，*，，

1.College of Aerospace Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China；

2.Key Laboratory of Unsteady Aerodynamics and Flow Control of Ministry of Industry and Information Technology，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

（Received 8 August 2021；revised 23 November 2021；accepted 20 December 2021）

Abstract: To predict the flow evolution of fish swimming problems，a flow solver based on the immersed boundary lattice Boltzmann method is developed. A flexible iterative algorithm based on the framework of implicit boundary force correction is used to save the computational cost and memory，and the momentum forcing is described by a simple direct force formula without complicated integral calculation when the velocity correction at the boundary node is determined. With the presented flow solver，the hydrodynamic interaction between the fish-induced dynamic stall vortices and the incoming vortices in unsteady flow is analyzed. Numerical simulation results unveil the mechanism of fish exploiting vortices to enhance their own hydrodynamic performances. The superior swimming performances originate from the relative movement between the“merged vortex”and the locomotion of the fishtail，which is controlled by the phase difference. Formation conditions of the“merged vortex”become the key factor for fish to exploit vortices to improve their swimming performance. We further discuss the effect of the principal components of locomotion. From the results，we conclude that lateral translation plays a crucial role in propulsion while body undulation in tandem with rotation and head motion reduce the locomotor cost.

Key words：immerse boundary；lattice Boltzmann method；complex deformable boundary；fluid-fish interaction；hydrodynamic mechanism；bionic propulsion

0 Introduction

The natural characteristics of fish swimming such as high maneuverability，long endurance abili?ty，and strong adaptability to the environment are beyond the reach of traditional underwater vehicles.These differences may be due to the excellent per?formances from the evolution of fish，including com?plex movement patterns，special epidermal charac?teristics，and perception of vortices in the natural en?vironment. The hydrodynamic mechanism of fish swimming has attracted widespread attentions from researchers and engineers who try to apply these mechanisms in the design of underwater vehicles［1-4］.Exploring the mechanism of fish exploiting vortices in the natural environment to improve their hydrody?namic performance has become a central issue for improving and enhancing the propulsion and endur?ance capabilities of aquatic bionic vehicles［5-6］.

The early research approaches on this subject are mainly theoretical analyses，including the resis?tive force theory［7］，the reactive force theory［8］，the waving plate theory［9］，and the potential flow theo?ry［10］. They are based on simple geometry and single motion patterns，and ignore the effect of fluid vis?cosity. From the 21st century，numerous models based on viscous incompressible fluids have been es?tablished to understand the mechanism of the inter?action between the unsteady flows and various aquatic swimmers in the natural environment. Two main simplified models of fish-like motion are rigid flapping and flexible undulatory models. The flap?ping model［11］was originally used to describe the aerial flights of birds and insects then extended to de?scribe the aquatic swimming of fishes. It is a combi?nation of pitching and heaving.Gopalkrishnan et al.［12］investigated the hydrodynamic performance of foil with the flapping model behind the D-shaped cylin?der. The results showed that the incoming vortices of the Kármán vortex street were repositioned，and their strength changed by the flapping foil. Akhtar et al.［13］established a simplified multi-fin model using two rigid finite-thickness flat plates with the flapping motion pattern，and investigated the effect of the vortex generated by the dorsal fin vibration on the thrust and efficiency of the downstream caudal fin.Karbasian et al.［14］numerically simulated foil swim?ming using an improved flapping model，and the re?lationship between the kinematics of foil and flow structure around the foil was analyzed. The abovementioned studies used the flapping model to de?scribe fish-like swimming，whereas the effect of flexible bodies was not considered.

Most of the streamlined fish adopt an undulato?ry mode to generate propulsion. Barrett et al.［15］studied the force and power during the motion of the robotic fish in the undulatory mode of propulsion.Experimental data showed that at the same propul?sive velocity，robotic fish with undulatory mode of propulsion can reduce the power consumption com?pared with rigid robotic fish. Beal et al.［16］studied the motion patterns of lifeless fish in a vortex wake.The results showed that the energy passively ex?tracted by the inanimate fish from the vortex would provide greater propulsion for its own motion，which caused it to move upstream. It was concluded that fish selectively extracted energy from the in?coming vortices by actively altering their own loco?motion. Considering the factors affecting the opti?mal hydrodynamic performance，the numerical re?sults proposed by Shao et al.［17］and Xiao et al.［5］confirmed that within certain parameter ranges，the incoming vortices increased the thrust generated by fish with a single undulatory mode of propulsion.However，these simplified models only considered the undulation，but neglected the flapping model.

For the locomotion of swimmers，it is recog?nized that there is no optimal motion pattern，but the combination of different motion patterns may im?prove the performance of swimmers. Bergmann et al.［18］proposed a model that considered coupled ki?nematics. Flapping and undulation motions were put into the locomotion of foil at the same time，and the effect of velocity coupling on its performance was numerically analyzed.In each time step，the locomo?tion of the foil consisted of active undulation and pas?sive flapping，and the rigid flapping was determined by the Newton laws of mechanics using the hydrody?namic forces and moment acting on the fish surface.Using the coupled kinematic model，Zhu et al.［19］numerically studied the adaptive behaviors of a fully self-propelled smart swimmer in complex vortex en?vironments. Akanyeti et al.［20］expanded a series of experimental research results of Liao et al.［21-23］and proposed a kinematic analytical model to describe the Kármán gait. This analytical model showed that the real locomotion of trout in the wake of D-shape cylinder was the superposition of pitching，heaving，undulation，and heading motion. Stewart et al.［24］studied the kinematics of the Kármán gait in the vor?tex wake behind the tandem cylinder，and found that the wavelength of the vortex wake and the body wavelength of the Kármán gait fish satisfied a linear correlation. Recently，Li et al.［6］simplified the Kármán gait model via the combination of heaving and undulatory motions，and the effects of flow con?trol parameters on the hydrodynamic performance of the fish were investigated numerically.

The existing numerical models do not consider the complete locomotion of live fish in the vortex en?vironment. The main contribution of the presented study is that the complete Kármán gait motion pat?tern is used to investigate the hydrodynamic mecha?nisms，which involves the pitching，heaving，undu?latory，and heading motions. Additionally，the hy?drodynamic interaction between the dynamic stall vortex induced by fish and the incoming flow vortex plays a key role in the hydrodynamic performance of the fish. Although numerous experimental and nu?merical studies have been undertaken，the focus is to understand the kind of flow control parameters that improve the hydrodynamic performance of fish.However，the relationship of the flow structure in?duced by the locomotion of fish and the incoming vortices is not well understood. The factors of flow pattern around the foil that influence the extraction of energy by fish from the incoming vortex have not been systematically investigated.

Using the Kármán gait kinematic analytical model to force the foil locomotion and place it be?hind the cylinder，the simulation of the fish swim?ming in the vortex street wake is performed based on the immersed boundary lattice Boltzmann meth?ods（IB-LBM）［25］. Benefiting from the common fea?tures of the Cartesian grid，the lattice Boltzmann method was combined firstly with the immersed boundary［26-27］in 2004［28］. Subsequently，several im?proved IB-LBM have been developed［29-32］. At pres?ent，IB-LBM has been proven to be an effective nu?merical simulation method for simulating the flow around complex deformable bodies［33-34］. In this work，a flexible iterative algorithm is employed to improve the implicit velocity correction IB scheme，so that the deformable moving boundary can be pro?cessed based on the desirable computational cost and memory.

The rest of this paper is organized as follows.Section 1 describes the physical model. Section 2 elaborates the computational model and perfor?mance parameters. Section 3 includes the numerical results and discussion. The last section of this paper is the conclusion.

1 Physical Model

The flow that goes past a fish adopting the Kármán gait periodically generates a pair of alternat?ing counter-rotating vortices from the sides of the fish head and the caudal fin. These kinematic vorti?ces interact with the incoming vortices to create a mysterious nonlinear coupled dynamic system. As the subject of this paper，the interaction between the incoming vortex and the vortex shedding from the swimming fish is investigated using a simpler model. The flow configuration is shown in Fig.1.The NACA0012 airfoil with the chord lengthLis placed directly behind the cylinder with the diameterD，whereL=2D. The distance between the center of the cylinder and the tip of the fish is equal to 2L.Numerical results show that the cylinder-foil dis?tance 2Lcan ensure that the vortex flow generated by the upstream cylinder remains periodicity and cannot be disturbed by the fluctuation of the down?stream foil.

Fig.1 Flow configuration composed of a stationary cylinder and a streamlined fish

The computational domain size is 10D×25D.The Poiseuille velocity distribution is used as the in?let boundary condition，and the free outflow condi?tion is set as the outlet boundary condition. The up?per and lower boundaries adopt stationary wall con?ditions. The cylinder and the fish body employ the IB model to force the non-slip condition. The Reyn?olds number is calculated based on the diameter of the cylinderD，the inflow velocityU，and the fluid viscosityν，whereReof all simulations discussed here are 100.

We use the Kármán gait analytical model pre?sented by Akanyeti et al.［20］to force the foil locomo?tion. The parameters and traveling wave equation expressions of the analytical model are listed in Ta?ble 1. In Table 1，φ0is the phase angle of heaving motion，fis the frequency of fish vibration，andλis the fish body wavelength，whereλ=1.25λwakeandf=fvsis satisfied（λwakeandfvsare the cylinder wake variables）. A series of instantaneous postures of fish within a kinematic cycle is shown in Fig.2. These postures agree well with the experimental results of Ref.［21］.

Table 1 Description of the Kármán gait analytical model

Fig.2 Fish instantaneous posture during a kinematic cycle

2 Computational Model

2.1 Numerical method

The split-forcing scheme Lattice Boltzmann equation （LBE）［35］with the Bhatnagar-Gross-Krook（BGK）collision operator is expressed as

wherefαis the density distribution function（DDF）along theα-direction at lattice nodexand timet.The discrete velocity setcαcouples time and space in the LB model. It guarantees that after Δt，the par?ticle at the lattice nodexprecisely reaches the adja?cent lattice nodex+cαΔt. It has been proven that for the internal node of computational domain，the splitforcing LBE is second-order accurate in space and time［36-37］. The collision operator represents the ad?vection offα，it is linear and related to the equilibri?um DDFfeqα. Limiting the Hermite expansion offeqαto the second-order，the hydrodynamics macroscop?ic laws can be guaranteed［38］. The expression offeqαcan be written as

whereωαis the weighted coefficient associated withcα. In this paper，the D2Q9 model is employed，and the details can be found in Ref.［39］.csis sound speed，Δx2/Δt2. Δxand Δtare the lattice space size and the time step size，respectively. In the isothermal LB model，c2sis the proportional coef?ficient between pressurepand densityρ，In the IB-LBM environment，the discrete force dis?tribution functionFαessentially describes the contri?bution from the boundary，which is a function of the momentum forcingf. Like expansion offeqα，the Hermite expansion ofFαis restricted to the secondorder.Its expression is given that

Depending on the multi-scale Chapman-Ensk?og expansion，the split-forcing LBE is able to recov?er the Navier-Stokes equations involving the mo?mentum forcing for solving the flow problems of in?compressible viscous fluid past the immersed bodies.

The macroscopic density and momentum of flu?id can be calculated in a down-top manner

The kinematic viscosity of fluid，v，is denoted by the relation timeτas

In the IB-LBM scheme，fluid particles are de?scribed by the uniform Eulerian lattice nodesx，and the immersed boundary is represented by a series of Lagrangian pointsX（s，t）.Xis the location of im?mersed boundary.xandsare the Eulerian and La?grangian coordinates. Transformation between the Lagrangian variables and Eulerian variables can be controlled by the Dirac delta function. A hat-func?tion［40］is used to discrete the Dirac delta function and construct the interpolation functionD. It is ex?pressed as

Inspired by Wu and Shu et al.［33-34］，the velocity correctionδU（X，t）can be used to enhance the noslip condition and distributed into the surrounding Eulerian fluid nodes to constructf（x，t）and updateu（x，t）by Eq.（5）.

According to Eq.（1），the numerical computa?tion process of implicit velocity correction can be di?vided into four steps：The collision，the first-forc?ing，the streaming，and the second-forcing. Pushing a new time step，the first three steps are to pre-cal?culate the density and velocity of fluid particles.Then，the fourth step is to calculate the velocity cor?rection of fluid particles with the no-slip boundary as the constraint condition. In this work，the secondforcing step is constructed in an iterative manner based on the multi-direct force scheme［25］. The im?proved IB-LBM algorithm is reflected in the use of a simple and flexible sub-iteration instead of solving the linear equation，thereby reducing the computa?tional cost［34］. The sub-iteration procedure is ex?pressed as

where Δlis the length of the boundary element. The iteration initial conditionu（0）=u*=∑cαfα/ρis used in the above iteration procedure；and the superscriptnindicates the number of IB iteration steps. To avoid additional computational cost，the iteration pa?rameter?is used to flexibly control the number of sub-iterations. When ||δU（n）||∞≤?is satisfied，the sub-iterations process is over. Analogously，Dash et al.［41］has also established the flexible forcing nonslip constraint by involving a new iteration parame?ter.

The velocity correctionδuat the Eulerian nodes can be calculated by

The direct-forcing formula at Eulerian nodes can be expressed as

The hydrodynamic forceHacting on the boundary element can be evaluated by Eq.（15）without complicated tensor calculations

2.2 Performance parameters

The propulsion performance is quantified by the mean thrust coefficientT. To ensure thatT＞0 when the flow provides a propulsion contribution to the fish within a certain time range［0，t］，Tis defined as

We also quantify the energy consumption level of the fish using the mean power coefficientP. Sim?ilarly，Pis calculated by

where -His the local force exerted by the fish on the fluid；uthe velocity of the fish and it can be set using the Kármán gait analytical model.

3 Results and Discussion

3.1 Numerical validation

To verify the present calculational model，we perform the simulation of the uniform flow past a stationary circular cylinder without foil behind the cylinder. As a benchmark case，this flow has been investigated based on other numerical schemes［32，34，42］. The comparison between the calcu?lation results of the present method and the numeri?cal results from the existing literature can serve as the verification of this method. The computational domains of 40D×40Dis used，and the center of the cylinder is positioned at（20D，20D）. The uniform mesh is applied into the entire computational do?main benefiting from excellent parallelism properties of IB-LBM. The lattice density ofD=50Δxand the Lagrangian point density ofΔxare used.The initial densityρ0and velocityu0are set to be 1.0 and（0.1，0），respectively. TheReis calculated based on the inlet velocityU，D，andv，Re=UD/v.

The drag coefficientCDand the lateral force co?efficientCLcan be calculated usingCD=FD/（0.5ρ0U2D） andCL=FL/（0.5ρ0U2D），whereFDandFLare thex-direction andy-direction compo?nents of the total force acting on the surface of the cylinder. The dimensionless lengthLwof recirculat?ing wake is calculated by usingLw=L/D. The Strouhal NumberStare calculated usingSt=fvsD/u∞，wherefvsis the frequency of vortex shedding.As shown in Table 2，the agreement with the avail?able results demonstrates the ability of the present model to predict the steady and unsteady flow evolu?tion.

Table 2 Comparison of the available results and the present results for flow over a stationary cylin?der at Re=20,40,100

In order to further verify the validation of the present IB-LBM used to deal with the problem of the flow around the moving boundary，the simula?tion of a NACA0015 airfoil that heaves and pitches simultaneously in a uniform flow［43-44］is carried out.The pitching and heaving motion，θ(t)=θ0sin(2πft) andH(t)=H0sin(2πft+π/2 )， are forced by the IB model atRe=1 100 andH0=Lf，whereH0is the heaving amplitude；θ0the pitching amplitude；andLfthe chord length of the airfoil.The simulations with the same numerical parame?ters used by Kinsey et al.［43］and Wu et al.［44］are per?formed，and two set of parameters are set as：θ0=76.33°，f*=0.14 andθ0=60°，f*=0.18. The mean drag coefficientD，the peak of the lift coefficientL，and the power extraction efficiencyηare listed in Table 3. The consistency between the current simulation results and the results of the existing liter?ature［43-44］verifies that the present IB-LBM used in this work is suitable for the flow around a moving boundary.

Table 3 Parameters of the flows over an oscillating NA?CA0015 airfoil at Re=1 100,H0=Lf,and xp=L/3

3.2 Fish swimming in the cylinder wake

In this work，based on the model shown in Fig.1，we study the hydrodynamic mechanism of fish exploiting vortex from the perspective of flow patterns. These experimental studies show that the hydrodynamic optimized properties of fish are relat?ed to the relative movement of the Kármán gait and the incoming vortices，and the relative movement is described using the phase differenceφ.

In order to control the relative motion between the vortical flow produced by the upstream station?ary cylinder and the downstream Kármán gait foil，we define the foil-vortex phase differenceφ. First，the simulation of flow pass through a single station?ary cylinder is performed to generate the stable vor?tex sheet wake atRe=100. When the vortical flow reaches a stable state，we use the obtained wake variables（the wake wavelength and the vortex shed frequency）to set the kinematic parameters of the Kármán gait model. Then，a stationary foil is placed behind the cylinder with the downstream dis?tance 2L. After the vortical flow reaches stable again，we take the state of the flow field where the minus vortex center（ωzL/U＜0）arrives at the foil center as the initial state of the unsteady flow simula?tion. Last，the foil begins to perform the Kármán gait motion with the initial phaseφ0，and we define the foil-vortex phase differenceφof the simulation equal toφ0.

To investigate the effect ofφon the periodic features of the hydrodynamic forces experienced by the swimming foil in the cylinder wake，a series of the simulations are performed under differentφcon?ditions. Fig.3 shows the mean thrust coefficient as a function of the time series. In Fig.3，for most of the ten differentφ，theT(T) curve is approximately a straight line，which implies that the flow evolution is periodic，because the surrounding fluid provides approximately equal propulsion to the fish in each ki?nematic cycle.Trepresents the vortex shedding peri?od. On the contrary，the fluctuating and stronger re?sistances are captured atφof 36° and 180°，which demonstrates that the fish cannot capture energy from the aperiodic flow inducted by the Kármán gait at such phase difference. We further conducted a se?ries of simulations with subdividedφaround 36° and 180°. The results show that whenφis in the range of 27° to 45°，or 171° to 189°，the flow could pro?vide aperiodic resistance for the fish. As a counterexample，these phase differences should be prevent?ed in term of optimizing the propulsion system.

Fig.3 Cycle-by-cycle T(T ) curves for φ of 0°, 36°, 72°,108°, 144°, 180°, 216°, 252°, 288°and 324°

For the flow with the periodic feature，the pro?pulsion performance and the locomotion cost are quantified using the mean thrust coefficient and the mean power coefficient for five oscillation cycles.Fig.4 showsT(5T)andP(5T)as a function ofφfrom 0° to 360°，excluding the interval from 27° to 45° and from 171° to 189°. In Fig.4（a），the dashed line represents the dividing line between thrust and resistance. We observe that the maximum thrust is obtained atφof 0° and 198°，whereas the strongest resistance is encountered atφof 54° and 252°. On the other hand，the minimum power cost is needed atφof 18° and 216° while atφof 108° and 288° the maximal power cost is required to hold the Kármán gait. From the perspective of propulsion system de?sign，it is desirable to achieve greater propulsion based on less energy consumption，which shows that the relative movement mode with larger energy consumption must be prevented. Atφof 0°，the hy?drodynamic performance of the fish is desirable. We assume that the Kármán gait withφof 0° is the real locomotion mode taken by fish in the wake of Kármán vortex streets.The discussion on the hydro?dynamic mechanisms of fish exploiting vortices will be carried out underφof 0°condition.

Fig.4 The mean thrust and power coefficient for φ of 0°,36°, 72°, 108°, 144°, 180°, 216°, 252°, 288° and 324°

As the reference case，we conduct a simulation with the fixed fish holding station behind the cylin?der atRe=100. Temporal thrust and lateral force coefficient for the two cases are illustrated in Fig.5.For the fixed fish，the fish will encounter resistance almost in a whole cycle time. This result explains why the fixed fish hardly obtains propulsion from the flow. For the swimming fish，the amplitude of thrust curve is amplified，and the fish captures thrust from the flow for most of the cycle time，which results in the maximum mean thrust coeffi?cient. In Fig.5（b），for the two cases，the nature of temporal lateral force coefficient curves is almost the same，except for their phase. This could cause that the Kármán gait has the same mean lateral force to the fixed fish. These results imply that the Kármán gait aids the fish to obtain the optimal pro?pulsion performance，but its effect on the lateral force is weak.

Fig.5 Time histories of CT and CL for the fixed fish and the fish adopting the Kármán gait with φ of 0°

The time-series of the Kármán gait and instan?taneous vortex distribution around the fish are plot?ted in Fig.6. We define the incoming vortex acting on the fish as“vortex 1”. The downward locomo?tion of the fish head causes significant splitting of a kinematic vortex，which locates on the upper side of the body. The generation of vortex is ending when the lowest position of lateral translation is reached，and the upward motion of the fish head begins. The upward motion of the fish head creates an inverted kinematic vortex on the underside of the body. The two counter-rotating kinematic vortices are called“vortex 2”. The same formation is done by the cau?dal fin，and a pair of counter-rotating kinematic vor?tices are produced（called“vortex 3”）. It can be ob?served that the relative movement of the fish loco?motion and the incoming vortices is consistent with the experimental results of Liao［23］.

Fig.6 Instantaneous vortex distribution around the fish in one kinematic cycle for φ of 0°

To understand the reason why the optimized propulsion performance is obtained，we perform Fourier spectral analysis to study the intrinsic nature of unsteady hydrodynamic force. As shown in Fig.7，the distributions of fundamental harmonics both forCDandCLare like the two cases. The fre?quency of the kinematic vortices induced by the Kármán gait is equal to the frequency of the incom?ing vortices. The major difference is the amplitude of the fundamental harmonics. Moreover，the contri?bution of the subharmonics is cut in the spectrum ofCD. For the fixed fish，the amplitude of the funda?mental harmonics is related to the intensity carried by the incoming vortices. Therefore，for the swim?ming fish，the larger amplitude demonstrates that the intensity of incoming vortices has increased.This may also explain why the frequency of the Kármán gait is adjusted to be equal to the frequency of the Kármán vortex street. Considering the charac?teristics of Fig.6 and Fig.7，we believe that under the conditions of the same rotation direction and fre?quency，“vortex 2”enhances the energy carried by“vortex 1”，and“merged vortex”is derived from the enhanced“vortex 1”.

Fig.7 Fourier spectra of CD and CL of the fixed fish and the fish adopting the Kármán gait with φ of 0°

We further plot the Kármán gait and instanta?neous vortex distribution around the fish atφof 54°and 180° in Fig.8 and Fig.9，respectively，to illus?trate the origin of resistance. In Fig.8，the main dis?crepancy is that the fishtail moves away from the“merged vortex”when the“merged vortex”slides near the fishtail. This process shows that a pair of stronger“vortex 3”can be generated. This means that the tendency of the fishtail to move toward the“merged vortex”plays a key role in the process of fish exploiting vortices，but this role is negative.

Fig.8 Instantaneous vortex distribution around the fish in one kinematic cycle for φ of 54°

Fig.9 Instantaneous vortex distribution around the fish in one kinematic cycle for φ of 108°

In addition，in Fig.9，we see that the vibration of the fish head produces“vortex 2”，and its direc?tion of rotation is opposite to“vortex 1”. From the mean thrust coefficient mentioned above （in Fig.3），we infer that“vortex 2”with opposite direc?tion of rotation to that of“vortex 1”will disturb the periodic properties of flow evolution，thus the fish cannot extract energy from the drifting vortex.

Last，the contribution of the four components of Kármán gait on fish swimming performance is as?sessed. The locomotion is demonstrated by utilizing the four traveling wave equations in the wake of the cylinder atRe=100. The fixed fish and the Kármán gait withφof 0° are considered as the two compari?sons. We also calculateT(5T) andP(5T) for each motion in Table 4. From the results （see Fig.10），we conclude that propulsion mainly origi?nates from lateral translation. However，it requires a greater energy consumption to hold the locomo?tion. Rotation，undulation，and head motions coop?erate with each other to reduce energy consumption of lateral translation motion. Therefore，hydrody?namic performance in fish swimming combines high propulsion and the desired locomotion cost. Four components work together to help fish achieve the desired hydrodynamic performance.

Table 4 T(5T )and P(5T )for different states

Table 4 T(5T )and P(5T )for different states

State Stationary Kármán gait Translation Rotation Undulation Head images/BZ_116_1409_2005_1442_2054.pngT(5T )-0.191 0.232 0.259-0.112-0.323-0.206 images/BZ_116_1409_2005_1442_2054.pngP(5T )—1.954 2.491 0.117 1.135-0.021

Fig.10 Time histories of CT for translation, rotation, undulation and head motion and the fixed fish and the Kármán gait as two comparisons

4 Conclusions

Based on IB-LBM，a flow solver that flexibly copes with complex deformable moving boundaries and controls the computational cost is developed to solve the problems of fish swimming in the vortex environment. Controlling the Kármán vortex street and the fish locomotion to model the nonlinear dy?namic system，the hydrodynamic mechanism of the fish exploiting vortices to improve their swimming performance is discussed in terms of propulsion and energy consumption. Numerical results illustrate the reason for the experimental conclusion that the fish adjusts the frequency of vibration to the frequency of the incoming vortex is that the moving vortex with the same frequency and rotation direction induced by the fish head will enhance the intensity of the in?coming vortex. The movement tendency of the fish tail towards the“merged vortex”determines the fish to extract energy from the vortex. Four compo?nents of the Kármán gait locomotion work together to help fish obtain the optimized propulsion based on a desired energy consumption.

The proposed flow solution only involves the effect of the deformable boundary on the flow evolu?tion，that is，the moving boundary is actively con?trolled by the kinematics analytical model. For fluidstructure interaction dynamics，the effect of hydro?dynamic force on the fish locomotion should be con?sidered. In future work，we will develop a flow solu?tion model that considers the influence of hydrody?namic force on the kinematic model of fish，and an?ticipate to provide insights for engineering applica?tions.

AcknowledgementsThis work was supported by the Prior?ity Academic Program Development of Jiangsu Higher Edu?cation Institutions（PAPD）. The authors realize that the time available for a review of such an ambitious subject are limited and，thus，regretfully，we are unable to cover many important contributions. The authors would like to acknowl?edge the following people for their assistance：GAO He，YANG Limin，and LI Longfei. They are members of the Key Laboratory of Unsteady Aerodynamics and Flow Con?trol of Ministry of Industry and Information Technology，Nanjing University of Aeronautics and Astronautics.

AuthorsMs. TONG Ying is a Ph.D. candidate in the De?partment of Aerodynamics，Nanjing University of Aeronau?tics and Astronautics（NUAA），majoring in fluid mechan?ics. Her research areas include computational fluid dynam?ics，immersed boundary lattice Boltzmann method，bionic propulsion，and fluid-structure interaction dynamics.

Prof. XIA Jian received his B.S. and Ph.D. degrees in aero?space engineering from NUAA, Nanjing, China, in 1992 and 1998, respectively. He worked as a post?doctoral re?searcher in College of Mechanical and Aerospace Engineer?ing at University of California Irvine and a visiting scholar at the University of Texas at Arlington before assuming his cur?rent position at NUAA. His research areas include computa?tional fluid dynamics, fluid?structure interaction, aerodynam?ic optimization design, and calculate aeroacoustics.

Author contributionsMs. TONG Ying designed the study，conducted the analysis，interpreted the results，and wrote the manuscript. Prof. XIA Jian contributed to the discussion and background of the study. Dr. CHEN Long complied the models. Mr. XUE Haotian contributed to the literature collection and data post-processing. All authors commented on the manuscript draft and approved the submission.

Competing interestsThe authors declare no competing interests.