Study on the Hydrodynamic Performance of Typical Underwater Bionic Foils with Spanwise Flexibility

applied sciences

Article Study on the Hydrodynamic Performance of Typical Underwater Bionic Foils with Spanwise Flexibility

Kai Zhou 1,* ID , Junkao Liu 2 and Weishan Chen 2 1 College of Mechanical and Electronic Engineering, Shandong Agricultural University, Taian 271018, China 2 State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China; [email protected] (J.L.); [email protected] (W.C.) * Correspondence: [email protected]; Tel.: +86-451-8641-7891 (ext. 123); Fax: +86-451-8641-6119

Received: 22 September 2017; Accepted: 26 October 2017; Published: 30 October 2017

Abstract: Bionic foils are usually similar in shape to the locomotive organs of animals living in fluid media, which is helpful in the analysis of the motion mode and hydrodynamic mechanisms of biological prototypes. With the design of underwater vehicles as the research background, bionic foils are adopted as research objects in this paper. A geometric model and a motion model are established depending on the biological prototype. In the model, two typical bionic foils—a NACA foil and a crescent-shaped foil—are chosen as research objects. Simulations of the bionic foils are performed using a numerical method based on computational fluid dynamics software. The hydrodynamic forces acting on the foils and flow field characteristics behind the foils are used to analyze the propulsion performance and hydrodynamic mechanism. Furthermore, a spanwise flexibility model is introduced into the motion model. Next, the hydrodynamic mechanism is further analyzed on the basis of hydrodynamic forces and flow field characteristics with different spanwise flexibility parameters. Finally, an experimental verification platform is designed and built to verify the reliability of the numerical results. Agreement between the experimental and numerical results indicates that the numerical results are reliable and that the analysis of the paper is reasonable.

Keywords: bionic foil; propulsion performance; biological swimming; ﬂow ﬁeld characteristics

1. Introduction In nature, through natural selection over thousands of years, creatures have developed the ability to swim or fly in fluid media such as air or water. In contrast, the propulsion performance of conventional screw propellers, which are widely used, is poor in comparison with that of animals. Therefore, a study of the propulsion modes of swimming or flying creatures has important significance to underwater propulsion. In the mechanical discipline, biomimetics has become a research focus [1]. Moreover, biomimetics is of vital importance in underwater exploration, ocean exploitation, national security and other important domains. Through technological advancement, different bionic propulsors have been successfully developed [2–4]. Unfortunately, the propulsion performance of these bionic propulsors cannot match that of their biological prototypes, which highlights the necessity of further study of propulsion mechanisms. The mechanism research plays an important role and will boost the development of the principle prototype. To reveal the mechanism of high-efficiency propulsion, scholars at home and abroad have devoted considerable effort with different research methods [5–7]. However, both the prototype-based experimental results and the numerical results achieved are far from the expected performance. Taking the high-efficiency propulsor developments as the research background, this paper chooses typical foils as the research objects and focuses on the propulsion mechanism of bionic foils. In decades past, computer technology has developed immensely. Accordingly, numerical simulation based on computational fluid dynamics (CFD) has been widely used in studies of biological propulsion.

Appl. Sci. 2017, 7, 1120; doi:10.3390/app7111120 www.mdpi.com/journal/applsci Appl. Sci. 2017, 7, 1120 2 of 15

New numerical methods were proposed during the past two decades and the calculation precision has been improving [8]. Compared with other methods, the flow field characteristics and hydrodynamic forces at any time can be easily extracted. Such flow field characteristics are important in the analysis of hydrodynamic mechanism. Furthermore, the numerical simulations require less time and can be easily conducted. Therefore, the CFD method is widely adopted to solve the Navier-Stokes equations, which has become an important method in the visual simulation of fluids. However, in biofluid dynamics research, fluid-structure interaction (FSI) problems are universal [9–11]. For this kind of problem, interactions exist among the viscous fluid, deformable body and free-moving boundary [12–14]. Therefore, these problems are difficult to solve. The previous studies of underwater biological propulsion usually focused on the locomotion of biological prototypes [15,16]. And biological observation was usually used in these studies. These biological prototypes, like tunas, usually use the common “body/caudal fin” swimming mode and have advantages of high propulsive efficiency and high speed. In these studies, detailed flow field characteristics or hydrodynamic forcescannot be provided. Through technological advancement, numerical simulation of underwater biological propulsion was conducted by scholars. Different biological prototypes were studied as research subjects. The detailed flow field characteristics and hydrodynamic forces can be provided with numerical simulations. For example, Borazjani and Sotiropoulos carried out fluid-structure interaction simulations to investigate the hydrodynamic mechanism in carangiform swimming [15,16]. However, the experimental method is still necessary because of the accuracy of numerical calculation. Besides that, the numerical simulation of self-propulsion is usually difficult to conduct. So, the swimming velocity of biological prototypes was usually set as a constant in previous numerical studies [17,18]. To simplify the problem of biological propulsion, the wings or fins of animals living in fluid media can be abstracted to bionic foils [19–22]. Study of the hydrodynamic mechanism of bionic foils would help to reveal the hydrodynamic mechanism of biological prototypes [23–25]. Additionally, bionic foils can easily be applied to the existing underwater propellers. Related research has been conducted aiming at the problem. However, the foils were usually modeled in two dimensions or simplified to simple three-dimensional shapes [26–28]. Li et al. mainly studied the wake topology of forked and unforked fish-like plate with numerical method [10]. Izraelevitz et al. studied experimentally the effect of adding an in-line oscillatory motion to the oscillatory heaving and pitching motion of flapping foils that use a power down stroke [19]. In addition, the foils in previous studies were usually regarded as rigid bodies. As is widely known, the locomotive organs of animals are flexible. The flexibility would influence the propulsion performance. Especially, spanwise flexibility merits more attention and further study because of the high aspect ratio of the bionic foil. As is shown in [29], a water tunnel study of the effect of spanwise flexibility on the thrust, lift and propulsive efficiency of a rectangular foil oscillating in pure heave was performed. Zhu et al. numerically examined the performance of a thin foil reinforced by embedded rays resembling the caudal fins of fish [30]. The numerical studies of spanwise flexibility on complex shaped foils were few. In [31], the influences of the motion parameters on the propulsion performance of a bionic caudal fin were analyzed by extracting performance parameters. In this paper, a more in-depth analysis is conducted aiming to further reveal the influences of spanwise flexibility by extracting the vortex patterns in the wake. To investigate the propulsion mechanism of bionic foils, this paper establishes a numerical model of propulsion of bionic foils and proposes a corresponding numerical method. First, according to the biological observations, a geometric model and a kinematics model of bionic foils are established. Next, the proposed numerical method is used to accomplish numerical simulations and study the propulsion mechanism of a NACA foil and a crescent-shaped foil. Additionally, a self-propulsion model of a thunniform bio-inspired model is adopted to verify the numerical model and solution approach and to further analyze the hydrodynamic mechanism of the thunniform mode. After that, the effects of spanwise flexibility on the propulsion performance are studied. By extracting the flow field characteristics in the wake and hydrodynamic forces acting on the foil, the hydrodynamics Appl. Sci. 2017, 7, 1120 3 of 15 mechanism of these effects is analyzed. To verify the reliability of the numerical results, verification experimentsAppl. Sci. 2017, 7 are, 1120 conducted. 3 of 15

2. Materials and Methods

2.1. Geometric Model The thunniform mode mode features features hi highgh efficiency efficiency and and high high speed. speed. Many Many fish fish species species that that have have a asingular singular ability ability to to swim swim adopt adopt this this mode, mode, such such as as sharks, sharks, swordfish swordfish and tuna. The The fish fish species utilizing thethe thunniform thunniform mode mode mainly mainly generate generate thrust thrust force byforce swinging by swinging their caudal their fins. caudal Specifically, fins. theSpecifically, caudal fins the achievecaudal fins thrust achieve force thrust through force a motion through that a motion contains that translational contains translational motion following motion thefollowing tail and the rotation tail and motion rotation around motion the around tail. Exceptthe tail. for Except the motion for the ofmotion the caudal of the fin,caudal the motionfin, the ofmotion the fish of the body fish existsbody exists mainly mainly in the in rearthe rear part part and and front front part part of of the the fish fish body body isis relatively rigid. rigid. Moreover, a crescent-shaped caudal fin fin is the most common shape among thunniform swimmers. For example, a crescent-shaped caudal fin fin can provid providee ninety percent percent of of the the thrust thrust force force for for a a tuna, meanwhile pectoral and pelvic finsfins play rolesroles onlyonly inin controllingcontrolling thethe attitude.attitude. A crescent-shapedcrescent-shaped caudal fin fin with a high aspect ratio is conducive to the shedding of the vortex rings, which in turn reduces thethe energy energy consumption consumption during during the sheddingthe shedding process process and achieves and achieves high propulsive high propulsive efficiency. Inefficiency. this section, In this as section, shown inas Figureshown1 in, three-dimensional Figure 1, three-dimensional geometric geometric models are models established are established with the thunniformwith the thunniform group as thegroup bionic as the prototype. bionic prototype. Fins that work Fins asthat auxiliary work as organs, auxiliary such organs, as pectoral such and as pelvicpectoral fins, and are pelvic not takenfins, are into not account. taken into The accoun geometrict. The model geometric contains model three contains parts: three head, parts: trunk head, and caudaltrunk and fin. caudal The lengths fin. The of theselengths three of these parts three are 0.28 parts m, are 0.4 0.28 m and m, 0.120.4 m m, and respectively. 0.12 m, respectively.

(a) (b)

Figure 1. Three-dimensional geometric model: (a) thunniform bio-inspired mode; (b) crescent-shaped foil. Figure 1. Three-dimensional geometric model: (a) thunniform bio-inspired mode; (b) crescent-shaped foil.

The cross-sectional shape of the geometric model is shown in Figure 2 and the curve equations are definedThe cross-sectional as follows: shape of the geometric model is shown in Figure2 and the curve equations are deﬁned as follows: 12√ 22 zx1 12 0.282 2 z1 = 35350.28 − x 543 2 zxxxx  29.245  30.624  9.4243  0.3853 2 z2 =2 −29.24x + 30.62x − 9.424x + 0.3853x  0.06764x2 0.096 (1) −0.06764x2 + 0.096 (1) zxxx  55.36322  70.42  28.35  3.64 3 3 2 2 z3 = −55.36x322+ 70.42x − 28.35x + 3.64 zxxx4  426.8 643.5 325.1 54.92 3 2 2 z4 = 426.8x − 643.5x + 325.1x − 54.92 √6 22 yx1 6 0.282 2 y1 = 35350.28 − x yxxxx 14.6254 15.31 4.712 3 0.1927 2 y = 2 −14.62x5 + 15.31x4 − 4.712x 3 + 0.1927x2 2 (2)  0.03382x 0.048 (2) −0.03382x + 0.048 1 yx3  1  0.52 y3 = − 2020(x − 0.52) Appl. Sci. 2017, 7, 1120 4 of 15 Appl.Appl. Sci. Sci. 2017 2017, ,7 7, ,1120 1120 44 of of 15 15

Figure 2. Cross-sectional shape of the geometric model. FigureFigure 2. 2. Cross-sectional Cross-sectional shape shape of of the the geometric geometric model. model.

InInIn thethe the process processprocess ofofof geometric geometricgeometric modeling, modeling,modeling, the thethe section sectsectionion shape shapeshape perpendicular perpendicularperpendicular to incoming toto incomingincoming flow is flowflow defined isis defineddefinedas elliptical asas ellipticalelliptical surface. surface.surface. In the process InIn thethe ofprocessprocess geometric ofof gege modelingometricometric modelingmodeling of the crescent-shaped ofof thethe crescent-shapedcrescent-shaped foil, firstly foil,foil,z3 function firstlyfirstly zz3and3 functionfunctiony3 function andand yy3 in3 functionfunction Equations inin Equations (1)Equations and (2) (1)(1) are andand used (2)(2) to areare construct usedused totothe constructconstruct incomplete thethe incompleteincomplete three-dimensional three-three- dimensionaldimensionalgeometry of geometrygeometry the foil, after ofof thethe that foil,foil, a curved afafterter thatthat surface aa curvedcurved defined surfacesurface by z4 defineddefinedfunction byby in z Equationz44 functionfunction (1) inin is EquationEquation used to cut (1)(1) the isis usedusedgeometric toto cutcut themodel.the geometricgeometric Finally, model.model. the three-dimensional Finally,Finally, thethe three-three- geometricdimensionaldimensional model geometricgeometric of the crescent modelmodel foilofof thethe which crescentcrescent is similar foilfoil whichwhichas the isis caudal similarsimilar fin asas of thethe thunniform caudalcaudal finfin mode ofof thunniformthunniform is completed. modemode isis completed.completed. InInIn nature,nature, nature, locomotivelocomotive locomotive organsorgans organs withwith with differentdifferent different shapshap shapeseses existexist exist inin in thethe the animalanimal animal kingdom.kingdom. kingdom. InIn In particular,particular, particular, thethethe animals animals animals that that that swim swim swim or or or fly fly fly in in in fluid fluid fluid media media media usually usually usually have have have locomotive locomotive locomotive organs organs organs with with with similar similar similar cross cross cross sections sections sections alongalongalong the the the flow flow flow direction. direction. direction. The The The force force force acting acting acting on on on locomoti locomoti locomotiveveve organs organs organs can can can be be be regarded regarded regarded as as as the the the resultant resultant resultant force force force ofofof forceforce force componentscomponents components actingacting acting onon on crosscross cross sections.sections. sections. Here,Here, Here, aa flappingflapping a flapping foilfoil foilmodelmodel model isis adoptedadopted is adopted toto representrepresent to represent thethe generalgeneralthe general formform formofof thesethese of theselocomotivelocomotive locomotive organs.organs. organs. TheThe crosscross The sectionssections cross sections ofof thethe bionic ofbionic the foils bionicfoils adoptedadopted foils adopted inin thisthis paperpaper in this havehavepaper samesame have shapeshape same asas shape thethe NACA asNACA the NACA (National(National (National AdvisoAdviso Advisoryryry CommitteeCommittee Committee forfor Aeronautics)Aeronautics) for Aeronautics) airfoilairfoil airfoil model.model. model. AsAs shownshownAs shown inin FigureFigure in Figure 3,3, thethe3, the NACA0013NACA0013 NACA0013 airfoilairfoil airfoil isis chosen ischosen chosen asas as thethe the crosscross cross section,section, section, wherewhere where LL isLis is thethe the chordchord chord length,length, length, BBB isisis thethe the lengthlength length alongalong along thethe the spanwisespanwise spanwise directiondirection direction andand and OOO isisis thethe the referencereference reference point.point. point. TheThe The referencereference reference pointpoint point isis is setset set aa a quarterquarterquarter ofof of thethe the chordwisechordwise chordwise lengthlength length awayaway away fromfrom from leadingleading leading edge.edge. edge.

FigureFigureFigure 3. 3. 3.Three-dimensional Three-dimensionalThree-dimensional geometric geometric geometric model model model of of of the the the NACA NACA NACA foil. foil. foil.

With this NACA foil as the research object, the results of the study would contribute to revealing WithWith this this NACA NACA foil foil as as the the research research object, object, the the results results of of the the study study would would contribute contribute to to revealing revealing the mechanism of biological propulsion. In addition, a propulsion system with a NACA foil can fully thethe mechanism mechanism of of biological biological propul propulsion.sion. In In addition, addition, a apropulsion propulsion system system with with a aNACA NACA foil foil can can fully fully absorb the advantages of different bionic prototypes and can be applied to existing propulsion systems. absorbabsorb the the advantages advantages of of different different bionic bionic prototypes prototypes and and can can be be applied applied to to existing existing propulsion propulsion systems. systems.

2.2.2.2.2.2. MotionMotion Motion ModelModel Model ForForFor thethe the thunniformthunniform thunniform bio-inspiredbio-inspired bio-inspired model,model, model, thethe the motimoti motiononon containscontains contains swingingswinging swinging ofof of thethe the caudalcaudal caudal finfin ﬁn andand and wavingwavingwaving of of of the the the body. body. body. Based Based Based on on on the the the geometric geometric geometric model, model, model, th th theee waving waving waving of of of the the the body body body is is is abstracted abstracted abstracted to to to a a atraveling traveling traveling wavewavewave thatthat that transferstransfers transfers fromfrom from headhead head toto to tailtail tail ofof of thethe the bodybody body withwith with graduallygradually gradually increasedincreased increased amplitude.amplitude. amplitude. TheThe The travelingtraveling traveling wavewavewave cancan can bebe be writtenwritten written as:as: as: y(yxtx, t,,sin2) = A Axt(x, t) sin(2π ff − kxkx) (3) yxt,,sin2 Axt   f kx   (3)(3) Appl. Sci. 2017, 7, 1120 5 of 15 where y is the displacement along the y direction, A is the amplitude, k is the wave number, f is the frequency and x is the x coordinate. The wave number can be given as: Appl. Sci. 2017, 7, 1120 5 of 15 k = 2π/λ (4) where y is the displacement along the y direction, A is the amplitude, k is the wave number, f is the frequency and x is the x coordinate. The wave number can be given as: where λ is the wavelength of the traveling wave of the body. The amplitude A contains two parts and the location term a1 and time term a2 can be writtenk  2/ as: (4) where λ is the wavelength of the traveling wave of the body. The amplitude A contains two parts a (x) = c + c x + c x2 ≤ x ≤ L and the location term a1 and1 time term0 a2 can1 be written2 as: 0 f (5)

2 ax1012( )t c cxcx1  2 π t 0  x Lf (5) t − 2π sin t 0 ≤ t ≤ t0 a2(t) = 0 0 (6) t ≥ t  tt121  0  sin 0 tt0 at() tt2 where Lf is the length of body, t0 is the2 startup 00 time and the c terms with subscripts are the coefficients(6)  1 tt of the amplitude envelope. The time terma2 is used to accomplish 0 a smooth start. For the crescent-shaped caudal fin and the NACA foil to imitate the forward movement of where Lf is the length of body, t0 is the startup time and the c terms with subscripts are the coefficients the bionicof the prototype, amplitude envelope. the foil movesThe time forward term a2 is at used a constant to accomplish speed a smoothU. In addition, start. the foil undergoes heave motionFor they(t )crescent-shaped and pitch motion caudalθ( fint) at and the the same NACA time. foil Theto imitate heave the motion forward is movement translational of the motion transversebionic to prototype, the direction the foil of moves propulsion forward and at a theconstant pitch speed motion U. In is addition, angular the motion foil undergoes around heave a spanwise axis. Exceptmotion fory(t) the and towing pitch motion motion, θ the(t) at translational the same time. motion The heave of the motion foil isexpressed is translational as: motion transverse to the direction of propulsion and the pitch motion is angular motion around a spanwise axis. Except for the towing motion, the transly(t) =ationalh cos motion(2π f t) of the foil is expressed as: (7) yt() h cos(2 ft ) (7) Here, h is the amplitude of the heave motion. Additionally, the tangent angle of trajectory θm can Here, h is the amplitude of the heave motion. Additionally, the tangent angle of trajectory θm can be described as: . be described as: y(t) θm(t) = arctan (8) yt U (t ) arctan  m   (8) The angle α(t) is the angle of attack and can be writtenU as: The angle α(t) is the angle of attack and can be written as: α(t) = αmax sin(2π f t) (9) ()tft max sin(2  ) (9) where αmax is the amplitude. The definitions of relevant parameters are shown in Figure4. The role where αmax is the amplitude. The definitions of relevant parameters are shown in Figure 4. The role of of heaveheave motion motion is is to to accomplish accomplish thrashthrash motion motion and and the the role role of ofpitch pitch motion motion is to isadjust to adjust the angle the angleof of thrashthrash motion. motion.

Figure 4. Definitions of relevant motion parameters. Figure 4. Definitions of relevant motion parameters. 2.3. Flexibility Model 2.3. Flexibility Model In nature, the locomotive organs of animals that swim or fly in fluid media are usually flexible. InTherefore, nature, theflexibility locomotive should organsbe modeled of animals during the that propulsion swim or of fly locomotive in fluid media organs arein the usually study of flexible. Therefore,propulsion flexibility of bionic should foils. be However, modeled the during modeling the and propulsion solving of of passive locomotive deformation organs are in difficult. the study of To simplify the problem, a defined flexible deformation model is imposed on the foil during the propulsion of bionic foils. However, the modeling and solving of passive deformation are difficult. motion. In addition, the defined flexible deformation model can be easily applied to foil propulsion, To simplifyto enhance the problem, the propulsion a defined performance flexible deformationof the system. modelBecause is of imposed the high onaspect the foilratio during of the bionic the motion. In addition,foil, spanwise the defined flexibility flexible merits deformation increased modelattention can and be further easily appliedstudy. In to the foil spanwise propulsion, flexibility to enhance the propulsion performance of the system. Because of the high aspect ratio of the bionic foil, spanwise Appl. Sci. 2017, 7, 1120 6 of 15 Appl. Sci. 2017, 7, 1120 6 of 15

flexibility merits increased attention and further study. In the spanwise flexibility model, a phase model, a phase difference γ is introduced into the definition of α(t). Bases on Equation (9), the angle difference γ is introduced into the definition of α(t). Bases on Equation (9), the angle of attack on the of attack on the edges is rewritten as: edges is rewritten as:

α1(t)1() =tftαmaxmaxsin sin(2(2 π f t − )γ ) (10)(10) Appl. Sci. 2017, 7, 1120 6 of 15 TheThe angles of of attack attack of of the the center center points points are are still still calculated calculated according according to Equation to Equation (9). This (9). Thisdefinition definitionmodel, leadsleads a tophase a tophase difference a phase difference γ is difference introduced between into between the the definition pitch the pitchof motion α(t). motionBases of on the Equation of center the (9), center points the angle points and the and edge the of attack on the edges is rewritten as: edgepoints points at the at same the same chordwise chordwise position. position. The Theshape shape description description of the of theflexible flexible NACA NACA foil foil is shown is shown in ()tft sin(2  ) inFigure Figure 5.5 With. With different different mathematical mathematical curve curve1 forms, forms,max the the coordinates ofof pointspoints atat thethe edgeedge(10) and and center center locationslocations cancan The bebe usedangles to toof construct constructattack of the a a centercurve curve points equation equation are still and and calculated using using thisaccording this curve curve to function,Equation function, (9). the theThis coordinates coordinates of ofother other points pointsdefinition at atthe leads the same sameto a phasechordwise chordwise difference position position between canthe can pitch be be calculated.motion calculated. of the Figurecenter Figure points 6 showsshows and the thethe edge shapeshape ofof the the points at the same chordwise position. The shape description of the flexible NACA foil is shown in crescent-shapedcrescent-shapedFigure 5. foilfoilWith withwith different aa typicaltypical mathematical phasephase curve difference.difference forms, the. Specifically,Specifically,coordinates of points thethe thrashthrash at the edge surfacesurface and center isis convexconvex whenwhen thethe phasephase locationsdifferencedifference can be isis positive.usedpositive. to construct Conversely,Conversely, a curve equation thethe thrashth andrash using surfacesurface this curve isis concaveconcave function, whenthewhen coordinates thethe phasephase of differencedifference isis negative.negative.other points at the same chordwise position can be calculated. Figure 6 shows the shape of the crescent-shaped foil with a typical phase difference. Specifically, the thrash surface is convex when the phase difference is positive. Conversely, the thrash surface is concave when the phase difference is negative.

FigureFigureFigure 5.5. ShapeShape 5. Shape descriptiondescription description of of ofthe thethe flexible ﬂexibleflexible NACA NACANACA foil. foil.foil.

(a) (b)

Figure 6. Shape description of the flexible crescent-shaped foil: (a) γ = 20°; (b) γ = ◦−20°. ◦ Figure 6. Shape description( ofa) the flexible crescent-shaped foil:(b) (a) γ = 20 ;(b) γ = −20 . 2.4. Numerical Solution Figure 6. Shape description of the flexible crescent-shaped foil: (a) γ = 20°; (b) γ = −20°. 2.4. NumericalThe Solution URANS (Unsteady Reynolds Averaged Navier Stokes) model is adopted in the numerical simulations. Using this model, the Navier-Stokes equations used to describe the hydrodynamics of 2.4. NumericalThe URANSfluids Solutionare restructured (Unsteady as: Reynolds Averaged Navier Stokes) model is adopted in the numerical simulations. Using this model, the Navier-Stokes equations used to describe the hydrodynamics of 2 '' The URANS (Unsteady ReynoldsUUii Averaged1 Navierp Stokes)uuij model is adopted in the numerical fluids are restructured as: UUij  (11) simulations. Using this model, the txNavier-Stokesjiijj equations x  xxx used to describe the hydrodynamics of fluids are restructured as: 2 ∂u0u0 ∂Ui ∂ U 1 ∂p ∂ Ui i j + U U =−i + ν − (11) i j  0 (12) ∂t ∂xj xi ρ ∂xi ∂x2 i∂xj ''∂xj UU1 p uuij iiUU  where ρ is the density, ν is the kinematic viscosityij and p is the pressure. The subscript i and j represent (11) txjiijj∂U   x  xxx the three coordinate directions. Here, the dependenti = variables0 are not only a function of the space (12) coordinates but also a function of time. Therefor∂xe,i the simulated velocity in URANS model can be split to the time-averaged part U and the fluctuationsU u. To close the equations, the Reynolds stresses where ρ is the density, ν is the kinematic viscosity andi 0p is the pressure. The subscript i and j represent term, the last term in Equation (11), can be expressed with the mean velocity gradient via the eddy (12) xi the three coordinateviscosity. The directions.eddy viscosity Here,can be therepresented dependent by k (turbulent variables kinetic are energy) not only and ε a (the function velocity of the space coordinateswhere ρ isand the but length density, also scale a function νof is turbulent the kinematic of time.motion), Therefore, viscositythat is the thek-andε turbulence simulated p is the model.pressure. velocity The numericalThe in URANS subscript simulations model i and can j represent be split are carried out by Fluent (ANSYS Inc., Canonsburg, PA, USA) software. The Finite Volume Method tothe the three time-averaged coordinate partdirections.U and theHere, fluctuations the dependentu. To closevariables the equations, are not only the Reynoldsa function stresses of the space term, thecoordinates last term inbut Equation also a function (11), can of be time. expressed Therefor withe, thethe meansimulated velocity velocity gradient in URANS via the eddy model viscosity. can be Thesplit eddy to the viscosity time-averaged can be representedpart U and the by fluctuationsk (turbulent u kinetic. To close energy) the equations, and ε (the the velocity Reynolds and stresses length scaleterm, of the turbulent last term motion), in Equation that is (11), the k-canε turbulence be expressed model. with The the numerical mean velocity simulations gradient are via carried the eddy out byviscosity. Fluent (ANSYSThe eddy Inc., viscosity Canonsburg, can be PA, represented USA) software. by k (turbulent The Finite kinetic Volume energy) Method and (FVM) ε (the is adoptedvelocity and length scale of turbulent motion), that is the k-ε turbulence model. The numerical simulations are carried out by Fluent (ANSYS Inc., Canonsburg, PA, USA) software. The Finite Volume Method Appl. Sci. 2017, 7, 1120 7 of 15 to discretize the Navier-Stokes equations into expressions that are first-order in time and second-order in space. The tetrahedral grid is used to divide the computational domain. The simulation is a dynamic process and the boundaries of the foil are changing constantly. The dynamic mesh method reconstructs the mesh at each time step and User-Defined Functions (UDF) linked to Fluent define the motion of the moving boundaries. For a single crescent-shaped foil separated from the fish body or NACA foil, the simulationsAppl. Sci. are 2017 started, 7, 1120 from the static state and then the foil is towed forward and moves7 of 15 according to the motion model defined above. For the thunniform bio-inspired model, the self-propulsion model is adopted.(FVM) is adopted In other to discretize words, th thee Navier-Stokes thunniform equations bio-inspired into expr modelessions attemptsthat are first-order to generate in thrust time and second-order in space. The tetrahedral grid is used to divide the computational domain. by swingingThe thesimulation caudal is finsa dynamic and wavingprocess and the the body. bounda Asries a result,of the foil the are hydrodynamic changing constantly. forces The acting on the body anddynamic caudal mesh fin method achieve reconstructs forward the propulsion mesh at each oftime the step thunniform and User-Defined bio-inspired Functions model.(UDF) At each time step,linked the hydrodynamic to Fluent define forces the motion are extracted of the moving and boundaries. then the position For a single and crescent-shaped attitude of the foil thunniform bio-inspiredseparated model from are the calculated fish body or according NACA foil, to the the simu hydrodynamiclations are started forces. from the Without static state taking and then into account the foil is towed forward and moves according to the motion model defined above. For the the flexibility,thunniform the bionic bio-inspired foils canmodel, be treatedthe self-propulsion as rigid bodies.model is Inadopted. these cases,In other the words, translational the and rotationalthunniform motion of bio-inspired the bionic model foils attempts can be definedto generate by thrust DEFINE_CG_MOTION by swinging the caudal fins function. and waving In addition, DEFINE_GRID_MOTIONthe body. As a result, function the hydrodynamic is adopted forces todefine acting on the the motion body and of caudal the flexible fin achieve foils forward and thunniform bio-inspiredpropulsion model. of Using the thunniform this function, bio-inspired the deformation model. At each velocity time step, of the each hydrodynamic grid point forces can beare defined in extracted and then the position and attitude of the thunniform bio-inspired model are calculated each timeaccording step. to the hydrodynamic forces. Without taking into account the flexibility, the bionic foils can Duringbe thetreated calculation, as rigid bodies. the SIMPLEIn these cases, algorithm the tran ofslational the pressure-speed and rotational motion amending of the bionic method foilsis adopted and the k-canε turbulence be defined by model DEFINE_CG_MO is adopted.TION To function. ensure In the addition, accuracy DEFINE_GRID_MOTION of the calculations, function the maximum number ofisiteration adopted to steps define is the 50 motion and every of the flexible motion foils period and thunniform contains bio-inspired 200 time steps. model. The Using wall this boundary function, the deformation velocity of each grid point can be defined in each time step. condition is setDuring at the the solidcalculation, boundaries the SIMPLE and algorith symmetrym of the boundary pressure-speed conditions amending are method imposed is on the boundariesadopted of the and fluid the domain. k-ε turbulence To exclude model is the adopted. influence To ensure of the the wall accuracy effect, of the calculations, computational the domain is set sufficientlymaximum large. number To of verify iteration the steps grid is 50 independency, and every motion choosing period contains thrust 200 velocitytime steps. as The the wall assessment index, threeboundary numerical condition simulations is set at the with solid differentboundaries grid and sizesymmetry are carried boundary out conditions and GCI are (grid imposed convergence on the boundaries of the fluid domain. To exclude the influence of the wall effect, the computational index) wasdomain calculated. is set sufficiently It is found large. that To the verify grid the size grid adopted independency, in the choosing paper canthrust meet velocity both as thethe precision requirementassessment and computational index, three numerical efficiency simulations requirement. with different Besides grid size that, are thecarried simulation out and GCI with (grid a 50% time step is carriedconvergence out and index) only was 6% calculated. of the differenceIt is found that between the grid size the adopted big time in the step paper and can the meet small both time step is found. Similarthe precision verification requirement processes and computational are conducted efficiency beforerequirement. numerical Besides simulationsthat, the simulation of bionic foils. with a 50% time step is carried out and only 6% of the difference between the big time step and the For example,small the time computational step is found. Similar domain verification and grids processes of the are thunniform conducted before bio-inspired numerical modelsimulations are shown in Figure7. Theof bionic domains foils. ofFor bionic example, foils the arecomputational discretized domain into aboutand grids 8.0 of× the10 thunniform5 grid cells bio-inspired and thedomain of thunniformmodel bio-inspired are shown in model Figure 7. is The discretized domains of intobionic about foils are 6.0 discretized× 106 grid into about cells. 8.0 Additionally, × 105 grid cells a refined 6 mesh is adoptedand the domain near the of thunniform boundaries bio-inspired of the solid model domain. is discretized Toavoid into about repetition, 6.0 × 10 thegrid computationalcells. Additionally, a refined mesh is adopted near the boundaries of the solid domain. To avoid repetition, domain andthe meshcomputational generation domain of and the mesh bionic generation foils are of the not bionic described foils are here. not described here.

(a) (b)

Figure 7. Computational domain and grids of the thunniform bio-inspired mode: (a) computational Figure 7. Computationaldomain; (b) grids. domain and grids of the thunniform bio-inspired mode: (a) computational domain; (b) grids. Relevant kinetic parameters should be calculated. For bionic foils, the chord Reynolds number and Strouhal number are defined by Relevant kinetic parameters should be calculated. For bionic foils, the chord Reynolds number UL and Strouhal number are defined by Re  (13) UL Re = (13) 2 fhν St  (14) 2Uf h St = (14) U Appl. Sci. 2017, 7, 1120 8 of 15 where ν is the fluid kinematic viscosity. The Strouhal number is an important parameter in biological swimming models. Many fishes can maintain high propulsion performance when the value of the Strouhal number is approximately 0.3. In addition, a dimensionless method of forces and torque is used [19] and the method can be written as:

−Fx(t) Cx(t) = (15) 0.5ρU2S

Fy(t) Cy(t) = (16) 0.5ρU2S

Mθ(t) Cm(t) = (17) 0.5ρU2SL

Here, Fy is the transverse force, Fx is the thrust force, Mθ is the torque and S is the projected area of the foil. As shown in [19], the propulsive efﬁciency η can be written as:

1 Z T Po(t) = Fx(t)Udt (18) T 0

1 Z T h . . i Pe(t) = Fy(t)y(t) + Mθ(t)θ(t) dt (19) T 0 P η = o (20) Pe where Po is the output power, Pe is the input power and T is the period of the motion.

3. Results and Discussion

3.1. Rigid Foil The geometric parameters and motion parameters of the thunniform bio-inspired model are ◦ assigned as follows: Lf = 0.16 m, λ = 1.0Lf, αmax = 20 , h = 0.5L and f = 1.25 Hz. Here, Lf is the length of the thunniform bio-inspired model and L is the chord length of the caudal fin. The thunniform bio-inspired model moves from the static state. Figure8 shows the velocity time-histories of the thunniform bio-inspired model. Here, u-fish is the velocity along the propulsion direction and v-fish is the velocity transverse to the propulsion direction. Initially, u-fish is zero and then increases gradually, until it finally reaches dynamic stability. Furthermore, u-fish and v-fish are not constant, but oscillate around a constant value. The process for reaching dynamic force balance is gradual. Figure9 shows the vortex pattern in the wake. There are vortex rings behind the thunniform bio-inspired model. The vortex ring consists of a train of inverted hairpin-like vortices braided together such that the legs of each vortex are attached to the head of the preceding vortex. Specifically, Figure9b shows the vorticity magnitude and velocity vector in two-dimensional space. The vortices shed from the thunniform bio-inspired model are arranged in two ranks in the wake, which can generate a jet stream. As a result, the thunniform bio-inspired model obtains a corresponding reaction force. The arrangement and shape of vortex pattern are similar with those found in nature and agree well with other research [15]. Hence, the numerical model and solution approach adopted in this paper are reliable. Appl. Sci. 2017, 7, 1120 9 of 15 Appl. Sci. 2017, 7, 1120 9 of 15 Appl. Sci. 2017, 7, 1120 9 of 15 0.025 0.025

0.000 0.000

-0.025 -0.025 u-fish -0.050 u-fish v-fish Velocity (m/s) -0.050 v-fish Velocity (m/s)

-0.075 -0.075

-0.100 -0.100 0246810121416 0246810121416Time (s) Time (s) Figure 8. Velocity time-historiestime-histories ofof thethe thunniformthunniform bio-inspiredbio-inspired mode.mode. Figure 8. Velocity time-histories of the thunniform bio-inspired mode.

(a) (b) (a) (b) Figure 9. Vortex pattern in the wake of the thunniform bio-inspired mode: (a) three-dimensional case; Figure 9. Vortex pattern in the wake of the thunniform bio-inspired mode: (a) three-dimensional case; Figure(b) two-dimensional 9. Vortex pattern case. in the wake of the thunniform bio-inspired mode: (a) three-dimensional case; ((bb)) two-dimensional two-dimensional case. case. The geometric parameters and motion parameters of the NACA foil are assigned as follows: h = The geometric parameters and motion parameters of the NACA foil are assigned as follows: h = L = 0.055The geometricm, B = 0.3575 parameters m, αmax = and25°, St motion = 0.3 and parameters Re = 11,000. of the As NACA shown foilin Figure10a, are assigned the asforces follows: and L = 0.055 m, B = 0.3575 m, αmax = 25°, St =◦ 0.3 and Re = 11,000. As shown in Figure10a, the forces and htorque= L = 0.055acting m, onB the= 0.3575 NACA m ,foilαmax fluctuate= 25 , St periodic= 0.3 andally.Re In= 11,000.one cycle As showntime, the in Figureresultant 10 a,force the forcesof the torque acting on the NACA foil fluctuate periodically. In one cycle time, the resultant force of the andtransverse torque force acting in ona motion the NACA period foil is fluctuatezero and periodically.the thrust force In is one positive. cycle time, That theis, the resultant NACA forcefoil can of transverse force in a motion period is zero and the thrust force is positive. That is, the NACA foil can theobtain transverse forward force thrust in aforce motion and period eliminate is zero transverse and the thrustforce. To force verify is positive. validity That and is, feasibility the NACA of foilthe obtain forward thrust force and eliminate transverse force. To verify validity and feasibility of the canmethod, obtain the forward geometry thrust parameters force and and eliminate motion paramete transversers force.are set To to verifybe consistent validity with and those feasibility of other of method, the geometry parameters and motion parameters are set to be consistent with those of other theexperimental method, the study geometry [19]. Furthermore, parameters and the motion mean C parametersx is 0.58, the are mean set to C bey is consistent 0.0065 and with the those thrust of experimental study [19]. Furthermore, the mean Cx is 0.58, the mean Cy is 0.0065 and the thrust otherefficiency experimental is about study47.0%. [ 19Compar]. Furthermore,ing with the meanexperimentalCx is 0.58, study, the mean the curvesCy is 0.0065 of force and coefficients the thrust efficiency is about 47.0%. Comparing with the experimental study, the curves of force coefficients efficiencyhave similar is abouttrends 47.0%. and the Comparing performance with parameters the experimental compare study,well with the it curves (Cx = 0.604, of force Cy = coefficients 0.0082, η = have similar trends and the performance parameters compare well with it (Cx = 0.604, Cy = 0.0082, η = have49.1%). similar Moreover, trends the and forces the performanceand torque curves parameters of the comparecrescent-shaped well with foil it are (Cx shown= 0.604, inC Figurey = 0.0082, 10b. 49.1%). Moreover, the forces and torque curves of the crescent-shaped foil are shown in Figure 10b. ηThe= 49.1% curves). Moreover,are similar the to those forces of and the torque NACA curves foil. Additionally, of the crescent-shaped three-dimensional foil are shown vortex in patterns Figure 10 inb. The curves are similar to those of the NACA foil. Additionally, three-dimensional vortex patterns in Thethe wake curves of are the similar NACA to foil those and of thecrescent-shaped NACA foil. Additionally, foil are shown three-dimensional in Figure 11. The vortex vortex patterns pattern in has the the wake of the NACA foil and crescent-shaped foil are shown in Figure 11. The vortex pattern has wakesimilar of arrangement the NACA foil and and shape crescent-shaped with that of foilthunniform are shown bio-inspired in Figure 11 model,. The vortex which patternalso compare has similar well similar arrangement and shape with that of thunniform bio-inspired model, which also compare well arrangementwith other researches and shape [32–34]. with thatContinuous of thunniform vortex bio-inspiredrings can be found model, in which the wake, also but compare the vortex well rings with with other researches [32–34]. Continuous vortex rings can be found in the wake, but the vortex rings otherbehind researches the crescent-shaped [32–34]. Continuous foil are vortex more ringscomp canlex be because found inof the the wake, more but complex the vortex geometry. rings behind The behind the crescent-shaped foil are more complex because of the more complex geometry. The thegeometric crescent-shaped parameters foil and are motion more complexparameters because of the ofcrescent-shaped the more complex foil are geometry. assigned The as follows: geometric h = geometric parameters and motion parameters of the crescent-shaped foil are assigned as follows: h = parameters0.06 m, L = 0.15 and m, motion αmax = parameters25°, St = 0.3 ofand the Re crescent-shaped = 45,000. foil are assigned as follows: h = 0.06 m, ◦ 0.06L = 0.15m, Lm, = 0.15αmax m,= 25αmax, =St 25°,= 0.3 St and= 0.3Re and= 45,000. Re = 45,000. Appl. Sci. 2017, 7, 1120 10 of 15 Appl. Sci. 2017, 7, 1120 10 of 15 Appl. Sci. 2017, 7, 1120 10 of 15 4 6 6 C 4 C x x 5 C 3 C 5 Cxy Cxy 3 4 Cy CCy m m 4 C 2 C 3 m 2 m 3 2 1 2 1 1 1 0 0 0 0 -1 -1 -1 -1 -2 Torque and force coefficient force and Torque -2 Torque and force coefficients -2

Torque and force coefficient force and Torque -3

Torque and force coefficients -2 -3 -3 -4 -3 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 -4 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 0.5 1.0 1.5 2.0 2.5Time 3.0 (s) 3.5 4.0 4.5 5.0 5.5 1.2 1.6 2.0 2.4 2.8Time 3.2(s) 3.6 4.0 4.4 4.8 5.2 Time (s) Time (s) (a) (b) (a) (b) Figure 10. Forces and torque curves: (a) NACA foil; (b) crescent-shaped foil. Figure 10. Forces and torque curves: ( a) NACA foil; ( b) crescent-shaped foil.

(a) (b) (a) (b) Figure 11. Three-dimensional vortex patterns in the wake: (a) NACA foil; (b) crescent-shaped foil. Figure 11. Three-dimensional vortex patterns in the wake: ( a) NACA foil; ( b) crescent-shaped foil. 3.2. Flexible Foil 3.2. Flexible Foil 3.2. Flexible Foil This section focuses on the influences of spanwise flexibility. Figure 12 shows the performance This section focuses on the influences of spanwise flexibility. Figure 12 shows the performance curvesThis versus section flexibility focuses phase on the differe influencesnce of of the spanwise NACA flexibility.foil. The case Figure with 12 γshows = 0° stands the performance for a rigid curves versus flexibility phase difference of the NACA foil. The case with γ = 0° stands for a rigid curvesNACA versusfoil. Additionally, flexibility phase an elliptic difference curve, ofa hyperb the NACAolic curve foil. Theand casea parabolic with γ curve= 0◦ stands are used for as a curve rigid NACA foil. Additionally, an elliptic curve, a hyperbolic curve and a parabolic curve are used as curve NACAfunctions foil. respectively. Additionally, The an figure elliptic shows curve, that a hyperbolic spanwise curveflexibility and acan parabolic enhance curve the arepropulsive used as functions respectively. The figure shows that spanwise flexibility can enhance the propulsive curveefficiency functions slightly respectively. within a certai Then figure range, shows but the that mean spanwise thrust flexibility force coefficients can enhance can theachieve propulsive higher efficiency slightly within a certain range, but the mean thrust force coefficients can achieve higher efficiencyvalues when slightly the flexibility within a phase certain difference range, but is posi thetive. mean Moreover, thrust force the elliptic coefficients curve can is more achieve conducive higher values when the flexibility phase difference is positive. Moreover, the elliptic curve is more conducive valuesto enhancement when the flexibilityof the propulsive phase difference efficiency, is positive.but less conducive Moreover, to the enhancement elliptic curve of is morethe mean conducive thrust to enhancement of the propulsive efficiency, but less conducive to enhancement of the mean thrust toforce enhancement when the phase of the difference propulsive is positive. efficiency, And but th lessere conducive is little difference to enhancement between the of the performance mean thrust of force when the phase difference is positive. And there is little difference between the performance of forcethe hyperbolic when the phasecurve differenceand the parabolic is positive. curve. And In there addition, is little the difference NACA foil between can achieve the performance both higher of the hyperbolic curve and the parabolic curve. In addition, the NACA foil can achieve both higher themean hyperbolic thrust force curve and and propulsive the parabolic efficiency curve. when In addition,the flexibility the NACAphase differ foil canence achieve is between both 10° higher and mean thrust force and propulsive efficiency when the flexibility phase difference is between 10° and mean20°. Figure thrust 13 force shows and the propulsive three-dimensional efficiency whenvortex the pattern flexibility in the phase wake difference of the NACA is between foil for 10 typical◦ and 20°. Figure 13 shows the three-dimensional vortex pattern in the wake of the NACA foil for typical 20flexibility◦. Figure phase 13 shows differences the three-dimensional for the elliptic curve. vortex The pattern figure inshows the wake that the of thevortex NACA rings foil are for stretched typical flexibility phase differences for the elliptic curve. The figure shows that the vortex rings are stretched flexibilityand the volume phase differencesis larger when for theγ = elliptic20° compared curve. Theto γ figure = −20°. shows A stronger that the jet vortex stream rings can be are generated stretched and the volume is larger when γ = 20° compared to γ = −20°. A stronger jet stream can be generated andbetween the volume the vortex is larger ranks when whenγ γ= = 20 20°◦ compared and, in turn, to γ the= − reaction20◦. A stronger force acting jet stream on the can foil be is generatedenhanced between the vortex ranks when γ = 20° and, in turn, the reaction force acting on the foil is enhanced betweenaccordingly. the vortexAs a result, ranks the when propulsionγ = 20◦ performanceand, in turn, theis enhanced. reaction force This actingresult indicates on the foil that is enhancedspanwise accordingly. As a result, the propulsion performance is enhanced. This result indicates that spanwise accordingly.flexibility can As influence a result, the propulsionshedding of performance vortex rings, is further enhanced. influence This result the arrangements indicates that spanwiseof vortex flexibility can influence the shedding of vortex rings, further influence the arrangements of vortex flexibilityrings and caneventually influence lead the to shedding a change of of vortex propulsi rings,on performance. further influence Furthermor the arrangementse, positive ofvalues vortex of rings and eventually lead to a change of propulsion performance. Furthermore, positive values of ringsflexibility and eventuallyphase difference lead to are a changemore conducive of propulsion to enhancing performance. the propulsion Furthermore, performance positive values than are of flexibility phase difference are more conducive to enhancing the propulsion performance than are flexibilitynegative values. phase The difference geometric are moreparameters conducive and motion to enhancing parameters the propulsion of the NACA performance foil are assigned than are as negative values. The geometric parameters and motion parameters of the NACA foil are assigned as negativefollows: h values. = 0.05 m, The L geometric= 0.1 m, B parameters= 0.3 m, αmaxand = 25°, motion St = 0.3 parameters and Re = 20000. of the NACA foil are assigned as follows: h = 0.05 m, L = 0.1 m, B = 0.3 m, αmax = 25°, ◦St = 0.3 and Re = 20000. follows: h = 0.05 m, L = 0.1 m, B = 0.3 m, αmax = 25 , St = 0.3 and Re = 20000. Appl. Sci. 2017, 7, 1120 11 of 15

Appl. Appl.Sci. 2017 Sci. 2017, 7, 1120, 7, 1120 11 of 1511 of 15

0.750.75 0.4550.455 Appl. Sci. 2017, 7, 1120 11 of 15 0.70 0.70 0.4500.450 Parabola Parabola Hyperbola 0.650.750.65 0.455 Hyperbola 0.4450.445 Ellipse 0.60 Ellipse 0.600.70 0.450 Parabola 0.440 0.55 0.440 Hyperbola 0.550.65 0.50 0.4350.445 Ellipse 0.60 0.435 0.50 Parabola 0.45 0.4300.440 0.450.55 ParabolaHyperbola efficiency Propulsive 0.40 0.430 Hyperbola Ellipse efficiency Propulsive 0.425 0.400.50Mean thrust force coefficients 0.435 0.35 Ellipse 0.425 Mean thrust force coefficients Parabola -40 -20 0 20 40 0.350.45 -40 -20 0 20 40 0.430 -40 -20Phase difference 0 (°) 20 Hyperbola 40 efficiency Propulsive -40Phase -20 difference 0 (°) 20 40 0.40 (a) Ellipse 0.425 (b)

Mean thrust force coefficients Phase difference (°) Phase difference (°) 0.35 Figure-40 12. Performance -20(a) curves 0 versus 20 phase 40 difference of the NACA-40 foil: -20(a()b mean) 0thrust force 20 40 Figure 12.coefficients;Performance (b) propulsive curves versus efficiency. phase difference of the NACA foil: (a) mean thrust force coefficients; Figure 12. PerformancePhase curvesdifference versus (°) phase difference of the NACA foil:Phase (a) differencemean thrust (°) force (b) propulsive efficiency. coefficients; (b) propulsive(a) efficiency. (b) Figure 12. Performance curves versus phase difference of the NACA foil: (a) mean thrust force coefficients; (b) propulsive efficiency.

(a) (b)

Figure 13. Three-dimensional vortex pattern in the wake of the NACA foil: (a) γ = 20°; (b) γ = −20°. (a) (b) Figure 14 shows the performance curves versus flexibility phase difference of the crescent- Figure 13. Three-dimensional vortex pattern in the wake of the NACA foil: (a) γ = 20°; (b) γ = −20°. ◦ − ◦ Figureshaped 13. Three-dimensional foil. The elliptic(a) curve vortex is more pattern conducive in the to wake enhancing of the the NACA propulsive foil:(b) (efficiency,a) γ = 20 which;(b) γ is= 20 . similar to the result for the NACA foil and the curves of the mean thrust force coefficients have similar Figure 14 shows the performance curves versus flexibility phase difference of the crescent- FiguretrendsFigure 14 showsas those13. Three-dimensional theof the performance NACA foil. vortex In curvescontrast, pattern versus the in thecrescent-shaped wake flexibility of the NACA phasefoil can foil: differencemaintain (a) γ = a20°; relatively of (b the) γ = crescent-shaped −high20°. shaped foil. The elliptic curve is more conducive to enhancing the propulsive efficiency, which is foil. The ellipticpropulsive curve efficiency is more value conducive while achieving to enhancing a larger mean the thrust propulsive force within efficiency, a relatively which large range is similar to the similarofFigure flexibility to the 14result phaseshows for difference. thethe NACA performance Natural foil and caudal curvesthe curves fins versus of offish the flexibilityusually mean have thrust phase a shapeforce differ coefficientssimilarence to of that thehave of crescent-the similar result for the NACA foil and the curves of the mean thrust force coefficients have similar trends as trendsshapedmodel as foil. those when The ofthe ellipticthe flexibility NACA curve phase foil. is Indifferencemore contrast, conducive is positithe crescent-shaped ve.to Theenhancing results indicate the foil propulsive can that maintain reasonable efficiency, a relatively spanwise which high is thosepropulsive ofsimilar theflexibility NACAto the efficiency can result foil. enhance for Invalue the contrast, the NACAwhile propulsion achieving thefoil and crescent-shapedperformance the a larger curves of mean ofa crescent-shaped the foil thrust mean can forcethrust maintain withinfoil force significantly. acoefficientsa relatively As havelargeshown high similar range propulsive efficiencyoftrends flexibilityin value Figure as those while phase15, theof theachieving difference.vortex NACA rings foil. aNatural in larger the In wakecontrast, caudal mean of the fins thrustthe crescent-shaped crescent-shapedof fish force usually within foil have foil under a relatively cana shape typical maintain similar flexibility large a relatively to range thatphase of high the flexibility phasemodelpropulsive difference.differences when efficiency the Naturalhave flexibility forms value caudal similar phase while fins todifference achieving those of fishof the is a usually positilargerNACAve. mean foil, have The so thrustresults afurther shape force indicate analysis similar within thatis nota to reasonablerelatively thatdescribed of thelarge here.spanwise model range when The geometric parameters and motion parameters of the crescent-shaped foil are assigned as follows: the flexibilityflexibilityof flexibility phasecan phase enhance difference difference. the propulsion is positive.Natural performance caudal The results fins ofof fish indicatea crescent-shaped usually that have reasonable a foil shape significantly. similar spanwise to Asthat flexibilityshown of the can in Figureh = 0.06 15, m, the L = vortex0.15 m, ringsαmax = 25°,in the St =wake 0.3 and of Rethe = 45000.crescent-shaped foil under typical flexibility phase enhancemodel the when propulsion the flexibility performance phase difference of a crescent-shaped is positive. The results foil significantly. indicate that reasonable As shown spanwise in Figure 15, differences have forms similar to those of the NACA foil, so further analysis is not described here. the vortexflexibility rings can in enhance the wake the propulsion of the crescent-shaped performance of0.31 foila crescent-shaped under typical foil flexibility significantly. phase As shown differences Thein Figure geometric0.50 15, the parameters vortex rings and inmotion the wake parameters of the crescent-shaped of the0.30 crescent-shaped foil under foil aretypical assigned flexibility as follows: phase have forms similar to those of the NACA foil, so further analysis is not described here. The geometric hdifferences = 0.06 m, L have= 0.15 forms m, αmax similar = 25°, toSt those= 0.3 andof the Re NACA= 45000.0.29 foil, so further analysis is not described here. 0.45 y h parametersThe geometric and motion parameters parameters and motion of theparameters crescent-shaped of the0.28 crescent-shaped foil are assigned foil are assigned as follows: as follows:= 0.06 m, ◦ 0.31 L = 0.15h =m, 0.06α maxm,0.40 L= =25 0.15, Stm, =αmax 0.3 = and25°, StRe = =0.3 45,000. and Re = 45000.0.27

0.50 0.30 Parabola 0.26 Parabola 0.35 Hyperbola 0.290.31 Hyperbola 0.45 y Ellipse 0.25 Ellipse 0.50 efficiencPropulsive 0.280.30 0.30 0.24 0.40 thrust Mean force coefficients 0.270.29

y 0.45 0.23 -40 -20 0 20 40 -40 -20 0 20 40 Parabola 0.260.28 Parabola Phase difference (°) Phase difference (°) 0.35 Hyperbola Hyperbola 0.40 0.250.27 (a) Ellipse (b) Ellipse Propulsive efficiencPropulsive Parabola 0.30 0.240.26 Parabola Mean thrust Mean force coefficients Figure 14. Performance curves versus phase difference of the crescent-shaped foil: (a) mean thrust 0.35 Hyperbola Hyperbola force coefficients; (b) propulsive efficiency. Ellipse 0.230.25 Ellipse -40 -20 0 20 40 efficiencPropulsive -40 -20 0 20 40 0.30 Phase difference (°) 0.24 Phase difference (°) Mean thrust Mean force coefficients (a) 0.23 (b) -40 -20 0 20 40 -40 -20 0 20 40 Figure 14. Performance Phase curves difference versus (°) phase difference of the crescent-shaped Phase difference foil: (a) (°)mean thrust force coefficients; (b) propulsive(a) efficiency. (b) Figure 14. Performance curves versus phase difference of the crescent-shaped foil: (a) mean thrust Figure 14. Performance curves versus phase difference of the crescent-shaped foil: (a) mean thrust force coefficients; (b) propulsive efficiency. force coefﬁcients; (b) propulsive efﬁciency. Appl. Sci. 2017, 7, 1120 12 of 15 Appl. Sci. 2017, 7, 1120 12 of 15 Appl. Sci. 2017, 7, 1120 12 of 15

(a) (b)

Figure 15. Three-dimensional vortex pattern in the wake of the crescent-shaped foil: (a) γ = 20°; (b) γ = −20°. Figure 15. Three-dimensional(a) vortex pattern in the wake of the crescent-shaped(b) foil: (a) γ = 20◦; (b) γ = −20◦. Figure4. Experiments 15. Three-dimensional and Discussion vortex pattern in the wake of the crescent-shaped foil: (a) γ = 20°; (b) γ = −20°. To verify the reliability of the numerical results, an experimental verification platform is 4. Experiments and Discussion 4. Experimentsdesigned and and built. Discussion The bionic foil can be driven by three servo motors and hydrodynamic forces are Tomeasured verifyverify thethrough the reliability reliability a force of sensor. theof numericalthe The numerical servo motors results, results, are an controlled experimental an experimental through verification an upper verification computer platform andplatform is designed a is motion control card. Under this reasonable control method, the bionic foil can move according to the anddesigned built. and The built. bionic The foil bionic can be foil driven can be by driven three servo by three motors servo and motors hydrodynamic and hydrodynamic forces are measuredforces are throughpreset a forcemotion sensor. model. The Figure servo 16 shows motors installation are controlled photos through of the bionic an upper foils. The computer bionic foils and are a motion measuredplaced through in the center a force of the sensor. water Thetank. servo The median motors fi lterare algorithmcontrolled is adoptedthrough in an the upper data processingcomputer and a control card. Under this reasonable control method, the bionic foil can move according to the preset motionof thecontrol original card. data. Under Figure this 17 shows reasonable the experiment control almethod, results and the numerical bionic foil results can move of the NACAaccording foil to the motionpresetand motion model. the crescent-shaped model. Figure 16Figure shows foil. 16The installationshows geometric installation parameters photos ofphotos and the motion bionic of the parameters foils. bionic The foils.chosen bionic The in foils this bionic aresection placed foils are in theplaced centerare in consistent the of thecenter water with of the tank.parameters water The tank. mediandefined The medianin filter numerical algorithm filter simulations algorithm is adopted isof adoptedrigid in the foils. datain theThe processing datasampling processing of the originalof thefrequency original data. Figuredata.of the Figure force17 shows sensor 17 shows the is set experimental the to 50experiment Hz and results theal bionicresults and foils numerical and are numerical colored results green results of to the ofmake NACA the themNACA foil andfoil theand crescent-shapedthereadily crescent-shaped observable. foil. The foil. The hydrodynamic The geometric geometric parametersforces parameters are chos anden andas motion the motion assessment parameters parameters index. chosen It chosencan inbe thisseen in this sectionthat section are the overall trends of hydrodynamic forces are consistent between numerical results and experimental consistentare consistent with parameterswith parameters defined defined in numerical in numerical simulations simulations of rigid foils. of rigid The sampling foils. The frequency sampling of the forceresults. sensor Because is set of toinstrumental 50 Hz and error, the bionic there are foils smal arel-scale colored oscillations green to in makethe experimental them readily curve. observable. It frequencyindicates of thethat forcethe numerical sensor ismethod set to can50 Hzcorrectly and thedescribe bionic the foils change are processcolored of green hydrodynamic to make them The hydrodynamic forces are chosen as the assessment index. It can be seen that the overall trends of readilyforces observable. acting on theThe bionic hydrodynamic foils. forces are chosen as the assessment index. It can be seen that hydrodynamicthe overall trends forces of hydrodynamic are consistent forces between are numericalconsistent resultsbetween and numerical experimental results results. and experimental Because of instrumentalresults. Because error, of instrumental there are small-scale error, there oscillations are smal inl-scale the experimentaloscillations in curve. the experimental It indicates curve. that the It numericalindicates that method the numerical can correctly method describe can thecorrectly change describe process the of hydrodynamic change process forces of hydrodynamic acting on the bionicforces acting foils. on the bionic foils.

(a) (b)

Figure 16. Installation photo of the bionic foils: (a) NACA foil; (b) crescent-shaped foil.

(a) (b)

Figure 16. Installation photo of the bionic foils: ( a) NACA foil; ( b) crescent-shaped foil. Appl. Sci. 2017, 7, 1120 13 of 15 Appl. Sci. 2017, 7, 1120 13 of 15

(a) (b)

Figure 17. Comparison of experimental results and numerical results: (a) NACA foil; (b) crescent- Figure 17. Comparison of experimental results and numerical results: (a) NACA foil; (b) crescent- shaped foil. shaped foil. The mean thrust force coefficient of the NACA foil measured by experiment is approximately The0.58, mean which thrust is close force to the coefficient numerical ofresult the (0.60). NACA Similarly, foil measured the mean bythrust experiment force coefficient is approximately of the 0.58, whichcrescent-shaped is close tofoil the measured numerical by experiment result (0.60). is approximately Similarly, the 0.44, mean which thrust compares force well coefficient with the of the numerical result (0.47). It can be seen that the experimental results compare well with the numerical crescent-shaped foil measured by experiment is approximately 0.44, which compares well with the results under certain parameter settings. It indicates that the numerical method adopted in the paper is numericalreliable result under (0.47). similar It parameter can be seen settings. that theTherefore, experimental the numerical results results compare in the paper well withare reasonable the numerical resultsand under the analysis certain would parameter provide settings. theoretical It indicates reference thatfor revealing the numerical the hydrodynamic method adopted mechanism. in the paper is reliable under similar parameter settings. Therefore, the numerical results in the paper are reasonable and the5. Conclusions analysis would provide theoretical reference for revealing the hydrodynamic mechanism. In this paper, the hydrodynamic mechanism of bionic foils is studied via numerical simulations. 5. ConclusionsDuring the simulations, two typical bionic foils are chosen as research objects. Specifically, the NACA Infoil this represents paper, thethe general hydrodynamic form of locomotive mechanism organs of bionicand the foils crescent-shaped is studied viafoil is numerical derived from simulations. the Duringcaudal the simulations, fins of thunniform two typicalswimmers. bionic Additionally, foils are chosenthe self-propulsion as research model objects. of the Specifically, thunniform the bio- NACA inspired model is adopted to verify the numerical model and solution approach and to analyze the foil represents the general form of locomotive organs and the crescent-shaped foil is derived from hydrodynamic mechanism of the thunniform mode. The numerical results show that vortex rings the caudalexist in fins the of wake thunniform of the bionic swimmers. foils or thunniform Additionally, bio-inspired the self-propulsion model. The ranks model of of vortices the thunniform can bio-inspiredinduce a model jet stream is adopted and thrust to forces verify can the be numerical obtained in model return. and In addition, solution spanwise approach flexibility and to analyzeis the hydrodynamicconsidered in mechanismthe simulations of theand thunniformthe results indica mode.te Thethat numericalspanwise flexibility results showcan influence that vortex the rings exist inshedding the wake of ofvortex the bionicrings, further foils or influence thunniform the arra bio-inspiredngements of model. the vortex The rings ranks and of eventually vortices can lead induce a jet streamto a change and thrustof the propulsion forces can performance. be obtained inFurthe return.rmore, In the addition, thrust force spanwise is enhanced flexibility significantly is considered in thewhen simulations the flexibility and phase the results difference indicate is positive. that Ad spanwiseditionally, flexibility the NACA canfoil can influence achieve both the sheddinghigher of mean thrust force and higher propulsive efficiency when the flexibility phase difference is between vortex rings, further influence the arrangements of the vortex rings and eventually lead to a change 10° and 20°. When the flexibility phase difference is positive, the crescent-shaped foil can achieve of thehigher propulsion thrust force performance. and maintain Furthermore, relatively high the propulsive thrust forceefficiency. is enhanced To verify the significantly reliability of when the the flexibilitynumerical phase results, difference an experimental is positive. verification Additionally, platform the NACAis designed foil and can built. achieve The bothexperimental higher mean ◦ thrustresults force andcompare higher well propulsive with the numerical efficiency results, when thewhich flexibility indicates phase that differencethe numerical is between results are 10 and 20◦. Whenreliable the and flexibility the analysis phase of the difference paper is reasonable. is positive, the crescent-shaped foil can achieve higher thrust force and maintain relatively high propulsive efficiency. To verify the reliability of the numerical Acknowledgments: The research has been financially supported by the National Natural Science Foundation of results,China an experimental(No. 50905040) and verification State Key Laboratory platform of is Robotics designed and andSystem built. (HIT TheNo. SKLRS200801C). experimental results compare well with the numerical results, which indicates that the numerical results are reliable and the analysis Author Contributions: All authors discussed the contents of the manuscript and contributed to its preparation. of theKai paper Zhou is contributed reasonable. to the research idea and the framework of this study. Junkao Liu and Weishan Chen contributed to the research plan and the result analysis. Acknowledgments: The research has been financially supported by the National Natural Science Foundation of China (No.Conflicts 50905040) of Interest: and The State authors Key Laboratorydeclare no conflict of Robotics of interest. and System (HIT No. SKLRS200801C).

Author Contributions: All authors discussed the contents of the manuscript and contributed to its preparation. Kai Zhou contributed to the research idea and the framework of this study. Junkao Liu and Weishan Chen contributed to the research plan and the result analysis. Conﬂicts of Interest: The authors declare no conﬂict of interest. Appl. Sci. 2017, 7, 1120 14 of 15

References

1. Wu, T.Y. Fish Swimming and Bird/Insect Flight. Annu. Rev. Fluid Mech. 2011, 43, 25–58. [CrossRef] 2. Wen, L.; Wang, T.M.; Wu, G.H.; Liang, J.H. Hydrodynamic investigation of a self-propelled robotic fish based on a force-feedback control method. Bioinspir. Biomim. 2012, 7, 36012. [CrossRef][PubMed] 3. Yan, Q.; Wang, L.; Liu, B.; Yang, J.; Zhang, S. A novel implementation of a flexible robotic fin actuated by shape memory alloy. J. Bionic Eng. 2012, 9, 156–165. [CrossRef] 4. Zhou, C.; Low, K.H. Design and locomotion control of a biomimetic underwater vehicle with fin propulsion. IEEE/ASME Trans. Mechatron. 2012, 17, 25–35. [CrossRef] 5. Szymik, B.G.; Satterlie, R.A. Changes in wingstroke kinematics associated with a change in swimming speed in a pteropod mollusk, Clionelimacina. J. Exp. Biol. 2011, 214, 3935–3947. [CrossRef][PubMed] 6. Wang, S.Z.; Zhang, X.; He, G.W. Numerical simulation of a three-dimensional fish-like body swimming with finlets. Commun. Comput. Phys. 2012, 11, 1323–1333. [CrossRef] 7. Zhu, X.; He, G.; Zhang, X. Flow-mediated interactions between two self-propelled flapping filaments in tandem configuration. Phys. Rev. Lett. 2014, 113, 238105. [CrossRef][PubMed] 8. Deng, H.B.; Xu, Y.Q.; Chen, D.D.; Dai, H.; Wu, J.; Tian, F.B. On numerical modeling of animal swimming and flight. Comput. Mech. 2013, 52, 1221–1242. [CrossRef] 9. Wu, J.; Shu, C. Simulation of three-dimensional flows over moving objects by an improved immersed boundary—Lattice Boltzmann method. Int. J. Numer. Methods Fluids 2012, 68, 977–1004. [CrossRef] 10. Li, G.J.; Zhu, L.D.; Lu, X.Y. Numerical studies on locomotion perfromance of fish-like tail fins. J. Hydrodyn. 2012, 24, 488–495. [CrossRef] 11. Zhu, L.; He, G.; Wang, S.; Miller, L.; Zhang, X.; You, Q.; Fang, S. An immersed boundary method based on the lattice Boltzmann approach in three dimensions, with application. Comput. Math. Appl. 2011, 61, 3506–3518. [CrossRef] 12. Tian, F.B.; Luo, H.X.; Zhu, L.D.; Liao, J.C.; Lu, X.Y. An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments. J. Comput. Phys. 2011, 230, 7266–7283. [CrossRef] [PubMed] 13. Unger, R.; Haupt, M.C.; Horst, P.; Radespiel, R. Fluid–structure analysis of a flexible flapping airfoil at low Reynolds number flow. J. Fluids Struct. 2012, 28, 72–88. [CrossRef] 14. Tian, F.B.; Lu, X.Y.; Luo, H.X. Propulsive performance of a body with a traveling-wave surface. Phys. Rev. E 2012, 86, 16304. [CrossRef][PubMed] 15. Borazjani, I.; Sotiropoulos, F. On the role of form and kinematics on the hydrodynamics of self-propelled body/caudal fin swimming. J. Exp. Biol. 2010, 213, 89–107. [CrossRef][PubMed] 16. Borazjani, I.; Sotiropoulos, F. Numerical investigation of the hydrodynamics of carangiform swimming in the transitional and inertial flow regimes. J. Exp. Biol. 2008, 211, 1541–1558. [CrossRef][PubMed] 17. Wolfgang, M.J.; Anderson, J.M.; Grosenbaugh, M.A.; Yue, D.K.P.; Triantafyllou, M.S. Near-body flow dynamics in swimming fish. J. Exp. Biol. 1999, 202, 2303–2327. [PubMed] 18. Zhu, Q.; Wolfgang, M.J.; Yue, D.K.P. Three-Dimensional flow structures and vorticity control in fish-like swimming. J. Fluid Mech. 2002, 468, 1–28. [CrossRef] 19. Izraelevitz, J.S.; Triantafyllou, M.S. Adding in-line motion and model-based optimization offers exceptional force control authority in flapping foils. J. Fluid Mech. 2014, 742, 5–34. [CrossRef] 20. Hu, J.; Xiao, Q. Three-dimensional effects on the translational locomotion of a passive heaving wing. J. Fluids Struct. 2014, 46, 77–88. [CrossRef] 21. Han, R.; Zhang, J.; Cao, L.; Lu, X. Propulsive performance of a passively flapping plate in a uniform flow. J. Hydrodyn. 2015, 27, 496–501. [CrossRef] 22. Tang, C.; Lu, X. Self-propulsion of a three-dimensional flapping flexible plate. J. Hydrodyn. 2016, 28, 1–9. [CrossRef] 23. Kim, D.; Gharib, M. Characteristics of vortex formation and thrust performance in drag-based paddling propulsion. J. Exp. Biol. 2011, 214, 2283–2291. [CrossRef][PubMed] 24. Shao, X.M.; Pan, D.Y. Hydrodynamics of a flapping foil in the wake of a D-section cylinder. J. Hydrodyn. 2011, 23, 422–430. [CrossRef] 25. Cubero, S.N. Design concepts for a hybrid dwimming and walking vehicle. Procedia Eng. 2012, 41, 1211–1220. [CrossRef] Appl. Sci. 2017, 7, 1120 15 of 15

26. Mantia, L.M.; Dabnichki, P. Effect of the wing shape on the thrust of flapping wing. Appl. Math. Model. 2011, 35, 4979–4990. [CrossRef] 27. Mantia, L.M.; Dabnichki, P. Influence of the wake model on the thrust of oscillating foil. Eng. Anal. Bound. Elem. 2011, 35, 404–414. [CrossRef] 28. Mantia, L.M.; Dabnichki, P. Added mass effect on flapping foil. Eng. Anal. Bound. Elem. 2012, 36, 579–590. [CrossRef] 29. Heathcote, S.; Wang, Z.; Gursul, I. Effect of spanwise flexibility on flapping wing propulsion. J. Fluids Struct. 2008, 24, 183–199. [CrossRef] 30. Zhu, Q.; Shoele, K. Propulsion performance of a skeleton-strengthened fin. J. Exp. Biol. 2008, 211, 2087–2100. [CrossRef][PubMed] 31. Zhou, K.; Liu, J.K.; Chen, W.S. Numerical study on hydrodynamic performance of bionic caudal fin. Appl. Sci. 2016, 6, 15. [CrossRef] 32. Bhalla, A.P.S.; Griffith, B.E.; Patankar, N.A. A forced damped oscillation framework for undulatory swimming provides new insights into how propulsion arises in active and passive swimming. PLoS Comput. Biol. 2013, 9, e1003097. [CrossRef][PubMed] 33. Bhalla, A.P.S.; Bale, R.; Griffith, B.E.; Patankar, N.A. A unified mathematical framework and an adaptive numerical method for fluid–structure interaction with rigid, deforming and elastic bodies. J. Comput. Phys. 2013, 250, 446–476. [CrossRef] 34. Neveln, I.D.; Bale, R.; Bhalla, A.P.S.; Curet, O.M.; Patankar, N.A.; Maclver, M.A. Undulating fins produce off-axis thrust and flow structures. J. Exp. Biol. 2014, 217, 201–213. [CrossRef][PubMed]

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).