Xi HE, Qinfeng GUO, Jinjun WANG

Fluid Mechanics Key Laboratory of Education Ministry, Beihang University, Beijing 100083, China

KEYWORDS Aerodynamic forces;Flexible wings;Fluid structure interaction;Kinematic analysis;Lift-enhancement mechanism;Wind tunnel tests

Abstract The kinematic characteristics of flexible membrane wing have vital influences on its aerodynamic characteristics. To deeply explore the regularities between them, time-resolved aerodynamic forces and deformations at different aeroelastic parameters and angles of attack (α) were measured synchronously by wind tunnel experiments.The membrane motion can be mainly divided into two states at α>0° with various lift-enhancement regularities: Deformed-Steady State (DSS)at pre-stall, and Dynamic Balance State (DBS) at around stall and post-stall. Besides, the mean camber, maximum vibration amplitude, and lift coefficient almost reach their maxima simultaneously within the DBS region.By introducing momentum coefficient Cμ of membrane vibration,positive correlation among amplitude, momentum and lift is successfully established, and the liftenhancement mechanism of membrane vibration is revealed.Moreover,it is newly and surprisingly found that at different vibration modes,the maximum vibration amplitude and root mean square of vibration velocity present positive and linear correlation with different slopes, and their chordwise locations are basically consistent. Therefore,novel ideas for active control of flexible wing are proposed: by controlling the vibration amplitude, frequency, and mode, while selecting the specific chordwise locations for intensive excitation,Cμ can be efficiently increased.Ultimately,the aerodynamic performance will be improved.

1. Introduction

With the rapid development and wide application of Micro Air Vehicles (MAVs), requirements for their aerodynamic performance are growing. Advanced MAVs are developing in the direction of intelligence, high-efficiency, and multitasking.Wing is the key component which determines the performance of MAVs. Traditional rigid fixed wing is designed based on specific flight tasks and environmental conditions,and can only achieve the optimal aerodynamic performance at limited design points. Therefore, it has an inherent defect in adaptability for complex environments(such as gust,turbulence, and near-ground). In the past two decades, as a novel bionic concept,1–6flexible membrane wing has been favored in the field of aircraft design due to its high maneuverability and low weight. Unlike traditional rigid fixed wing, flexible membrane wing can change its shape under aerodynamic loads,thereby greatly improving the environmental adaptability of aircraft.

The aerodynamic advantage of flexible membrane wing results from its Fluid-Structure Interaction (FSI) characteristics. Generally, FSI affects the aerodynamic performance of membrane wing from two aspects: mean deformation and unsteady vibration. When there exists an angle of attack between the membrane wing and the incoming flow, pressure difference will generate between the upper and lower wing surfaces, causing mean deformation to the leeward side. Compared with a rigid wing, the mean deformation can increase wing camber and cause the wing more streamlined, thus leading to increased lift and delayed stall.7–10

Besides, the membrane wing will vibrate near the mean camber under certain inflow conditions, and show various unsteady vibration states with the change of angle of attack(α), Reynolds number (Re), stiffness, and aspect ratio. In the early experimental studies of Galvao et al.11and Song et al.12they both found that the flexible membrane exhibited standing-wave vibration modes along chordwise direction during the FSI. The vibration order ascended with increasing Re.Rojratsirikul et al.13,14investigated the unsteady vibration characteristics in the range of Re=5.31×104～1.06×105,α=0°～30° and discovered different dynamic responses at various α. They also found that changing the pre-strain and excess length of the membrane is equivalent to changing the stiffness of the membrane,which greatly affects the membrane kinematics. Bleischwitz et al.15analyzed aspect ratio effect on the flexible membrane wing at Re=6.75×104. It was indicated that the effective α of the membrane wing with small aspect ratio was reduced due to the strong tip vortex downwash effect, so the stall at large α was suppressed. Meanwhile,the small aspect ratio membrane wing showed higher vibration order and frequency. Rojratsirikul et al.16also studied the overall dynamic responses of membrane wing with small aspect ratio at Re=2.43×104, 3.65×104, and 4.87×104.It was shown that the low aspect ratio wing was affected by both leading-edge vortex and tip vortex, which showed mixed chordwise and spanwise vibration modes at small α and second-order vibration mode at large α.Sun et al.17conducted numerical simulation of flexible membrane wing in laminar flow (Re=1×102–1×104) and found that with the increase of Re or α,the membrane wing first bifurcated from static state to periodic vibration due to the trailing-edge vortex shedding.Then the wing changed from period to chaos due to the leading-edge vortex shedding. All the studies above have shown that the unsteady vibration features of flexible membrane wing are restricted by considerable fluid and structure parameters, which further influence the aerodynamic characteristics.

To consider the effects of mean deformation and unsteady vibration separately, flexible membrane wing was compared with the equivalent rigid cambered wing in the experiments of Rojratsirikul et al.13Flexible wing performed higher lift and larger stall α, which was attributed to the interaction between membrane vibration and vortices. However, previous studies only generally indicated that the unsteady membrane vibration could further improve aerodynamic performance,lacking detailed analysis of the regularities between membrane vibration and lift, also lacking understanding of the effects of different vibration modes. In this paper, the kinematic and aerodynamic characteristics as well as their regularities are investigated by synchronous measurements of deformations and aerodynamic forces. The lift-enhancement mechanism of membrane vibration is revealed by Particle Image Velocimetry(PIV) measurement of flow fields. In addition, based on the effects of different vibration modes,novel ideas for active control of flexible wing are proposed, which can provide theoretical guidance for the development of flexible wing control techniques.

2. Experimental methods

2.1. Model design

The present experiment was carried out in the D6 wind tunnel at Beihang University (BUAA). It is a low-speed, openloop, closed-jet wind tunnel with a cross section of 430 mm(width)×500 mm (height). The experimental scheme is illustrated in Fig. 1. The wing model is composed of two rigid leading- and trailing-edge supports, a flexible membrane,and two endplates. The ends of the two supports are fixed on the endplates as the rigid framework of the membrane wing. The flexible membrane is attached between the leading- and trailing-edges to form a complete membrane wing. As shown in Fig. 1, the membrane wing is horizontally mounted in the test section. The wing has a span of 380 mm and chord length of c=120 mm, so the corresponding aspect ratio is 3.17 and larger than 3. According to Bleischwitz et al.15and Mizoguchi and Itoh18this aspect ratio can ensure the quasi two-dimensional (2D) property at the mid-span section of the membrane wing. The free stream velocities U∞are 5 m/s, 7.5 m/s, 10 m/s, and 12.5 m/s, resulting in the Reynolds number based on c of Re=4×104,6×104, 8×104, and 1×105, respectively. The free stream turbulent intensity Tu is less than 0.3% under the current operating conditions.

Fig. 1 Schematic diagram of present experiment.

Fig. 2 Membrane wing details.

2.2. Measurements

As shown in Fig. 1, aerodynamic forces of the flexible membrane wing are determined by two six-axis force sensors(ATI-Mini40) with a range of 20 N (calibration standard SI-20-1). The two sensors are respectively connected to the endplates on both sides of the membrane wing. The total aerodynamic forces are obtained by adding the data from both sides.Each measurement point is the average of data sampled at 10 kHz over 30 s. For all the operating conditions in the current experiment, the fluctuation frequencies of the aerodynamic forces are in the range of 101–102Hz. Therefore, the current measurement parameters can meet the requirements of high time-resolution and ergodicity. The uncertainty of force measurement is less than 1.25% of full-scale load with 95% confidence level.

In order to measure the instantaneous deformations of the membrane wing, a continuous laser with maximum output power of 8 W and a wavelength of 532 nm is utilized as the light source. The mid-span section of the membrane is illuminated by the laser sheet with the thickness of around 1.5 mm.Meanwhile, a Pco.dimax CS4 high-speed camera is employed to record the instantaneous deformations of the membrane wing section. The sampling frequency is 1 kHz, which is also time-resolved. The duration of deformation sampling is 4.2 s,so 4200 snapshots of instantaneous deformations can be captured at one time. The field of view is 140 mm×90 mm and the spatial resolution is 1464×932 pixels,resulting in the magnification of 0.096 mm/pixel. The measurements of aerodynamic forces and deformations are controlled by a MicroVec Micropulse-725 synchronizer, whose synchronization error is less than 0.25 ns.

The approach of recognizing membrane deformation is the same as our previous studies.19,21Firstly, the pixel matrix of the membrane position is obtained by selecting a binary threshold to convert the raw image to binary image.Secondly,a single-valued function in xtrandirection is attained by obtaining the boundary of the matrix. Finally, the instantaneous membrane curve is acquired by interpolation and polynomial fitting methods. The present approach can only recognize the 2D deformation at a certain membrane wing section, which can provide pixel accuracy with the resolution of 0.096 mm/pixel and uncertainty of around 0.08%c (0.096/120 ≈0.08%).

3. Membrane kinematics

3.1. Membrane deformation statistics

As presented in Fig. 3, different motion states of the membrane with the variation of П and α are discovered in the current study. Different state regions are represented by different symbols.At α=0°,the membrane initially has almost no tension and is in an unstable equilibrium state. At this time, disturbance in the incoming flow as well as the membrane gravity can cause the membrane to deviate from its initial equilibrium position and start vibrating about a certain camber,which is called membrane ‘‘buckling” instability in the study of Waldman and Breuer.22With the increase of α, the effect of membrane gravity is greatly reduced compared with the surging aerodynamic loads.

Fig. 3 Partition of membrane motion states.

Besides, the membrane motion at α>0° can be mainly divided into two states: Deformed-Steady State (DSS) and Dynamic Balance State (DBS). In the DSS region, the membrane wing bends under aerodynamic loads and maintains at the mean camber with negligibly small vibration, while in the DBS region, the membrane wing begins to vibrate coherently near the mean camber, showing specific vibration modes. It can also be discovered that the turning point from DSS to DBS occurs at the moderate angles of attack (α=12°–16°)for different П.The appearance of DSS and DBS in this experiment is similar to the numerical study of Li et al.23,24so this paper follows their nomenclatures for the two states.

Based on the membrane motion,the statistics of membrane deformation and vibration can be obtained. Fig. 4 shows the variation of mean membrane camber at the four tested П.Mean camber is the maximum displacement in ztrandirection after time-averaging the membrane positions of all snapshots.In this paper, mean camber is nondimensionalized by the chord length c and noted as C*. It can be seen that C*increases with the decrease of П at any α, because the increasing U∞leads to the increase of pressure difference between the upper and lower wing surfaces. To maintain the equilibrium condition of forces, the increased pressure difference is balanced by increased tension of the membrane and makes C*increase. On the other hand, the variation of C* with α is related to the specific membrane motion states,which are indicated by different symbols in Fig.4.In the initial stage of DSS,C* grows with the increase of α, which is consistent with the findings of Tiomkin and Raveh25in the stable state. However in the DBS region, C* increases first, then decreases with α,and finally remains constant.There is an obvious turning point between the DSS and DBS regions.

Vibration amplitude A* is calculated by the standard deviation of the membrane instantaneous position signals and nondimensionalized by the chord length c. The formula of A* is:

Fig. 4 Variation of mean membrane camber C*.

Fig. 5 Variation of maximum vibration amplitude A*max.

The variation of the maximum vibration amplitude A*maxof the membrane is displayed in Fig.5.A*maxis the maximum A* value along xtrandirection. Specifically, at α=0°, A*maxincreases with the decrease of П, because the increase of U∞causes larger flow disturbance,aggravating the buckling instability and vibration of the membrane. In the DSS region, the vibration of the membrane wing is very weak with negligible amplitude. Then the membrane vibration switches to the DBS region with a sudden increase of A*max. During the DBS, A*maxincreases first and then decreases with α for the four П. When the angles of attack are very large (α ≥28°),membrane vibration tends to be weak again.With the decrease of П, the maximum value of A*maxincreases and the corresponding angle of attack α（A*max)maxis delayed. Besides, the angles of attack for the maxima of C* and A*maxin the DBS region are listed in the second and third columns of Table 1. It is noteworthy that αC*maxis basically consistent withα（A*max)max,which indicates that C*and A*maxof the flexible membrane wing almost reach their maxima simultaneously in the DBS region at all tested П.

3.2. Frequency characteristics

According to the previous investigations,the membrane could show different vibration modes and frequencies during FSI.In this paper, the vibration frequencies of the membrane are obtained by Fourier Mode Decomposition (FMD), which was proposed by Ma et al.26Compared with the single-point Fast Fourier Transform (FFT), FMD can comprehensively reflect the vibration frequency characteristics along the whole chord of membrane, and has been successfully applied in previous studies.19,21Fig. 6 presents the dominant frequencies as well as the vibration modes of the membrane wing with the change of П and α. The dominant frequency f is nondimensionalized as the Strouhal number St=fc/U∞. In Fig. 6(a),when П=15.1, the variation of dominant frequency is flat and always maintains low values, because the membrane always vibrates in dominant mode 1 in the DBS region at П=15.1 (see the curve П=15.1 in Fig. 6(b)). However at other three П, the dominant frequency first keeps high values and then reduces to lower frequencies, indicating that vibration mode transition occurs in these П cases. From Fig. 6(b),it is clear that the transition from mode 2 to mode 1 occurs at П=11.5, while the transition from mode 3 to mode 1occurs at П=9.5 and П=8.2. The smaller the П is, the larger the range of α could maintain high frequencies and vibration modes.

Table 1 Angles of attack for the maxima of different parameters in DBS region.

Fig. 6 Dominant vibration frequencies of membrane wing and dominant vibration modes of membrane wing. Phase-averaged membrane deformations are examples showing mode 1,2,and 3 at α=14°for П=15.1,11.5,and 9.5 respectively,with fluctuations amplified by a factor of 3 for better display.

4. Aerodynamic characteristics

Fig.7 Variation of lift coefficients Cl of flexible membrane wing and rigid plate wing. Only the lift curve of rigid plate wing at Re=1×105 (the largest Re) is displayed for comparison due to its little dependence on П.

The lift coefficients CLof flexible membrane wing (shown by solid curves) and rigid plate wing (shown by dashed curve)are presented in Fig. 7. For the rigid plate wing, CLis found to show very little dependence on Reynolds number in our tested cases. As a result, only the lift curve of rigid plate wing at Re=1×105(the largest Re)is displayed for comparison in Fig. 7.

In general, compared with the rigid plate wing, the flexible membrane wing has obvious effect on lift-enhancement in a broad range of angles of attack. For the two cases of П=9.5 (in blue) and 8.2(in red)with high U∞, the variation of CLis obviously different in various membrane motion states. In the DSS region, CLincreases linearly with α at α ≤8°. After that, the slope of lift curve gradually decreases.However, in the DBS region, CLof flexible wing continues to increase. The turning points from DSS to DBS appear at α=14° and 16° for the two cases respectively. On the right side of the turning points (DBS region), the slope of lift curve increases suddenly, leading to a larger growth rate of CL. The stall of flexible wing occurs in the DBS region at all tested П.The maximum lift coefficient CLmaxand stall angle of attack αstallincrease with the decrease of П. The specific αstallare shown in the fourth column of Table 1. Hence, the decrease of П can delay the stall of flexible wing. By comparing the angles of attack in Table 1, striking similarities can be found that C*, A*max, and CLof the membrane wing obtain their maxima nearly simultaneously in the DBS region.

Fig. 8 shows the drag coefficients CDof the flexible and rigid wings. CDof the flexible wing is larger than that of the rigid wing at almost all α regardless of П,and especially when it reaches the DBS region,CDincreases sharply.This indicates that the flexible membrane wing has to pay the price of increasing drag while enhancing lift,and the membrane vibration will cause additional drag increment.

Fig.8 Variation of lift coefficients CD of flexible membrane wing and rigid plate wing. Only the lift curve of rigid plate wing at Re=1×105 (the largest Re) is displayed for comparison due to its little dependence on П.

Fig. 9 Variation of aerodynamic efficiency for flexible and rigid wings. Only the curves of rigid plate wing at Re=1×105 (the largest Re) are displayed for comparison due to its little dependence on П.

From the perspective of flight mechanics, the two significant aerodynamic parameters, lift-to-drag ratio (CL/ CD)and power factor (CL3/2/CD), represent the range and power efficiencies respectively. Accordingly, the two parameters are further analyzed in this section. The variation of lift-to-drag ratios for the flexible and rigid wings is firstly shown in Fig. 9(a). The lift-to-drag ratios reach the maximum values at α=6°for rigid wing while at α=6°or 8°for flexible wing.The lift-to-drag ratio variation of rigid wing is severe and the curve is relatively steep. When 0°≤α ≤6°, the lift-to-drag ratio increases rapidly from about 0.5 to the maximum; when α>6°,the lift-to-drag ratio decreases quickly.In contrast,the lift-to-drag ratio variation of the flexible wing is gentler with plumper curves, especially at П=11.5, 9.5, and 8.2. When α>6°, the lift-to-drag ratios of the flexible wing decrease slowly from the maximum values until they coincide with the curves of the rigid wing at α=24°. It can be concluded from Fig. 9(a) that the flexible membrane wing can significantly increase the lift-to-drag ratios in the range of 6°<α ≤24°(belonging to moderate and large α), which is a great advantage over the rigid plate wing.When α>24°,the lift-to-drag ratios of the flexible and rigid wings completely coincide, implying that the flexible wing no longer has advantages in this poststall α range. In addition, different from the rigid wing, the lift-to-drag ratios of the flexible wing are greatly affected by П. The improvement of lift-to-drag ratios at high α is better for the smaller П. The variation of power factors for the flexible and rigid wings is further displayed in Fig.9(b).Compared with the rigid wing, α with the maximum power factor of the flexible wing is slightly delayed by around 2°. Similar to the lift-to-drag ratios in Fig. 9(a), the power factor curve of the rigid wing is steeper, while the curves of flexible wing are gentler. At П=11.5, 9.5, and 8.2, the flexible wing can greatly improve the power factors at all measured α, which is also a unique advantage of the flexible wing.In summary,the flexible wing can improve the range and power efficiency of aircraft and has the potential to improve the maneuverability of aircraft at large α.

5. Lift-enhancement mechanism of membrane vibration

According to the analyses above,CLis dominated by the mean camber due to the negligible vibration in the DSS region,while in the DBS region, CLis further increased by the combined effects of the mean camber and instantaneous vibration. C*,A*max, and CLalmost reach their maxima simultaneously within the DBS region in all П cases. Consequently, the liftenhancement of flexible membrane wing at DBS can be attributed to two aspects of contribution. Firstly, CLincreases with the mean camber, which is consistent with the rule of traditional rigid airfoil. Secondly, CLfurther increases with the membrane vibration amplitude. For the second aspect, the mechanism seems to be less intuitive than the former. Therefore,this section will focus on the lift-enhancement mechanism of membrane vibration.

Referring to Bohnker and Breuer,27the membrane vibration momentum coefficient Cμallcan be defined for quantitative analysis of the membrane motion influence. The formula of Cμallis as follows:

where urmsis the root mean square (rms) value of the locally measured vibration velocity of the membrane; x and y are the chordwise and spanwise coordinates respectively; Cμallis the entire momentum coefficient after the 2D surface integral and is nondimensionalized by the free stream dynamic pressure.Since the object of this paper is a quasi-2D flexible membrane wing, Eq. (2) can be simplified as the momentum coefficient per unit span Cμ, which is expressed as.

As stated in Section 2.2, the sampling frequency of membrane deformation is 1 kHz, and 4200 snapshots of instantaneous deformations can be captured at one time. Due to the high time-resolution, it is feasible to calculate the instantaneous membrane vibration velocity:

where u is the local vibration velocity of the membrane,and Δt is the time interval of deformation sampling (Δt=10-3s in this study). As a result, 4199 instantaneous membrane vibration velocities can be obtained. Then urmsis calculated by.

Because of the negligible membrane vibration in the DSS region, this paper only provides the variation of Cμat α=0° and in the DBS region, which is shown in Fig. 10. It can be seen that Cμin the DBS region increases first and then reduces with the increase of α.Besides,with the decrease of П,the rise of maximum Cμis obvious and the angle of attack corresponding to maximum Cμ(denoted as α（Cμ)max) performs a tendency to postpone.α（Cμ)maxfor all П cases are listed in the fifth column of Table 1. By comparing the angles of attack for the different maxima in Table 1,similarities can be discovered among αC*max,α（A*max)max,αstall,and α（Cμ)max,particularly the high consistency between α（A*max)maxandα（Cμ)max, indicating that Cμis closely related to membrane vibration. It means that Cμmay be applied to explain the lift-enhancement mechanism of membrane vibration. Hence, this issue will be discussed below.

Fig.10 Variation of membrane vibration momentum coefficientCμ.

According to Eq. (3), urmsis an important intermediate variable determining the value of Cμ. As a result, the relationship between A*maxand the maximum urms(nondimensionalized by U∞and denoted as u*rmsmax) is firstly analyzed in Fig. 11. It is shown that at each П, A*maxis positively correlated with u*rmsmaxunder most operating conditions except for a few cases at α=0°.Furthermore,considering the significant influence of the membrane vibration modes on the amplitudes, it can be surprisingly found based on vibration modes that all the points in Fig. 11 collapse on three straight lines with different slopes, corresponding to the first- to thirdorder standing wave vibration modes of the membrane. Even the few singularities at α=0° also fall on the line of the first-order vibration mode. The discovery above means that A*maxand u*rmsmaxpresent various positive and linear correlation because of different vibration modes,which is one of the new findings in this paper. The slopes (k) of these three linear fitting lines are 0.29, 0.19, and 0.15 respectively with the coefficients of determination for fitting (R2) all greater than 0.984.

Fig. 11 Relationship between A*max and maximum urms. The first- to third-order vibration modes of the membrane are shown by dashed lines with different colors.

Fig. 12 Relationship between A*max and Cμ.

Subsequently,the direct relationship between A*maxand Cμis investigated in Fig. 12. Overall, the variation regularity is similar to Fig. 11. Except for α=0°, A*maxand Cμof the membrane are also positively correlated at each П. However,referring to Eq.(3),this positive correlation is no longer linear,but approximately quadratic.The quadratic polynomial fitting curves at the first-and third-order vibration modes are given in Fig. 12, which further verifies the quadratic relationship between A*maxand Cμ. Because the cases of the secondorder vibration mode are rare in the current experiment, it is not accurate to conduct quadratic polynomial fitting. Hence,the fitting curve of the second-order mode is not shown in Fig. 12. Nevertheless, it can be deduced that the secondorder modal fitting curve should be intermediate between the first-order and the third-order curves.

As initially stated, this section attempts to explain the liftenhancement mechanism of membrane vibration by Cμas a medium,and has found the highly positive correlation between A*maxand Cμ. Therefore, the relationship between Cμand CLis further analyzed in Fig. 13. The increasing direction of a in the DBS region at each П is shown by black arrows. It is displayed that with the increase of a, both of Cμand CLare also positively correlated at poststall.

Fig. 13 Relationship between Cμ and CL.

So far, the introduction of Cμas well as the flow field data in this paper can provide a reasonable explanation for the liftenhancement of membrane vibration. Compared with the amplitude, Cμdirectly reflects the transfer of momentum and energy in the system. The increase of Cμmeans that the membrane vibration transfers more energy to the flow field in the process of FSI.This energy could accelerate the fluid in the leeward surface separation region, increase the turbulent kinetic energy and Reynolds shear stress over the wing surface,enhance the mixing between the low-momentum fluid near the wing and the high-momentum freestream, thus overcome the adverse pressure gradient and suppress the flow separation,and finally increase the lift. Besides, it is remarkable that this positive correlation and the fitting analyses at different vibration modes in Fig. 11 and Fig. 12 can provide guidance for the development of novel active control techniques of flexible membrane wing. In the design process, Cμof the membrane can be raised as much as possible by controlling the three parameters of membrane amplitude, vibration frequency,and vibration mode to improve the lift of the membrane wing.

Fig. 14 Flow fields around rigid plate wing as well as flexible membrane wing at Re=6×104 and α=16°.

6. Vibration at post-stall angles of attack

The linear relationship between u*rmsmaxand A*maxhas been explored from Fig. 11. Indeed, their chordwise locations are also found to be consistent.As shown in Fig.15,the horizontal and vertical coordinates are the chordwise locations of u*rmsmaxand A*max, which are denoted as x(u*rmsmax) and x(A*max), respectively. The black solid line represents the straight line with a slope of 1, and the two black dashed lines represent the error band with a dimensionless chordwise location resolution of±0.02.It can be seen that most of the points fall into the error band of the line,which indicates that x(u*rmsmax)and x(A*max) are basically consistent in most cases of the DBS region. This is also a new finding in this paper and can provide another idea for the active control of flexible wing:the centralized locations of x(u*rmsmax) and x(A*max) can be selected for intensive excitation to efficiently control the membrane motion.

Moreover,it can be observed from Fig.15 that the points in the red circle deviate from the error band. By searching and comparison, these points are found to approximately correspond to the post-stall conditions at α>24° where flexible wing no longer improves the lift-to-drag ratio in Fig. 9(a).Consequently, there is a great difference between x(u*rmsmax) and x(A*max) at this post-stall α range.

The dominant frequencies of lift,drag,and vibration of the membrane wing (taking П=11.5 as an example) are plotted in Fig. 16. When α>24°, there is no obvious dominant frequencies of lift and drag, which means that the coupling between aerodynamic forces and membrane deformation disappears due to the weak and disordered vibration of the membrane.This may be the reason why the aerodynamic advantage of the flexible wing is greatly deteriorated when α>24°.

Fig. 15 Relationship of x(u*rms max) and x(A*max). Black solid line (k=1) represents the straight line with a slope of 1.

Fig. 16 Dominant frequencies of aerodynamic forces and vibration of membrane wing (taking П=11.5 as an example).

7. Conclusions

Time-resolved synchronous measurements on aerodynamic forces and deformations of a quasi-2D flexible membrane wing were conducted in a wind tunnel.The experimental model is a simplified leading- and trailing-edge supported membrane wing with angles of attack ranging from α=0° to α=30°.The results are mainly divided into four aspects: membrane kinematics, aerodynamic characteristics, lift-enhancement mechanism of membrane vibration, and active control ideas of flexible wing.

(1) In the current study, П and α are the two significant parameters determining the performance of flexible membrane wing.Specifically,the categorization of membrane motion states is dominantly affected by α. At α=0°, the membrane is in buckling instability state.At α>0°,the membrane motion can be mainly divided into two states: Deformed-Steady State (DSS) at prestall, and Dynamic Balance State (DBS) at around stall and post-stall. In the DSS region, the membrane wing bends under aerodynamic loads and maintains at the mean camber with negligibly small vibration. Within the DBS region, the membrane wing begins to vibrate coherently near the mean camber, performing specific vibration modes. Besides, the turning point from DSS to DBS, mean camber, maximum vibration amplitude and highest vibration mode are determined by П. With the decrease of П, the four variables show an increasing tendency at certain α.

(2) The aerodynamic characteristics are distinctly influenced by the membrane kinematics. With the increase of α, in the DSS region,the lift coefficient enhancement is dominated by the mean camber,while in the DBS region,CLis further increased by the combined effects of the camber and instantaneous vibration with a larger growth rate.The stall of membrane wing occurs in the DBS region at all tested П, whilst with П decreasing, due to the increased mean camber and vibration amplitude, the membrane wing yields higher CLmaxand postponed stall.Compared with the rigid plate wing, the flexible membrane wing has obvious effects on lift-enhancement and stall delay,and can greatly increase the lift-to-drag ratios of 6°<α ≤24°,which plays important roles in improving the aerodynamic efficiency and maneuverability of the wing at moderate and large α.

(3) It is discovered that the mean camber, maximum vibration amplitude, and lift almost reach their maxima simultaneously within the DBS region in all П cases.To interpret this phenomenon, the momentum coefficient Cμof membrane vibration is utilized in this paper.Accordingly, the positive correlation among amplitude,momentum, and lift is successfully established, and the lift-enhancement mechanism of membrane vibration is revealed combined with our previous flow field data.The increase of amplitude leads to the increase of Cμ,which means that the membrane vibration transfers more energy to the flow field in the process of FSI.This energy can increase the turbulent kinetic energy and Reynolds shear stress over the wing surface, thus suppress the flow separation, and finally enhance the lift.

(4) Moreover, it is newly found that in different vibration modes, the maximum vibration amplitude and root mean square of vibration velocity present positive and linear correlation with different slopes, and their chordwise locations are basically consistent at α ≤24°. However at α>24°, the membrane vibration tends to be disordered, and the aerodynamic advantage of the passive deformation of flexible wing is greatly deteriorated.Based on the new findings,novel ideas for active control of flexible wing are proposed: by controlling the amplitude, frequency, and vibration mode of the membrane,and selecting the specific chordwise locations for intensive excitation, Cμcan be efficiently increased as much as possible, ultimately improving the aerodynamic performance of the flexible wing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors acknowledge the financial support from the National Natural Science Foundation of China (Nos.11761131009 and 11721202).