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.     

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.     

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。     

蜻蜓非均匀柔性前翼扑动飞行的气动性能研究

为了探究柔性对于蜻蜓前翼在扑动向前飞行时的气动性能, 本文根据蜻蜓前翼的实际参数建立蜻蜓前翼模型, 提出了两种柔性分布方式即均匀柔性分布和沿蜻蜓前翼弦向的变柔性分布. 本文通过STAR-CCM+软件, 首先采用重叠网格和双向流固耦合技术, 用于实现蜻蜓前翼的扑动流固耦合, 其次通过改变蜻蜓前翼固体区域的杨氏模量函数从而实现蜻蜓前翼的两种不同柔性分布. 结果表明, 在均匀柔性分布条件下, 柔性翼在杨氏模量较小时的升力系数和阻力系数曲线的变化规律滞后于刚性翼半周期并且给飞行增加阻力, 但是随着杨氏模量的逐渐增加即柔性逐渐减小, 蜻蜓前翼受到的阻力减小, 获得的推力增加且推力给予蜻蜓前飞的动量增量、加速度以及时均推力系数先增加后减小. 在合理的非均匀柔性分布条件下, 柔性翼显著提高推力系数峰值和时均推力系数, 在扑动前飞时, 给予蜻蜓前翼较大的动量增量以及加速度. 两种柔性分布方式的蜻蜓前翼与刚性翼对比之下, 蜻蜓前翼在柔性为非均匀柔性分布时可以获得更好的气动性能.      

Stereo PIV measurement of a finite, flapping rigid plate in hovering condition

Qualitative and quantitative flow visualizations were performed on a flapping rigid plate to establish a quantitative method for flow observation and evaluation of the force in the near field of a flapping wing. Flow visualization was performed qualitatively with dye visualization and quantitatively with velocity measurements using stereo particle image velocimetry (PIV) on three planes near the tip of the plate along its chord and oriented normally. By ensemble averaging the velocity fields of the same phase angles, they represent a portion of the volume near the tip. Measurements were conducted with two flapping frequencies to compare the flow structure. The second invariant of the deformation tensor visualized the leading edge and mid-chord vortices around the plate appearing due to flow separation behind the plate while other vortical structures were visualized by streamlines. These structures appear to be related to the dynamics of the leading edge vortex. Force analysis by integrating the phase-averaged velocity field within a chosen control volume showed increases in the maxima of the magnitudes of the non-dimensional unsteady force terms on the edge of the plate at the angles after the end of each stroke. The non-dimensional phase-averaged momentum flux was similar for both flapping frequencies.     

Experimental study of the response of a flexible plate to a harmonic forcing in a flow

Most aquatic animals propel themselves by flapping flexible appendages. To gain insight into the effect of flexibility on the swimming performance, we have studied experimentally an idealized system. It consists of a flexible plate whose leading edge is forced into a harmonic heave motion, and which is immersed in a uniform flow. As the forcing frequency is gradually increased, resonance peaks are evidenced on the plate's response. In addition to the forcing frequency, the Reynolds number, the plate rigidity and the forcing amplitude have also been varied. In the range of parameters studied, the main effect on the resonance is due to the forcing amplitude, which reveals that non-linearities are essential in this problem.     

The flexibility effect of a plate according to various angles of attack in a free-stream

This paper presents a computational fluid–structure interaction analysis for a flexible plate in a free-stream to investigate the effects of flexibility and angle of attack on force generation. A Lattice Boltzmann Method with an immersed boundary technique using a direct forcing scheme model of the fluid is coupled to a finite element model with rectangular bending elements. We investigated the effects of various angles of attack of a flexible plate fixed at one of the end edges in a free-stream at a Reynolds number of 5000, which represents the wing flapping condition of insects and small birds in nature. The lift of the flexible plate is maintained at the large angle of attack, whereas the rigid plate shows the largest lift at angles of attack around 30–40° and then drastic reductions in the lift at the large angle of attack. If we consider the efficiency as the lift divided by the drag, the flexible plate shows better efficiency at angles of attack greater than 30° compared to the rigid plate. The better performance of the flexible plate at large angles of attack comes from the deformation of the plate, which produces an interaction between the trailing edge vortex and the short edge vortex. The horseshoe-shaped vortex produced by a large vortex interaction at the trailing edge side has an important role in increasing the lift, and the small projection area due to the deformation reduces the drag. Furthermore, we investigate the role of flexibility on the lift and the drag force of the rectangular plate in a free-stream as the Reynolds number increases. Whenever a large vortex interaction at the trailing edge side is shown, the efficiency of the rectangular plate is improved. Especially, the flexible plate shows better efficiency as the Reynolds number increases regardless of the angle of attack.     

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.     

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。     

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.     

Experimental study on interaction of aerodynamics with flexible wings of flapping vehicles in hovering and cruise flight

Flapping wings are promising lift and thrust generators, especially for very low Reynolds numbers. To investigate aeroelastic effects of flexible wings (specifically, wing’s twisting stiffness) on hovering and cruising aerodynamic performance, a flapping-wing system and an experimental setup were designed and built. This system measures the unsteady aerodynamic and inertial forces, power usage, and angular speed of the flapping wing motion for different flapping frequencies and for various wings with different chordwise flexibility. Aerodynamic performance of the vehicle for both no wind (hovering) and cruise condition was investigated. Results show how elastic deformations caused by interaction of inertial and aerodynamic forces with the flexible structure may affect specific power consumption. This information was used here to find a more suitable structural design. The best selected design in our tests performs up to 30% better than others (i.e., less energy consumption for the same lift or thrust generation). This measured aerodynamic information could also be used as a benchmarking data for unsteady flow solvers.     

Effect of spanwise flexibility on flapping wing propulsion

A water tunnel study of the effect of spanwise flexibility on the thrust, lift and propulsive efficiency of a rectangular wing oscillating in pure heave has been performed. The thrust and lift forces were measured with a force balance, and the flow field was measured with a Particle Image Velocimetry system. Introducing a degree of spanwise flexibility was found to be beneficial. For Strouhal numbers greater than 0.2, a degree of spanwise flexibility was found to yield a small increase in thrust coefficient, and a small decrease in power-input requirement, resulting in higher efficiency. In this case, a moderately stronger trailing-edge vortex system was observed. Introducing a far greater degree of spanwise flexibility, however, was found to be detrimental. A large phase delay of the wing tip displacement was observed, leading to the root and tip moving in opposite directions for a significant portion of the flapping stroke. Vorticity of opposing sign was observed to be shed from the root and tip, resulting in a weak and fragmented vorticity pattern. The thrust coefficient was observed to be significantly reduced, and the efficiency diminished. It is noted that the range of Strouhal numbers for which spanwise flexibility was found to offer benefits overlaps the range found in nature, of 0.2<Sr<0.4. From a design aspect, flexibility may benefit flapping-wing Micro Air Vehicles both aerodynamically and in the inherent lightness of flexible structures.     

Falling cards and flapping flags: understanding fluid–solid interactions using an unsteady point vortex model

A reduced-order model for the two-dimensional interaction of a sharp-edged solid body and a high-Reynolds number flow is presented, based on the inviscid representation of the solid’s wake as point vortices with unsteady intensity. This model is applied to the fall of a rigid card in a fluid and to the flapping instability of a flexible membrane forced by a parallel flow.     

Hydroelastic analysis of gravity wave interaction with submerged horizontal flexible porous plate

The present study deals with the surface gravity wave interaction with submerged horizontal flexible porous plate under the assumption of small amplitude water wave theory and structural response. The flexible porous plate is modeled using the thin plate theory and wave past porous structure is based on the generalized porous wavemaker theory. The wave characteristics due to the interaction of gravity waves with submerged flexible porous structure are studied by analyzing the complex dispersion relation using contour plots. Three different problems such as (i) wave scattering by a submerged flexible porous plate, (ii) wave trapping by submerged flexible porous plate placed at a finite distance from a rigid wall and (iii) wave reflection by a rigid wall in the presence of a submerged flexible porous plate are analyzed. The role of flexible porous plate in attenuating wave height and creating a tranquility zone is studied by analyzing the reflection, transmission and dissipation coefficients for various wave and structural parameters such as angle of incidence, depth of submergence, plate length, compression force and structural flexibility. In the case of wave trapping, the optimum distance between the porous plate and rigid wall for wave reflection is analyzed in different cases. In addition, effects of various physical parameters on free surface elevation, plate deflection, wave load on the plate and rigid wall are studied. The present approach can be extended to deal with acoustic wave interaction with flexible porous plates.     

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.     

Simulation of incompressible viscous flows around moving objects by a variant of immersed boundary‐lattice Boltzmann method

A variant of immersed boundary‐lattice Boltzmann method (IB‐LBM) is presented in this paper to simulate incompressible viscous flows around moving objects. As compared with the conventional IB‐LBM where the force density is computed explicitly by Hook's law or the direct forcing method and the non‐slip condition is only approximately satisfied, in the present work, the force density term is considered as the velocity correction which is determined by enforcing the non‐slip condition at the boundary. The lift and drag forces on the moving object can be easily calculated via the velocity correction on the boundary points. The capability of the present method for moving objects is well demonstrated through its application to simulate flows around a moving circular cylinder, a rotationally oscillating cylinder, and an elliptic flapping wing. Furthermore, the simulation of flows around a flapping flexible airfoil is carried out to exhibit the ability of the present method for implementing the elastic boundary condition. It was found that under certain conditions, the flapping flexible airfoil can generate larger propulsive force than the flapping rigid airfoil. Copyright © 2009 John Wiley & Sons, Ltd.     

Bond graph modeling of a typical flapping wing micro-air-vehicle with the elastic articulated wings

In the present research, a new comprehensive model of a flexible articulated flapping wing robot using the bond graph approach is presented. The flapping kinematics of a two-section wing is introduced via the bond graph based approach on a hybrid mechanism providing amplitude and phase characteristics. The aerodynamic quasi-steady approach equipped with stall correlation is utilized according to the reduced flapping frequency and the angle of attack ranges. The local flow velocity and the wing position are calculated in both wing and body coordinates taking into account rotation and translation of the wing different parts. Estimation of the effective angle of attack is performed by calculating the instantaneous torque distribution on both wing sections. Aeroelastic modeling is employed in which the wing structure is assumed as an elastic Euler–Bernoulli beam at the leading edge with three linear torsional modes. In this novel integrated bond graph model, computation of the performance indices including the average lift and thrust, consumed and produced powers by flapping and mechanical efficiency are presented. Due to existence of the numerous geometric and kinematic parameters in articulated flexible flapping wing, such a model is essential for design and optimization. Consequently, an example of a typical parametric study and the results validation are carried out. It is indicated that the sensitivity of the bird performance to relative change in design variables would increase for out of phase flapping, second part stiffness, flapping amplitude, frequency and velocity respectively. It is interesting that by employing the reverse-phase flapping which is possible only via articulated wings, the maximum efficiency could be achieved. In addition, it is shown that adjusting the wing torsional stiffness is a crucial item in design of passive flapping robots. The key advantage of the two-section flapping wing is depicted as the controlling capability of the angle of attack in the outer part of the wing. Finally, the improved version of the bird is being addressed by approximately 15% progress in propulsive efficiency.

     

Passive control of a single flexible flag using two side-by-side flags

Numerical simulations using an improved version of the immersed boundary method are performed to explore a passive control concept for a single flexible flag in a viscous uniform flow. In order to control a single flag passively, we utilize the distinct dynamics of two side-by-side flags, characterized by in-phase and out-of-phase flapping modes depending on their spanwise gap distance. When the two side-by-side flags are in an in-phase flapping mode with a small spanwise gap distance, the flapping amplitude of a single downstream flag is highly enhanced due to synchronization between the vortices shed from the upstream and downstream flags. However, when the two upstream flags flap in an out-of-phase flapping mode with a large spanwise gap distance, the flapping of the single flag is significantly weakened with a reduction of the dominant flapping frequency. Because the upstream flags induce consecutive counter-rotating vortex pairs with a high frequency due to their flapping mode (out-of-phase state), relatively strong interaction with an upcoming vortex of the opposite rotational direction leads to flapping inhibition of the single flag. For an intermediate spanwise gap distance, the vortex-to-vortex interaction between the flags becomes more complicated, and a change of the flapping phases of the two side-by-side flags depending on streamwise gap distance between the upstream and downstream flags occurs. The interactions between coupled flags are documented through the root-mean-square cross-stream tail positions, frequency, drag coefficient, vorticity and pressure contours of the flags with varying non-dimensional parameters relevant to the problem. The proposed passive control concept of a single flag using two side-by-side flags is applicable to the development of energy harvesting systems to extract more energy and flapping control systems to suppress vibration.     

Effects of mass and chordwise flexibility on 2D self-propelled flapping wings

A self-propelled flexible flapping wing 2D numerical model undergoing a combined pitching and heaving motion is presented. Since such freely moving foil experiences zero net thrust, a definition of efficiency for this kind of problem is proposed and discussed against other formulations found in the literature. It is also shown that the deviation motion of wings such as that found in natural flyers is likely a consequence of the fluid–structure dynamics of the wings. The passive deviation motion observed in numerical simulations is either a consequence of a feathering mechanism referred to as rigid feathering or of the inertial displacement caused by the wing deformation. The effects of flexibility on the performance of the wing are also presented. It is found that flexibility may significantly enhance the efficiency in pressure-driven deformation cases. The rigid feathering mechanism is found to have an effect similar to that of the feathering caused by wing flexibility on the performances of pressure-driven deformation cases.     

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.