Chao ZHOU, Yanlai ZHANG, Jianghao WU

School of Transportation Science and Engineering, Beihang University, Beijing 100083, China

KEYWORDS

Abstract Introducing flexibility into the design of a vertically flapping wing is an effective way to enhance its aerodynamic performance. As less previous studies on the aerodynamics of vertically flapping flexible wings focused on the lift generated in a wide range of angle of attack·a 2D numerical simulation of a purely plunging flexible airfoil is employed using a loose fluid–structure interaction method.The aerodynamics of a fully flexible airfoil are firstly studied with the flexibility and angle of attack.To verify whether an airfoil could get aerodynamic benefit from the change in structure, partially flexible airfoil with rigid leading edge and flexible trailing edge were further considered.Results show that flexibility could always reduce airfoil drag while lift and lift efficiency both peak at moderate flexibility.When freestream velocity is constant,lift is maximized at a high angle of attack about 40° while this optimal angle of attack reduces to 15° in drag-balanced status. The airfoil drag reduction, lift augmentation as well as efficiency enhancement mainly attribute to the passive pitching other than the camber deformation. Partially deformed airfoil with the longest length of moderate flexible trailing edge can achieve the highest lift. This study may provide some guidance in the wing design of Micro Air Vehicle (MAV).

1. Introduction

Motivated by the agile flight of natural flying creatures,the flapping Micro Air Vehicle(MAV)has been a hot research area1,2in the last two decades and many protocols have been developed,such as Nano Hummingbird3invented by AeroVironment Inc., Delfly4by Delft University of Technology as well as Robobee5by Harvard University. In the initial stage of MAV design, designers mostly focused on mimicking wing motions of natural flyers but recently much more attention has been concentrated on the optimized design for lift augmentation, drag reduction as well as power efficiency improvement.One of the effective ways is to introduce flexibility into wing design.

The passive deformation of the wing due to structural flexibility is found beneficial for improving aerodynamic performance of both insects6,7and MAVs.8The insect wing deformation is usually divided into spanwise bending, spanwise twist as well as chordwise camber.6,7,9The wing deformation in spanwise bending and chordwise camber is beneficial for aerodynamic force increment while the spanwise twist leads to the increment in power efficiency. MAV wings, which consist of veins and membrane,imitate insect wings.However,due to the differences in wing structures and material properties,the aerodynamics of MAV wings is remarkably different from their counterparts in nature.Therefore,flapping MAV designers also investigated the relationship between aerodynamic performance and structures of the wing to find an optimal structural design.10–12All these works and findings provide us with a meaningful reference for the wing aerodynamic optimization by utilizing wing flexibility.

To obtain more accurate guidance in the wing design,some researchers also explored the quantitative relation between wing structural parameters and its aerodynamic performance. The spanwise stiffness of the wing was found three times higher than that in chordwise direction,13thus the chordwise deformation is much more severe than spanwise deformation. Consequently,more interests were focused on the effect of chordwise flexibility of the wing on its aerodynamic forces based on 2D airfoil models14–16combining plunging and pitching motions.In such a system, non-dimensional structural parameters,17,18that affect its aerodynamic performance are density ratio (ρ*) and stiffness(Π).How these two parameters influence the aerodynamic performance has been systematically analyzed. Tian et al.19conducted a fluid–structure interaction analysis of a plunging/pitching airfoil in cruising flight and considered the effect of ρ*and frequency ratio ω*(merely Π-dependent)on thrust and lift.They found that both thrust and propulsive efficiency peak at a frequency which is near one-third of the airfoil natural frequency while lift peaks at the flapping frequency that is slightly lower than the natural frequency.Clearver et al.20quantitatively measured the thrust enhancement by a partially flexible plunging airfoil in a water tunnel and found that the optimal Π for thrust enhancement is about 0.1–2.Quinn et al.16scaled the propulsive performance of heaving flexible panels with dimensionless parameters and found that the propulsive performance strongly depends on structural resonance when the trailing edge amplitude is maximized. Kang and Shyy21also explored the scaling law between propulsive performance and lift with airfoil structural property as well as kinematic parameters.

Till now,most of the studies on the aerodynamics of a vertically plunging flexible airfoil focused on the effect of flexibility on propulsive performance at zero angle of attack while less were on the airfoil’s capability in lift generation. However,despite the application in propulsion, a vertically flapping can also be used for lift generation. Natural insects like dragonfly,22butterfly23and drone-flies approximately flap their wings vertically24in cruising flight. Such a vertical plunging model is also applied in MAV designs, one of whom is the micro flapping rotary wing.25–28A flapping rotary wing,arranged anti-axisymmetric around a vertical shaft, is selfpropelled and the rotating motion is driven by thrust produced through wing vertically flapping. To further improve the design of unmanned flapping robots using vertically flapping wings, it is of great importance to further study the lift generation capability of such a vertical plunging airfoil system.

One of the key kinematics affecting the lift produced by a plunging airfoil is the pitching motion. Previous studies, especially numerical simulations, usually assumed that the airfoil underwent an ideal sinusoidal pitching motion.However,when designing MAVs, the precise realization of this ideal pitching motion requires complex mechanics which definitely increase the weight of aircraft.One alternative method is to fix the wing initial angle of attack and the pitching is achieved by passive deformation. To meet the requirements of wing design in this method, it is necessary to first understand the effect of initial angle of attack and flexibility on aerodynamic performance.

Vertically plunging flexible airfoil is usually a self-propelled system where the thrust mainly comes from the unsteady motion of Trailing Edge (TE). As an airfoil cruises and plunges, lift is produced and most of the lift is contributed from the Leading Edge(LE)area caused by the Leading Edge Vortex (LEV). Consequently, it is an interesting question whether an airfoil could get aerodynamic benefit from a configuration with rigid LE and flexible TE. To the author’s best knowledge,the airfoils studied in previous studies were usually fully flexible and an airfoil with both rigid and flexible components was seldom studied except Gursul et al.,14,29where a tear-drop airfoil with flexible TE was investigated. As the series work conducted by Gursul et al.14,29only considered the propulsive performance at zero angle of attack, we expand the largest angle of attack to a high value at 60° to address whether an airfoil can gain benefits in both thrust enhancement and lift augmentation by a combination of a rigid LE and flexible TE at non-zero angle of attack.

In this paper, the effect of flexibility on the aerodynamic performance of a periodically plunging flexible airfoil is numerically studied using a partitioned Fluid-Structure Interaction (FSI) method. Firstly, the overall aerodynamic performance of a fully flexible airfoil varying with airfoil flexibility and Angle of attack is addressed. Then, the instantaneous forces, deformation pattern and vortex structures are further analyzed to explain the difference in (cycle-averaged) aerodynamic forces as the airfoil becomes flexible. Finally, the aerodynamic performance of a partially flexible airfoil is also considered to address the effect of airfoil structural configuration on aerodynamic performance.

2. Models and methods

2.1. Models

In this paper, three airfoil models,i.e. a rigid airfoil(shown in Fig.1(a)),a fully flexible airfoil(shown in Fig.1(b))as well as a partial flexible airfoil with rigid LE and flexible TE(shown in Fig.1(c)),are studied.For the partial flexible airfoil,the length of rigid LE is donated ascR.

For simplification, a flat airfoil with half circles at LE and TE is used for simulation. Two Cartesian coordinate systems including an inertial coordinate systemOXYand a bodyfixed coordinate systemoxyare introduced to describe the periodical plunging motion and passive deformation of the airfoil (Fig.1). The originocoincides with the position of LE in the middle of upstroke.AxesOXandOYare parallel and vertical to the horizontal direction respectively. The body-fixed coordinate systemoxymoves with the airfoil and its axisoxis along the mean camber line when the airfoil is not deformed(the dotted line in Fig.1).

The airfoil is initially fixed at a constant angle of attack(α0)and then plunges vertically in a time-harmonic function against a constant horizontal freestream with a velocityU∞.The plunging motion inoYdirection is given as:

Fig.1 Schematic of airfoil models, coordinate systems and kinematics.

wherehdenotes the plunging displacement;h0andfrepresent the plunging amplitude and frequency, respectively;tis the physical time.The passive deformation of the airfoil is numerically solved using the FSI method introduced in Section 2.2.The airfoil kinematic is non-dimensionalized following our previous work. The mean velocity of plunging motionUref=4h0fand chord lengthcare used as the reference velocity and the reference length, respectively. The reference timeTrefis computed asc/Uref. Accordingly, plunging amplitude and freestream velocity can be non-dimensionalized ash0*=h0/cand advance ratioJ=U∞/Uref, respectively.

2.2. Methods

2.2.1. Fluid–structure interaction solving procedure

The aerodynamic problem of a flexible airfoil is a complex fully coupled FSI problem. A partitioned scheme,7,21,30,31also referred as a loose coupling for FSI simulation,is performed to compute the airfoil flexible deformation. This scheme has been widely and successfully applied to numerous FSI problems including plunging airfoils19,21and flapping wings.7,17,21In this partitioned scheme, the Computational Fluid Dynamic (CFD)solver and Computational Solid Dynamics (CSD) solver are called one after another Therefore,the partitioned scheme allows the use of existing mature CFD and CSD solvers and the development of a FSI interface for data exchange, i.e. aerodynamic pressures and deforming displacements,between the two solvers.In this paper,the FSI solver is built based on an in-house CFD solver and the ANSYS finite element program.A brief introduction of the algorithm is illustrated in Algorithm 1.

For FSI problems in a high-density environment like water,the numerical solution may encounter the difficulty in converging owing to the added-mass effect.32To overcome such a problem, an implicit scheme is used and in each physical time step, the CFD and CSD solvers are called one after another until the solution converges. Given that the implicit scheme requires more time and higher computational costs, two measures are taken to accelerate the convergence in every physical time step. The first measure is to predict the deforming displacement at the beginning of the physical time step,7thereby ensuring the initial deforming displacement approaches the final solution. The second measure is introducing the Aiken Δ2method33,34for computing ωmin Step (5) to accelerate the convergence of sub-iterations.

2.2.2. CFD solver

The equations governing the flow around the airfoil are the 2D incompressible unsteady Navier–Stokes equations and its dimensionless form written in the inertial coordinate system is given as follows:

whereuis the velocity vectors andpis the static pressure.Reis defined asUrefc/υ where υ is the kinematic viscosity coefficient.The artificial compressibility method developed by Rogers et al.35,36is employed to derive the velocity and pressure in the flow field. An O-shape grid (Fig.2) is used and a grid deformation technique is applied to the grid displacements.Details of the flow solver are provided in our previous study.37

Fig.2 Overview of computational grid.

Once the FSI solution in each physical time converges,aerodynamic forces (i.e., drag (D) and lift (L)) can be computed by integrating pressure and the viscous forces over the airfoil surface. The drag and lift coefficients can be computed as follows:

Then, the power consumption of the airfoil can be computed by

2.2.3. CSD solver

The deforming displacement of the airfoil is computed through a transient analysis using the commercial software Ansys in the body-fixed coordinate system. When the airfoil becomes very flexible, its transverse deforming displacement is significantly enlarged and the effect of non-linear deformation must be considered. Therefore, the large displacement and non-linear effect are always considered in the transient analysis for all cases in this study. The airfoil is simplified into a cantilever beam with forced displacement in LE and meshed by a beam element (i.e. beam 3). The implicit time integration algorithm in ANSYS,i.e.the Newmark method,is used to ensure numerical stability in the structural non-linear analysis.Nodes of the CSD grid are the same as that of the CFD grid on the airfoil surface to reduce numerical error during the transfer of aerodynamic pressure and structural displacement between the CFD and CSD solvers.

2.3. Method validation

2.3.1. Grid test

The computational code and the grid models are validated to ensure the rationality of the simulation.The grid density,computational domain size,and time step are firstly verified.In our latest literature,37the grids used to emulate a plunging rigid airfoil are tested. To further verify the grid-independence in the FSI simulations, we performed a grid test on the basis of our previous work.37Parameters in the tested case are selected asRe=350,h0*=0.1, α0=20°,J=0, Π=4.5 and ρ*=103.

Time courses ofCL,CDand TE deformation displacements(xTEandyTE)computed using three grids with different densities in a flapping cycle are plotted in Fig.3. The nodes of the three grids are 100×161 (in the normal and chordwise directions, respectively, for Grid 1), 151×241 (for Grid 2), and 200×325 (for Grid 3). The outer boundary is located 40caway from the airfoil surface. The first grid distances near the airfoil are 10-4,10-4and 5×10-5.400 time steps per flapping cycle are used in the computation.Non-dimensional time^tin Fig.3 is defined to describe the time in one plunging cycle.^t=0 and ^t=1 indicate the start of the downstroke and the end of the upstroke,respectively.Fig.3 shows that the computational results from Grid 1 deviate slightly from those based on the other two grids while the results from Grid 2 are almost the same with those from Grid 3.In order to capture the subtle structure in the flow field,the mesh density of Grid 3 is selected and the test of gird size and time step are further conducted based on Grid 3. The computational domain size is proposed to be 40cand 400 time steps per flapping cycle are sufficient.In summary,the preceding analyses indicates that Grid 3 with 400 time steps in one flapping cycle is appropriate and is therefore selected in this study.

2.3.2. Numerical code validation

Fig.3 Computed results of CL, CD and TE deformation displacements (xTE and yTE).

The computational code is validated to ensure its rationality.In previous studies,a code validation based on a plunging rigid airfoil model has already been conducted. To further test whether our code can simulate flexible airfoil aerodynamics,an experimental study conducted by Heathcote and Gursul14is chosen for comparison as a code validation. This case has been widely accepted for code test in various FSI problems.In this experiment, the effect of chordwise flexibility on propulsive characteristics at lowRe(Re=9000) was investigated using a plunging airfoil (Fig.4(a)) against a uniform flow. A plunge motion was imposed on the rigid teardrop-shaped leading-edge and the flexible tail dynamically deformed. The plunging amplitude was 0.194 m and the Strouhal number(St, defined as 2fh0/U∞) was 0.34. The chord length of LE was 30 mm and the length and elastic modulus of flexible plates were 60 mm and 2.05×1011Pa respectively. To ensure a varied flexibility, the plate thickness was varied. For code test, the deforming displacement ofyTEandh* of the ‘‘very flexible” airfoil withb*=0.56×10-3is computed and compared with the experimental results (Fig.4(b)). As plotted in Fig.4(b), the computed displacement agrees well with the experimental results. The slight variations are mainly attributed to the difference in the airfoil shape and the absence of the spanwise flow in a 2D simulation. It also should be noted that as the density of the TE is about 7200 kg/m3which is only 7.2 times larger than fluid density (i.e. 1000 kg/m3for water),the TE deformation is determined by both aerodynamic and inertia forces. The good agreement in deformation displacement between our code and literature also indicates the accuracy of our FSI method in force computing. In summary,our code is suitable for the present study.

Fig.4 Code validation using experimental results of Heathcote and Gursul.14

3. Results and discussion

3.1. Non-dimensional parameters

Non-dimensional parameters determining the FSI characteristics of a flexible airfoil can be separated into three types, i.e.kinematic parameters (i.e.Re,h0*,J, α0), material property parameters (i.e. Π, ρ*) as well as structural configuration parameters (cR). This paper concentrates on the effect of α0,stiffness and structural geometrical parameters on aerodynamics and the kinematic parameters are fixed atRe=1500,h0*=0.2,b*=0.01 andJ=1. TheSt number and reduced frequencyk=fc/U∞, derived from kinematics parameters,are 5 and 1.25, respectively. The density ratio was set at ρ*=103as the ρ* for a MAV wing moving in the air is at a magnitude ofo(103).cRranging from 0 tocwith an interval of 0.25care studied in this part to reveal the effect of airfoil structure on its aerodynamic forces. The Airfoil withcR=0 andcR=crepresents the fully flexible airfoil and the fully rigid airfoil, respectively. Details for the parameter space in this study is listed in Table 1.

In this paper, we firstly focus on the effect of Π and α0on the aerodynamic performance,especially drag,lift and lift efficiency, of a fully flexible plunging airfoil. Then, further investigations on how stiffness changes the airfoil aerodynamics are elucidated.Finally,a study of partially flexible airfoils with differentcRis conducted to answer whether plunging airfoil could benefit from the change in airfoil structure.

3.2. Effect of stiffness on aerodynamic performance of fully flexible airfoil

Table 1 Variables and its values of purely plunging airfoil.

Fig.5 D, L, p and L/p changing with Π and α0.

Most literature on the plunging airfoil focuses on propulsive characteristics at α0=0°. In order to further prove the rationality of our numerical results and expand our understanding of the propulsive characteristic of the purely plunging flexible airfoil, the airfoil propulsive efficiency at α0=0° is computed and discussed. The optimal stiffness for the highest propulsive efficiency of the vertically plunging airfoil at α0=0° has been revealed in previous literature. The optimal Π found by Clearver et al.36is 0.1–2 and that addressed by Dewey et al. is 0.5. For comparison, our results are presented in Table 2. It can be seen that propulsive efficiency peaks at about Π=1. Considering the difference in the models, our result matches reasonably with that from experiments. Furthermore,unlike the work done by Cleaver et al.38where only airfoil propulsive performance at α0=0° was considered,whereas our finding further suggests that Π ～1 is an optimal value for both thrust enhancement at α0=0°and lift augmentation and lift efficiency improvement at α0＞0°.

Table 2 Propulsive efficiency of fully flexible airfoil at α0=0°.

3.3. Instantaneous forces, deformation pattern and vertex structures

Instantaneous aerodynamic forces are tightly related to the airfoil deformation.Fig.7(c)schematics positive motions,passive deformation pattern as well as force vectors of the three airfoils in the 35th period. In this flapping period, two flexible airfoils always deform upward due to the aerodynamic force.Higher flexibility leads to severe deformation and the TE of the airfoil with Π=0.01 is sometimes even above the LE.From the TE displacement in the inertia frame presented in Fig.7(d) and (e), the TE of the very flexible airfoil almost moves horizontally, resembling a flag fluttering in the air.Unsteadiness caused by TE of the very flexible airfoil to the fluid around the wing is consequently greatly reduced and only small lift and drag is produced at Π=0.01 as shown in Fig.7(c). Despite the upward deformation, the TE of the moderate flexible airfoil moves in a probably sinusoidal mode and a phase lag to LE is observed.As shown in Fig.7(c),this passive deformation causes a larger resultant aerodynamic force.Besides, due to airfoil upward deformation, the much more resultant force is projected into the vertical direction while less into the horizontal direction, which also causes the lift augmentation and drag reduction.

Airfoil passive deformation leads to an equivalent pitching motion, where the mean Angle of attack and pitching amplitude is 20° and 15°, respectively (Fig.7(f)). This means Angle of attack is consistent with the optimal pitching kinematics for the maximum lift found on the basis of a flapping rotary wing.Therefore, airfoil pitching motion can also be produced effectively by airfoil passive deformation instead using transmission mechanism.

Fig.6 J, L as well as L/p at drag-balanced status changing with α0.

Fig.7 Aerodynamic forces and deformations of rigid airfoil (FA), very flexible airfoil (FA) and moderate FA.

Airfoil deformation can be divided into the chordwise chamber and passive pitching motion.6,7,9To elucidate which deformation pattern mainly accounts for the difference in aerodynamic performance as airfoil becomes flexible, aerodynamic forces of the airfoil with the only chordwise chamber and passive pitching are computed and compared with that of RA and FA. More specifically, an airfoil with only chordwise chamber [FA(C)] is a deformed airfoil plunging at fixed Angle of attack and its chamber follows the same function with the flexible airfoil. Airfoil with only passive pithing [FA(PP)] is a rigid flat airfoil following the passive pitching kinematics of flexible airfoil. The chordwise chamber and passive pitching motion are computed based on instantaneous shapes of the flexible airfoil. Rigid airfoil and the moderate flexible airfoil with Π=1 and α0=40° is chosen for discussion.Then, instantaneousCLandCD, as well as its mean values,produced by FA(C) and FA(PP) are computed and compared with that from FA and RA.Results are plotted in Fig.8.It can be seen that the time course ofCLandCDin 30th-40th flapping periods of FA(PP)resembles that of FA while time course ofCLandCDof FA(C)is much like with that of RA.Comparison ofC-Lfurther supports this finding. TheC-Lof FA(PP)(4.41) is approximately 26% larger than that of FA (3.49)while that of FA(C), i.e. 1.56, is only half of that of FA, it is almost twice than the difference between RA and FA(C).This suggests that the passive pitching of the airfoil due to deformation mainly accounts for the difference in aerodynamic forces production between RA and FA. From this point of view, a simplified passive pitching airfoil model can approximately characterize theC-Lof a fully flexible airfoil with Π=1 despite the slight overestimation, which may be useful for engineering application in lift estimation.

To further explain how flexibility affects airfoil aerodynamic performance, vortex field, described using the vorticity ω, around the aforementioned three airfoils at α0=40° are analyzed.Vortex fields in the 35th plunging period are plotted in Fig.9(a)–(c).As shown in Fig.9,the vortex field around the very flexible airfoil is much different from those around the others. As the very flexible airfoil is in streamline shape, the vortex generation is so weak that the aerodynamic pressure mainly concentrates on the LE, causing a small lift and drag.The vortex field around the rigid airfoil are similar with that around the moderate flexible airfoil and a strong LEV and Trailing Edge Vortex (TEV) can be observed in both cases.Due to the incoming freestream, their newly formed LEV firstly moves upstream for about one chord length, then reverses its moving direction and migrates to downstream following the incoming freestream. The LEV moving backward encounters the TEV, causing a strong vortex-vortex and vortex-airfoil interaction and leads to a complicated flow pattern around the airfoil. The similarity in the flow patterns of RA and FA (Π=1) thus results in the similarity in aerodynamic pressure on the surface (shown in Fig.9(c) and (d))and the instantaneous aerodynamic forces. As shown in Fig.9(c) and 9(d), at the beginning of upstroke at ^t=0 (or downstroke at^t=0.5),wake capture effect is much significant that the strong downwash (or upwash) produced by the last downstroke and upstroke introduces a comparatively uniform pressure distribution. While in the middle of upstroke (at^t=0.25) and downstroke (at ^t=0.75), the LEV and TEV have formed and thus a low-pressure region is resulted below the LEVs and TEVs, which becomes the main source for the high aerodynamic force.

Despite the complex vortex structure around airfoil, some distinct differences in the vortex field could still be observed between rigid airfoil and moderate flexible airfoil. It can be clearly seen from Fig.9(a)and(b)that those LEVs and TEVs around the moderate flexible airfoil are weaker and much closer to airfoil surface than the rigid airfoil. This is mainly because the passive deformation of moderate flexible airfoil is almost opposite to its plunging motion, thus the resultant velocity of flexible airfoil is greatly reduced, especially for the TE at the middle of half stroke, and the perturbation of the TE to the fluid around the wing is weaker compared to rigid airfoil.Consequently,a weaker and scattered TEV is produced by moderate flexible airfoil and thus could be easily influenced by other vortexes. A group of vortices, including the downward moving LEV and TEV, accumulate in the upper surface of the flexible airfoil.The differences in vortex structure caused by wing flexibility directly change the pressure distribution on the airfoil surface which thus manipulates the aerodynamic force production.As shown in Fig.9(c)and(d),due to a complicated vortex field around the flexible airfoil,the pressure distribution, especially pressure magnitude and direction, differs with wing flexibility.The pressure on the moderate flexible airfoil surface is at a larger magnitude and is much closer to vertical direction compared to that on the rigid airfoil surface.Therefore,the contribution of pressure to lift is increased while that to drag is decreased.This provides an explanation for the difference in lift and drag production discussed above as the airfoil becomes moderate flexible.

Fig.8 CL, CD and its mean values from airfoil with different deformation patterns.

Fig.9 Vortex field around airfoil and aerodynamic pressure on airfoil surface.

3.4. Effect of cR on airfoil aerodynamic forces

Fig.10 L and D produced by moderate flexible airfoil and very flexible airfoil at different cR and α0 (i.e. α0=20° and 40°).

Note that there are two cases significantly distinct from other cases for the partially flexible airfoil with very flexible TE. They are the cases withcR=0.75cbut α0=20° and 40°, respectively. For the former case, the airfoil produces thrust while for the latter one the airfoil’s lift is greatly enhanced.This phenomenon is consistent with the results from Clever et al.38and our aforementioned results on a fully flexible airfoil (seen in Fig.5(a)). Clever et al.38found the optimal Π,which is non-dimensionalized based on the length of flexible TE, for propulsive performance is about 0.5–2. For these two cases, Π redefined using the method in Clever et al. is about 0.64.This Π not only lies in the range of the Π found by Clever et al.for optimal propulsive performance,but also agrees with the optimal Π about 1 that is previously found for high lift and thrust generation based on fully flexible airfoils. This further proves that a Π around 1 may be an optimal value for both lift and thrust generation.

Vortex structures are further analyzed to explore the effect ofcRon airfoil aerodynamics. The moderate flexible airfoil at α0=40°is first considered and vortex structures withcR=0,0.5candcare plotted in Fig.11. As can be seen from Fig.11,the vortex structure and its evolution in thecR=0.5ccase is much closer to that in the rigid airfoil withcR=cand thus induce a similar pressure distribution on both airfoil surfaces.The differences of vortex structures incR=0.5candccases are mainly reflected in TEV. The flexible deformation of TE reduces the intensity of TEV as well as the downward jet of thecR=0.5ccase, compared with that in the RA. As the deformation of TE is not as significant as the plunging displacement,the changes in TEV do not lead to a severe change of aerodynamic pressure on the flexible airfoil.However,as the airfoil deforms upward, the pressure at TE is tilted vertically,leading to drag reduction and lift augment.

Fig.11 Vortex field and pressure distribution of moderate flexible airfoil in the 35th period at α0=40° but different cR.

4. Conclusions

(1) For the fully flexible airfoil, flexibility could always reduce airfoil drag while lift and lift efficiency both peak at moderate flexibility with the stiffness (Π) at 1. The maximum lift of a flexible airfoil is almost 2.8 times larger than that of a rigid airfoil as the trailing edge deformation enhances the resultant force and induces the tilting of aerodynamic forces towards lift direction.When the freestream velocity is constant, the initial angle of attack corresponding to lift peak is about 40°and reduces to 15° for airfoil in drag-balanced status.

Fig.12 Vortex field of very flexible airfoil in the 35th period at α0=40° but different cR.

(2) The airfoil drag reduction, lift augmentation as well as efficiency enhancement are mainly attributed to the passive pitching motion other than the camber deformation.A rigid plunging and pitching airfoil is an alternative model to estimate the aerodynamic performance of a purely plunging flexible airfoil in the initial design stage of a MAV wing.

(3) The fully flexible airfoil with moderate flexibility airfoil is found to produce lower drag and higher lift than partially flexible airfoils but the fully flexible airfoil with very high flexibility is just beneficial for drag reduction.Thus,a MAV wing is suggested to be moderately flexible and fully deformable to enhance its aerodynamic performance.

Acknowledgement

This research was primarily supported by the National Natural Science Foundation of China (No. 11672022).