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

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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].
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