Numerical model and hydrodynamic performance of tuna finlets

Jun-Duo Zhang, Wei-Xi Huang

AML,DepartmentofEngineeringMechanics,TsinghuaUniversity,Beijing100084,China

Keywords:Tuna Finlets Thunniform swimming Leading-edge vortex Vortex interactionw

ABSTRACT Finlets, a series of small individual triangular fins located along the dorsal and ventral midlines of the body, are remarkable specializations of tuna and other scombrid fishes capable of high-speed swimming.In this study, a symmetric model containing nine finlets of tuna is proposed to overcome the limitation of measurement without losing authenticity. Hydrodynamic performance along with three-dimensional flow structures obtained by direct numerical simulation are demonstrated to disclose the underlying hydrodynamics mechanism of finlets. Complex interactions of leading-edge vortices (LEVs), trialing-edge vortices(TEVs), tip vortices (TVs) and root vortices (RVs) are observed from the three-dimensional vortical structures around the finlets. Two more cases consisting of the 3rd to 9th (without the first two) and the 3rd to 7th (without the first two and the last two) finlets are also simulated to examine the effects of the first two and the last two finlets.

Tuna, a group of Scombrid fish, is capable of long-distance swimming at high speed [ 1 , 2 ]. This outstanding swimming ability of tuna draws attention from various fields and has been studied through different approaches. Dewar and Graham [3] and Donley and Dickson [4] investigated swimming kinematics of tuna by measurements of living tuna and summarized the trait of the thunniform mode, a locomotion mode named after tuna. Robotic model is also an effective tool to study the properties of tuna and thunniform swimming that can hardly be obtained from experiments on living specimen. Triantafyllou and Triantafyllou [5] built a robotic tuna to visualize the wake structures and discussed vortex control of tuna swimming. Drag reduction [6] and maneuvering and stability performances [7] were also experimentally investigated using a robotic tuna. In addition, Xia et al. [8] and Li et al. [9] numerically simulated tuna swimming under self-propulsion and carried out systematic parametric studies of the various kinematic parameters. Recently, Zhu et al. [10] designed a high-frequency robotic tuna based on experimental data from free-swimming tuna. This robotic tuna can achieve high tail beat frequencies and agree well with tuna kinematics. Particle image velocimetry (PIV) was conducted to visualize the wake structures and the leading-edge vortex (LEV) on the caudal fin of the robotic tuna.

To enhance swimming performance, tuna has evolved several adaptations, including streamlined body, crescent caudal fin and specializations like finlets and caudal keels [ 11 , 12 ]. Finlets are a series of small individual triangular fins located along the dorsal and ventral midlines of the body between the dorsal fin and the caudal fin, as illustrated in Fig. 1 , while the caudal keels are a pair of lateral keel-like structures along the caudal peduncle anterior to the caudal fin. Many studies focused on these subtle specializations to explore the effects on tuna swimming. To study the function of caudal keels, Wainwright and Lauder [13] performed experiments on simplified tuna-like foil and found that the caudal keels reduce the power consumption of swimming. Zhang et al. [14] numerically investigated the function of caudal keels on whole tuna model.Enhancement of thrust and efficiency on the caudal fin were detected with the presence of caudal keels. Yang et al. [15] simulated a self-propelled flexible plate with a keel-like structure to represent caudal fin and caudal keels, and found that the cruising speed and the propulsion efficiency are increased by the keels. Finlets, on the other hands, even draw more attentions and have been studied through various approaches in recent years. Nauen and Lauder[16] conducted experiments on living mackerel to qualify the kinematics of finlets and suggested that the finlets may direct flow to the caudal fin and thus contribute to the thrust. Nauen and Lauder[17] further visualized flow around the finlets using PIV and flow redirection by the finlets was confirmed. Zhu et al. [18] simulated a tuna-like model with finlets, which were simplified as coalescent rectangular plates. Their results indicated that the vortices interaction between finlets and caudal fin assumes a constructive mode and thus enhances the thrust. Wang et al. [19] numerically studied the swimming of a tuna model with similar coalescent finlets, and found that the finlets redirect the local flow but no improvement in hydrodynamic performance was observed.

Fig. 1. Illustration of the finlets (yellow), caudal keels (green) and the dorsal/caudal fins (blue) of tuna.

Recently, Wainwright and Lauder [13] clearly illustrated the morphology of finlets through micro-computed tomography on specimen of yellowfin tuna. By filming the locomotion of living tuna, they observed that finlets rotate about their roots based on the flow direction and generally mimic the motion of caudal fin with a phase difference. In addition, the more posterior finlets have larger flapping amplitudes. To test the performance of finlets,they conducted experiments on tuna-inspired foils with finlets, and discovered that the flexible finlets reduced the lateral force amplitudes and power requirement. On the other hand, Wang et al.[20] numerically investigated the hydrodynamics of finlets in yellowfin tuna. They established finlets a numerical model consisting of the 3rd to 7th finlets, which undergo heaving and pitching motions reconstructed from measurement of living tuna. The simulation results indicated that the finlets produce drag instead of thrust. They also showed that the interaction between finlets reduces drag and the pitching motion of the finlets decreases the power consumption. As the kinematics of finlets model in Wang et al. [20] are highly realistic, the heaving and pitching motions during the left stroke and the right stroke are naturally asymmetric, which leads to evident asymmetry in the instantaneous forces and wake structures. The asymmetric motion may also result in other unsteady mechanisms [ 21 , 22 ] and complicate the condition.In addition, due to the difficulty of measuring the free swimming tuna, only the 3rd to the 7th finlets were modeled by Wang et al.[20] whereas yellowfin tuna possesses nine finlets. In other words,the first two and last two finlets of tuna were not considered by Wang et al. [20] . However, the last two finlets may play more important role as they have larger motion amplitudes [13] . Therefore,hydrodynamic traits of the whole finlets of tuna under symmetrical condition is worth further examining.

The purpose of the present study is to numerically investigate the hydrodynamics of finlets in tuna swimming. An idealized symmetrical kinematic model of finlets is proposed based on measurements from the previous study [20] to reconcile conciseness and authenticity. The hydrodynamic performance and flow structures of the present finlets model are obtained through direct numerical simulation by employing the immersed boundary method. The effects of the first two and the last two finlets are discussed by comparison of models with different finlet numbers. Instantaneous flow structures are also demonstrated to elucidate the underlying mechanism of finlets.

To focus on the finlets, the body and other fins of tuna are ignored and only the finlets are considered in the present model. The shape of finlet is obtained from the profile of yellowfin tuna (Thunnusalbacares) [ 2 , 13 , 20 ] and the geometry of each finlet is identical in this study for concision. The chord length (c) at midspan of the finlet is selected as the characteristic length, as demonstrated in Fig. 2 . The locomotion of the finlets can be well represented by combining the heaving and pitching motions [20] . Therefore, the kinematics of the finlets are set as a combination of heaving motion in the transverse (z) direction and pitching motion about the root of the finlet along the spanwise (y) direction, as demonstrated in Fig. 2 . The previous studies [ 23 , 24 ] have shown that the local flow is converging to the narrowed posterior body outlines, i.e. parallel to the finlets in tuna swimming, as the upstream flow past the posterior body of the fish. Thus, the incoming velocityUis set to be parallel to the plane of finlets series and the streamwise (x) direction is defined the same as the incoming flow, as illustrated in Fig. 2 . The present model of finlets is consistent with the previous study [20] , in which the model has been proven to be appropriate for finlets simulation.

For numerical simulation, an analytical expression of finlet kinematics is required. Similar with Wang et al. [20] , Fourier series are employed for representation of the displacement of the finlet root and the pitching angle of the finlet in the present study. Based on the measurement by the previous study [20] , the first and third order Fourier series are applied to interpolate the heaving and pitching kinematics, respectively. Additionally, as noted above, the kinematics of finlets in Wang et al. [20] are realistic but naturally asymmetric, which results in obvious asymmetry in the forces and wake structures. Here a symmetrized kinematics based on measurement data from the previous study [20] is proposed in order to provide a symmetrical finlets model without losing authenticity.Under the ideal symmetrical condition, the amplitudes of the heaving and pitching motions should be the same during each half of the period, while the direction should be opposite. In other words,the expression of the symmetrized finlets kinematics should be odd harmonic. As a result, the even harmonic components, i.e. the even order terms, of the Fourier series are vanished. Thus, the symmetrized expressions of the heaving and pitching motions of finlets are described as, respectively,

wherei= 1,2,…,9 denotes the index of finlets,ω= 2π fdenotes the angular frequency withfthe tail-beat frequency, andai-fiare the coefficients of the Fourier series. The kinematics of the 3rd to 7th finlets can be fitted from the measured data [20] . However,the kinematic data of the first two and the last two finlets were not obtained by Wang et al. [20] due to the limitation of experiment. Thus, the Fourier series coefficients of the first two and the last two finlets are determined by extrapolation of the coefficients of the 3rd to 7th finlets. The Fourier series coefficients of all the finlets are listed in Table 1 .

The time histories of the heaving and pitching motions of the nine finlets are demonstrated in Fig. 3 a and 3 b, respectively. It

Fig. 2. Schematic of the present finlets model.

is noticed that the amplitudes of both the heaving and pitching motions are increased in sequence from the 1st to the 9th finlet.Meanwhile, the phases of the peak in the heaving and pitching kinematics also increase with the finlet number, that is, the motion of the posterior finlets lag behind the anterior ones. These features of the present finlets kinematics are consistent with the measurements of the previous study [20] . Additionally, as a result of symmetrization, the amplitude and phase of the present kinematic model during the left and right strokes of each finlet are exactly the same.

Fig. 4. Comparison of the finlets kinematics: ( a ) The present finlets model; ( b ) The snapshots of free-swimming tuna, reproduced with permission from Wainwright and Lauder [13] (? IOP Publishing. Reproduced with permission.).

Table 1 Fourier series coefficients of the finlets kinematics.

In order to examine the authenticity and adequacy of the symmetric kinematics of the whole nine finlets, the trajectories of the present finlets model at different instants are demonstrated in Fig. 4 a and compared with the finlets in free-swimming tuna recorded by Wainwright and Lauder [13] . From Fig. 4 a and 4 b,it is found that the motions of the present finlets model are in good agreement with the observation of living tuna. Particularly,the motion of the last finlet, highlighted in purple in both Fig. 4 a and 4 b, determined by extrapolation in this study, is well consistent with the free-swimming tuna. Therefore, the symmetrized kinematic model for the nine finlets proposed in this study is adequate for representing the locomotion of tuna finlets.

The fluid motion around the finlets is governed by the incompressible Navier–Stokes (N-S) equations and the continuity equation:

whereuis the velocity vector,tis the time,pis the pressure,Reis the Reynolds number, andfdenotes the momentum force utilized to implement the no-slip condition along the immersed boundary (IB). The velocity of the uniform incoming flowUis defined as the characteristic velocity and the midspan chord lengthcis selected as the characteristic length. Thus, the timetis nondimensionalized byc/U, and the pressurepand the momentum forcefare non-dimensionalized byρU2, whereρis the density of the fluid. In this study, the Reynolds number is defined asRe=Uc/ν, whereνis the kinematic viscosity of the fluid.

The flow field around the finlets is solved by employing the IB method, which was first proposed by Peskin [25] and has been widely adopted in biomimetic flow simulations, e.g., fishes [26–29] , insects [30–32] , and birds [33] . In the IB method, the finlets,i.e.the IB, are represented by a Lagrangian mesh, while the fluid motion is defined on a background Eulerian grid. For flows with complex/moving boundaries, the computational efficiency can be substantially improved by using a Cartesian Eulerian grid instead of a body-fitted mesh [ 34 , 35 ]. The interaction between the IB and the surrounding fluid (i.e., the no-slip condition along the IB) is enforced by introducing a momentum forcefto the Eq. (3) . Furthermore, the Eulerian and Lagrangian quantities are transformed by employing the smoothed delta functionδ[36] , i.e.

wherexis the Cartesian coordinates,sdenotes the curvilinear coordinates, and the IB points are described byX(s,t) with a corresponding Lagrangian velocityU(s,t) and Lagrangian forceF(s,t). In the present algorithm, the Lagrangian forceFis obtained explicitly with the penalty method (or feedback law), which was introduced in detail in the previous studies [ 37 , 38 ]. Fully implicit time advancement is applied with the Crank–Nicholson scheme for the convection and diffusion terms. The discretized N-S equations are solved by employing the fractional step method on the staggered Cartesian grid. The velocity and pressure are decoupled by utilizing the block LU decomposition in conjunction with the approximate factorization, which can be referred to Kim et al. [39] for more details. Although the present numerical method has been validated in the previous studies [ 37 , 38 , 40 ], a validation case is also performed to confirm the qualification of the present solver for heaving and pitching finlet simulation. A single finlet possessing the same geometry and kinematics with the first finlet from Wang et al. [20] is simulated under the same condition ofRe= 999.6 and the reduced frequencyk= 0.206. The drag coefficient obtained by the present method is shown in Fig. 5 a. It is found that the present result is in good agreement with that from the previous study [20] .

Fig. 5. Time histories of the force coefficients: ( a ) Comparison of the drag coefficient between the present result and the previous study [20] ; ( b ) Force coefficients obtained from different computational settings, where solid lines represent the results for the domain size 40 c × 20 c × 10 c with a grid size h / c = 0.078, dashed lines for 40 c × 20 c × 10 c with h / c = 0.039 (the present setting), dash-dotted lines for 80 c × 40 c × 20 c with h / c = 0.078, and circles for cases with the present setting at the third cycle while other results are all from the fourth cycle.

In this study, the finlets are simulated in a computational domain with size of 40c× 20c× 10c. The computational domain is discretized by a uniform Cartesian grid with grid sizeh/c= 0.078,while each finlet is resolved by an unstructured mesh with 516 triangular elements. The time step is set asΔt= 0.0 0 014c/U. A uniform flow with a constant velocityUis employed at the inlet boundary, while the convective boundary condition is applied at the outlet and slip condition is set at the side boundaries. The cases in the present study are conducted at the Reynolds numberRe=Uc/ν= 300 and the reduced frequencyk=fA/U= 0.206,wherefis the tail-beat frequency andA= 5.85cis the root amplitude of the 7th finlet according to the previous study [20] . Several cases are carried out to test the independency of the present computational setting. Force coefficients (non-dimensionalized by 1/(2ρU2c2)) obtained from simulations at different domain sizes and grid resolutions are shown in Fig. 5 b. The results obtained from cases using a larger domain size (80c× 40c× 20c) and finer grid (h/c= 0.039) show a good convergence with the present setting. Thus, the computational setting with domain size 40c× 20c× 10cand grid sizeh/c= 0.078 is selected in the following simulations. Furthermore, by comparing the force coeffi-cients of the 3rd and 4th cycles, high consistency of instantaneous forces is also observed in Fig. 5 b. Thus, the periodic steady state is achieved at the 4th cycle, and all the data in the following are obtained from the 4th tail-beat cycle.

Fig. 6. Comparison of the force coefficients of different arrays of finlets: Solid lines represent the 1st to 9th finlets (all nine finlets), dash-dotted lines denote the 3rd to 9th finlets (without the first two finlets) and dashed lines are for the 3rd to 7th finlets (without the first two and the last two finlets).

Overall, the total forces of all the three cases exhibit evident symmetry between the first half (the left-to-right stroke) and the second half (the right-to-left stroke) of the tail-beat cycle, as a result of the symmetrized kinematics. The total streamwise force of the finlets are positive throughout the entire tail-beat cycle. In other words, the finlets produce drag rather than thrust, which is consistent with the previous study [20] . In addition, the three arrays of finlets also show similar trends over time for the three components of forces. The streamwise force shows two major peaks during each half of the cycle, whereas two peaks and two valleys arise in the transverse force. The spanwise force is non-zero but relatively small in magnitude as compared with the other two components. As the finlets are connected to the body in the spanwise direction, which is relatively irrelevant to the hydrodynamic performance, only the streamwise and transverse forces are discussed in the following. The peaks and valleys of the forces in all the three cases show a high correlation with the motion of the finlets. While the finlets flaps from left to right during the first half of the tail-beat cycle, the pitching angle of the finlets increases to a maximum value, as seen in Fig. 3 b. This pitching motion increases the windward area in the streamwise direction but decreases the windward area in the transverse direction, which leads to a major peak in the drag and a local minimum in the transverse force. Subsequently, the finlets gradually pitch back to the middle position,resulting in decrease of windward area in the streamwise direction and thus the local minimum in drag. Meanwhile, the lateral windward area is increased as the finlets move back, and the transverse force is also increased. These traits of forces on finlets are qualitatively consistent with the previous study [20] . By comparing the forces of the three cases, it is found that the both the drag and the transverse force of the array containing the 3rd to 7th finlets(dashed lines in Fig. 6 ) are smaller than the other two models as a result of a smaller number of finlets. The forces of the model with nine finlets (solid lines) are generally larger than the other two cases due to the larger windward area. However, it is interesting to see that the peak value of the drag of the nine finlets is slightly smaller than that of the array including the 3rd to 9th finlets (dash-dotted lines) despite containing two more finlets.

Fig. 7. Three-dimensional vortical structures visualized by iso-surfaces of Q-criterion at t / T = 0.6 ( a )-( c ) and t / T = 0.7 ( d )-( f ): ( a )( d ) the 1st to 9th finlets (all nine finlets); ( b )( e )the 3rd to 9th finlets (without the first two finlets); ( c )( f ) the 3rd to 7th finlets (without the first two and the last two finlets).

Fig. 8. Three-dimensional vortical structures visualized by iso-surfaces of Q-criterion at t / T = 0.8. ( a ) The 1st to 9th finlets (all nine finlets); ( b ) The 3rd to 9th finlets (without the first two finlets); ( c ) The 3rd to 7th finlets (without the first two and the last two finlets).

To discuss the relation between the hydrodynamic performance and the flow structures, the instantaneous three-dimensional vortical structures of the three cases are visualized by using the Qcriterion [41] in Figs. 7 and 8 . As the locomotion of the finlets are symmetric in the present study, only the vortices during the rightto-left stroke (abbreviated as the left stroke) are demonstrated. It is clearly observed that complex vortices, including leading-edge vortices (LEVs), trialing-edge vortices (TEVs), tip vortices (TVs) and root vortices (RVs), shed from the finlets and interact with each other.

Specifically, at the early left stroke, the finlets start moving leftward from the right side, where the upper tips of the posterior finlets are not sheltered by the anterior finlets. Thus, TVs were formed as the upstream flow went through the tips of finlets, and then are stretched by the incoming flow as the finlets move leftward, as shown in Fig. 7 . Although the coalescence of the TVs already occurs, TVs shedding from the 2nd finlet (TV-2) to the 9th finlet (TV-9) can still be distinguished in Fig. 7 a. Besides interacting with each other, interaction between TV and finlet also can be observed as TV-8 encounters the leading edge of the 9th finlet and is separated into two parts as marked in Fig. 7 a and 7 b.Meanwhile, the RVs has already merged into a strong RV tube that rotates inverted with the TVs. In addition, the TEVs shed from the previous right stroke (TEV-R) are still present in the wake, which may interact with the caudal fin. By comparing the three different arrays of finlets in Fig. 7 a- 7 c, it is found that the vortical structures anterior than the 6th finlet are generally similar among the three cases, whereas the wakes posterior than the 7th finlet are evidently more concise in Fig. 7 a.

Fig. 9. Time histories of the drag ( a )( b ) and transverse force ( c )( d ) coefficients of each individual finlet for the three cases, where solid lines represent the 1st to 9th finlets(all nine finlets), dash-dotted lines denote the 3rd to 9th finlets (without the first two finlets) and dashed lines are for the 3rd to 7th finlets (without the first two and the last two finlets).

Subsequently, the finlets continue flapping leftward and pitching to larger angles of attack. LEVs are generated and developed as the finlets move at large angles of attack, as demonstrated in Fig. 7 d- 7 f. Although LEV often contributes to the flying [42] and swimming [ 14 , 43 ], the low pressure induced by the LEVs at the lee side of the finlets enhances drag at the present condition. Therefore, a peak arises in the drag of the finlets shortly afterwards.Figure 7 d exhibits that the LEVs occur on the finlets posterior than the 4th finlets (LEV-4), while LEVs are attached to all the finlets in Fig. 7 e and 7 f. On the other hand, TEVs are shed from the trialing edge of the finlets caused by the rotating motion [ 40 , 44 ]. Distinct interaction between TEV-3 and LEV-4 can be observed in Fig. 7 d,while complex interactions of TEVs, TVs and LEVs occur posterior than the 5th finlet in all three cases.

The TEVs generated during the left stroke (TEV-L) continually develop and further propagate into the wake as the demonstrated in Fig. 8 a. Meanwhile, the TEV-R formed from the previous half stroke is not dissipated yet. Thus, the complex TEV shed from the nine finlets remain enduringly in the wake, and interact with other part of tuna, e.g., caudal keels and caudal fin. Fig. 8 also reveal that the TEV-R of the array consisting of the 3rd to 7th finlets is much weaker than that of the other two cases, indicating that the presence of the last two finlets leads to a stronger TEV in the wake.

In order to further examine the effect of the interactions among finlets, the streamwise and transverse forces on each individual finlet from the three models are shown in Fig. 9 . It is found that the variation trends of the forces over time on each finlet are similar to that of the total force shown in Fig. 5 , although the amplitude and phase vary from finlet to finlet in the three cases. The more posterior finlets have larger force amplitude as a result of larger flapping extents, while the phase of the forces on the more anterior finlets are more advanced due to the more leading kinematics. It is noticed in Fig. 9 that the amplitude of both the drag and transverse force of the individual finlet in the model containing nine finlets (solid lines) are smaller than that of the corresponding finlet in the other two cases. Specifically, the drag on the 3rd and 4th finlets in the nine finlets case is significantly smaller than that in the other two cases as seen in Fig. 9 a. The peak value of the drag and the local maximum amplitude of the transverse force of the 5th to 9th finlets are slightly smaller than that of the other two cases. The drag decrement of the individual finlets explains the reduction of the major peak of the total drag in Fig. 5 .By comparing the nine finlets case (solid lines) and that containing the 3rd to 9th finlets (dash-dotted lines), it is found that the forces of the 3rd and 4th finlets are significantly affected by the first two finlets while the changes on the forces of the 5th to 9th finlets can also be observed. Comparison of the forces of the arrays of the 3rd to 9th finlets (dash-dotted lines) and the 3rd to 7th finlets (dashed lines) indicates that the last two finlets slightly decrease the drag and the transverse force amplitudes of the 7th finlet whereas no evident difference is found on the forces of filets anterior than the 7th finlet.

Fig. 10. Comparison of the spanwise vorticity distributions at t / T = 0.62 ( a )( b ) and t / T = 0.72 ( c )( d ): ( a )( c ) Array of the 1st to 9th finlets (all nine finlets); ( b )( d ) array of the 3rd to 9th finlets (without first two finlets).

Fig. 11. Comparison of the spanwise vorticity distributions at t / T = 0.74. ( a ) Array of the 1st to 9th finlets (all nine finlets); ( b ) Array of the 3rd to 9th finlets (without the first two finlets); ( c ) Array of the 3rd to 7th finlets (without the first two and the last two finlets).

The instantaneous spanwise vorticity distributions are demonstrated in Figs. 10 and 11 to further analyze the effects of the first two finlets and the last two finlets respectively. Figure 10 a and 10 b show the flow structures at the early left stroke. The leading edge of the 3rd finlet in the case without the first two finlets directly encounters the incoming flow and generates strong LEV, i.e.LEV-3 in Fig. 10 b, which also interacts with the 4th finlet and contributes to LEV-4 in Fig. 10 d. In the all nine finlets case, however,the 3rd finlet is sheltered by the first two finlets so that the LEV-3 in Fig. 10 a is much weaker than LEV-3 in the case without the first two finlets. As the finlets move leftward shown in Fig. 10 c and 10 d, the LEV-4 is enhanced by the strong LEV-3 in Fig. 7 d while the intensity of LEV-4 in Fig. 10 c is apparently lower for a much weaker LEV-3. As noted above, the weaker LEV leads to a smaller drag. Hence, the drag on the 3rd and 4th finlets is evidently decreased with the presence of the first two finlets. In addition, no obvious direct interaction between the first two finlets and the finlets posterior than the 4th finlets is observed in Fig. 10 . By comparing Fig. 11 a- 11 c, it is found that the TEV-7 in the case without the last two finlets are stronger than that of the other two cases.The previous studies [ 40 , 44 ] have shown that the TEV can also induce low pressure and enhance the pressure force like the LEV.Thus, the more intense TEV-7 results in a larger drag of the 7th finlet, and the last two finlets decrease the drag of the 7th finlet by suppressing TEV-7.

In the present study, a symmetrical model containing the whole nine finlets of the yellowfin tuna was proposed based on the previous measurement [20] for conciseness and authenticity. The trajectories of the present finlets kinematics were in good agreement with the previous observation of free swimming tuna [13] , and the hydrodynamic performance of the present finlets model are qualitatively consistent with the previous numerical study [20] . Three different arrays of finlets, i.e. the 1st to 9th, the 3rd to 9th and the 3rd to 7th, were directly simulated by adopting the IB method.By comparing the forces of the three cases, it is found that the first two finlets significantly decrease the drag of the 3rd and 4th finlets and slightly reduce the drag of the other posterior finlets, which leads to the decrease of the peak in the total drag even containing two more finlets than the case without the first two finlets. In addition, the last two finlets slightly affect the force of the 7th finlet,while no obvious effect was found on the more anterior finlets. The 2D and 3D vortical structures were illustrated to analyze the interactions of finlets. Complex vortices, including LEVs, TEVs, TVs and RVs, are shedding from the finlets and interacting with each other.The presence of the last two finlets also leads to a stronger TEV that remains longer in the wake, which may interact with other parts of tuna. The vorticity distribution showed that the first two finlets significantly reduce the LEV of the finlets and thus decrease the drag, while the last two finlets reduce the drag of the 7th finlet by suppressing the TEV. These findings in the present study provide more insights into the interaction mechanism of the finlets and also indicate the potential effect of interaction with other parts of tuna.

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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grant number 11772172 .