Fluid dynamics of a self-propelled biomimetic underwater vehicle with pectoral fins

Ningyu Li ,,Jiyun Zhung ,,,Yzhou Zhu ,,Gungsheng Su ,Yumin Su

a Science and Technology on Underwater Vehicle Laboratory,Harbin Engineering University,Harbin,Heilongjiang 150001,China

b School of Naval Architecture & Ocean Engineering,Jiangsu University of Science and Technology,Zhenjiang,Jiangsu 212003,China

Abstract Fluid dynamics of a self-propelled biomimetic underwater vehicle (BUV) with pectoral fins is investigated by an immersed boundary(IB) method.Typically,the BUV with a pair of pectoral fins starts from rest and attains a constant mean velocity as the mean longitudinal force is zero.The capability and accuracy of the IB method to deal with the interaction between the fluid and complex moving body are firstly validated.Then we carry out a parametric study to understand the effect of key governing parameters on the dynamic response of the BUV.It is found that with the increase of motion frequency or rolling amplitude,the pectoral fin propulsors can induce larger forward velocity so that the BUV takes less time to attain its stable periodic swimming state.Although the pectoral fin is a very complicated lifting surface,a linear relationship between forward Reynolds number (final swimming velocity is used as velocity scale) and frequency Reynolds number (product of motion frequency and fin chord length is used as velocity scale) can be established when the frequency Reynolds number is above a critical value.A linear relationship between forward Reynolds number and rolling amplitude is also found within the studied range of rolling amplitude.Furthermore,a small-density-ratio BUV is sensitive to the surrounding flow with more rapid evolution process of self-propulsion.Whereas,BUV with a large density ratio is more stable.The implications of the hydrodynamic analysis on the bio-inspired engineering design of BUV with pectoral fins are also discussed.© 2020 Shanghai Jiaotong University.Published by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Keywords: Fluid dynamics;Self-propelled;Underwater vehicle;Pectoral fin;Immersed boundary method;Biomimetic.

1.Introduction

Despite impressive innovations in underwater vehicles,both the military and scientific communities hope to benefit from high-performance vehicles,and the biomimetic propulsion system that applies principles abstracted from fish swimming has been increasingly used in the propulsion of underwater vehicles [1-6].The swimming categories of fish are usually named after the body and/or caudal fin propulsion mode and median and/or paired fin propulsion mode [7].Propulsion by pectoral fins fall in the category of the latter.Due to the relatively simple design and greater thrust,oscillating caudal fin has attracted an increasing interest from researchers 1,5,6,8-11].On the other hand,in nature,many kinds of fishes achieve agile maneuvering,thrust generation and forward swimming motion by the use of pectoral fins,such as rajiform,diodontiform and labriform fishes.As an extreme case,it should be noted that bird wrasse uses its pectoral fin as its sole propulsor as its tail fin is degraded.Additionally,turtles and penguins use pectoral finlike flippers to propel their bodies forward.Akhtar et al.[12]have found that vortex shedding from the upstream fin is capable of improving the thrust of the downstream tail fin.High propulsive efficiency of 89% is associated with Strouhal number within the optimal range (0.2 ~ 0.4) for manta that swims by its pectoral fins in Fish et al.[13].It should be noted that the optimal range of Strouhal number can control in non-homogeneous fluid referred to the study by Barak et al.[14].The wide application of underwater vehicles has promoted studies on the hydrodynamics of pectoral fins[15-17].

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Several studies focus on the thrust production and propulsive efficiency of the flapping foil,which can be seen as a simplified model of the pectoral fin.In this context,research is carried out by a tethered foil in a constant velocity free stream,e.g.,the foil is limited to a prescribed heaving and/or pitching motion without free movement in a longitudinal direction.Pedro et al.[18]used an arbitrary Lagrangian-Eulerian method to investigate the efficiency of a two-dimensional (2D) flapping foil and analyzed the effect of some parameters on the foil efficiency.Liu [19]have found that each foil of the double-foil propulsor produces greaterthrust than the single-foil configuration under the condition of moderate to heavy load.Dong et al.[20]used an immersed boundary (IB) method to study the effect of aspect ratio on vortex structures and hydrodynamic characteristics of flapping foils.Li et al.[21]developed a calculation approach for the efficiency of three-dimensional (3D) flapping foils and investigated the effects of kinematic parameters on the thrust and efficiency.For more studies on flapping foils in a uniform incoming flow,the reader is referred to Akhtar et al.[12],Triantafyllou et al.[22],He et al.[23],Esfahani et al.[24],Liu et al.[25]and Lu et al.[26].Summing up these studies,for a 2D foil,the wake is dominated by an inverse Kármán vortex street,but for a 3D foil it is dominated by two sets of oblique vortex rings.

Given that pectoral fins represent a more complex lift surface than flapping foils,which have been examined previously,it is expected that pectoral fins would show special hydrodynamic characteristics.In recent years,some studies on pectoral fins have been devoted to the labriform swimming mode.These investigations focus on the pectoral fin with prescribed rotational motion in a uniform incoming flow.Suzuki et al.[27]analyzed experimentally the propulsive performance of a robotic pectoral fin and the effect of kinematic parameters on its load characteristics.Shoele and Zhu[28]used a boundary element method to study the force production and fluid-fin interaction of a biomimetic pectoral fin during labriform swimming.For more researches on pectoral fins in a constant velocity free stream,the reader is referred to the papers [29-32].

In addition to the aforementioned,most of the existing researches on biomimetic pectoral fin propulsion either assume that the pectoral fin is immersed in a non-zero flow field,or deal with the fin performance in steady forward motion.These investigations have provided some insights into the fluid dynamics of fishlike aquatic locomotion,but are limited by the assumption of prescribed constant inflow/swimming speed.Numerical simulations based on this assumption miss one key characteristic of real aquatic locomotion,i.e.,the swimming speed is rarely constant,considering the fact that the average value of the hydrodynamic force applied on swimming object is not always zero.Moreover,the study on the fluid dynamics starting from a zero-initial-speed has some practical applications,such as the sudden movements in fish,self-propelled swimming of biomimetic underwater vehicle (BUV),etc.Therefore,more recently,attention has been drawn to the self-propelled locomotion under a coupled interaction between swimming object and viscous fluid around it,where the swimming speed is not prescribed first rather it is obtained as a solution.However,many studies in this area are concentrated on self-propelled oscillating foils/fins without any adjacent body.The investigations by Alben and Shelley[33],Lu and Liao [34]and Hu and Xiao [35]show that the symmetric flow at left and right edges of the foil with a heaving motion will break over a critical flapping frequency,producing a net thrust in the translational direction,and therefore resulting in a self-propelled locomotion.Hua et al.[36],Yeh and Alexeev [37],Lee and Lee [38],Olivier and Dumas [39],Pan et al.[40]and Zhu et al.[41]have examined the effect of flexibility on the self-propulsion of plates/foils,including propulsive velocity,passive pitching,vortex structures,etc.For more studies on self-propelled oscillating foils/fins,the reader is referred to the publications [42-44].The researches mentioned above provide some insights into the hydrodynamics of oscillating-based locomotion by fish fins,but are limited by the absence of the adjacent body.A self-propelled freeswimming foil/fin is unphysical,and it should be attached and propel a body that produces drag.A foil/fin that is propelling a body,compared to an isolated one,would be in a state of larger thrust.Thus,computations of flow past whole fish or BUV with robotic fins are desirable.Currently,fluid dynamics of a BUV self-propelled by pectoral fins is still open for study.Moreover,systematic parametric study of the dynamic response of BUV is badly needed to guide the practical application,since keeping one of the parameters constant while changing others in a repeatable manner is hard to achieve in real fish experiments [45-48].

In this paper,we perform a comprehensive study on dynamical behaviors of a micro BUV propelled by a pair of pectoral fins.Our attention is firstly focused on the acceleratingcruising process,where the BUV accelerates from static state to steady cruise.Following this,since the hydrodynamic performance strongly depends on kinematic and structural parameters,we further discuss the effects of motion frequency,rolling amplitude and density ratio.This allows us to address the practical question of how the dynamic response of the micro BUV with biomimetic pectoral fins is expected to change with changes in the above key parameters.It should be noted that the current numerical study addresses the limitation of most existing researches on self-propelled foils/fins which is the absence of the adjacent body,and the limitation of experiments on freely swimming fish due to the inability to vary key parameters systematically with live fish.Finally,we comment on the overall hydrodynamics of the self-propelled BUV with pectoral fins and the corresponding implications for the biomimetic engineering design.

Fig.1.The computational model of the pectoral fin.

2.Problem description and numerical method

2.1.Computational model and IB method

The pectoral fin of the BUV is modeled on the basis of the observed pectoral fin of black bass (Micropterussalmoides),as shown in Fig.1 with a maximum chord lengthCof 0.155 m and an aspect ratio of about 1.21.Black bass is selected for the present study because its pectoral fin is large enough to observe by a video camera and this kind of fish is easy to breed in an aquarium.Furthermore,the bass has been described as a generalist blessed with the functions of accelerators,cruisers and maneuvers in the study by Webb 49,50].

The low-aspect-ratio pectoral fins perform prescribed rolling and pitching motions.The BUV propelled by a pair of fins can freely swim forward in the longitudinal (x) direction,and therefore 3D effects and the fluid-body interaction are well embodied in this study.The length,width and height of the BUV main body areL=5.65C,W=1.15CandH=2.86C,respectively.The prescribed rolling motion of the fin (aroundxaxis) is defined as

whereφAis the rolling amplitude,fis the motion frequency of the fin andtis the time.Based on the experimental observation and analysis of the bass pectoral fin,Kato et al.27,51,52]and Wang et al.[17]have used a sinusoidal motion pattern to model the fin motion and investigated both experimentally and numerically the hydrodynamic performance of isolated single fin Herein we employ a pair of pectoral fins with sinusoidal motion pattern to propel a BUV.The prescribed pitching motion of the fin (aroundzaxis) is defined as whereθAis the pitching amplitude,andψis the phase angle between rolling and pitching.

The study of self-propelled locomotion 31,53]involves complex geometry and kinematics.IB methods make it possible to simulate flows with complex moving boundaries on fixed Cartesian grids (referred to the review paper [54]),and they have been given increasing attention in the research field.The IB method used in this work has been described in detail by Li and Su [55].Herein,we briefly describe some salient features of this numerical method.The equations governing this flow are the 3D unsteady,incompressible Navier-Stokes equations:

whereuis the fluid velocity,pthe pressure,ρthe fluid density andμthe dynamic viscosity.The above equations are discretized in the framework of a cell-centered arrangement of the primitive variables (u,p).CVandCSdenote the controlvolume and control-surface,respectively,andnis the unit vector normal to the control-surface.

A fractional step method with second-order accuracy,a variant of the projection method in [56]is employed for advancing the flow equations from time levelnton+1.The first sub-step of the method is to solve for an intermediate velocity field.The semi-discrete form of the fluid momentum equation is expressed as

where the second-order Crank-Nicolson scheme [56]is used,andΔtdenotes the time-step size,andU*andu*are the intermediate face-center velocity and cell-center velocity,respectively.The second sub-step of the method is the following pressure correction step:

wherep′=pn+1-pndenotes the pressure correction.The final face-center velocity at then+1 time level satisfies the following integral mass conservation equation:

Thereby we obtain the integral Poisson equation for the pressure correction given by

Fig.2.Grid distribution: (a) surface mesh with triangular elements used to represent the BUV with pectoral fins and (b) Cartesian volume grid of the flow field with IB.

Oncep′is calculated by numerically solving Eq.(7),the face-center velocity and cell-center velocity are updated respectively as

wherefcandccrepresent face-center and cell-center,respectively.

A multi-dimensional ghost-cell method is employed to impose the effect of the boundary on the flow,and the ghostcell equations which make the velocity and pressure boundary conditions prescribed are solved in a fully coupled manner with Eq.(3) for the neighboring fluid-cells.The unstructured mesh with triangular elements is used to discretize the surfaces of the BUV with pectoral fins,as shown in Fig.2 (a).This geometric representation is very fit for the wide variety of biological flow configurations,since it is flexible enough to handle arbitrarily complex boundaries.The unstructured surface grids of the body and fins are then “immersed” into the Cartesian volume mesh of the flow field,as displayed in Fig.2 (b).

In boundary-motion case,we need to move the boundary from its current location to a new location at every time-step.This is achieved by moving the nodes of the surface triangles,as what has been done in Li and Su [55].The following formula is used to update the position of a surface element vertex:

whereris the position vector of the vertex andupthe vertex velocity.

For non-self-propelled flapping foils/fins,the vertex velocity can be fully obtained through the equations of prescribed motions.However,in the current study,the BUV motion is coupled to the fluid,which adds to the complexity of the numerical simulation.The fin motion (taking the pectoral fin on the right side of the body for instance) is explained as

whererjointis the position vector of the joint,where the fin is linked to the body,ubis the forward velocity of the BUV,andiis the unit vector in the longitudinal (x) direction.The angular velocity of the pectoral finωis calculated using the angular velocity components of the rolling and pitching motions,andas

The motion of pectoral fins on both sides of the body is symmetrical about the midline of the body.The forward velocity of the body including fins as a whole is determined by the fluid force produced by the fluid-body interaction,and the corresponding ordinary differential equation is written:

wherembis the mass of the BUV andFxis the fluid force in the longitudinal (x) direction.Then Eq.(13) is integrated to obtain the forward velocity,

2.2.Parameter definition

The main kinematic and dynamic parameters are illustrated in this section.To quantify the prescribed fin motion,a frequency Reynolds number 34,43]is defined as

To quantify the induced forward velocity of the whole BUV,the forward Reynolds number is defined as

whereρbdenotes the density of the pectoral fin.The nondimensional forward velocity is calculated by

The longitudinal (x) force coefficient (including the net thrust produced by pectoral fins and the body drag) is expressed as

Table1 Results of grid and time-step sensitivity study at γ=200,σ=1 .0,φ A=30 °and θA=30 °.

whereABUVis the surface area of the BUV.

2.3.Sensitivity study and validation of the numerical method

2.3.1.Gridandtime-stepsensitivitystudy

The computational mesh consists of two parts: the BUV surface mesh with unstructured triangular elements and the flow field volume mesh with structured cuboidal elements,as displayed in Fig.2.The computational domain has dimensions of 10L(X) × 9L(Y) × 3.6L(Z).Based on our simulation experience and after conducting test simulations,we believe that such a computational domain is large enough to eliminate the effect of the boundaries of the flow field on simulation results.Three grid slices (perpendicular toX,YandZ,respectively) of the domain are plotted in Fig.2 (b),which present the gradual increase of the grid size from the BUV.Three different meshes are employed for independence study: from a coarse mesh (4.86 million elements) to a nominal mesh (9.79 million elements) and then to a fine mesh (19.56 million elements).We have carefully designed the grids in a cuboidal region around the BUV,so that the boundary layer has high resolution and has sharp interface characteristic of IB.It should be noted that for the nominal mesh the minimum element spacing sizes in this region areΔX=ΔY=ΔZ=0.55%C.To carry out time-step independence study,three levels of timediscretization number for one fin motion periodTare tested:100,200 and 400.

As shown in Table1,the time-step size ofT/200 provides time-accurate simulation results for both mean and instantaneous computed variables.In Table1,¯Uis the nondimensional mean forward velocity,ˆUis the fluctuation amplitude of the non-dimensional forward velocity,and ˆCxis the fluctuation amplitude of the longitudinal (x) force coefficient.Moreover,simulations show that the nominal mesh with 9.79 million elements yields satisfactory accuracy in space since the difference in results between the nominal and fine mesh are quite small,with variations of 0.9% for ¯U,0.5% for ˆU,whereas ˆCxmatches closely within 0.2%.As a comparison,the coarse mesh differs by 13.1% for ¯Ufrom the fine mesh.Thus,the nominal mesh and the time-step size ofT/200 are selected in the current study.

2.3.2.ValidationoftheIBmethod

In Li and Su [55],the flow solver has been extensively validated by simulating flow past stationary sphere,oscillating cylinder and flapping foil,and comparing results with published experimental and numerical ones.In this section,we conduct two additional validations that are directly linked to the current research subject.

Fig.3.Comparison with previous experimental study by Wang et al.on a pectoral fin in a uniform incoming flow of 0.25m/s (f=0.5 Hz).

Fig.4.Comparison with previous numerical research by Hu and Xiao on a foil with an aspect ratio of 1.0 (σ=4.0 and h=0.5 C f oil).

The first validation test is carried out on a pectoral fin with prescribed coupled rotational motion in a uniform incoming flow investigated by Wang et al.[17].The fin used in their experiment features the same geometry as in the present study.The mesh in the test case has minimum element spacing around the fin of 0.55%C,and 200 time-steps per flapping cycle is adopted.The time variation of the fin thrust coefficientCTFINduring one period is plotted in Fig.3.It shows that the current numerical results match well with the experimental ones of Wang et al.[17].

The second validation test is carried out on a 3D selfpropelled foil with a prescribed heaving motion while free movement in the longitudinal (x) direction is allowed [35].This is done to validate the accuracy of the IB method for a fluid-body interaction problem.As shown in Fig.4,the forward Reynolds number of the foil (Re=ρ¯ubCfoil/μ,Cfoilis the chord length of the foil) from the IB method is compared with the results from the commercial CFD package FLUENT in Hu and Xiao [35].The heaving Reynolds number of the foil (Reh=ρ(h f)Cfoil/μ,his the heaving amplitude) is varied from 45 to 140.The present simulation results are in good agreement with Hu and Xiao’s [35]results,with only slight underestimation of Re at small Rehand slightly overestimated Re at large Reh.This is because of the difference in numerical methods employed in both the studies.

Fig.5.Accelerating-cruising process for the BUV with γ=200,σ=1 .0,φ A=30 ° and θA=30 °.

3.Results and discussion

In this section,we provide a comprehensive description of the evolution process of a BUV starting from rest due to the fluid-body interaction,followed by a discussion about the effect of key parameters on the dynamic response of the micro BUV propelled by a pair of pectoral fins.

3.1.Accelerating-cruising process

We have conducted simulations of several cycles to achieve the fully developed status.In the process where the BUV accelerates from static state to steady cruise,the time histories of the non-dimensional forward velocityUand longitudinal force coefficientCxare depicted in Fig 5.For this analysis,we focus on the case withγ=200,σ=1.0,φA=30°,θA=30°andψ=90°,and similar plots for other combinations of governing parameters (not provided here) substantially show all the qualitative trends.The BUV accelerates from rest to an asymptotic mean forward velocity of ¯U=7.788 (¯ub=0.065L/s) and fluctuates with amplitude of 0.228 (0.0019L/s) as the average thrust is compensated by the average drag.The above qualitative characteristic is in line with published studies on self-propelled swimming 6,57-60].For instance in Li et al.[6],a thunniform swimmer with an oscillating caudal fin accelerated from rest to an asymptotic mean forward velocity of ¯ub=0.278L/s and fluctuates with an amplitude of 0.0011L/s.In Kern and Koumoutsakos [57],an anguilliform swimmer with reference motion pattern accelerated from rest to an asymptotic mean forward velocity of ¯ub=0.40L/s and fluctuates with an amplitude of 0.01L/s.

3.2.Motion frequency effect

In this section,we examine the effect of motion frequency on the dynamic response of the micro BUV propelled by a pair of pectoral fins.For this analysis,the rolling and pitching amplitudes are fixed at a value of 30°,and we will investigate the amplitude effect in Section 3.3.With the variation of the frequency,the frequency Reynolds number is in a range between 120 and 280,and similar values have been used in previous researches on flapping foil dynamical behaviors 43,61,62].Herein the density ratio is fixed equal to 1.0 since fish density is close to water density,and the effect of density ratio will be studied in Section 3.4.

Fig.6.Variation of non-dimensional forward velocity with time for different frequency Reynolds numbers.For all these cases,σ=1 .0,φ A=30 ° and θA=30 °.

Fig.7.Time variation of longitudinal force coefficient for different frequency Reynolds numbers.For all these cases,σ=1 .0,φ A=30 ° and θA=30 °.

Fig.6 presents the time variation of the non-dimensional forward velocity for different frequency Reynolds numbers during the whole evolution process.It can be observed that the pectoral fin propulsors can induce faster forward velocity at largerγand the micro BUV takes less evolution time to reach the stable periodic swimming state.

The variation of the longitudinal force coefficient with time for different frequency Reynolds numbers is shown in Fig.7,where the flow around the fin has reached a periodic stable status.The fluctuation amplitude of the longitudinal force increases withγ.Furthermore,the peaks for differentγappear at the nearly same location along the time(non-dimensionalized byT) axis,i.e.,the fluctuation phase of the longitudinal force is almost independent of the change inγ.Since induced forward velocityubcan be considered as a function of longitudinal forceFx(referred to Eq.(13)),larger force can result in faster velocity and this is consistent with the variation trend of forward velocity and force plots in Figs.6 and 7.

At the fully developed state,the relationship curve of forward Reynolds number versus frequency Reynolds number is depicted in Fig.8.It is seen that the Re becomes larger asγis increased.Interestingly,Re turns into approximately linear growth inγ,whenγis over a critical value of about 200.This linear relationship between Re andγis much expected for flapping foils,which have been explored theoretically and verified 33,43,63,64].It is encouraging that the same behavior is seen for a much more complicated lifting surface.It is significant to compare our discovery to biological data.Rohr and Fish [65]have obtained fluke-beat frequency as a function of length-specific swimming velocity (¯ub/L) by experimental observation on different kinds of cetaceans.The experimental data shows that in order to increase swimming velocity,these cetaceans need to increase their beat frequency.Interestingly,the swimming velocity increases at the same rate as the beat frequency,i.e.,the frequency is found to increase linearly with the increase of ¯ub/Lby aggregating all the 267 swimming kinematic data.From view of non-dimensionality,a constant reduced frequency (f*=fL/¯ub) is maintained.In the current work,as Re increase linearly with increasingγ,together with the definitions ofγand Re Eqs.(15) and ((16)),we obtain

Fig.8.Forward Reynolds number versus frequency Reynolds number.

Fig.9.Temporal variation of non-dimensional forward velocity for different rolling amplitudes.For all these cases,σ is equal to 1.0 with a fixed γ of 280 and a fixed θA of 30 °.

For the BUV,L/Cis a fixed value,suggesting a constant reduced frequency as shown by the biological kinematic data.

3.3.Rolling amplitude effect

In the above section,we vary motion frequency while keeping motion amplitude constant to obtain differentγand analyzed the dynamic response of the micro BUV.Inspired by the measurements of pectoral fin locomotion 30,66],the rolling amplitude should be limited within a finite range.Furthermore,in terms of fluid dynamics,if the rolling amplitude is too small,the size of the vortices that are important for wake pattern and thrust production will be comparatively smaller.On the contrary,excessive rolling amplitude will lead to energy dissipation primarily occurring in the vertical direction.Herein we test rolling amplitudes in-between 20°to 60°to study the effect of rolling amplitude on dynamical behaviors.

Fig.10.Forward Reynolds number versus rolling amplitude.

The time history of the non-dimensional forward velocity for different rolling amplitudes is shown in Fig.9.With the increase ofφA,the pectoral fin propulsors can induce faster forward velocity and the micro BUV takes less time to achieve its stable periodic swimming state.A similar phenomenon has been found in the above section,where the frequency effect is investigated,i.e.,qualitatively the rolling amplitude plays a similar role of the frequency.

Following this,study is pursued on the plot of forward Reynolds number versus rolling amplitude.As presented in Fig.10,an approximately linear relationship between Re andφAis found within the studied range of rolling amplitude.This linear relationship between Re andγhas been observed in the above section as well,and the same behavior is seen from the plot of Re versusφAfor the BUV propelled by a pair of pectoral fins.Recall from Lu and Liao [34]and Vandenberghe et al.[67]that for a self-propelled flapping foil,Re is nearly linear for a fixed frequency with increasing heaving amplitude.The present BUV show similar dynamical characteristics,although the pectoral fin is a more complex lifting surface in comparison with the flapping foil.It should also be noted that since the pectoral fin inherently will pivot about the root linked to the fish body,the rolling motion used in the current work is more realistic than the heaving motion used in those previous studies on the flapping foil.In contrast,the biological data of Rohr and Fish [65]do not show agreement with Lu and Liao [34],Vandenberghe et al.[67]and the present study.The experimental observation of Rohr and Fish [65]indicate that in order to increase swimming velocity,these cetaceans raise their fluke-beat frequency while keeping constant beat amplitude.This might be done for decreasing hydrokinetic energy loss induced by the increase of beat amplitude.Probably,this is also associated with some non-hydrodynamic factors,e.g.,it is difficult to achieve largeamplitude heaving or rolling in a narrow space or complex environment,and the approach to increase swimming speed by increasing frequency has a lower risk of injury compared to increasing amplitude,etc.

3.4.Density ratio effect

Previous researches on self-propelled flapping foils have shown that the foil evolution process depends on density ratio 33,43].We now move on to the effect of density ratio on dynamical behaviors of the micro BUV and for this discussion,density ratios selected are 1.0,2.0,4.0 and 8.0.This range is generally representative,since fish density is close to water density and the ratios of the densities of common engineered materials (e.g.,aluminum and steel) to water density are within the range of density ratios studied here.

Fig.11.Variation of non-dimensional forward velocity with time for various density ratios.For all these cases,γ=200,φ A=30 ° and θA=30 °.

Fig.12.Forward Reynolds number as a function of density ratio for different frequency Reynolds numbers.For all these cases,φ A=30 ° and θA=30 °.

Fig.11 shows the time variation of the non-dimensional forward velocity for various density ratios.Clearly it is seen that for large values of density ratio,the micro BUV takes more evolution time to attain the periodic stable status and the velocity profile has smaller fluctuation.This shows that the micro BUV with large density ratio is not as sensitive to the viscous fluid past it compared to small density ratio BUV.The micro BUV is more stable at largeσ,however it is relatively less flexible from the maneuverability aspect.If the BUV is designed to work flexibly in a narrow space or complex environment,or to speed up in a relatively short period of time,a smallσwould be a good choice.If the BUV is designed to work steadily in a relatively wide space,or to cruise in a long voyage,a largeσwould be a good choice.

More simulations are conducted for the micro BUV with different density ratios and frequency Reynolds numbers,as presented in Fig.12.Results show that Re remains constant nearly for a fixedγregardless of density ratio,and similar finding holds for the micro BUV with either large or smallγ.Additionally,under the same density ratio,Re increases withγ.

4.Conclusions

Numerical simulations are firstly carried out to study the fluid dynamics of a self-propelled BUV with pectoral fins.The fins undergo prescribed rolling and pitching motions in the viscous fluid.Modelling and simulation of the complex interaction of the BUV with surrounding viscous flow is achieved by the IB method,in which the induced selfpropulsion is purely determined by unsteady fluid forces under the fluid-motion coupling.

The computational results show that during the process of accelerating-cruising,the forward velocity increases gradually and eventually fluctuates with small amplitude aroundVsas the mean longitudinal force is zero.This qualitative characteristic is in agreement with previous investigations [57-60]on self-propelled locomotion.

Numerical simulations are also carried out to discuss the effect of key governing parameters on the dynamic response of the micro BUV with a pair of pectoral fins.With the increase of motion frequency,the pectoral fin propulsors can induce larger forward velocity and the BUV takes less time to attain its stable periodic swimming state.Interestingly,the forward Reynolds number as a solution of self-propulsion has approximately linear growth withγ,whenγis over a critical value of about 200.It is encouraging that the linear relationship between Re andγis seen for a much more complicated lifting surface than a flapping foil surface.Therefore,when we develop a biomimetic pectoral fin,γshould be larger than the critical value to make sure that the relationship between Re andγlies within the rapid linear growth region.

The current study shows that the rolling amplitude qualitatively plays a similar role of motion frequency.An approximately linear relationship between Re andφAis found within the studied range of rolling amplitude.

The evolution process becomes slower with increasing density ratio,however final mean swimming velocity remains constant nearly for a fixedγregardless of density ratio.The lighter BUV is more sensitive to the flow field around it than the heavier one.Thus,for the design of the BUV with robotic pectoral fins,we should comprehensively consider the maneuverability and stability to select a suitable density ratio.

IB methods have advantages over conventional body-fitted grid methods in biomimetic flow simulations with complex moving boundaries,and this has been demonstrated by the present study and a number of published studies16,20,55,58,68-73].However,more advanced algorithms for solving differential equations,such as reproducing kernel algorithm 74-76]and residual power series method [77],could be used in biomimetic flow simulations in the future.

This study contributes to the hydrodynamic analysis of the dynamic response of the BUV propelled by a pair of pectoral fins.In the current work,the transverse force generated by the left fin is cancelled by that generated by the right fin,together with the symmetry of the body,thus leading to no transverse and yawing motion.It is ideal to consider more degrees of freedom for simulations of flow past the BUV.This will be one of our future research subjects.

Declaration of Competing Interest

None.

Acknowledgments

This work has been supported by National Natural Science Foundation of China (Grant Nos.51809059,51709136);Project funded by China Postdoctoral Science Foundation(Grant No.2018M631915).The authors are grateful to the anonymous referees and to the editor for their suggestion and help in significantly improving the manuscript.The authors also thank Sharath Yerneni at the University of Michigan for checking and modifying the language errors.