Numerical Investigation on the Propulsive Efficiency of Flapping Airfoil with Different Up-Down Plunge Models

27TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES

Numerical investigation on the propulsive efficiency of flapping airfoil with different up-down plunge models

Xingwei Zhang*，Chaoying Zhou ** Shenzhen Graduate School of Harbin Institute of Technology Xili Shenzhen University Town, Shenzhen, 518055, China

Keywords: flapping wing, plunge model, unsteady flow, vortex street

Abstract number of studies carried out. However, how the different plunging motions alter the forces and A systematic study of the influence of up- hence the trust and lift on a flapping wing when down plunge models with different time ratios of the wing is in a forward flight is still not fully down-stroke duration to up-stroke duration on understood. This work attempts to investigate the the performance of a forward flight NACA0014 influences of up-down plunge motions on the airfoil is carried out through numerical solutions aerodynamic performance of a forward flight of two-dimensional incompressible Navier- airfoil by analyzing the aerodynamic Stokes equations using a multigrid mesh method. characteristics of a forward flight NACA0014 Special attention is paid to the vortex shedding airfoil with different up-down plunge motion. mechanism of nonsymmetrical motion and the Numerous experimental as well as influence of oblique attack angle. The attack numerical studies on the flapping motion during angle is selected from -5° to 10° with an interval forward flight of flyers have been reported in of 2.5° and for each attack angle seven different open literatures, pertaining to the forces, control- time ratios are examined. For the cases with the ability and the propulsive characteristics of same reduced frequency and the plunge flapping wings (e.g. Weis-Fogh, 1956; Zarnack, amplitude, the vortex pattern in the wake and the 1988; Dudley and Ellington, 1990; Tobalske and traveling direction of the vortex street are Kenneth, 1996; Willmott and Ellington, 1997; predominantly determined by the plunge motion. Schilstra and Van Hateren, 1999; Lai and The nonsymmetrical plunge motion adjusts the Platzer,1999; Hoon, Seung Y etc, 2004: Tian etc, time of the leading edge vortex separation which 2006). Previous experimental studies have found may results in higher propulsion than the that wake pattern of forward flight is associated symmetrical motion. For each plunge motion, the with the Strouhal number, Reynolds number, traveling direction of the vortex street is also amplitude of plunge and the attack angle of wing. affected by the oblique attack angle which It has been noted that the down-stroke duration is causes a significant change in propulsion and lift. different from that of up-stroke for some The present work confirms that both the trust creatures. However, how the duration difference and lift of the flapping airfoil can be improved influences the aerodynamic performance is still by a nonsymmetrical plunge motion. unclear. To capture all the relevant details of the influence in the flow just by experimental 1 Introduction observations is very difficult, and it is not feasible to find a representative wing kinematics The unsteady viscous fluid flow around a of the free forward flight to predict the likely flapping wing is of practical importance and has aerodynamic consequences of control. Besides received a great deal of attention. Creature flying experimental investigations and theoretical can be very instructive in disclosing the analysis, numerical simulation has become mechanisms of such flow and there have been a another very effective approach and many

**E-mail: [email protected] 1 Xingwei Zhang，Chaoying Zhou

computational studies have been carried out to ⎧ n=m ⎫ study the flapping mechanisms of a flapping h = ⎨a0 + ∑()an cos()n ∗ w∗t + cn ⎬c ⎩ n=1 ⎭ airfoil having the same duration time for up and ⎧n=m ⎫ down strokes [e.g. Tuncer and Platze, 1996; + ⎨ ∑()bn sin()n ∗ w∗t + d n ⎬c (1) Lewin et al, 2003; Zhu et al 2008]. It has been ⎩ n=1 ⎭ found that the separation of the leading edge where t is the time, a0, an, bn and dn are constants, vortices at low heaving frequencies leads to w denotes the flapping frequency defined by diminished thrust and the efficiency decreases 2π/T and T is flapping period. The effect of for higher frequencies when the airfoil plunges different time ratio τ between down and up in a symmetrical pattern. stroke duration is analyzed in figure 1 where τ = From the above discussion, it is obvious 0.60, 0.80, 0.82, 1.00, 1.22, 1.26 and 1.68 are that there have been relatively less studies on the selected. Though the kinematics contains many flapping mechanics of a plunge motion with individual idiosyncrasies, the variation on the different up and down stroke duration time. velocity and acceleration between the different Accordingly, the present work aims to increase half strokes can be always successive. the fundamental understanding of a flow across a When τ =1.0, the wing motion follows a NACA0014 airfoil plunging with equal or symmetrical pattern. The displacement is limited unequal up-stroke and down-stroke duration. A to the cosinusoidal function identified as one finite volume method is applied to solve the two- special Fourier equation which keeps all the dimensional time dependent incompressible coefficients at zero expect a = h = 0.4 . Navier-Stokes equations. It is expected that the 1 0 present results are of fundamental use in exploring the aerodynamic features of the flow around a flapping wing and in getting into physical understanding of creature flying mechanisms.

2 Description of the model

2.1 Kinematics The inspiration of the present type of heaving with different time ratio between down- stroke and up-stroke duration comes from the kinematics of some insects [1, 3, 6] such as the Fig. 1 Kinematics of the airfoil with different ratio desert locust, the hawkmoth etc. with forward τ between down and up stroke duration flight. The experimental data of wing-beats with different duration times for down-stroke and up- 2.2 Forces and power stroke is fitted in a form of Fourier series as described by equation (1). Some conversion The period-averaged consumption power factors have been used to ensure that the down- rate P is defined as stroke begin at the upper branch of the amplitude 1 T dSn and to keep all the stroke amplitude with 2h0c P = ∫0 Fn ()t dt , (2) where h0 is non-dimensional amplitude and c is T dt the chord length of the wing. where F (t) represents the instantaneous The displacement h with different up-down n generated total force components in the normal plunge duration can be expressed by Fourier direction of the wing surface and dS dt equation as follows n

2 Numerical investigation on the propulsive efficiency of flapping airfoil with different up-down plunge models

denotes the traveling speed of the airfoil as it executes the plunge motion. The period-averaged thrust force can be evaluated as

1 T Fx = ∫ Fx ()t dt , (3) T 0 (a) where Fx ()t represents the instantaneous generated force components in x-direction of the wing surface. The period-averaged input power coefficientδ is determined by P δ = , (4) ⎛ 1 2 ⎞ ⎜ ρU ∞cs⎟U ∞ ⎝ 2 ⎠ (b) where ρ is density of fluid, s the surface area Fig. 2 (a) Schematic configuration of rigid NACA0014 per unit spanwise length of the airfoil and U ∞ airfoil; (b) Close-up views of the computational grid free stream velocity. The period-averaged thrust power The dimensionless governing equations for two-dimensional, unsteady, incompressible fluid coefficientε is given by flow can be expressed as follows F U 1 T x ∞ ∂u ε = = CThm = ∫0 ()− Cd dt , i ⎛ 1 2 ⎞ T = 0 , (7) ⎜ ρU ∞cs⎟U ∞ ∂xi ⎝ 2 ⎠ ∂u ∂u 1 ∂p ∂ ⎡ ∂u ⎤ (5) i + u i = − + ⎢v i − u 'u ' ⎥ whereC is the mean thrust coefficient and C j i j Thm d ∂t ∂x j ρ ∂xi ∂x j ⎣⎢ ∂x j ⎦⎥ the drag coefficient. (8)

The propulsive efficiency λ is defined as 2 ∂u ∂u j u 'u ' = kδ − v ( i + ) (9) ε i j ij λ = . (6) 3 ∂x j ∂xi δ where u is the velocity vector, the pressure p is 2 ' ' 3 Governing equations and solver normalized by ρU , ρuiu j is the Reynolds stress, δ is the Kronecker Delta number. As shown in Fig.2, viscous flow over an ij airfoil with chord c of infinite span undergoing a At the inflow, a uniform profile for the plunging motion with a constant initial angle is streamwise velocity is prescribed. A fully considered. The computational domain is 18c in developed outflow boundary condition is given horizontal direction and 10 c in vertical direction. at the downstream boundary. A symmetry After a great deal of grid independent tests, a boundary condition is applied on the upper and grid system with 4.2×104 elements is chosen and lower boundaries of the computational domain. a non-dimensional time step 0.0025 is selected. The non-slip boundary conditions are specified In each time interval, a rigid grid method is used at the airfoil surface. The calculation starts in the internal domain around the airfoil and a assuming the fluids is initially at rest. The 4 dynamic deformable remesh method in Reynolds number is kept at 10 . The Strouhal accordance with Geometric conservation laws number of current models is 0.255. The method [14] is applied in the outer field. conservation equations subjected to the

3 Xingwei Zhang，Chaoying Zhou

aforementioned boundary conditions are solved of the change with the plunging time ratio for using the shear stress transport (SST) turbulence different angleα is different. The plunge motion model coupled with the conformal-hybrid grids has a significant effect on the thrust when the method [15]. The PISO algorithm [16] is wing flight forward with fixed inclination. employed to deal with the coupling between the Whether an asymmetrical plunge motion pressure and velocity, and the third-order produces higher thrust than a symmetrical one is accurate QUICK scheme [17] is used to discrete affected by the initial angle α 0 . The similar the convective terms. The residual smoothing phenomenon was experimentally observed by approach is also applied to accelerate the Willmott and Ellington [6]. They found that the convergence in solutions in each time step. hawkmoth increases in forward speed tended to be accompanied by an increase in stroke angle 4 Results and discussion and a decrease in body angle. Current study The present numerical code is first validated implies that the thrust force can be changed by by comparing the experimental results of Tuncer adjusting the nonsymmetrical character of and Kaya [18] and other numerical result of motion when the angle of attack can not be Miao and Ho [19]. The comparison is show in changed. Fig. 3 where the drag coefficient is plotted against the non-dimensionlal time t′ = tU ∞ c for flow over the airfoil spending the same time in the up- and down-stroke duration. It can be seen that the present calculation is reasonably accurate which confirms that the numerical code is reliable for predicting the fluid forces for the flow over a flapping wing with a plunging model.

(a)

Fig. 3 Comparison of drag coefficient with results of Tuncer et al. and Miao et al. In order to facilitate the comparison on the effect of different plunging models, the reduced frequency k =wc/U∞ is kept at 2. Fig. 4(a) shows the variation of the averaged thrust coefficient CThm against the attack angle α for different plunging time ratios. It can be observed immediately from the figure that the (b) nonsymmetrical motions produce bigger average Fig. 4 The period-averaged thrust coefficients (a) and thrust than the symmetrical motion when the vertical force coefficients (b) with respect to the angle α attack angleα is zero. It also seen that the trend when k=2.

4 Numerical investigation on the propulsive efficiency of flapping airfoil with different up-down plunge models

The variation of the mean vertical force the experimental research of Von Kármán and coefficients CV against the attack angle α for Burgers [20]. different plunging time ratios is given in Fig. To investigate the aerodynamic 4(b). The mean vertical force produced in one performance of the nonsymmetry plunging cycle by the symmetrical motion is found to be motion, the vorticity structure in the near wake is increase with the attack angle from -5° to 10°. presented in Fig. 6. The counterclockwise For the nonsymmetrical plunge motion, the vorticity marked with red colour and clockwise curves are no longer monotonically increase. The vorticity with blue one so that the structure of the changes of the period-averaged input power reverse Von Kármán or classical Von Kármán coefficients δ and the propulsive efficiency λ streets can be easily identified. It is noted that the for different plunging time ratios are plotted in vortex patterns in the near wake for different Fig. 5(a) and 5(b) respectively. The averaged plunging motions are quite different. The figures power coefficient appears to have only a very show the evolution behavior of the leading edge little change with the attack angle for each vortices (LEV) and the trailing edge vortices plunging motion (see Fig. 5(a)). The propulsive (TEV) of the airfoil for different plunge motions. efficiency of the symmetrical movement can be There is a clear correlation observed from the improved by keeping the cross-section with a figures that the vortex patterns in the near wake small angle with respect to the incoming flow decided by the formation and the growth of both when the wing travels at a uniform speed (Fig. the LEV and the TEV, and also affected by the 5(b)). The propulsive efficiencies of the interaction between them. This is quite similar to nonsymmetrical plunge motions are not always that observed numerically by Lewin and Haj- high because the flight needs some compensation hariri using a symmetry plunge model. inputs for the longitudinal forces which results in Gopalkrishnan et al. [21] had similar vertical oscillations so as to meet a constant experimental observations. Also the present speed forward flight. study explores that the fate of the LEV and how In the symmetrical plunging case, the wake it interacts with the TEV can be adjusted by the structure is associated with the amplitude h and input motion of the flapping wing. The number the reduced frequency k when the initial angle is of vortices shed per shedding cycle can be two or zero as suggested by Lewin and Hai-hariri [12]. more which can be adjusted by the The present results show that there are always unsymmetrical plunge motion even when the thrusts existed with the opposite direction of the reduced frequency and amplitude keep constant, incoming flow. The alternate vortex shedding of While the analogous results of the symmetrical the model corresponds to the reverse Von plunging appear at a higher Strouhal number. Kármán vortex street which is in agreement with

(a) (b) Fig. 5 Interrelationship between the input power coefficient δ and the propulsive efficiency λ with respect to angle α.

5 Xingwei Zhang，Chaoying Zhou

(a) τ =1.22 (b) τ =0.82

(e) τ =0.60 (f) τ =1.68 Fig.6 Vorticity contour plot for nonsymmetry plunge with different down-up time ratio when α =0° (blue: clockwise, corresponding to negative vorticity values).

It is found that for the same plunge motion symmetrical plunge motion. The attack angle has the influence of the attack angle on the significant effect on the traveling direction of the formation，the separation time of the vortices vortex street. For instance, when τ =1.26, the and the structure of the vortex street are quite formation of vortex shedding are quite similar limited. The important effect of the attack angle except the inclination of the vortex street (see is on the evolution of the vortex street in the near Fig. 7) which also happens for a symmetric wake. It is seen that as the attack angle increases plunge motion. Additionally, the intensity of traveling direction of the vortex street changes vortices in the near wake changes as the angleα greatly. Therefore once the plunge motion is increases. These results reveal that the vortex designed the corresponding vortex structure is derived from the interaction between the LEV determined. Another interesting finding is that and TEV is predominantly determined by plunge when τ = 0.8, 0.82, 1.68, the LEV remains motion and affected by the attack angle. The attached to the surface of the airfoil for each thrust and lift are decided by the formation, stroke which results in a higher lift as shown in shedding and evolution of vortices which can be Fig. 4(b). This phenomenon is similar to the greatly affected by the plunge motion. numerical study of Wang [22] for the 6 Numerical investigation on the propulsive efficiency of flapping airfoil with different up-down plunge models

(a) α = -5° (b) α = -2.5°

(e) α = 7.5° (f) α = 10°

Fig.7. Vorticity contour plot for nonsymmetry plunge with down-up time ratio τ =1.26 when the wing at highest position. (blue: clockwise, corresponding to negative vorticity values). The present results show that there are always thrusts existed with the opposite direction of the 5 Conclusions incoming flow where the reverse Von Kármán A systematic study of the influence of up- vortex street appears. This is in agreement with down plunge motion with different time ratios on the experimental research of Von Kármán and the performance of a forward flight NACA0014 Burgers. airfoil is carried out in a numerical simulative It is found that the nonsymmetry of a method. The wake structure and the plunge motion is one of significant factors to corresponding propulsive performances of the improve the performance of the flapping wing. airfoil have been studied. With a symmetrical The asymmetry of the plunge affects the plunge motion, the wake structure of the airfoil formation of leading (LEV) and trailing (TEV) is associated with the plunge amplitude and the edge vortices and their interaction, including the reduced frequency when the initial attack angle direction of motion of the vortex sheet. The is zero, as suggested by Lewin and Hai-hariri. combination of these effects can lead to higher

7 Xingwei Zhang，Chaoying Zhou

propulsive efficiency and thrust for some [9] Park, H.C. and Lee, S.Y. and Lim, S.M. and Lee, S.K. unsymmetric flapping cases. For each and Yoon, K.J. and Goo, N.S. Design and demonstration of flapping wing device powered by unsymmetric plunge, the propulsive efficiency is LIPCA. Smart Structures and Integrated Systems, also related to attack angle. These results suggest 2004. that the thrust performance and the loads of the [10] Tian, X., Iriarte-Diaz, J., Middleton, K., Galvao, flapping vehicles can be improved by mean of R.,Israeli, E., Roemer, A.,Sullivan, A., Song, A., adjusting the nonsymmetrical factors such as the Swartz, S. and Breuer, K. Direct measurements of the time ratio between up-stroke and down-stroke kinematics and dynamics of bat flight. Bioinspiration & Biomimetics, Vol. 1, pp 10-18, 2006. duration over a forward flight. Future studies [11] Tuncer, I.H. and Platzer, M.F. Thrust generation due will be considering the effect of deformation to airfoil flapping. AIAA Journal. Vol.34, No. 2, pp caused by load distribution along the wing 509-515, 1996. surface, and computation on the three- [12] G. C. Lewin and H. Haj-hariri. Modelling thrust dimensional rigid model to investigate the generation of a two-dimensional heaving airfoil in a spanwise flow is also necessary. viscous flow. J. Fluid Mech, Vol. 492, pp.339-362, 2003. [13] B. L. Zhu, T.H. Xiao, H.S. Ang, X.C.Zhou. Acknowledgement propulsive effects of same frequency dual flapping a airfoils in a micro air vehicle. J T singhua Uuiv, Vol. The work described in this paper was 48, pp 298-301, 2008. supported by a grant from National Natural [14] Lesoinne, M. and Farhat, C. Geometric conservation Science Foundation of China (Project number laws for flow problems with moving boundaries and No. 90715031). The financial support is deformable meshes and their impact on aeroelastic computations. Computer Methods in Applied gratefully acknowledged. Mechanics and Engineering, Vol. 134, No.1-2, pp 71- 90, 1996. [15] A. Brandt. Multi-level Adaptive Computations in References Fluid Dynamics. Technical Report AIAA-79-1455, AIAA, Williamsburg, VA, 1979. [1] Weis-Fogh, T. Biology and physics of locust flight. Ⅱ.Flight performance of the desert locust. Biological [16] Issa, RI. Solution of the implicitly discretized fluid Sciences, Vol.239, No. 667, 1956. flow equations by operator-splitting, Journal of Computational Physics, Vol 62, pp 40-65, 1985. [2] Zarnack, W. The effect of forewing depressor activity on wing movement during locust flight. Biological [17] Leonard, BP. A stable and accurate convective Cybernetics, Vol. 59, No. 1, pp 55-70, 1988. modelling procedure based on quadratic upstream interpolation. Int. J. Mech. Sci. Vol. 16, pp 183-308, [3] Roland Ennos, A. The kinematics and aerodynamics 1976. of the free flight of some diptera. Journal of Experimental Biology, Vol. 142, pp 49-85, 1989. [18] Tuncer, I.H. and Kaya, M. Thrust generation caused by flapping airfoils in a biplane configuration. Journal [4] Dudley, R. and Ellington, C. P. Mechanics of of Aircraft. Vol. 40, pp 509-515, 2003. forward flight in bumblebees. Ⅰ.Kinematics and [19] Miao, J.M. and Ho, M.H. Effect of flexure on morphology. Journal of Experimental Biology, Vol. aerodynamic propulsive efficiency of flapping flexible 148, No. 1, pp 19-52, 1990. airfoil. Journal of Fluids and Structures, Vol. 22, No. [5] Tobalske, W. and Kenneth, P. Dial. Flight kinematics 3, pp 401-419, 2006. of black-billed magpies and pigeons over a wide range [20] Von Kármán, Burgers J. M.. Aerodynamic theory, of speeds. The Journal of Experimental Biology, Vol. Springer, Berlin, pp280-310, 1935. 199, pp 263-280, 1996. [21] Gopalkrishnan, R., Triantafyllou, M. S., Triantafyllou, [6] Willmott*.A.P. and Ellington, C. P. The mechanics of G. S. & Barrett, D. Active vorticity control in a shear flight in the hawkmoth manduca sexta. Ⅰ.Kinematics flow using a flapping foil. J. Fluid Mech, Vol. 274, pp of hovering and forward flight. The Journal of 1–21, 1994. Experimental Biology, Vol. 200, pp 2705-2722, 1997. [22] Wang, Z.J. Vortex shedding and frequency selection [7] Schilstra, C. and Van Hateren, J.H. Blowfly flight and in flapping flight. Journal of Fluid Mechanics, Vol. optic flow Ⅰ.Thorax kinematics and flight dynamics. 410, pp 323-341, 2000. The Journal of Experimental Biology, Vol. 202, pp 1481-1490, 1999. [8] Lai, J.C.S., Platzer, M.F. Jet characteristics of a plunging airfoil. AIAA J.Vol.37, No. 12, pp1529-1537, 1999 8 Numerical investigation on the propulsive efficiency of flapping airfoil with different up-down plunge models

Contact Author Email Address

*E-mail:[email protected] **E-mail:[email protected] Fax: +86 0755 26033866,

Copyright Statement The authors confirm that they, and/or their company or organization, hold copyright on all of the original material included in this paper. The authors also confirm that they have obtained permission, from the copyright holder of any third party material included in this paper, to publish it as part of their paper. The authors confirm that they give permission, or have obtained permission from the copyright holder of this paper, for the publication and distribution of this paper as part of the ICAS2010 proceedings or as individual off-prints from the proceedings.