143Available online at www.sciencedirect.comSCIENCE; doiREcT.(IIDJournal of HydrodynamicsSer.B, 2006,18(2): 143-147sdlj.chinajournal.net.cnNUMERICAL ANALYSIS OF THE UNSTEADY FORCE IN INSECTFORWARD FLIGHT*ZHA Xiong-quan, LU Xi-yun, YIN Xie-zhenDepartment of Mechanics and Mechanical Engineering, University of Science and Technology of China, Hefei230026, China, E-mail: xlu@ ustc .edu.cn(Received Apr.11, 2005)ABSTRACT: The objective of this study is to get into physicalcomplexity and expensive cost of modeling fluidinsights to the unsteady force and the relevant mechanisms inflows in three dimensions, it is reasonable to employforward flight of insects. Unsteady force in the forward flight2D simulations. to study the basic mechanisms inwas studied, based on a virtual model problem of a foil withflapping flight B. 4. Further, it is noted that mostoscillating translation and rotation in a uniform flow, bsolving the two. dimensional incompressible Navier Stokesnumerical predictions on unsteady force have beenequations with a finite element method. The effects of typicalcarried out for hovering flight, but not for forwardparameters, including the advance ratio, the inclined angle offlight.stroke plane， the stroke amplitude, and the amplitude o1pitching angle of attack, on the forces and the flow structuresoscillating translation and rotation in a uniform flow,were analyzed.s employed to model insect forward flight. TheKeywords: unsteady flow, vortex dynamics, thrust and lift,unsteady force and the relevant mechanisms irflapping motion, biomechanicsforward flighttwo-dimensional incompressible Navier-Stokesequations with a finite element method. Systematic1. INTRODUCTIONcomputations have been carried out to elucidate theAlthough the quasi -steady analysis provides aeffects of some typical parameters on the forces andsimple framework for quantitative prediction of thethe flow structures.forces in insect flight, the predicted forces and powerrequirements of insects in hovering or forward flightcannot prove that insects use the steady-state principle.2. MATHEMATICAL FORMULATION ANDThus, it is needed to get into physical insights to theNUMERICAL METHODunsteady mechanisms of force generation in insectAs shown in Fig.l for the sketch, the chordlength of the foil c and the velocity U related to .the oscillating translation (defined in the following)flapping wings, computational fluid dynamics haare used as the length and velocity scales, respectively. .been developed to solve the Navier-Stokes equationThenthnon-dimensionalincompressiblearound a moving wing. The unsteady flow around aNavier- Stokes equations are given asmodel fruit fly wing conducting flapping motions wasstudied to explore the force behavior in forward flightdu+i.Vu=-Vp+-V2i(1). Alternatively, a two-dimensional (2D) simulationReof the problem can be taken to study the basicmechanisms in flapping flightOwing to the* Project supported by the Innovation Project of the Chinese Academy of Sciences (Grant No: KJCX SW-L04), the NationalNatural Science Foundation of China (Grant No: 10332040), and the Hundred-Talent Prooram of the r "hinece Arademy of Sciences.Biography: ZHA Xiong- quan (1979-), Male,Master中國(guó)煤化工’HCNMH G.144amplitude of pitching angle.V.u=0(2)No-slip velocity boundary condition is used onthe surface of the foil based on the motions describedwhere u is the velocity vector，p is the pressure,in Eqs. (3) and (4). A uniform velocity U。is set atand Re is the Reynolds number, defined asthe upstream far boundary and can be expressed by aRe=Uc/v with V being the kinematic viscosity.non-dimensional parameter "，i.e., the advance ratioTo clearly express the problem and simplify theJ =U。1U. On the exterior far boundary, a pressurecomputation, two coordinate systems are used in Fig.condition dp1dnp =0 is prescribed, where n,1. One is an inertial frame (referred as X- andY - axis), and the other is an instantaneous inertialindicates the unit vector in the normal direction of theboundary.frame (referred as x- andy一axis). TheA fractional-step velocity correction method isinstantaneous inertial frame is fixed to the foil. For theused to solve Eqs. (1) and (2). The finite elementinstantaneous inertial frame fiting to the foil, thespatial discretization is performed using the Galerkinx- and y- axis must be changed with time buweighted residual method. Thediscretizedstill be fixed to the inertial frame at any instant. Then,formulation was described by Kovacs and Kawahara 5the grid system can always be attached to the foil, andin detail. The present computational code has beenthe time derivative term in Eq. (1) should be carefullyverified by our previous workcalculated in a proper way.In this study, an O-type computational domainwith a far boundary about 20c is used and theunstructured mesh is employed in the finite elementcalculation. The number of element for the presentcalculation is 10 approximately, and the time step is0.001.入1!3. RESULTS AND DISCUSSIONBased on experimental visualizations of typicalinsect flying, the parameters are typically chosen asfollows. The Reynolds number is Re= 100, the advanceratio J = 0.2-0.6， the stroke plane angle isFig. 1 Sketch of an oscilating translation and rotation foil inβ = 30°-60', the stroke amplitude Am =2.5-5, thea uniform flow. The downstroke phase is indicatedamplitude of pitching angle of attack 0,m =35*-65",by the gray llipses and the upstroke by the emptyellipsesand the position of rotational axis X, = 0.5. AnThe translation position of the rotating axis of theelliptic foil with the thickness ratio λ =0.125 is used.3.1 Effect of stroke plane anglefoil is represented asFigure 2 shows the mean lift cofficient C,_A,and thrust coefficient Cp. C, decreases and CA(t)= ." [cos(2rft)+1](3)2increases with the increase of β . Meanwhile, whilewhere f and Am represent the frequency andC[ increases and Cr decreases with the increaseof J for the same β. To deal with the propulsiveamplitude of oscillating translation. Similar to thperformance of a foil in forward flight, as shown inprevious study叫，the reference velocity U isFig.2(b), a thrust (i.e， CT >0) is generated atdefinedas U =2fAmJ=0.2 over the range of β considered here.The pitching motion of the foil, i.e, the foilorientation with respect to the X - axis, is describedHowever, at high flight speed, e.g, J =0.6, Crachanges its sign from negative (i.e., drag force) topositive (i.e., thrust force) around β=47° with thea(t)=π/2+ β +oxm sin(2πf)(4)increase of β . This behavior is consistent with theprevious predic中國(guó)煤化工to about 60°,where β is the stroke plane angle and Xm is theespecially at mYHCN MH GJ =0.4) and.145 .high speed (e.g.. J =0.6)川positive thrust in the upstroke. In Fig. 3(a), C, inthe downstroke is even larger, and Cr in the1=0.2= 0.4upstroke is negative but is of small magnitude.Therefore，the mean lift is contributed by thedownstroke，and the upstroke hanegativecontribution. Similarly, in Fig. 3(b), the magnitudevariation of Cr decreases with the increase of β.Cp is negative in the downstroke and positive with304(large magnitude in the upstroke. Thus, the mean thrustβis mainly contributed by the upstroke.(a2T782Fig.2 Mean lift cofficient C，and thrust cofficientCrfor Am =2.5and Q, =45°β=30°Fig. 4 Vorticity contours during one stroke for J = 0.4,β=60"Am =2.5, 0xm=45'"， β =45°ofThe time-dependent force is closely related tothe vortex structures near the foil. To elucidate themechanism of the generation of the lift and thrust, Fig.-50.250..754 shows the vorticity contours in one stroke. During1the downstroke， a stronger leading-edge vortex isgenerated over the upper surface of the foil. Thus, asshown in Fig. 3(a), a higher C[ is formed. During0fthe upstroke, there is a stronger leading edge vortexover the lower surface of the foil. Correspondingly, ahigher Cp is contributed by the vortex structures.3.2 Effect of stroke amplitudeThe mean lift and thrust coefficients are shown1050.250.75in Fig. 5. At low speed J = 0.2, with the increase ofAm，C，increases and Cr decreases due toFig. 3 Time-dependent lift coefficient C，and thrustdelayed stall mechanism during the stroke. However,coefficient CT during one stroke for J = 0.4,at medium and high speeds, e.g.，J =0.4 and 0.6, asAm =2.5and 0m =45°e flow stall occurs， C， first increases, thenTo analyze the force behavior during the stroke,the time-dependent lift coefficient C，and thrustdecreases around Am =4.5 for J =0.4 and aroundAm= 3.5 foJ= 0.6. . Correspondingly， Crcoefficient Cr in one stroke are shown in Fig. 3.Positive lift is mainly produced in the downstroke andapproaches中國(guó)煤化工1， stall atMYHCNMH G.146larger Am values. At high speed J =0.6, Cr isalways negative for Am =2.5-5, which means thatthe stroke motions used for these parameters may beinapplicable to the flight with high speed. Meanwhile,it is reasonably predicted that Cl increases and Crdecreases with the increase of J for the same Am .0.2500.75J=0.22.J=0.4l0一心0.5A.-5-- 0.25~(bFig.6 Time- dependent lift and thrust coefficients during onestrokefor J =0.4，β =45° and 0xm =45°一一1=0412!■0.6b)Fig. 5 Mean lift and thrust cofficients for β = 45° and0,m =45°930- 4506lFigure 6 shows the time-dependent lift and thrusta)cofficients in one stroke. Basically, positive lift ismainly produced in the downstroke and positive thrustn the upstroke. In Fig. 6(a)， C[ is mainlycontributed by the downstroke.' I he variation (兒magnitude of C[ becomes slower to some extentwith the increase of Am during the stroke. As shownin Fig. 6(b), the variation of magnitude of Crdecreases with Am during the upstroke.406x3.3. Effect of amplitude of pitching angleThe mean lift and thrust coefficients are shownFig. 7 Mean lift and thrust cofficients for β = 45°inFig. 7 for am =35*-65° It is observed that C,and Am =2.5decreases with the increase of 0m . Cr firstincreases somewhat to a peak around 0m = 40", thenFigure 8 shows the time-dependent lift and thrustcoefficients in one stroke. The magnitudeof CLdecreases with a m .中國(guó)煤化工YHCNMH G.147during the downstroke decreases with the increase of ratio, the inclined angle of stroke plane, the stroke0m. It is noted that Cp is negative with small valueamplitude, and the amplitude of pitching angle ofin the downstroke and positive with large magnitudeattack, on the unsteady forces and vortical structureshave been analyzed. Basically, positive lift is mainlyin the upstroke， especially at 0 = 40°produced in the downstroke and positive thrust in thecorresponding to the peak valueof Cr in Fig. 7. Theupstroke. The time-dependent force is closely relatedrelevant vortex structures near the foil are similar toto the vortex structures near the foil. Thecharacteristics revealed are helpful in getting intothese in Fig. 4.physical insights to the mechanisms in insect forwardflight.-____ ..50"----- a。= 60°REFERENCES1] SUN Mao and WU Jiang hang. Aerodynamic forcegeneration and power requirements in forward flight in afruit fly with modeled wing motion [J]. Journal ofExperimental Biology, 2003, 206: 3065-3083.0.250.50.752] WANG J, BRICH J. M. and DICKINSON M. H.a)Unsteady forces and flows in low Reynolds hoveringflight: two-dimensional computations Vs robotic wing!experiments [J]. Journal of Experimental Biology,2004, 207: 449-460.[3] WANG J. Dissecting insect flight []. Annual Reviewof Fluid Mechanics, 2005, 37: 183-210.4] RAMAMURIT R. and SANDBERG W. C. Athree-dimensional computation study of thaerodynamic mechanisms of insect flight [J]. Journal ofExperimental Biology, 2002, 205: 1507-1518.5] KOVACS A. and KAWAHARA M. A finite elementscheme based on the velocity correction method for thesolution of the time-dependent incompressibleb)Navier-Stokes equations [J]. International Journal forNumerical Methods in Fluids, 1991, 13: 403-423.Fig. 8 Time-dependent lift and thrust coefficients during one6] LIAO Qin， Dong Gen-jin and LU Xi-yun. Vortexformation and force characteristics of a foil in the wakestrokefor J =0.4，β =45° and Am =2.5of a circular cylinder [J]. JourFluids andStruetures, 2004, 19: 491-510.! ueures,zo7]LI Dan-yong，Nan-sheng, LU Xi-yun and YinXie-zhen. Force characteristics and vortex shedding of a4. CONCLUDING REMARKSpitching foil in shear flows [J]. Journal ofNumerical analysis on the unsteady force inHydrodynamics, Ser. B, 2005, 17(1): 27-33.insect forward flight has been carried out based on a8] YANG Yan, LU Xi-yun and Yin Xie- zhen. Propulsivevirtual model problem by solving the two-dimensionalperformance and vortex shedding of a foil in flappingincompressible Navier-Stokes equations. The effectsmotion []. Journal of Hydrodynamics, Ser. B, 2003,15(5): 7-12.of some typical parameters, including the advance中國(guó)煤化工MHCNMH G.