,

(State Key Laboratory of Hydrodynamics,China Ship Scientific Research Center,Wuxi 214082,China)

Abstract:The ground wall effect on the propulsive performance of a flapping foil was numerically investigated by using overlapping grid method at a moderate high Reynolds number of 1.0×104.The numerical results show that both the fluid dynamics and flow structures of the flapping foil are tremendously affected by the ground wall.When the hydrofoil is arranged close to the ground wall,a distinct thrust enhancement(at high St number)and a lift reinforcement can be acquired,compared with the case without ground wall.Moreover,a crescent vortex loop was observed as the result of mutual interaction between the flapping foil and the ground wall.

Key words:flapping foil;ground wall effect;overlapping grid;propulsive performance;vortex structure

0 Introduction

Flapping foils are a kind of foils undertaking pitching motion with respect to the spanwise axis and heaving motion along vertical direction synchronously and there often exists a phase lag between these two motions[1-2].According to Pedro’s work[3],the phase lag is often set as 90°with consideration of producing larger thrust.There is a wide range of applications for flapping foils and the most popular one is in bionic propulsion.Compared with the conventional fixed-wing air vehicle or propeller-based underwater vehicle,the shining merits of a flapping foil serving as the propulsive unit are low-noise and good maneuverability[4].The development of micro air vehicles and small unmanned underwater vehicles has led to a growing interest in the mechanism of flapping foils.Numerous studies concerning flapping foils’propulsive performance have been conducted by scholars and researchers from both home and abroad and their studying point includes analyzing the forces and flow structures of a single flapping foil or several ones[5-6],examining the role of aspect ratio of flapping foils[7-8],researching the function of body’s flexibility in flapping foil-based motions[9-10]and investigating the propulsive performance of flapping foils under steady or unsteady flows[11],etc..

The researches focus mainly on the hydrodynamic performance of flapping foils in an unbounded flow field,failing to take the ground wall effect into account.However,in the real natural world,whether they are birds,insects or fishes,their fins and the tails are connected with their bodies[12].Besides that,it often happens that the fishes or birds move next to the ground wall and the existence of ground wall will surely produce subtle effects upon their motion characteristics.Therefore,the study into the wall effect upon flapping foil’s propulsive performance shall be much helpful in providing physical insight into the mechanism hidden in those high efficiency creatures.

However,current existing studies concerning the ground wall effect on flapping foils are very limited.The pioneering work was conducted by Moryossef and Levy[13]who numerically simulated a two-dimensional airfoil undergoing vertically oscillating motion near the ground.Similar work was also conducted by Gao and Lu[14]and the difference was their physical model altering from NACA foil to elliptical foil.Later on,Wu et al[15]conducted numerical simulation upon ground wall effect by using the developed immersed boundary-lattice Boltzman method at a low Reynolds number of 150 and their study concentrated on the effect of the distance between the flapping foil and the ground wall on flapping foil’s propulsive performance.Truong et al[16]measured the force behaviors and investigated the flow patterns of a single flapping foil of beetle during take-off in their experiments.Recently,Amin et al[17]experimentally investigated the wall effect on the forces exerted by the fluid on a propulsive foil.

The above works focus mainly on two-dimensional flapping foil.However,there exist significant differences in both hydrodynamic behavior and flow structure between them.Apart from that,the numerical simulation mentioned above were conducted at relative low Reynolds numbers in consideration of computation cost while the typical Reynolds number in the real world lies in the range of 104-106,which is far greater than the one studied in their work.According to Asharf and Deng’s work[18-19],the Reynolds number plays a vital role in a flapping foil’s propulsive performance.

In this article,an overlapping grid method is adopted to investigate the ground wall effect upon flapping foils’propulsive performance atRe=1.0×104.A three-dimensional NACA0012 hydrofoil executing combined motion of harmonic heaving and pitching motion is selected in this study.After specifying the Reynolds number and the phase lag between heaving and pitching as well as the motion amplitudes,including the heaving amplitude and pitching amplitude,the effects of vertical distance between the center of hydrofoil and the ground wall and the motion frequency of the flapping foil are investigated.According to the numerical results obtained,the ground wall effect on flapping foil’s hydrodynamic forces and flow structures is investigated.

1 Numerical modelling and validation

1.1 Kinematics

As mentioned above,the physical model selected in this paper is a three-dimensional NACA0012 hydrofoil undergoing combined motion of heaving motionh(t)along theYdirection and pitching motionθ(t)with respect toZaxis with the same motion frequency,as shown in Fig.1.HereXprepresents the distance alongXdirection between the leading edge of the hydrofoil andZaxis and its value is set asXp=C/4,BandCare the span length and chord length of the hydrofoil,takingCas the characteristic length and its value is set as 0.1 m,the aspect ratio can be calculated asRA=B/C,the currentRAvalue is set as 2.0 with consideration of larger thrust force to be produced[20].

Fig.1 Physical model of flapping foil

The perspective view of flow over a flapping foil with ground wall effect is shown in Fig.2 and the combined heaving-pitching motion can be expressed as

Fig.2 Perspective view of flow over a flapping foil with ground wall effect

whererepresents the vertical distance between the center of foil and ground wall,h0is the mean distance,hmis the heaving amplitude,θmis the pitching amplitude,θ0is the mean pitching degree,ψis the phase difference between heaving and pitching motion,andfis the motion frequency.Since the main focus of this study is to investigate ground wall effect of the flapping foil,other parameters except for the flapping frequency are set constant asψ=90°,hm=0.25C,θ0=0° andθm=30°.Based on the free stream velocityU∞,heaving amplitudehmand the dimensionless frequency,Strouhal number can be defined asSt=2hmf/U∞.

1.2 Numerical method and validation

Considering that the position of the foil is not in a static state,the hydrofoil’s flapping motion can be attributed to typical moving boundary issue.The conventional method to solve this tricky problem is to adopt absolute reference frame coupling with the dynamic mesh or sliding mesh technique,which requires much computation resource and the computation is extremely time-consuming[21].The overlapping grid method is now well developed to handle the moving boundary problem,which involves two sets of grids,the background grid and overlapping grid.The former grid remains static and merely the overlapping grid undertakes the corresponding motion.The information exchange is realized through the interface between these two sets of grids.The difficulty in grid generation is reduced and the relative motion among the grids is free.Thus,an overlapping grid can conveniently simulate the foil flapping problem under different conditions,and each move does not need to regenerate a grid.A detailed description of the numerical method can be found in Bank’s work[22-23].Apparently,the latter method is feasible and stable,and is therefore utilized in the current study together with the User Defined Function(UDF)feature of the commercial software ANSYS FLUENT.

A sufficiently large 3D cuboid computation domain,setting as the background grid,is presented in Fig.3(a).The whole region is constructed using structured mesh and the domain size,takingh0=Cas example,is(x,y,z)=(15C,5C,10C).Fig.3(b)demonstrates the overlapping domain,which is also set as cuboid-like shape and the domain size is(x,y,z)=(1.5C,0.5C,2.5C),just a little larger than the hydrofoil model with the purpose of minimizing the number of grids to the best extent.As for the grid size of the background domain,a non-uniform mesh is adopted,the rotational domain,located at the center of the background domain with the size of(x,y,z)=(3C,1.5C,3.0C),which is fine and uniform with the same spacing of 0.01C,and the other grid size along three directions shows a linear growth with the growing ratio 1.08.The grid size of the overlapping domain is uniform with the same grid size of 0.01Cwith the purpose of convenient and efficient information exchange between these two sets of domains.The total grid number reaches 1.5 million.

Fig.3 Sketch of computation domain

The grid independence test results are presented in Tab.1,showing 8 numerical simulation results,including four sets of grids with different grid number and two differentStnumbers,whereCT-MeanandCL-Meanrepresent the time-averaged thrust force coefficient and lift force coefficient respectively,and the corresponding expressions can be seen in Eq.(2),whereFTandFLrepresent the thrust force and lift force experienced by the flapping foil correspondingly,Tis a complete time period,t0is random motion moment.Other parameters are set as,h0=C,U∞=0.1 m/s,time stepΔt=0.001 s.It can be easily seen from Tab.1 that the third set of mesh with a total of 1 200 000 nodes is fine enough to achieve an accurate result and is therefore chosen in the following simulation.

Tab.1 Grid independence test results

In order to further testify the accuracy of the numerical method employed in the study,the thrust force coefficient with different pitching amplitudes are also presented in Fig.4,and good agreement with the results in Ref.[24]is shown in the figure.Corresponding parameters are set asf=1.0 Hz,hm=0.30C,ψ=90°.

Fig.4 Comparison of thrust force coefficient with results of Ref.[24]

2 Simulation results

After validatation of the numerical method employed in this article,the ground wall effect on the propulsive performance of the flapping foil is analyzed in this chapter.The mean distanceh0/Cis set as 0.3,0.6,0.8,1.0 and 1.5 respectively and theStnumber is selected as 0.1,0.25 and 0.5 at each specified distance.

2.1 Hydrodynamic performance due to ground wall effect

The force behaviors of the flapping foil due to ground wall effect are presented in Fig.5.The mean value of the drag force coefficient and lift force coefficient at different vertical distances andStnumbers are shown in Fig.5(a)and Fig.5(b)respectively.To make a comparison,the force behaviors of a flapping foil with no ground wall effect is also presented and the corresponding result can be seen in Fig.5(c)and Fig.5(d).

Fig.5 Force behaviors of flapping foil with ground wall effect

As can be seen from Fig.5,the existence of ground wall produces much influence on the flapping foil’s force behaviors to a different extent under several ground wall distances.With the increase of the ground wall distance,the ground wall effect reduces gradually and when the ground wall distance exceeds 1.5C(as seen in Fig.5(c)and(d)),the influence is too tiny to be observed,thus we may roughly assume that the ground wall effect disappears ath0=1.5C.

As for the mean value of drag force coefficient(CD-Mean),there exists little difference among several ground wall distances at lowStnumbers(St=0.10),and the corresponding value ofCD-Meanhovers around zero.With the increase of theStnumber(St=0.25),the value ofCD-Meanowes a negative value,meaning that a positive thrust force is produced and the smaller the ground wall distance is,the higher the thrust force will be.That phenomenon is more obvious under a higherStnumber(St=0.50)and the value ofCD-Meanincreases by 48.5% with the ground wall distance reducing from 1.5Cto 0.3C.Similarly,the mean value of the lift force coefficient(CL-Mean)also shows a corresponding growth when the ground wall distance alters from 1.5Cto 0.3Cunder differentStnumbers and the higher theStnumber is,the faster theCL-Meanincreases.In view of this,it can be concluded that the existence of ground wall helps enhance the thrust force generated by the flapping foil and improve the lift force experienced by the flapping foil.

To further explore the difference of the flapping foil’s force behavior under different ground wall distances,the time histories of the drag force and lift force coefficients in single motion period under five ground wall distances and three typicalStnumbers are presented in Fig.6.

Fig.6 Time history of force coefficients in single motion period

As seen from Fig.(6),there exist two peaks ofCDin single motion period and the changing period ofCDis half of the motion period.As forCL,a single peak emerges and that changing period ofCLcoincides with the flapping foil’s motion period under different ground wall distances andStnumbers.This is in good consistence with that of the flapping foil in an unbounded flow.As forSt=0.10,with the ground wall distance’s decrease,the peaks and crests ofCDandCLshow a corresponding growth and the growing amplitudes are approximately equal,thus the mean value ofCDandCLshows little difference among various ground wall distances.When it comes to a higherStnumber(St=0.25),the peaks ofCDunder smaller ground wall distances show a smaller growth while the crests ofCDpresent an obvious increase,resulting in an obvious growth ofCD-Mean.As forCL,that situation has turned into the opposite side.The peaks ofCLshow much growth while the crests do not change much,making theCL-Meanpresent an evident growth with the ground wall distance altering from 1.5Cto 0.3C.The situation is quite obvious with the highestStnumber(St=0.50).

2.2 Vortex structures due to ground wall effect

Apart from the hydrodynamic performance,the vortex structures would also be affected as a result of ground wall effect.As shown in Fig.7,vortex structures at the rear part of the flapping foil withSt=0.50 under four typical ground wall distances are illustrated in three different perspectives(t=3.0T).Considering that there exists little difference in force behaviors of the flapping foil when the ground wall distance alters fromh0=∞toh0=1.5C(shown in Fig.5(c)and(d)),we roughly take the vortex structures ath0=1.5Cto approximate the corresponding flow patterns out of ground wall effect.

Fig.7 Vortex structures at rear part of flapping foil at different perspectives(St=0.50)

As can be seen in Fig.7(a),two complete vortex loops emerge at the rear part of the flapping foil and the vortex loop moves toward the center line of the flapping foil.The vortex length along the flow direction is similar to that of the flapping foil while the span length of the vortex loop shows a gradual decrease and that phenomenon is similar to that of the flapping foil in an unbounded flow,as reported in Ref.[25].With the ground wall distance’s decrease(shown in Fig.7(b)),the vortex structures show a distinct difference.Instead of approaching to the center line,the vortex shows a gradual deviation from the center line for the sake of the ground wall distance’s existence and the span length of the vortex loop is slightly longer than that of the flapping foil while the vortex length along the flow direction remains unchanged.Besides that,it can be also found in the side perspective that the vortex loop at the lower part has been compressed to be oblate as a result of the ground wall’s constraint.As the ground wall distance keeps decreasing(0.6Cand 0.3C),the vortex loops substantially deviate from the center line with the chord length,and the span length of the vortex loop presents a corresponding increase,finally forming a shape of a crescent.Similarly,the vortex loop at the lower part has been compressed more severely.Consequently,there is an obvious angle between the ground and the center line of the vortex street,that phenomenon is similar to what was reported in Cheng and Luo’s work[26].

3 Conclusions

The propulsive performance of a three-dimensional flapping foil is numerically investigated in this paper with consideration of ground wall effect.To carry out the simulation,the overlapping grid method is adopted in conjunction with the User Defined Function(UDF)feature of the commercial finite-volume code ANSYS FLUENT,and the Reynolds number is chosen as 1.0×104,which is extremely close to that in real nature.After specifying the motion amplitudes and phase lag,we concentrate on the investigation of the distance between the ground wall and the flapping foil together with the motion frequency.Here we briefly summarize the results reached above.

(1)The force behaviors of flapping foils,including drag force and lift force,are tremendously affected by the ground wall.When foils are placed in close proximity to the ground wall,a distinct thrust enhancement(at a highStnumber)and a lift reinforcement can be acquired,compared with the case without ground wall effect.

(2)The vortex structure at the rear part of a flapping foil is also strongly influenced by the ground wall.With the decrease of the ground wall distance,the vortex loop starts to deviate from the center line of the flapping foil and the corresponding span length shows an obvious growth,finally forming a shape of a crescent.