A parametric investigation of the propulsion of 2D chordwise-flexible flapping wings at low Reynolds number using numerical simulations

This paper presents a numerical investigation of the effects of chordwise flexibility on flapping wings at low Reynolds number. The numerical simulations are performed with a partitioned fluid–structure interaction algorithm using artificial compressibility stabilization. The choice of the structural dimensionless parameters is based on scaling arguments and is compared against parameters used by other authors. The different regimes, namely inertia-driven and pressure-driven wing deformations, are presented along with their effects on the topology of the flow and on the performance of a heaving and pitching flapping wing in propulsion regime. It is found that pressure-driven deformations can significantly increase the thrust efficiency if a suitable amount of flexibility is used. Significant thrust increases are also observed in zero pitching amplitude cases. The effects of the second and third deformation modes on the performances of pressure-driven deformation cases are discussed. On the other hand, inertia-driven deformations generally deteriorate aerodynamic performances of flapping wings unless the behavior of the wing deformation is modified by the presence of sustainable superharmonics in a way that produces slight improvements. It is also shown that wing flexibility can act as an efficient passive pitching mechanism that allows fair thrust and better efficiency to be achieved when compared to a rigid pitching–heaving wing.     

柔性扑翼的气动特性研究

以往扑翼的气动力计算研究都很少考虑扑翼的柔性,而在鸟的扑翼动作中,在外加气动力和鸟自身的扑动力作用下,扑翼的柔性变形相当大。本文在原有匀速刚性模型的基础上,提出考虑了扑翼扑动速率变化和形状变化的扑翼分析模型,使之更接近鸟翼柔性扑动真实情况。通过计算分析气动特性发现,控制适当的话,柔性变形能大大改善扑翼的气动性能。本文通过模拟鸟扑翼的柔性运动,计算了时柔性扑翼气动力以及平均升力系数和平均推力系数随着扑动角、倾斜角等参数变化的情况,从而从气动的角度解释了为什么鸟在不同的飞行阶段扑翼规律各不相同,并为柔性扑翼飞行器的设计提供了理论依据。     

Lift enhancement through flexibility of plunging wings at low Reynolds numbers

In this paper the combined effect of two mechanisms for lift enhancement at low Reynolds numbers are considered, wing oscillations and wing flexibility. The force, deformation and flow fields of rigid and flexible low aspect ratio (AR=3) and high aspect ratio (AR=6) wings oscillating at a fixed post-stall angle of attack of 15° and amplitude of 15% of chord are measured. The force measurements show that flexibility can increase the time-averaged lift coefficient significantly. For low aspect ratio wings the maximum lift coefficient across all Strouhal numbers was C_l=1.38 for the rigid wing as opposed to C_l=2.77 for the flexible wing. Very similar trends were observed for the high aspect ratio wings. This increase is associated with significant deformation of the wing. The root is sinusoidally plunged with small amplitude but this motion is amplified along the span resulting in a larger tip motion but with a phase lag. The amount it is amplified strongly depends on Strouhal number. A Strouhal number of Sr_c=1.5 was selected for detailed flow field measurements due to it being central to the high-lift region of the flexible wings, producing approximately double the lift of the rigid wing. For this Strouhal number the rigid wings exhibit a Leading Edge Vortex (LEV) ring. This is where the clockwise upper-surface LEV pairs with the counter-clockwise lower-surface LEV to form a vortex ring that self-advects upstream and away from the wing's upper surface. Conversely the deformation of the flexible wings inhibits the formation of the LEV ring. Instead a strong upper-surface LEV forms during the downward motion and convects close to the airfoil upper surface thus explaining the significantly higher lift. These measurements demonstrate the significant gains that can be achieved through the combination of unsteady aerodynamics with flexible structures.     

Influence of wing flexibility on the aerodynamic performance of a tethered flapping bumblebee

The sophisticated structures of flapping insect wings make it challenging to study the role of wing flexibility in insect flight. In this study, a mass-spring system is used to model wing structural dynamics as a thin, flexible membrane supported by a network of veins. The vein mechanical properties can be estimated based on their diameters and the Young's modulus of cuticle. In order to analyze the effect of wing flexibility, the Young's modulus is varied to make a comparison between two different wing models that we refer to as flexible and highly flexible. The wing models are coupled with a pseudo-spectral code solving the incompressible Navier–Stokes equations, allowing us to investigate the influence of wing deformation on the aerodynamic efficiency of a tethered flapping bumblebee. Compared to the bumblebee model with rigid wings, the one with flexible wings flies more efficiently, characterized by a larger lift-to-power ratio.     

Aerodynamic Coefficients and Deformation Measurements on Flexible Micro Air Vehicle Wings

This paper documents the elastic deformations and corresponding aerodynamic coefficients of flexible wings used for micro air vehicles (MAVs). These low-aspect ratio wings, developed and fabricated at the University of Florida, incorporate an elastic latex membrane skin covering a thin carbon fiber skeleton. The wings were tested in a unique low-speed wind tunnel facility integrating a visual image correlation (VIC) system with a six-component strain gauge sting balance. Model characteristics are presented, along with the appropriate specimen preparation techniques and wind tunnel instrumentation. The static response characteristics, including full-field displacements and plane strain measurements, for three distinct MAV wing designs are presented. The full-field deformation results show how passive wing flexibility preferably affects aerodynamic performance when compared to a rigid model of similar geometry.     

On the aerodynamic characteristics of hovering rigid and flexible hawkmoth-like wings

Insect wings are subjected to fluid, inertia and gravitational forces during flapping flight. Owing to their limited rigidity, they bent under the influence of these forces. Numerical study by Hamamoto et al. (Adv Robot 21(1–2):1–21, 2007) showed that a flexible wing is able to generate almost as much lift as a rigid wing during flapping. In this paper, we take a closer look at the relationship between wing flexibility (or stiffness) and aerodynamic force generation in flapping hovering flight. The experimental study was conducted in two stages. The first stage consisted of detailed force measurement and flow visualization of a rigid hawkmoth-like wing undergoing hovering hawkmoth flapping motion and simple harmonic flapping motion, with the aim of establishing a benchmark database for the second stage, which involved hawkmoth-like wing of different flexibility performing the same flapping motions. Hawkmoth motion was conducted at Re = 7,254 and reduced frequency of 0.26, while simple harmonic flapping motion at Re = 7,800 and 11,700, and reduced frequency of 0.25. Results show that aerodynamic force generation on the rigid wing is governed primarily by the combined effect of wing acceleration and leading edge vortex generated on the upper surface of the wing, while the remnants of the wake vortices generated from the previous stroke play only a minor role. Our results from the flexible wing study, while generally supportive of the finding by Hamamoto et al. (Adv Robot 21(1–2):1–21, 2007), also reveal the existence of a critical stiffness constant, below which lift coefficient deteriorates significantly. This finding suggests that although using flexible wing in micro air vehicle application may be beneficial in term of lightweight, too much flexibility can lead to deterioration in flapping performance in terms of aerodynamic force generation. The results further show that wings with stiffness constant above the critical value can deliver mean lift coefficient almost the same as a rigid wing when executing hawkmoth motion, but lower than the rigid wing when performing a simple harmonic motion. In all cases studied (7,800 ≤ Re ≤ 11,700), the Reynolds number does not alter the force generation significantly.     

蜻蜓滑翔时柔性褶皱前翅气动特性分析

蜻蜓是自然界优秀的飞行家,滑翔是其常见且有效的飞行模式.蜻蜓优异的飞行能力来源于其翅膀的巧妙结构,褶皱是蜻蜓翅膀上最为显著的结构之一,不仅提高了翅膀的刚度,还改变了其气动特性,而飞行过程中柔性翅膀会产生变形是蜻蜓翅膀的另一特性.为揭示蜻蜓在滑翔时,柔性褶皱前翅的变形,探究褶皱和柔性的共同作用对其气动特性的影响,基于逆向工程,依据前人的测量数据和研究成果,通过三维建模软件建立了蜻蜓三维褶皱前翅的计算流体力学(computational fluiddynamics,CFD)模型和计算结构力学(computational structuralmechanics,CSD)模型,并通过模态分析验证了此模型有足够的精度.基于CFD方法和CFD/CSD双向流固耦合计算方法分别对蜻蜓滑翔飞行时刚性和柔性褶皱前翅的气动特性进行了数值模拟,结果表明,柔性褶皱前翅受气动载荷后,翅脉和翅膜产生形变,柔性前翅上下表面压力差相较于刚性前翅减小了,从而其升力和阻力也减小了,而在大攻角时,变形后的前缘脉诱导出比刚性前翅更强的前缘涡.因此在攻角小于10$^\circ$时刚性前翅的气动特性优于柔性前翅,继续增大攻角,柔性前翅的气动特性则优于刚性前翅.前翅受载后气动响应时间短,翅尖的变形最大,仅仅产生了垂直于翅膀所在平面方向上的变形,而没有发生扭转,翼根处受到应力最大,褶皱上凸部分承受蜻蜓滑翔时前翅的主要载荷.      

Force production and asymmetric deformation of a flexible flapping wing in forward flight

Insect wings usually are flexible and deform significantly under the combined inertial and aerodynamic load. To study the effect of wing flexibility on both lift and thrust production in forward flight, a two-dimensional numerical simulation is employed to compute the fluid–structure interaction of an elastic wing section translating in an inclined stroke plane while pitching around its leading ledge. The effects of the wing stiffness, mass ratio, stroke plane angle, and flight speed are considered. The results show that the passive pitching due to wing deformation can significantly increase thrust while either maintaining lift at the same level or increasing it simultaneously. Another important finding is that even though the wing structure and actuation kinematics are symmetric, chordwise deformation of the wing shows a larger magnitude during upstroke than during downstroke. The asymmetry is more pronounced when the wing has a low mass ratio so that the fluid-induced deformation is significant. Such an aerodynamic cause may serve as an additional mechanism for the asymmetric deformation pattern observed in real insects.     

Passive flow control by membrane wings for aerodynamic benefit

The coupling of passive structural response of flexible membranes with the flow over them can significantly alter the aerodynamic characteristic of simple flat-plate wings. The use of flexible wings is common throughout biological flying systems inspiring many engineers to incorporate them into small engineering flying systems. In many of these systems, the motion of the membrane serves to passively alter the flow over the wing potentially resulting in an aerodynamic benefit. In this study, the aerodynamic loads and the flow field for a rigid flat-plate wing are compared to free trailing-edge membrane wings with two different pre-tensions at a chord-based Reynolds number of approximately 50,000. The membrane was silicon rubber with a scalloped free trailing edge. The analysis presented includes load measurements from a sting balance along with velocity fields and membrane deflections from synchronized, time-resolved particle image velocimetry and digital image correlation. The load measurements demonstrate increased aerodynamic efficiency and lift, while the synchronized flow and membrane measurements show how the membrane motion serves to force the flow. This passive flow control introduced by the membranes motion alters the flows development over the wing and into the wake region demonstrating how, at least for lower angles of attack, the membranes motion drives the flow as opposed to the flow driving the membrane motion.     

Effects of unsteady deformation of flapping wing on its aerodynamic forces

Effects of unsteady deformation of a flapping model insect wing on its aerodynamic force production are studied by solving the Navier-Stokes equations on a dynamically deforming grid.Aerodynamic forces on the flapping wing are not much affected by considerable twist,but affected by camber deformation.The effect of combined camber and twist deformation is similar to that of camber deformation.With a deformation of 6% camber and 20°twist(typical values observed for wings of many insects),lift is increased bv 10%～20%and lift-to-drag ratio by around 10%compared with the case of a rigid flat-plate wing.As a result.the deformation can increase the maximum lift coefficient of an insect.and reduce its power requirement for flight.For example,for a hovering bumblebee with dynamically deforming wings(6?mber and 20°twist),aerodynamic power required is reduced by about 16%compared with the case of rigid wings.     

Calculating the effect of gusts on an arbitrary thin wing

A numerical method of calculating the unsteady flow about a thin wing moving in an ideal incompressible medium is developed on the basis of the lifiting surface scheme. The variation of the boundary conditions on the wing surface with time and coordinates may be arbitrary. Therefore, the method makes it possible to examine the aperiodic motion of a wing as a rigid body, consider any wing deformations, analyze the wing entry into a gust, study the effect of a weak shock wave on the wing, etc. In addition, practically no limitation is imposed on the shape of the thin lifting surface: the method is applicable to monoplane wings of any planform, to annular wings, to systems of similar wings, etc.Studies in which the effect of a gust on a wing is analyzed have been reviewed in [1, 2]. Without dwelling on this review, we note that at subsonic speeds an effective solution of the problem has been obtained only for a profile.The author wishes to thank E. P. Kapustina for working the examples.     

Experimental investigation of the effect of chordwise flexibility on the aerodynamics of flapping wings in hovering flight

Ornithopters or mechanical birds produce aerodynamic lift and thrust through the flapping motion of their wings. Here, we use an experimental apparatus to investigate the effects of a wing's twisting stiffness on the generated thrust force and the power required at different flapping frequencies. A flapping wing system and an experimental set-up were designed to measure the unsteady aerodynamic and inertial forces, power usage and angular speed of the flapping wing motion. A data acquisition system was set-up to record important data with the appropriate sampling frequency. The aerodynamic performance of the vehicle under hovering (i.e., no wind) conditions was investigated. The lift and thrust that were produced were measured for different flapping frequencies and for various wings with different chordwise flexibilities. The results show the manner in which the elastic deformation and inertial flapping forces affect the dynamical behavior of the wing. It is shown that the generalization of the actuator disk theory is, at most, only valid for rigid wings, and for flexible wings, the power P varies by a power of about 1.0  of the thrust T. This aerodynamic information can also be used as benchmark data for unsteady flow solvers.     

VISCOELASTIC CONSTITUTIVE MODEL RELATED TO DEFORMATION OF INSECT WING UNDER LOADING IN FLAPPING MOTION

Flexible insect wings deform passively under the periodic loading during napping flight. The wing flexibility is considered as one of the specific mechanisms on improving insect flight performance. The constitutive relation of the insect wing material plays a key role on the wing deformation, but has not been clearly understood yet. A viscoelastic constitutive relation model was established based on the stress relaxation experiment of a dragonfly wing (in vitro). This model was examined by the finite element analysis of the dynamic deformation response for a model insect wing under the action of the periodical inertial force in flapping. It is revealed that the viscoelastic constitutive relation is rational to characterize the biomaterial property of insect wings in contrast to the elastic one. The amplitude and form of the passive viscoelastic deformation of the wing is evidently dependent on the viscous parameters in the constitutive relation.     

Numerical simulation of a flapping four-wing micro-aerial vehicle

A three-dimensional numerical simulation of a four-wing (two wings on each side, one on top of another) flapping micro-aerial vehicle (FMAV), known as the Delfly micro, is performed using an immersed boundary method Navier–Stokes finite volume solver at Reynolds numbers of 5500 (forward flight condition). The objective of the present investigation is to gain an insight to the aerodynamics of flapping wing biplane configuration, by making an analysis on a geometry that is simplified, yet captures the major aspects of the wing behavior. The fractional step method is used to solve the Navier–Stokes equations. Results show that in comparison to the Delfly II flapping kinematics (a similar FMAV configuration but smaller flapping stroke angles), the Delfly-Micro flapping kinematics provides more thrust while maintaining the same efficiency. The Delfly-Micro biplane configuration generates more lift than expected when the inclination angle increases, due to the formation of a uniform leading edge vortex. Estimates of the lift produced in the forward flight conditions confirm that in the current design, the MAV is able to sustain forward flight. The potential effect of wing flexibility on the aerodynamic performance in the biplane configuration context is investigated through prescribed flexibility in the simulations. Increasing the wing׳ spanwise flexibility increases thrust but increasing chordwise flexibility causes thrust to first increase and then decrease. Moreover, combining both spanwise and chordwise flexibility outperforms cases with only either spanwise or chordwise flexibility.     

扑翼柔性及其对气动特性的影响

以往对扑翼气动特性的研究基本上都是基于简单的匀速刚性模型,但是通过大量观察不同飞鸟的扑翼动作发现,该模型与鸟翼的实际扑动还有很大差别。鸟翼不但上扑段和下扑段所需时间不同,而且在扑动过程中,鸟翼的形状无论沿弦向或展向都存在着相当大的柔性变形。本文在原有匀速刚性模型的基础上,加入了扑动速率变化和形状变化的影响,得出新的变速柔性扑翼分析模型,使之更接近鸟翼柔性扑动的真实情况。通过对比计算发现,柔性变形对扑翼的升力与推力都有着显著影响,如果控制得当,柔性变形能大大改善扑翼的气动性能。     

How a flexible tail improves the power extraction efficiency of a semi-activated flapping foil system: A numerical study

The improvement of power extraction of a semi-activated flapping foil system via the use of a flexible tail is numerically investigated in this work. A NACA0015 airfoil arranged in a two-dimensional laminar flow synchronously executes a forced pitching motion and an induced plunging motion. A flat plate attached to the trailing edge of the foil is utilized to model a tail, and thereby they are considered as a unit for the purpose of power extraction. The tail is either rigid or deformable. At a Reynolds number of 1100 and the position of the pitching axis at third chord, the effects of the mass and flexibility of the tail as well as the frequency of pitching motion on the net power extraction are systematically examined. It is found that compared to the foil with a rigid tail, the efficiency of net power extraction for the foil with a deformable tail can be improved. Based on the numerical analysis, it is indicated that the enhanced power extraction, which is caused by the increased lift force, directly contributes to the net efficiency improvement. Moreover, owing to high enhancement of power extraction, a flexible tail with high flexibility is recommended in the semi-activated flapping foil based power extraction system.     

Nondimensional frequency scaling of aerodynamically-tensioned membranes

Membrane wings have applications that involve low Reynolds number flyers such as micro air vehicles. The time-averaged and time-dependent deformations of the membrane affect the aerodynamic characteristics of the wing, primarily in the region beyond the maximum aerodynamic efficiency of the wing. This paper investigates an appropriate nondimensional vibration frequency scaling of a spanwise tensioned membrane with free (unattached) leading and trailing edges at low Reynolds numbers relative to nondimensional aeroelastic parameters. Silicone rubber membranes with varying spanwise pre-tension, aerodynamic tension (due to wing angle-of-attack and flow dynamic pressure), modulus of elasticity, span, and thickness are studied. Experimental results are compared to a proposed scaling that simplifies the aerodynamic loading as a uniform pressure distribution acting on the membrane. Data is further compared and discussed relative to previous published results of membrane wings with finite wing spans (three-dimensional flow) and fixed (rigid) leading edges.     

A simulation-based study on longitudinal gust response of flexible flapping wings

Winged animals such as insects are capable of flying and surviving in an unsteady and unpredictable aerial environment. They generate and control aerodynamic forces by flapping their flexible wings. While the dynamic shape changes of their flapping wings are known to enhance the efficiency of their flight, they can also affect the stability of a flapping wing flyer under unpredictable disturbances by responding to the sudden changes of aerodynamic forces on the wing. In order to test the hypothesis, the gust response of flexible flapping wings is investigated numerically with a specific focus on the passive maintenance of aerodynamic forces by the wing flexibility. The computational model is based on a dynamic flight simulator that can incorporate the realistic morphology, the kinematics, the structural dynamics, the aerodynamics and the fluid–structure interactions of a hovering hawkmoth. The longitudinal gusts are imposed against the tethered model of a hovering hawkmoth with flexible flapping wings. It is found that the aerodynamic forces on the flapping wings are affected by the gust, because of the increase or decrease in relative wingtip velocity or kinematic angle of attack. The passive shape change of flexible wings can, however, reduce the changes in the magnitude and direction of aerodynamic forces by the gusts from various directions, except for the downward gust. Such adaptive response of the flexible structure to stabilise the attitude can be classified into the mechanical feedback, which works passively with minimal delay, and is of great importance to the design of bio-inspired flapping wings for micro-air vehicles.     

Three-dimensional effects on the translational locomotion of a passive heaving wing

This study was carried out on a three-dimensional wing with a freedom in translational direction under a prescribed up and down heaving motion. The investigation focused on how the system kinematics and structural parameters affect the dynamic response of a wing with a relatively small span length. The induced wing motion is a result of the system stability breakdown, which has only been observed by previous researches in the two-dimensional case. The results obtained indicate that the evolution of the wing locomotion is controlled not only by the flapping frequency but also influenced by the system inertia as well as the wing aspect ratio and density ratio. Moreover, initial perturbation effect on wings flexibility plays a role in the evolution development.     

An efficient method for nonlinear aeroelasticy of slender wings

This paper aims the nonlinear aeroelastic analysis of slender wings using a nonlinear structural model coupled with the linear unsteady aerodynamic model. High aspect ratio and flexibility are the specific characteristic of this type of wings. Wing flexibility, coupled with long wingspan can lead to large deflections during normal flight operation of an aircraft; therefore, a wing in vertical/forward-afterward/torsional motion using a third-order form of nonlinear general flexible Euler–Bernoulli beam equations is used for structural modeling. Unsteady linear aerodynamic strip theory based on the Wagner function is used for determination of aerodynamic loading on the wing. Combining these two types of formulation yields nonlinear integro-differentials aeroelastic equations. Using the Galerkin’s method and a mode summation technique, the governing equations will be solved by introducing a numerical method without the need to adding any aerodynamic state space variables and the corresponding equations related to these variables of the problem. The obtained equations are solved to predict the aeroelastic response of the problem. The obtained results for a test case are compared with those of some other works and show a good agreement between results.