateral distance decreases to 0.5L(Fig.11(d)), the split vortex streets behind each tadpole are combined into a reverse Kármán vortex street again.Therefore,columns of positive and negative vortices appear alternately in the wake of paralleled tadpoles.As the lateral distance decreases, the vortex structures of wake flow field may experience merging, splitting and recombining.But for fish schooling, the vortex-pair rows or the vortex rows in the wake rank along the streamwise direction[11].In other words, vortices in the wake of fish schooling are in a regular pattern without merging splitting and recombining.

In addition, the strength of the total backward jets and the pressure distribution in the wake of schooling tadpoles change relatively little with the decreasing lateral distance (as seen in Figs.12, 13).This may account for the tiny variations in the hydrodynamic performance.

3.2.2 Effect of the undulating frequency

In this section, we simulate tadpoles swimming in schools at a lateral distanceS=0.5Lwith different undulating frequencies.The effect of the undulating frequency on the hydrodynamic performance of the tadpoles schooling is similar to that on the solitary tadpole swimming.The thrust and power consumption both increase with the increasing undulating frequency.The optimal propulsive efficiency is also achieved when the resultant force in the swimming direction is approximate to zero.

Fig.11 (Color Online) Contours of vorticity zΩ at the midplane z=0 of tadpoles schooling with different lateral distance

Observing the vorticity field at the mid-planez=0 of tadpoles schooling in Fig.14, it is found that columns of positive and negative vortices appear alternately in the wake of paralleled tadpoles.When the undulating frequency is lower (f*=1.0, 1.8), the wake flow field is similar to the Kármán vortex street and strong forward jets are formed which contribute to the drag force.At the non-dimensional frequencyf*=2.4, the cores of positive and negative vortices lie nearly in line and the forward and backward momentums of the fluid field in the horizontal direction almost compensate for each other.Hence,the resultant force on the tadpole body in the swimming direction is close to zero.When the undulating frequency is higher (f*=3.0, 4.0), the wake flow field is similar to the reverse Kármán vortex street and strong backward jets are formed which contribute to the thrust force.

3.2.3 Effect of the Reynolds number

Fig.12 (Color Online) Contours of velocity Ux at the midplane z=0 of tadpoles schooling with different lateral distance

Fig.13 (Color Online) Contours of pressure at the mid-plane z=0 of tadpoles school swimming with different lateral distance

The real tadpoles are confined to a low Reynolds number world (102-104), mainly resulting from ecological or physiological factors.Although the kinematic parameters of our tadpole model are based on the real tadpole in natu re, the highest Reynolds number here is exten ded to 106because most underwater robots swim at hig h Reynolds number.When the Reynolds number is 106, the flow is turbulence.In this subsection, the tadpoles swimming in side-by-side arrangement(f*=4.0,S=0.5L) withRe=800, 2 000 and 106are separately simulated.

Fig.14 (Color Online) Contours of vorticity Ωz at the midplane z=0 of tadpoles schooling at t/ Τ=0.325 with different undulating frequencies

As seen from Fig.15, with the Reynolds number increasing, thrust gets larger and larger.Nevertheless,with the Reynolds number increasing, the lateral force decreases and so does the lateral power consumption.Hence，the total power input reduces as the Reynolds number becomes high (Fig.15(b)).As listed in Table 1, with the increase of the Reynolds number, tadpoles in side-by-side configuration gain higher propulsive efficiency.

As shown from the vorticity field at the mid-planez=0 of tadpoles schooling in the laminar and turbulent flow (Figs.16(a), 16(b)), respectively, it can be found that the merging of two reverse Kármán vortex streets generated by the adjacent tadpoles seems more disordered and the backward jets are much stronger in in turbulent flow.Based on the theorem of momentum, the rate of change of the momentum with time may reflect the force.From Fig.17, the high velocity region near the tail in the turbulent flow is stronger than that in the laminar flow.So the momentum change between the two moments in the turbulent flow is larger than that in the laminar flow.Hence, the higher thrust is obtained in the turbulent flow.

Fig.15 (Color Online) In one undulating period, performance of tadpoles swimming in schools at different Reynolds numbers (f*=4.0, S=0.5L)

Table 1 The propulsive efficiency of tadpoles schooling at different Reynolds numbers

Fig.16 (Color Online) Contours of vorticity zΩ at the midplane z=0 of paralleled tadpoles swimming in an in-phase mode at two Reynolds numbers

3.2.4 Effect of the blunt snout

Fig.17 (Color Online) At two typical moments 0.325T and 0.675T, contours of velocity Ux at the mid-plane z=0 of paralleled tadpoles swimming in an in-phase mode at two Reynolds numbers

Because of the blunt snout, the shape of a tadpole is quite different from that of a streamlined fish.It is interesting to investigate the effect of the blunt snout on the propulsive performance and instantaneous wake dynamics.As discussed before, a pair of positive and negative vortices is formed behind the blunt snout.In this part, a streamlined model based on the tadpole model is built as shown in Fig.18.The width in theydirection and the height in thezdirection of the streamlined model’s snout are the same as those of a tadpole.Thus the horizontal profile of the streamlined model is based on the NACA0020 airfoil.Comparison of hydrodynamic performance between the tadpole and the streamlined model undulating like tadpoles including a solitary swimming and schooling side-by-side atS=0.5Lare presented in Fig.19, Table 2.The thrust on the streamlined model is always less than that on the tadpole.Besides, the propulsive efficiency of the streamlined model is lower than that of the tadpole.However, the streamlined model consumes less energy compared with the tadpole.

In addition, we also found that the hydrodynamic performance of streamlined models while schooling changes little compared with a solitary streamlined model swimming, differing from that in the literature[19].The reason is that our streamlined model is not a real fish.The relative flat snout of our model can strongly weaken the channel effect.It further proves that the flat snout is the reason for the limited effect of tadpoles’ schooling on the hydrodynamic performance.

Fig.18 Schematic diagram of wavy bodies

Fig.19 (Color Online) Performance of the tadpole model and streamlined model undulating in tadpole mode,including solitary swimming and school swimming at f*=4.0

Table 2 Propulsive efficiency of the tadpole and streamlined model undulating in tadpole mode at f*=4.0

From the vorticity field at the typical moment 0.325T, four vortices are formed in the wake of the undulating streamlined model in one period, which is similar to the tadpole.Nevertheless, the boundary layer separation was only found near the tail of the streamlined model, and no vortices shed from the streamlined model’s snout.As seen from Figs.20(a),20(b), the vortex pair shed from the tadpole’s blunt snout can enhance the boundary layer along the tadpole’s tail as well as the intensity of vortices shed from the tail.Hence, the vortex streets in the wake of schooling streamlined models are dissipated much earlier.As a result, compared with the tadpole, the backward jets in the wake of the schooling streamlined models are obviously weaker (Figs.21(a),21(b)), which explains why the thrust on the streamlined model is much less than that on the tadpole.

Fig.20 (Color Online) Contours of vorticity Ωz at the midplane z=0 of wavy bodies swimming like tadpoles at t/ Τ=0.325 ( f*=4.0)

Fig.21 (Color Online) Contours of velocity Ux at the midplane z=0 of wavy bodies swimming like tadpoles at t/ Τ=0.325 ( f*=4.0)

Besides, the vortex pair shed from the tadpole’s snout can also improve the pressure distribution on the surface of the tail.As seen from the pressure distribution contours in Fig.22, the vortex pair leads to the lower pressure zone along the curved tail, which exhibits suction force on the tail.This helps contribute to the thrust generated by the tadpoles.

Fig.22 (Color Online) Contours of pressure at the mid-plane z=0 for wavy bodies swimming like tadpoles at t/ Τ=0.325 ( f*=4.0)

4.Conclusions

In the present study, a 3-D study on a solitary tadpole swimming and tadpoles swimming side-by-side in an in-phase mode is carried out.The hydrodynamics and vortex structures in the flow field are analyzed.The following conclusions are obtained.

(1) When a solitary tadpole swims, two vortex rings are shed in each undulating period.However, the vortex rings are obviously asymmetric, shaped like“C”, as the resultant force on the tadpole is drag.When the resultant force in the swimming direction approaches zero, the axes of the vortex rings are nearly vertical to the swimming direction.Distorted vortex rings are found when the resultant force on the tadpole is thrust.When the tadpole model obtains the optimum propulsive efficiency, its swimming speed and undulating frequency are close to the values observed in nature[2].

(2) When tadpoles swim side-by-side in an inhase mode, they gain a small hydrodynamic advantages compared with a solitary tadpole swimming, which may be one of the reasons why tadpoles rarely, if ever swim, side by side for any amount of time or distance in nature.When the tadpoles are getting closer in the parallel swimming,the vortex structure of the wake flow field may merge,split and recombine.However, the pressure distribution and the total strength of the backward jets in the field change little and so does the hydrodynamic performance.

(3) Both the thrust and power consumption of the tadpole in parallel schooling increase with the increasing undulating frequency.However, the maximum propulsive efficiency also obtained at the frequency where the resultant force is nearly zero.

(4) As the Reynolds number increases, parallel schooling tadpoles can achieve higher thrust with lower energy cost.Hence, the propulsive efficiency increases as the Reynolds number increases.The backward jets are much stronger in turbulent flow.

(5) Due to the non-streamlined shape, a pair of positive and negative vortices is shed from the blunt snout of the tadpole.This vortex pair can strengthen the vortex intensity in the wake and improve the pressure distribution.Hence, compared with the streamlined model with the same thickness as a tadpole, the thrust and propulsive efficiency of the tadpole are higher.

The results obtained in this study are helpful to understand hydrodynamics and flow structures for the flow over a solitary tadpole and the flow over schooling tadpoles.However, the mechanism in tadpoles’ natural swimming in schools is not fully resolved.Future work should focus on the natural movement of tadpoles swimming in schools.

In addition, tadpoles rarely, if ever swim, side by side for any amount of time or distance in nature.Why tadpoles in nature don’t swim like the models suggested they should is not clear.Apart from a few hydrodynamic benefits, behavioral and ecological factors need to be taken into consideration.For example, are tadpoles swimming side by side more likely to attract predators? Do tadpoles have the neural and muscular capability to swim side by side at a good speed for any length of time? Before anyone goes off to build in-phase schooling robotic tadpoles, it would be good to know why the locomotor mode suggested by the models is rare in nature.