A review of bird-like flapping wing with high aspect ratio

Changchuan XIE, Nongyue GAO, Yang MENG, Yue WU, Chao YANG

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

KEYWORDS Aeroelasticity;Bird-like flapping wing;Flapping motion;High aspect ratio;Modeling;Nonlinear

Abstract Bird-like flapping-wing vehicles with a high aspect ratio have the potential to fulfill missions given to micro air vehicles,such as high-altitude reconnaissance,surveillance,rescue,and bird group guidance, due to their good loading and long endurance capacities. Biologists and aeronautical researchers have explored the mystery of avian flight and made efforts to reproduce flapping flight in bioinspired aircraft for decades. However, the cognitive depth from theory to practice is still very limited. The mechanism of generating sufficient lift and thrust during avian flight is still not fully understood. Moving wings with unique biological structures such as feathers make modeling, simulation, experimentation, and analysis much more difficult. This paper reviews the research progress on bird-like flapping wings from flight mechanisms to modeling.Commonly used numerical computing methods are briefly compared.The aeroelastic problems are also highlighted.The results of the investigation show that a leading-edge vortex can be found during avian flight.Its induction and maintenance may have a close relationship with wing configuration, kinematics and deformation. The present models of flapping wings are mainly two-dimensional airfoils or threedimensional single root-jointed geometric plates,which still exhibit large differences from real bird wings. Aeroelasticity is encouraged to consider the nonignorable effect on aerodynamic performance due to large-scale nonlinear deformation. Introducing appropriate flexibility can improve the peak values and efficiencies of lift and thrust, but the detailed conclusions always have strong background dependence.

1. Introduction

Many aircraft-related inspirations are obtained from flying creatures in nature, and various examples of bionic applications on aeronautics have emerged.1-6Birds are one of the‘‘flying masters”among flying creatures due to their physiological structure and flying skills and have been flying on Earth for more than one million years.7For example, the albatross has excellent flying skills and flight performance. Their characteristics include relatively high-aspect-ratio wings and good energy execution and planning routes during long-distance flight.8-10This bird can increase flight altitude according to the changeable wind field and modify its attitude to obtain an Angle of Attack (AOA) most suitable for gliding to realize the longest flight endurance.11

Early researchers usually found and summarized the characteristics by observing the flight of real birds. Rayner12-13proposed the vortex theory of hovering and avian flight. Pennycuick and Tucker were considered as the first two researchers to study bird flight and migration using an optimization method.8They obtained the shapes, dimensions, and parameters of the aerodynamics of different kinds of birds, such as auks, albatrosses, and petrels, through observation.14-16Pennycuick17summarized the ‘U’-type power line of avian flight,establishing the existence of a flying speed that can achieve the minimum energy consumption.These birds fly in two ways:gliding and flapping. In fact, large birds such as the albatross may have insufficient muscular strength to support continuous dynamic flight.Thus,they apply a no-power flying mode,gliding or soaring, to save energy during flight. According to the research report of Pennycuick,15the lift coefficient under minimum energy consumption and the lift coefficient under average flying speed of different kinds of high-aspect-ratio birds are collected in Fig. 1. The upper line (red) represents the lift coefficient curve of different kinds of birds flying at the minimum energy consumption speeds that fitted according to the data shown in red dots.The lower line(blue)represents the lift coefficient curve of different kinds of birds flying at the average speeds that fitted according to the data shown in blue squares.The flying modes of WAN,BBA,GHA and MAC are soaring and flapping, while Fg, CAP, PRN and WIL have only flapping modes.18Logarithmic coordinates are used following the suggestion given by Rayner.19The results show that the aspect ratio is correlated with the lift coefficient to some extent among these birds. A larger aspect ratio is always combined with a higher lift coefficient. The scientific names15of birds and their abbreviations are shown in Table 1.

Fig. 1 Effect of the aspect ratio on the lift coefficient.

Table 1 Scientific names and abbreviations of the species used in graphs.

Large birds usually have a larger aspect ratio, and their basic metabolic power may account for a smaller percentage of the total energy consumption than that of small birds.20Flying creatures’ energy is consumed to overcome flight drag,which includes friction drag, pressure drag and induced drag;maintain muscular activities; and support basic metabolism.Friction drag is caused by the viscosity of air.Relative motion exists between the surface of the moving object and the directcontact fluid. The friction caused by the relative motion leads to a velocity gradient in the boundary flow.Pressure drag(also called as wave drag) is generated due to the flow separation,which significantly changes the pressure distribution over the surface. The friction drag and pressure drag both constitute parasitic drag caused by the boundary layer. Induced drag reveals the drag cost of generating the lift needed to support the body weight in the air. When the wing produces lift, the free wake falling from the wing tip induces a downward velocity in the spanwise profile, which causes drag because of the decrease in the effective AOA.21Parasitic power and induced power are defined as the dot product of the project length of parasitic drag and induced drag on the motional direction and the velocity rate, respectively. Schmidt-Nielsen20used the energy(Calorie)consumption of the unit mass moving unit distance to calculate the energy consumption rate during animal locomotion.

In which the result is expressed in unit. cal means Calorie,the unit of heat. And hr represents hour, the unit of time. Eq.(1) eliminates the time parameter, so the energy consumption in this context has no relationship with motional velocity.This indicates that Eq. (1) assumes that the motion rate of the object remains constant, or some representative velocity rate is chosen during analysis. Research shows different energy consumption under different flying velocity rates. Tucker22found the existence of a flying velocity that can achieve the minimum energy consumption according to the velocityenergy consumption curve of parrots, which was measured experimentally. When comparing birds under different masses and scales, the energy-consuming efficiency under velocity based on the minimum or maximum aerodynamic power can be chosen as the criterion,23-24which is written as Eq. (2). J means Joule, the unit of energy.

Figs.2 and 3 show the flying energy consumption of different kinds of birds calculated by Eq.(2).In both figures,upper lines (red) represent energy consumption curves of different kinds of birds flying at the maximum sum of induced power and parasitic power that fitted according to the data shown in red dots, and lower lines (blue) represent energy consumption curves of different kinds of birds flying at the minimum sum of induced power and parasitic power that fitted according to the data shown in blue squares. The descending trend of energy consumption caused by an increasing aspect ratio indicates that birds with a higher aspect ratio can carry the same load for longer distances with less energy. Compared with Fig. 2, the data points in Fig. 3 add the consideration of the energy consumed by basal metabolism. This part also implies energy consumption while birds are not flying. Therefore, the energy-consuming efficiency in Fig. 3 is higher than that in Fig. 2. It can be found from Fig. 3 that after adding the consideration of basic metabolic energy consumption, all the data points are nearly distributed on a line in doublelogarithmic coordinates. Usually, double-logarithmic coordinates are employed to show the statistical results, which are described by the linear regression method with formula log y=a+b log x.For example,this method was used in Refs.18 and 20 to build the relationship among bird span, body mass and energy-consuming efficiency. The relationship between the aspect ratio and energy consumption revealed the possibility of realizing high-aspect-ratio bird-like flapping-wing aircraft.

Although researchers have carried out a great deal of study on bioinspired flapping wings in the past several decades,most of the efforts have been on insects or small-sized birds,such as hummingbirds, which have emphasized their capabilities for clandestine reconnaissance and scouting missions.25-26However, the bird-like flapping-wing vehicles usually belong to Micro Air Vehicles (MAVs) with upper limitations of 1 m and 2 kg for wingspan and weight, or even micro Unmanned Aerial Vehicles (μUAVs) with upper limitations of 2 m and 5 kg.27The aerodynamic and structural effects under this scale may probably not be the same as those of insect flight,although similar wing kinematics and structures are used. In addition, compared with insect-like MAVs, bird-like flapping-wing vehicles, which can carry more equipment and stay longer in the air, have more advantages in completing environmental monitoring, high-altitude reconnaissance, or other missions.27Some small birds(except hummingbirds)also have the ability to hover.28

Fig. 2 Comparison of energy consumption without basic metabolism.

Fig. 3 Comparison of energy consumption, including basic metabolism.

Based on a literature survey,it was found that several main problems in the bird-like flapping wing with a high aspect ratio field have still not been addressed clearly, which include the following: (A) How can well-behaved wing kinematics be designed with ideal aerodynamic performances? (B) How can rapid fine numerical analysis be carried out? (C) How can the structural deformation of bird-like flapping wings be considered to make the response closer to the real situation?

The research literature on bird-like flapping wings is rich,and this review only highlights specific cases to discuss the characteristics of different modeling methods and solutions in the bird-like flapping-wing field. The main conclusions are summarized based on the previous studies. The shortcomings are also discussed to provide a reference for bioinspired research in the near future. The rest of the paper is organized as follows:

The unsteady aerodynamic effect is overviewed briefly in Section 2.Bird-like flapping-wing kinematic modeling methods that are widely used are described in Section 3. Then, aerodynamic models with different degrees of fidelity and common structural modeling and analysis methods are presented in Sections 4 and 5. The aeroelastic problems that influence flight performance through structural deformation are also mentioned in Section 6. Next, the influence of bird-like wing kinematics, wing configuration and airfoil and the structural flexibility are summarized in Section 7.Finally,major research opinions on bird-like flapping wings with high aspect ratios are presented in Section 8.

2. Unsteady aerodynamic effect

Compared with the hovering insect flight29or the hovering and intermittent hummingbird flight,30-31there are notable differences between avian flight and them, especially the oceanic bird flight.32Hovering is a kind of ability to stay still in the air, mainly used by small-scale flying creatures but seldom by medium- or large-scale flying creatures. As mentioned in Section 1, birds mainly use gliding mode or flapping mode to fly. As a result, a high-lift mechanism such as clap and fling,rapid pitch rotation, and passive pitching may not exist under this situation.33

In some early analyses, quasi-steady theory was mostly used to estimate the aerodynamic effect of flapping wings.17,34-35However,studies have shown that dynamic stalls appearing on bird-like flapping wings cannot be ignored.Through the delayed stall mechanism, lift and thrust can be increased significantly. This unsteady procedure has a close relationship with the Leading-Edge Vortex’s (LEV’s) separation and development. Lighthill36theoretically analyzed large birds and found that using this mechanism can generate additional lift in some of the flight stages, which shows the important role this high-lift mechanism plays in avian flight.

Usherwood et al.37measured the pressure load on the Canadian goose wing during its takeoff and observed that wing tip pressure reaches two peaks throughout the plunging procedure, which may be influenced by the LEV. Videler32found that two flow patterns occurred on the wings of gliding birds: conventional attached flow and LEV flow. Hubel and Tropea38found the appearance of LEV when testing a goose-size flapping model in a wind tunnel through flow visualization. Figs. 438and 538show the flapping-wing model and the vortex and vorticity field, respectively. The observing slice positions in Fig. 4 are dimensionless positions of spanwise coordinates y divided by chord length c. k and Re means reduced frequency and the Reynolds number respectively in Fig. 5. Prominent LEVs can be observed at the outer middle section of the wing through(a)to(d)in Fig.5.Muijres et al.28studied the hovering flight of passerines and found that they utilized LEVs during the downstroke to increase lift.They also hypothesized that the elasticity of the wing may help keep the LEV from separating.Razak and Dimitriadis39experimentally concluded that high lift can be generated in pure flapping oscillations with the development of LEV, while the dynamic drag coefficient remained within the static drag coefficient range.

Kruyt et al.40experimented and analyzed the revolution of LEVs on revolving wings with aspect ratios of 2-10. They found that the LEV separated outboard on higher-aspectratio wings at high AOA but remained at lower AOA instead.However,they chose revolution as the wing kinematics,which is not fully available for bird-like flight mode. Thielicke and Stamhuis studied the effects of airfoil thickness and camber on the generation of LEV under different Strouhal number.It is found that the increase of thickness and camber may reduce the strength of LEV. In addition, the conclusion that the LEV exists only in slow-speed flapping flight rather than cruising flight has also been made.41

Fig. 4 Flapping-wing model with visualization planes and measurement position along the span.38

Linehan and Mohseni42focused on the alula of birds and explored the spanwise position of alula and its connection with inducing LEV. They found that the alula may induce and stabilize an LEV on the outer portion of the wing in deeply stall situations or help leverage the LEV lift during gliding.

Fig. 5 Vector and vorticity fields showing the flow at different positions along the span during the mid-downstroke at a static AOA of 8 deg.38

Through observation,it has been found that birds in nature that are adapted to long migration distances usually have relatively high-aspect-ratio wings.43Researchers have focused more on their unsteady aerodynamic effect of flight in recent years. On the one hand, a higher-aspect-ratio wing can obtain a better aerodynamic efficiency and reduce the induced drag44in the gliding or soaring mode, presenting a similar flying situation to fixed-wing aircraft. In addition, a high-aspect-ratio wing with dynamic soaring technology can efficiently utilize the energy in the wind field to keep flying.45-47On the other hand, researchers have found that increasing the wing aspect ratio48and controlling its structural flexibility to a certain extent49can improve the aerodynamic benefit of flapping wings. Relative research progress is overviewed in Section 6.

Many experimental or numerical studies have been carried out to simulate insect and small bird flight in threedimensional(3D)space.50-52However,studies focusing mainly on birds are still rare.How to design the kinematics of flapping wings with high aspect ratios to realize the control of dynamic stall generation and to obtain higher aerodynamic efficiency,the influences of wing configuration and structural flexibility on the induction and maintenance of LEVs construct two main less involved fields in bird-like flapping-wing studies and have high research value.

3. Kinematic models

Different from the fixed wing, the aerodynamic effect of flapping wings can produce lift and thrust simultaneously. The basic flight mechanism of the flapping wing is described clearly in Otto Lilienthal’s book.53In this book, the author vividly explains the effectiveness of flapping flight, which provides a powerful illustration for researchers to understand flapping flight. A review of the flapping flight of birds can be found in the literature of Altshuler et al.54They analyzed the feasibility of flapping flight depending on three main mechanisms from a biological perspective: wing rotational velocity around shoulders,wing shape,and AOA relative to flow.These works have further developed and elucidated the theory of flapping flight.

Birds are vertebrates, whose wings usually have three rotational joints from the wing root to the wing tip.55-57These joints have different forms of motion during the flapping flight.Several studies on its function have been conducted by biologists and aerodynamic scientists.Brown58and Hedrick et al.59carried out indoor flight experiments of pigeons and parrots,respectively. According to the stroboscopic images recorded in the experiment, the motion forms of different sections of the wing are quite different. Tobalske60studied the motion forms of different sections of the wing under various wind speeds. According to the experimental results, the joint structures of birds is conductive to changing the sweep angle and switching different flapping postures during flight.

In addition, the influence of bird feathers on in-flight wing morphing, flow control, attitude control, is extremely complicated. Biologists have discovered that feathers are not directly connected to bones,but are implanted into skin and connected with complex muscle, tendon and the nervous system.32Matloff et al.61investigated the movement of feathers with skeletons in birds flight from a biological point of view and found that the feathers generate passive compliant folding due to the skeletons’ movement. All these findings indicate the great challenge in accurate modeling of bird-like flapping-wing kinematics.

Inspired by the locking mechanism of the friction barbules,Wissa et al.62designed a kind of fixed-wing model with adhesive patches to mimic the attachment and separation of feathers in flight for the first time. The results show that the birdlike feather-morphing wing can improve the flight performance,especially in the gust environment.Di Luca et al.63proposed a novel morphing wing with artificial feathers that can change the wing geometry to satisfy different aerodynamic environments. The experiment reveals that folding feathers can reduce the drag coefficient while expanding feathers can increase the lift coefficient. Similar conclusions have also been made in a more recent study by Chang et al.64

Wu and Popovic´65described a kind of bird-like model for kinematics animation with feathers. Feathers have been considered for bending compliance. However, kinematics animation extracted from the realistic does not have a distinct physical model.Inspired by Wu’s work,Colognesi et al.66formulated a biomechanism model based on the anatomy of a bird, and modeled the bones and feathers of the wing. The model is shown in Fig. 6.66In their multi-body system, many feather-like models are articulated to the three parts of the wing. The three segments of the flapping wing are arm, forearm and hand with the length of la，lfand lh, respectively.The length of three primary (outside) and four secondary feathers(inside)are lp1-lp3，ls1-ls4,respectively.The feathers are coupled to each other by spring constraints,so that all the feathers can move smoothly with the folding and expansion of the wing.Based on this model,the wing kinematics can be further studied according to biological observations. It should be realized that because feathers have a vital impact on flight,and bird wings have flexible folding and shrinking,it is not enough to describe the kinematics of the skeleton alone.

With gradually deeper insight into the essential nature of bird-like flapping wings, the kinematics based on multi-body models, such as single-segment wings, two-segment wings,and multi-segment wings, have been widely utilized in the existing researches.

To further understand the forces generated at different positions of the wing structure during certain motions, it is necessary to begin analysis from the basis of the kinematic model with the perspective of multi-body dynamics.

Three kinds of coordinate systems are constructed: the wing-fixed coordinate system with superscript w, the bodyfixed coordinate system with superscript b,and inertial coordinate system with superscript i.

Fig. 6 Multibody model of a bird-like flapping wing.66

Fig. 7 Definition of the three coordinates and the vector expression of an arbitrary structure point Pk on an arbitrary part of a twosegment flapping-wing half model sketch.

We have presented the kinematic description of the arbitrary wing-structure point Pk.In the same way,if Q is an arbitrary point on the body, its kinematic description can be written as

Eqs. (4) and(5) describe the position of the arbitrary point on the flapping wing and body in a vector-operation way.They include the natural structural position, the relative motion of the wing to the body,the movement of the aircraft, the deformation of the wing, and the deformation of the body. All the vectors and matrices are time varying. Fig. 7 shows the vector expression of the spatial position composition of an arbitrary structure point Pkon an arbitrary part of a two-segment flapping-wing half model sketch. The undeformed wing is shown in yellow lines. The deformed wing is drawn with blue lines. Pkis an arbitrary wing-fixed point, and P＇kis its undeformed position. All the vectors are written in bold. Vectors described in wing-fixed coordinates and body-fixed coordinates are shown in red, and vectors described in inertial coordinates are shown in black. The dashed black vector is the object that needs to be expressed eventually.

Some theoretical models are based on the theory of multibody dynamics.68-70Liu et al.69modeled a two-segment avian wing to numerically simulate the real kinematics of some largescale birds and built corrected, bird-like flapping-wing kinematics for further analysis. Verstraete et al.70proposed a gull-like,flapping-wing model with a high aspect ratio to study nonlinear and unsteady aerodynamics. The kinematics of the two-segment morphing wing is described using multi-body theory.

The linkage model is another kind of multi-body kinematic model that is widely used in practical engineering.For the real bird-like flapping-wing model that can fly successfully, using linkages is the main way to realize the active flapping kinematics. Therefore, the linkage is widely applied to the design and analysis of an ornithopter. Fig. 871shows the gear box and linkage design sketch of the single-segment and two-segment flapping wing, in which variables l1-l4represent the length of the four bars in four-bar linkage in the left part of Fig. 8(b) and (c) while the actual designed values are written on the right. The single-segment flapping-wing model is designed to realize the flapping motion, and the two-segment flappingwing model is designed to realize the flapping and folding motion. The word ‘ornithopter’ is used to represent the birdlike flapping-wing prototypes or bird-like robots specifically in the following paragraphs of this paper.

Many single-segment-wing ornithopters are designed with a single bar.This kind of ornithopter mostly adopt the form of a leading-edge rigidelastic rod equipped with a membrane wing and is sometimes supplemented by a structural reinforcement skeleton. The main features of this ornithopter are the relatively light structural weight, with the premise of meeting the strength requirement,and the simple four-bar linkage relationship between the gear actuator and the flapping wing. Some representative ornithopters are discussed in Refs. 72-75.

On the basis of many previous prototypes, the multisegment-wing ornithopter gradually began to be explored by more researchers. This kind of ornithopter usually use the girder-type bar with a bird-like airfoil as the wing rib, and its girder-type bar is a part of the linkage.

To realize the combined motion of multi-segment flapping wings, two design methods are usually adopted: The first is a multi-segment and multi-actuator structure, which can easily control the segmented motion of a flapping wing through their respective actuator. However, the increase in the number of actuators will increase the weight of the model. The second is the multi-segment and single-actuator structure. It is noted that the single actuator corresponds to the single wing of the ornithopter here and that the number of actuators corresponding to a pair of flapping wings is not limited to only one.Although the desired weight can be ensured by using a single actuator,the complexity of the flapping-wing mechanical kinematic model will increase.It is apparent that the complexity of the connecting rod motions increases with the number of segments in flapping wing model. Some representative ornithopters are discussed in Refs. 71,76-79.

Lin et al.73designed an ornithopter by using a four-bar linkage.The aerodynamic performances were analyzed according to the results of a wind tunnel experiment. Mueller et al.72proposed a single-crank-driven ornithopter to investigate the effect of flapping-wing compliance on aerodynamic characteristics. Jiang et al.76designed a new, gull-like, flapping-wing ornithopter with two-segment wings that can flap and fold and successfully tested its flight ability. They also derived the kinematic model of a four-rod linkage according to the ornithopter and validated the flapping motion in MSC.ADAMS. Widhiarini et al.74proposed two bird-like flapping-wing prototypes to compare the aerodynamic generation performance and to realize autonomous flight control.The difference between the two prototypes lies in the connecting rod structure between the flapping wing and actuator, i.e.,the single- and double-crank driver systems. Ryu et al.71focused on optimization and force measurement experiment of two kinds of bird-like prototypes. Through the experiment,two merits of the flapping and folding motion have been concluded: a shorter wing-stroke period and higher average lift.

For more information, readers can reference Ref. 80, and Section 2.1 of Ref. 81. Although certain flying ornithopters with link-based wing kinematic models have been successful,a large gap between the model size of the bird-like flapping wing and its real object still exists, especially when imitating large-scale birds with high aspect ratios, such as the gull and the albatross.

Fig. 8 Design sketch of the gear box and the four-bar-linkage actuator.71

Those 3D models can be further separated into three parts:a single wing and a pair of wings with and without a body. A single wing is a kind of simple model that can reflect flapping motion.However,this model ignores the wing-wing and wingbody mutual influence, especially when they are close to each other. Some high-lift vortex effects caused by a pair of wings are not suitable for analysis in this kind of model. The model of a pair of wings without a body reveals wing-wing effects more exactly but still excludes the wing-body effect. For a more complex model with a body, both mutual influences and vortex effects can be considered. The added mass due to the attitude change can also be included.82In addition,further flight dynamics analysis can be carried out after introducing flight parameters.

By further simplifying the model, the airfoil can represent the bird-like flapping motion as a two-dimensional (2D)model, because the high-aspect-ratio model can disregard the spanwise 3D influence to some extent.

Fig. 9 shows that employing airfoil active motion to represent the kinematics of a flapping wing may be a powerful simplified method.Fig.9(a)shows a 3D high-aspect-ratio flapping wing model with airfoil. The location of the elastic axis of the wing are displayed in gray dash line. Fig. 9(b)shows the kinematic and aerodynamic parameters of cross section with red boundary at the spanwise station R in Fig. 9(a). The motion of the airfoil can be decomposed into three parts: back and forth along x parallel to the flow,pitch and plunge along z perpendicular to the flow and pitching torsion θ about the rotation center. The effective AOA α considers the relative horizontal flow speed U∞and ˙x, as well as the pitching and plunging velocity ˙z.The force on the airfoil can be decomposed into lift L in the vertical direction, drag D in the horizontal backward direction and moment M about the rotation center.Forward thrust is generated when drag D is negative.This can be a typical dimension reduction model and is usually used in single-segment infinite-span numerical flapping-wing models.It can also express the chordwise profile’s aerodynamic effect of a wing.

Fig. 9 Dimension reduction 2D airfoil model extracted from a 3D high-aspect-ratio flapping wing model.

Given that these multi-body models and link-based models are more complex than the models shown in Fig. 9, the kinematic models depicted in Fig. 9 are still widely utilized in current research on flapping wings, especially for high-aspectratio flapping wing.The simplified model still retains the main characteristic of the flapping effect,which is helpful to explore the mechanism of flapping of large birds in a relatively simple and ideal way.

In this section, diverse kinematic models are highlighted.The main difference among them lies in the modeling accuracy.Different analysis purposes lead to different choices regarding the complexity of the kinematic model.

4. Aerodynamic models

Aerodynamic models with different computing fidelities have had many successful applications in previous studies on flapping-wing flight. The computing method is generally chosen based on the following factors:44,83the required calculation time, the required computing accuracy and the stage of the project or the properties of the study. Low-fidelity aerodynamic models mainly include the quasi-steady aerodynamic model84and lift coefficient function model.85-87Flow separation, viscous effect and compressibility are not considered in this kind of model. Although these algorithms are classified as low-fidelity methods,they are widely used with conventional fixed-wing aircraft.44Medium-fidelity aerodynamic models mainly include strip theory considering 2D effects88-90and panel methods considering 3D effects.91-96Panel methods are a widely used aeroelastic analysis method in industry, among which the Vortex Lattice Method (VLM) adjusted to timedomain simulation can be used for static and dynamic analysis. This method can also solve the coupling problem of flapping-wing motion and structural flexible deformation.97-98High-fidelity aerodynamic methods include calculation methods based on the inviscid Euler equations and viscous Navier-Stokes (N-S) equations. The Euler solutions do not consider the viscous effect, flow separation and special problems for low Reynolds-number flight, which has relatively large limitations, so they can only be applied to analyze large birds in attached flow flight with slow flapping frequencies at moderate amplitudes.99The N-S solutions can reflect the flow field characteristics in more detail.100The main approaches based on the N-S equation include the Reynolds-Averaged Navier-Stokes (RANS) model, Unsteady Reynolds-Averaged Navier-Stokes (URANS) model, Large-Eddy Simulation(LES) model and hybrid RANS-LES model. With technological advancements, these approaches have been widely employed to analyze nonlinear aerodynamic characteristics of the whole bionic MAV field in recent years,not only limited to bird-like MAV.101-104For more information on the research progress of the aforementioned models and other Computational Fluid Dynamics (CFD) technologies, please refer to Refs. 105-107.

Dimensionless parameters, such as the Reynolds number,reduced frequency and Strouhal number, are commonly used in the flapping-wing field. The degree of unsteadiness of the flow is characterized by the reduced frequency k.

in which f is the flapping frequency, c is the chord length, and U∞is the freestream speed. The reduced frequency k denotes the ratio of the space scale of fluid disturbance and chord length.Ames et al.108pointed out that the fluid can be considered as quasi-steady flow and the wake effect can be disregarded when k satisfies 0 ≤k ≤0.03. For 0.03 ≤k ≤1, in contrast, the fluid becomes unsteady. The additional mass can be ignored in the present context,but the wake effect cannot.Refs.39 and 108 also refers to this state as quasi-unsteady flow.The fluid is fully unsteady when 1 ≤k,in which case the additional mass should be taken into account.

Another dimensionless parameter that is usually used is the Strouhal number. This number characterizes the vortex dynamics of the wake and shedding behavior of vortices of a flapping wing in forward flight.109It is estimated through the following equation

in which A denotes double the flapping amplitude,corresponding to the stroke of the flapping wing. Theoretical and experimental data reveal that the propulsion efficiency of the flapping wing reaches a peak point in a certain range of St values for any given flapping motion.For a group of specific flapping motions, 0.2 ＜St ＜0.4 is always satisfied when the propulsion efficiency is optimized.110

4.1. Low-fidelity models

Low-fidelity aerodynamic models are usually based on the quasi-steady condition or lift coefficient function. Under the condition of ignoring the wake effect, these models predict aerodynamic force at a certain time through polynomials including translational, rotational, and other motional state variables and their first derivatives over time. Munk111predicted that the contribution of rotational motion to aerodynamic force can be ignored when the 2D airfoil performs small amplitude rotational and translational motions. However, aerodynamic experts have recognized the unsteady aerodynamic effects of wing flutter. The work of Theodorsen112and Fung113improved the quasi-steady-state model of flutter and predicted that the critical axis is located at a distance of 3/4 chord length from the leading edge. Reid,114Silverstein and Joyner,34Halfman115and Farren116et al.provided experimental verification for this prediction and established that the aerodynamic force generated by an oscillating airfoil placed in the steady fluid is different from the theoretical results in the steady state. These experiments at a high Reynolds number simulated the inviscid condition and confirmed that the improved quasi-steady model is suitable for simulating birdlike flapping-wing models.

The quasi-steady or lift coefficient function model provides a convenient quantitative computing method, which is easy to apply to system energy and power analysis and easily embedded into the flight dynamic control model.Studies have shown that the quasi-steady model can predict flapping-wing aerodynamic effects at medium and low Reynolds numbers,similar to insect flight conditions.84In the absence of high-fidelity simulation means,it is helpful for bird-like flapping-wing studies to evaluate the effects of different parameters, such as wing area and aspect ratio, on flight character based on dimensional analysis using the low-fidelity equation.Tennekes summarized the relationship among cruise speed,weight and wing load.He produced a widely related graph that includes birds, insects and artificial aircraft,117which can compare the relationship between species with prominent size differences.

4.2. Medium-fidelity models

Medium-fidelity aerodynamic models compute aerodynamic forces based on strip theory or the panel methods.In the aerodynamic simulation of a flapping wing with a high aspect ratio,the viscous effect of flow can be disregarded at a low Reynolds number,and the compressibility of air flow can be neglected at low speeds. In the flight environment of a large bird, the discretization method based on potential flow is an ideal analysis method. These methods are fully introduced in some textbooks.21,118-119Although viscosity is not considered, the results or trends of the Unsteady Vortex Lattice Method(UVLM), one of the panel methods, are consistent with other more accurate simulations or experiments.

Lambert et al.120combined the UVLM with three kinds of load estimation techniques to predict aerodynamic forces.Compared with a rectangular flapping-wing model experiment,the predicted lift can be reasonable while the drag is not very ideal. Persson et al.121compared the UVLM with the highfidelity CFD simulation of flapping-wing flight in detail,showing that the UVLM produces accurate results for additional flow energy and even maintains the trend correlation in the presence of flow separation.Roccia et al.51studied the application of UVLM at low Reynolds numbers, and the calculated results were found to be in good agreement with the experimental data, which can be used for flow field simulation and lift prediction based on the actual wing motion of fruit flies or small birds. Their computing method of LEV strength and motion was the same as that for calculating the trailingedge vortex.The Kutta condition was used to calculate vortex strength at the edge of separation and applied to the flow separation of highly swept delta wings.118,122This is because the reversal of the flapping direction leads to only a single vortex system.33,39,52,123That is, there is no vortex shedding process caused by the accumulation of vortex strength in the flapping process.Medium-fidelity models can provide better simulation results than low-fidelity methods. Based on the efficiency of this kind of calculation method, some research works have applied the method to the design of bird-like flapping wings in flight.124-128

However, various kinematic or geometric parameters, as well as the flow field characteristics,may easily affect the aerodynamic performance of a flapping wing, and it is difficult to establish the direct relationship between these factors and aerodynamic performance.

Benefitted from the progress of data induction and analysisrelated technologies in recent years,data-driven methods have been applied to modify existing medium-fidelity models and even to directly predict aerodynamic forces.129-130The accuracy of these methods shows good agreement with the experimental results.

Lee et al.131established a generic and adaptable quasisteady model to rapidly predict aerodynamic force for the MAV design process. The deviation in their model results can be controlled within a certain range compared with the numerical simulation results. Han et al.132proposed a semiempirical, quasi-steady, aerodynamic model to extract the aerodynamic coefficients of a flapping wing model. The proposed model was proved to improve the accuracy in estimating aerodynamic forces and moments compared with the experimental results.

Although insect-like flapping wing was chosen as the object in their research, the data-driven method can be a reference and applied to bird-like model to compensate the effects related to sensitive design parameters. Aerodynamic modeling with correction based on empirical values will also constitute potential research work with future practical application value.

Nguyen et al.133proposed a kind of fast prediction method to avoid formulating a relationship between results and parameters and to estimate the unsteady aerodynamic force based on the neural network. This method can also measure the required energy. In the research, a three-layer, artificial neural network has been designed.The results of 4000 random cases obtained from an extended UVLM model that considers the LEVs effect have been selected to train the neural network.The predictions agree well with the simulating results, and the computing time for the optimization process is reduced considerably.

A key component of the abovementioned method is system identification. Classical identification methods are represented by the least squares method and maximum likelihood estimation, while the modern identification methods are represented by neural-network, fuzzy logic and wavelet network. One of the obvious advantages of these methods is that the correlation between flow field input data and output data can be established on the basis of considering flow properties with complexity to a certain extent and that the accuracy of prediction results can be controlled within an acceptable range.Another advantage of these methods is that they can in conjunction with the subsequent optimization process, which is very conducive to more comprehensive research from a design,optimization and control perspective, especially to reduce the computational cost in optimization.

In addition, research adopted the semi-empirical modified,medium-fidelity aerodynamic model that uses data fusion method should belong to the category of multi-fidelity models.However, limited by the scope of this review, this model will not be separately classified and introduced.More details about the data-driven aerodynamic models can be found in a recent review of Kou and Zhang.134

4.3. High-fidelity models

The theoretical basis of high-fidelity aerodynamic models can be divided into the Euler equation, which does not consider viscosity,and the Navier-Stokes equation,which does consider viscosity.44,135For a more comprehensive simulation environment of a flapping wing,the N-S equation is usually chosen to simulate a turbulent flow field.136-137The most accurate approach for simulating turbulent flow is Direct Numerical Simulation (DNS) which solves the full N-S equations. DNS can predict almost all nonlinear aerodynamic phenomena44and can be performed to study the unsteady vortex behavior in the flapping-wing process. Srigrarom138numerically simulated the flow around an albatross flapping wing based on unsteady 3D compressible N-S equations and solved the acoustic characteristics of a complete flapping period. Chang et al.104explore the aerodynamic performance of a bird-like bionic flapping wing by using an unsteady flow solver. The dynamic hybrid generation technique is applied for multibody separation and complex configuration with a moving boundary. De et al.139employed the 3D, unsteady, laminar,and incompressible N-S equations to solve the aerodynamics to analyze the differences in the aerodynamics among five flapping wings.

However, DNS can be computationally expensive and is only applied to low-Reynolds-number flows over simple geometry.105The bird-like flapping-wing motion with a high aspect ratio is a relatively slow process because of the low flapping frequency,140which renders the analysis time and computational resources of each case very expensive when investigating several flapping periods. As a result, the RANS model, which focuses on steady-state flow,has also been chosen for bird-like flapping-wing aerodynamic analysis. Chen et al.141solved the RANS equation to obtain the unsteady aerodynamic characteristics of flapping wings with a high aspect ratio. Yang et al.101researched the effects of a rectangular wing’s flexibility on the flapping aerodynamic performance through the preconditioned RANS model. Nevertheless, as mentioned in Section 2, it is not enough to compute only the steady flow as the unsteady aerodynamic effect is very significant in the flight of birds. To obtain the unsteady vortex information, the LES approach is also adopted to explore flow separation and the formation process of large eddies. Gordnier and Visbal142employed the implicit LES approach to solve the dynamicstall problem of a flapping thin airfoil. In a following study,Gordnier et al.143also utilized the same approach to study the interaction between unsteady aerodynamics and flexible wing structure dynamics.Kondo et al.144revealed the relationship between flow patterns with different AOAs and the aerodynamic performance of an owl-like wing through the LES approach.

Based on the LES approach, a hybridization of the RANS method and LES method is another promising method that balances the access of separated turbulent flow field information and computing cost. Durrani145used the hybrid RANSLES approach to simulate unsteady flow around a flapping wing with mesh deformation to compare vortex development and structural displacement of models with different chordwise flexibilities and varying thickness during a flapping motion.Li et al.146developed an aeroelastic framework, including the RANS-LES model, to describe the aerodynamic performance that can be applied to a flexible multibody bionic model.However, their research object is a bat-like flapping wing. Considering the generality of the framework and the approximation of a bird-like and bat-like flapping wing model147, their work is still informative in the study of the bird-like flapping wing.

Although the LES model can capture a large amount of flow field eddy information, it may still be time-consuming and computational storage-space-consuming in flapping-wing flows. For the hybrid RANS-LES model, the problem of the interface between the RANS region and the LES region remains. An alternative is the URANS model. Unger et al.148-149studied the evolution of separated bubbles in a flapping-wing model at a low Reynolds number. They used an empirical separated prediction linear model fully coupled with the URANS algorithm and gave the simulation results of flapping motion in air under specific parameters. They further studied the fluid flow around a flexible seagull-like airfoil.150

5. Structural dynamics and fluid-structure interaction models

A model that considers only the rigid motion of the flapping wing and ignores the structural elastic deformation can simplify the research object, but the real flapping wing usually has the coupling process of rigid motion and elastic deformation.

Structural dynamic models are mainly divided into three categories according to the research background:

(A) Pure structural dynamic model,which is used to analyze the response of a structure under known loads. The loads can be constant or time-varying with a known function.

(B) Structural dynamic model considering the influence on aerodynamics, which is mostly utilized for the modeling and solution of aerodynamic coupling. This kind of model needs to advance iteratively between aerodynamics and structural dynamics and the comprehensive model is referred to Fluid-Structure Interaction (FSI).Generally, FSI refers to the interdependence between fluid mechanics and structural mechanics. The fluid characteristics are determined by the shape of the structure and motion, while the movement and deformation of the structure are also affected by the fluid force acting on it. This iteration will be introduced in detail later in the paper. Taking the structural dynamics coupled with UVLM as an example, since the solution of the UVLM represents the aerodynamic force as the discrete force in discrete time,151most structural dynamic models are discrete models that need to be interpolated at the aerodynamics/structure interface.

(C) Structural dynamic model considering rigid body motion. On the basis of (B), this kind of model is commonly applied in flight dynamic analysis.To study flight trajectory or flight stability, it is necessary to introduce the influence of rigid body motion into the dynamic model, which is embodied in various forms of inertial forces and often renders the dynamic model more complex.152

On the basis of (C), if the control design is further carried out, the dynamic model will be constructed according to the selected linear or nonlinear control method in most cases.For the majority of bird-like flapping-wing models, choosing the linear control method is enough,153-154and the model is written into a state-space format. In addition, if the optimization design is further carried out,the reduced-order model may be the first choice for reducing the amount of iterative computation.155-156

For a bird-like flapping-wing model without cross-sectional design, it is convenient to use simple structural dynamic models, such as the Euler-Bernoulli beam,151multi-segment spring model,157-158and flexible plate or membrane model.159-160For a discrete model or a more complex model, a high-fidelity structural Finite Element Method (FEM) will be utilized to obtain the mass matrix and stiffness matrix or the final solution. Each of the models has its own merits. Thus, we should pay attention to the diversification of structural dynamics models for different research purposes. When selecting an appropriate model, the current modeling background, accuracy of the model, and expected computational cost should be considered. It would not always be optimal to choose a complex structural dynamic model.

Obviously,the third structural dynamic model is the closest to the real flight situation of bird-like flapping wings. According to the different scales of rigid body motion and deformation, the current flapping-wing dynamic models can be divided into small-scale and large-scale rigid body motion and linear and nonlinear elastic deformation. The distinction of small-scale and large-scale rigid body motion does not have a clear definition according to the existing literature. Thus,small-scale motion is defined as the ratio of wing oscillating or vibrating amplitude h to reference chord length c,and when the dimensionless vibrating amplitude h/c ≤0.5 (i.e. the flapping angle γ ≤26.6°)can be regarded as an approximate measurement criterion here. For the angular motion, the pitching angle α ≤10°. Large-scale rigid body motion is defined as the usual flapping motion of the wing that is beyond the above scope in this paper. These are discussed in the following section. The FSI computing methods will be introduced briefly in Section 5.4.

5.1. Small-scale rigid body motion

This kind of flapping kinematics generates enough aerodynamic force by exciting modal oscillation on the flexible wing surface, which usually attracts attention in fixed-wing flutter research or flapping kinematic research.161

One of the earliest studies about propelling by an oscillating airfoil or airfoil-aileron combination was proposed by Garrick.162The analysis was based on inviscid flow with objects undergoing small pitch and heave motion. The author also ignored the LEV effect. Anderson et al.163optimized the propulsive performance of a harmonically oscillating foil and found that the highest propulsive efficiency measured in experiments can reach 87%.However,some chosen working conditions of the oscillating airfoil are beyond the range of the small-scale motion defined in the present paper.

Most of the studies are on the level of airfoils or 3D rectangular wings, so deformation is not considered. Only a small number of studies analyzed the linear deformation of the wing from the perspective of flutter. Pendaries et al.35applied the quasi-steady model to high-aspect-ratio High-Altitude Long Endurance (HALE) UAVs and developed propulsion by wing flapping through the first bending mode.This kind of flapping wing is also called a vibrating wing because the disturbance producing an effective aerodynamic force mainly comes from modal vibration. Shao et al.164excited a fixed-wing model and found that natural first-order bending and twisting should be designed closed when utilizing modal vibration to produce lift and thrust.

5.2. Large-scale rigid body motion with linear elastic deformation

This kind of flapping kinematics has been applied to most of the current research because large-scale motion can cause a large disturbance of the flow and then influence the complexity of the flow field.This modeling method can simulate the actual flow separation, LEV and spanwise flow effect. Moreover, the real flapping-wing models used in most studies also exhibit the coupling phenomenon of large-scale motion and linear elastic deformation.165-166The structural dynamic simulation of rigid-elastic coupling can be conveniently solved by various kinds of self-developed numerical computing programs or commercial software such as MSC.ADAMS. Pfeiffer et al.167computed the free flight trajectory of flexible multibody flapping-wing aircraft and physically analyzed the trim longitudinal flight of an ornithopter. Based on that study, Lee et al.168-169analyzed the effect of flapping frequency on flight dynamics when considering the linear deformation of the wing and tail.They also analyzed the influence of the wing flexibility on the trim flight dynamics and stability.

5.3. Large-scale rigid body motion with nonlinear elastic deformation

For the sake of model simplicity, most of the previous studies assumed that the elastic deformation of the flapping-wing structure is a small quantity. However, a larger deformation of the flapping wing that has reached nonlinear structural deformation usually exists on the wing surface of birds and may produce aerodynamic effects that cannot be ignored.

Bird-like flapping-wing deformation can be divided into active deformation and passive deformation. Active deformation refers to the shape change driven by muscle movements during flight.16As mentioned in Section 3, the kinematics of bird wings that involving the combined motion of different wing segments and feathers can be regarded as the active deformation caused by geometric shape morphing.In contrast,passive deformation refers to the structural shape change caused by nonmuscular movements during flight, which is mainly produced by the aerodynamic forces and inertial loads acting on the wing structure.54,170Winzen et al.171observed the passive deformation of the natural wing of a barn owl in complex aerodynamic environment through the Particle-Image Velocimetry (PIV) method. The highly flexible wing structural adaptation to optimal aerodynamic performance have been researched.It is found that large structural deflection appeared at the trailing edge.In the following studies,they used the PIV method and force/force moment measurement to explore the connection between the local structural deformation and aerodynamic forces.172The experiment results showed that the flexibility of the wing causes obvious wing deformation, and the unsteady trailing-edge deflection excited a high-frequency structural response which increases the complexity of the flow field. However, both of their researches chose the low-speed hunting gliding flight mode as the research state and did not consider the flapping motion.

The wings of birds have high flexibility. When birds flap their wings, the velocities of feathers near the joint are relatively small, while distal feathers have higher local velocities.Combined with the increase in shaft diameter and barb density, distal feathers contribute most of the aerodynamic force in the flapping mode.173Therefore,the aerodynamic force acting on the wing will lead to both large deformation of the total wing structure and obvious local deformation of feathers.The linear relationship between displacement and strain is no longer applicable under this circumstance, and it is necessary to build nonlinear model that can describe large displacement correctly.

Lentink et al.174conducted a wind tunnel experiment of a flexible swift model and recorded the forces and images under different AOAs and wind speeds. The front view of different states is shown in Fig.10.174Nonlinear structural deformation of the model occurs under different wind speeds.Orlowski and Girard175derived a nonlinear dynamic model that considered the wing’s mass and the effect of the inertial coupling term.Significant differences of the model behavior have been found compared with the model neglected those factors. Grauer and Hubbard176developed a nonlinear multi-body dynamic model based on an ornithopter. The necessity of building nonlinear model has been emphasized from a perspective of multimission controlling realization. Paranjape et al.152established a nonlinear aeroelastic model of a tailless bird-like flappingwing model to analyze the steady-state performance and flight stability. They concluded that the performance can be improved when using moderate elasticity compared with rigid wings.They also discussed the relationship between wing dihedral angles and turning performance. Kodali et al.177emphasized the necessity of using geometrically nonlinear structural model when the wing deformation is extremely large although the linear beam model is adopted in their current research.

Large structural deformation also appears on high-aspectratio ornithopters and a nonlinear structural model should be used to describe the deformation precisely. Larijani178set up a nonlinear aeroelastic model based on a kind of ornithopter to analyze the aerodynamic characteristics of a flapping wing. They tested several state variables from a scaling model and a full-scale model. The predicted results using the nonlinear Newmark method are accurate compared with experimental data. Reichert95researched the kinematics of Snowbird, a human powered ornithopter,with a span of nearly 32 m.After optimizing the wing kinematics, the vertical displacement of the wing tip in flight could reach 3.9 m,and its twisting amplitude was 40°. It has been found that the origin for the high propulsive efficiency was the larger area swept out by the flapping motion of the wing due to its large deformation.

Although the flapping-wing structure with nonlinear elastic deformation is more in line with the real situation, few studies have considered the structural nonlinearity in dynamic modeling in the field of bird-like aircraft.

5.4. Fluid-structure interaction

To obtain both aerodynamic and structural results under different situations mentioned above,one of the most widely used methods,FSI,is chosen to iteratively compute the whole state during flight. The FSI algorithm is primarily divided into monolithic coupling and staggered coupling.44,119

In monolithic coupling FSI algorithms, the equations of fluid, structure, and mesh movement are solved simultaneously. A notable advantage of using monolithic coupling is the high robustness of its solver.However,this kind of method has various challenges from problem formulation to numerical discretization, which is seldom used in current research.

In staggered coupling FSI algorithms, two kinds of coupling methods, loosely coupled and closely coupled, are commonly used. These two kinds of methods solve the fluid and structural equations separately and are coupled in an interface model. This approach uses different solvers, but the solutions are coupled into one single module with exchange of information taking place at the interface or the boundary via an interface module, thereby making the entire computational aeroelasticity model tightly coupled.179The staggered coupling FSI algorithm can be separated into solving fluid equations and solving structural equations, which can ensure the accuracy of the nonlinear problem. It can also combine different structural and fluid algorithms without the requirement of a flow field and structure sharing the same mesh generation scheme, which makes it widely used.180

Fig. 10 Front view of the elastic deformation of the swift model with different AOAs and wind speeds.174

Shi et al.181derived a fully coupled FSI model based on the N-S solver and nonlinear beam theory to study the aerodynamic performance of an airfoil with controllable Young’s modulus. For a more detailed explanation of FSI, please refer to the Refs.182-183. Zhu49proposed a kind of FSI algorithm.Each iteration of this algorithm corresponds to a time step in fluid mechanics computation and several time steps in structural simulation computation, which can achieve highprecision convergence with fewer iterations. It shows that to improve the stability of numerical solutions, a smaller time step is needed to solve the structural dynamics equation than to solve the fluid dynamics equation. Afonso et al.44pointed out that the aerodynamic grid needs to be finer than the structural mesh to simulate the small-scale flow field structure.

6. Aeroelastic problems

The aeroelastic problems of the bird-like flapping wing that concerned by researchers mainly include two aspects: the aeroelastic stability during flapping and the influence of flapping-wing flexibility on aerodynamic characteristics or dynamic performances.

For the former, foci on bird-like flapping-wing self-excited vibration have been explored preliminarily. This topic will be reviewed in Section 6.1. The flexible effects of flapping wings will be highlighted in Section 6.2.

6.1. Aeroelastic stability and flight stability

As mentioned in Section 4.2, the flapping frequencies of birds are much lower than those of insects,especially those of largescale birds with high aspect ratios. According to the experimental formula of the flapping frequencies of birds, except hummingbirds,140there is

in which f represents the flapping frequency, m denotes the mass of the bird. Therefore, the flapping frequencies of large birds with high aspect ratio may be close to their wings structural natural frequencies,184which increase the complexity of the aeroelastic problem. Spedding et al.185investigated the management mechanism of aerodynamics for small-scale birds in a potentially unstable flow regime and suggested that the conclusions may not applicable to a large-scale bird-like study because the Reynolds number varies with the scale of bird.As a result, aeroelastic characteristics of insects, or even small birds, may differ from those of large birds.

Paranjape et al.152stated that although the flexible wing has the hidden danger of aerodynamic divergence and flutter,MAVs are usually assumed to fly at speeds low enough so that aeroelastic instability will not be excited.Although Ho et al.186has emphasized the important role of flutter in flapping flight nearly two decades ago, the point of view in Ref. 152 shows that most current bird-like researches directly avoid the problem of aeroelastic divergence through parameter adjustment.Nevertheless, recent researches on the flapping mechanism of bird-like flapping wings do have learned a few from the perspective of self-excited vibration, a kind of phenomenon that can lead to aeroelastic problem.

Fig. 11 Lift and drag on the self-excited flapper under different AOAs.187

Fig. 12 Flow visualization at AOA = 5 deg.187

Curet et al.187explored the‘flutter’of the bird-like flappingwing model from a biological perspective. They discovered that,unlike the aeroelastic divergence of traditional aerial systems such as the fixed wing, rotor wing or turbine blade, the self-excited vibration of the bird-like flapping-wing model can significantly improve the mean lift coefficient after reaching the critical speed of flutter. Fig. 11187shows the lift and drag on the self-excited flapper under different AOAs and obvious changes have obtained near U*=1.0 in all cases, in which U*is defined as the ratio of the actual wind speed and the aeroelastic critical speed. Fig. 12187shows the flow visualization of the non-flapping and flapping state at AOA = 5°.The flapping wing is drawn in white dash line and the arrow on the right side of each snapshot indicates the movement direction of the main wing. Four representative relative times in a flapping cycle, T = 0, 0.25, 0.5, and 0.75, are chosen in Fig. 12(b). Compared with Fig. 12(a), an apparent pitch and plunge motion existed on the flapping wing which is behind the main wing. They also hypothesized that passive flapping motion may arise in the wings of gliding birds based on the observation of instability that occurs in a Reynolds number regime for animal flight.

Except for the classical flutter,which is essentially a kind of self-excited vibration and usually occurs in fixed wing, resonance phenomenon also plays a significant role in bird-like flapping wing due to the proximity of the flapping frequency and the natural frequency of the wing structure. Several researches have focused on the influence of resonance on flight stability. However, it is worth noting that the resonance and flutter are essentially different.

Srigrarom and Chen188researched near-resonance albatross-like flapping-wing models and evaluated the aerodynamic characteristics through experimental species. Their results showed that thrust can be constantly generated, while lift is sinusoidally changing. Kodali et al.177proposed a twoway coupled solution to study the effect of spanwise flexibility from the perspective of aeroelasticity. The analytical results show good agreement with numerical and experimental results,even for lower frequencies. Two models including a mediumaspect-ratio passerine model and a high-aspect-ratio goose model,are used to analyze the aerodynamic performances considering the spanwise flexibility.Both models obtained optimal performance near the structural first-order resonance, which indicated that birds may utilize resonance to generate thrust.

These findings may be distinct compared with those of Ramananarivo et al.189who found that insect flyers may always stay below the resonance point to improve performance even though the structural resonances will not be invalidated.Nevertheless, they also recognized that due to the lack of relevant literature, more works about the frequencies and resonance of birds and insects should be carried out before drawing any conclusions.

Taha et al.190first explored the vibrational stabilization of large-scale insects from the theoretical level. It is worthy to note that the ‘stability’ according to their research is not the state description of a structural dynamic system, but a flight dynamic system. They also suggested that it is necessary to study other creatures containing oscillatory locomotor mechanisms, such as bird wings, to achieve MAV stability without feedback.

The advised work has been carried out earlier by Iosilevskii.191-192The theoretical derivation indicated that large flapping amplitude and low flapping frequency may induce the resonances of the wing and body. As a result, it is necessary to stabilize the system through sweeping motion of the flapping wing, which is different from the conclusion reached for insect-like flapping wings.

6.2. Effects of flapping-wing flexibility on aerodynamic and flight dynamic performance

A flexible bird-like flapping wing with a relatively high aspect ratio also produces a favorable aeroelastic effect. This kind of flapping-wing aircraft utilizes coupling bending and twisting of flexible wings to produce sufficient lift and thrust during flight.157,193-194The response of bending and twisting caused by inertial force generated from flapping motion makes the aeroelastic-tailored wing have a higher effective pitch and plunge AOA relative to flow.195

As mentioned in Section 5.3, the structural deformation and feather deformation are both existed when flapping a bird-like flapping wing. However, the consideration of aeroelasticity of flapping wing feathers is very difficult. Most of the previous studies chose the simplified geometric wing, such as the rectangular wing, as the object.

The elastic deformation can be divided into deformation caused by chordwise flexibility and spanwise flexibility.It usually uses z-direction up and down flapping or θ-angle twisting with pitching as the flapping-wing driving mode when considering chordwise flexible deformation, which is shown in Fig.9(b).196-197The up and down flapping with angle γ is chosen to drive the flapping wing when considering spanwise flexible deformation.198

Many studies have focused on the analysis of the effect of flapping wings with different chordwise flexibilities.Heathcote and Gursul199conducted a water tunnel experiment under constant amplitude. They used flapping-wing models that have rigid teardrop leading edges and different flexible trailing edges to discover the effect of chordwise flexibility.The result shows that an increase in the thrust coefficient and propulsive efficiency are observed on the airfoil when the optimal flexibility is chosen.Prempraneerach et al.200tested airfoil flapping under different Strouhal numbers. The results revealed that compared with the rigid foil, the efficiency of the flexible foil improves by 36% when using the optimum flapping process and flexibility.Zhou et al.201carried out aerodynamic analysis of flat airfoils with different chordwise flexibilities and computed the aerodynamic characteristics of a fully flexible flat model for the first time. The results showed that flexibility could reduce airfoil drag, while the lift and lift efficiency both remained at moderate flexibility.Kim et al.202studied the aerodynamic performance of a biomimetic flexible flapping wing.PIV was used to show that the coupling of aerodynamic force and chordwise flexibility caused a stall delay. Chordwise flexibility was found to have a prominent effect on the unsteady aerodynamic effect of flapping wings. Unger et al.150analyzed the relationship between the chordwise flexibility of a seagulllike airfoil and its aerodynamic performance. The results showed that the airfoil decambered as its stiffness reduced,and this led to the decrease of lift coefficient and propulsion efficiency. The proper design of temporal adaptive stiffness of the airfoil may help improve the efficiency.

An unignorable 3D aerodynamic effect occurs in the flapping-wing aerodynamic or flight dynamic performance when considering spanwise flexibility. The research of Heathcote et al.203showed that the thrust coefficient and efficiency can be increased slightly under a certain degree of spanwise flexibility when the Strouhal number is larger than 0.2. A medium-strength trailing-edge vortex can be observed at the same time. However, it has been proven that excessively high spanwise flexibility can be harmful because of the excessively large phase delay of the wingtip displacement. Based on this study, Chimakurthi et al.204used the same model to further analyze the effect of spanwise flexibility in different parameter environments through an N-S solver coupled with a structural solver. They considered that spanwise flexibility can have a good impact on thrust generation, and increasing the reduced frequency can improve the thrust generation of flexible wings.Aono et al.198also continued to use the flapping-wing model in the Ref.203.Their experimental results showed that the phase lag of the wingtip response caused by spanwise flexibility is the key factor in generating thrust and spanwise flexibility can have a favorable effect on thrust when the response phase lag is less than 90°. In addition, the increase in effective AOA caused by slight wing deformation due to spanwise flexibility is another main factor to enhance thrust.Kodali et al.177analyzed the effects of the spanwise flexibility on locomotion based on two kinds of bird-like models. The numerical results showed that maximum thrust and minimum power can be achieved at the same time.

In addition, the chordwise and spanwise flexibility of the flapping wing are both considered in some studies. Lin et al.73test the lift and thrust of an ornithopter.It is observed that the flexibility of the wing can cause a saturate of lift or even decrease it. The spanwise large deformation causes the reduction of the lift force.Yang et al.101studied the coupled effect of chordwise and spanwise flexibility of flapping wings by CFD.They concluded that chordwise deformation can enhance the aerodynamic performance, while spanwise flexibility should be restricted within a certain range to achieve the same objection.

Researches that studied the influence of the flexibility of bird-like flapping wing on aerodynamic or flight dynamics characteristics mainly focuses on the influence on lift coefficient, thrust coefficient and propulsion efficiency. In addition,the role flexibility plays in the delayed stall mechanism under high AOA is also often explored.Most studies stated that optimum chordwise or spanwise flexibility can improve thrust coefficient and propulsion efficiency, while too-high flexibility will have adverse effects. Nevertheless, the coupling of chordwise and spanwise flexibility is still a very profound problem.The ‘competitive’ relationship between them makes it necessary to pay special attention to balance the weights of the two in the design process of bird-like flapping wing, i.e., constraints that achieve one optimization may not make another optimal. Therefore, there is still a long way to go to improve the FSI coupling model and apply it to the analysis of aeroelastic problems,and even the subsequent optimization.In particular, FSI algorithm using nonlinear structural model. We have made many efforts to optimize models and improve the precision of the aerodynamic and structural dynamic simulation of geometrically nonlinear fixed wings with relatively fast numerical computing speed.205-210This would be helpful to explore aeroelastic models employing high-precision fastcomputing numerical methods adapted to bird-like flappingwing models in the near future.

7. Main factors influencing the aerodynamic performance of bird-like flapping wings

After investigating the relevant literature, it can be concluded that the aerodynamic performances of high-aspect-ratio flapping wings are mainly affected by the following three aspects:

(1) Flapping kinematics. The interaction between the flapping wing and the air has the characteristics of large disturbance and instability. Different flapping kinematics determine different aerodynamic effects produced during the upstroke and downstroke processes. For a given flappingwing mechanism with n Degrees of Freedom (DOFs), the trajectory of the 2n-DOF state-space equation is chosen to describe the position and attitude changes of the mechanism during its flapping process.

Soueid et al.211optimized the motion of a designated airfoil through sensitivity functions.They concluded that the effective AOA should be maintained at approximately 11°. According to the experiment of Lambert et al.,120when the pitching phase leads the flapping phase by 90°,the effective AOA of the wing is relatively small to maintain a mainly laminar flow on the wing and a maximum thrust to a large extent. In contrast,when the pitching phase lags the flapping phase, a high effective AOA may occur, and a high transient lift is generated.Lang et al.212studied the aerodynamic performances of an owl-like airfoil undergoing bioinspired flapping kinematics.As a result, the model generates sufficient lift with a lower power consumption. Rival et al.213experimented with a rectangular wing with plunging kinematics. Through PIV, they found that the LEV can be stabilized by tuning the kinematics carefully without the necessity of a spanwise flow.

Therefore, it is very important to design and optimize the flapping kinematics well to obtain ideal aerodynamic performances close to those of flying creatures.

(2) Wing configuration. The parameters that describe the plane shape of the flapping wing include the span,aspect ratio,taper ratio, etc. Parameters that described the airfoil include relative thickness, maximum camber, and maximum camber position. Taking the plane shape as the projection and scanning the airfoil,the configuration of the wing can be obtained.

A more complex wing configuration, the curve surface model, is usually modeled by introducing changeable chord lengths or leading-edge and trailing-edge spin lines. Ghommem et al.124studied the shape optimization of a flapping wing model using curve surface simplification in forward flight mode. The results showed that several parameters, the aspect ratio of the wing, the arcs of the leading and trailing edges and their respective curvatures, that may affect flight performances.

By using a more accurate model of the airfoil that can show geometric parameters in detail instead of a flat plate or a plate with camber, the numerical simulation result can be closer to the experimental data or the real result, and the aerodynamic performance brought by the existence of an optimal airfoil can be reflected.The research of Razak and Dimitriadis39proposed that there are differences in sensitivity to the flapping Reynold number between symmetric and asymmetric airfoils,affecting the generation of LEVs under high AOA. Lang et al.212also compared an owl-like airfoil with two standard NACA airfoils from the perspective of flow field structure development.Ramananarivo et al.214proposed a kind of optimization algorithm that can be applied to flapping propulsion and FSI analysis and conducted shape tailoring of airfoils.

(3)Structural flexibility.In the actual flapping kinematics of bird wings, visible spanwise and chordwise flexible deformation both exist. The studies mentioned in Section 6 have revealed that structural flexibility has notable influences on the aerodynamic performance of flapping wings, positive or negative.Therefore,it is necessary to consider the appropriate structural flexibility in the flapping-wing structural design stage to obtain the optimal lift, propulsive efficiency, and unsteady aerodynamic effect.

8. Conclusions

In this article, a number of previous efforts to promote the development of bird-like flapping wings have been highlighted.Despite efforts, studies choosing birds with high aspect ratios as objects are still very rare compared with those on insects and hummingbirds. There are still vast unknown areas to be explored based on existing observations.

(1) The energy consumption of flapping flight has a close relationship with wing kinematics and the aspect ratio.There exists a minimum energy consumption to maintain flapping flight for flapping wings with different scales. The minimum energy consumption is reduced by increasing the aspect ratio.

(2) The bird-like flapping wing can induce and keep the LEV during flapping and gliding. The dynamic stall mechanism depending on the LEVs can explain the high-lift mechanism. The generation and development of LEVs are determined by the flapping kinematics and wing structures. As a result, optimizing the wing kinematics to keep the AOA in an appropriate range may realize a better aerodynamic effect by utilizing the dynamic stall mechanism. However, wing structures,referring in particular to feathers,with specific functions such as holding the LEV, may be difficult to model.Therefore, it is necessary to find a rational surrogate model.In addition,the generating and developing mechanism of LEV still needs further exploration,which may help researchers fundamentally understand avian flight.

(3) The bird-like flapping-wing model with more than one joint that is usually employed in ornithopter design has not been considered extensively in theoretical kinematic design. However, suitable motional cooperation among different parts of the wing may result in good aerodynamic and flight performance.

(4) A bird-like flapping wing with a relatively high aspect ratio exhibits nonlinear structural deformation. The wing deformation can both change the local AOA of the fluid and influence the structure of the flow field vortex system.In addition,the wing deformation under different degrees of flexibility can be either positive or negative to the aerodynamic effect according to the aeroelastic analysis. However, there are few bird-like flapping wings with high aspect ratios,and the nonlinear characteristics are always ignored in the present research, which should be considered in future studies.In addition,the conclusion about the influences of structural elasticity on the aerodynamic performance still has a very strong background dependence, and this needs further discussion not only based on experiments or numerical simulations but also from the perspective of the theoretical formula.

(5) High-fidelity CFD is widely used for flapping foil simulation because it can visualize the flow field structure and is accurate for unsteady aerodynamic analysis.However,the time costs are enormous. The UVLM has a much faster computing speed but does not consider the viscosity problem. Some extended UVLM methods considering LEVs,51,215-216and some improved UVLM methods based on semi-empirical modification or machine learning have been proposed,but they still need to be further discovered in the direction of highprecision fast aerodynamic computation. The decision to use the N-S solver or the UVLM depends in part on the specific research objective and should be carefully considered.

(6) The time-advanced staggered coupling algorithm is a common method to solve linear and nonlinear FSI problems. However, the staggered coupling algorithm exhibits prominent hysteresis of force or displacement at a time step, which may lead to error accumulation. The time step needs to be short enough to ensure that the set of state variables approximately satisfies both the fluid equation and the dynamic equation. Too many time steps in a flapping period are disadvantageous to process-based kinematic optimization. Therefore, it is worth developing a high-precision nonlinear FSI algorithm (i.e., the monolithic coupling algorithm) to analyze the response of a flexible flapping wing. This approach does have difficulties due to various problems in the derivation of the theoretical equation.(7) Wind tunnels are the main experimental platform for flow field analysis, or the aerodynamic performance evaluation of birds or bird-like flapping wings depends on the flow field display technology.However,this platform has limitations such as constant speed and horizontal flow direction. To simulate a more realistic flight environment,wind tunnel test technology to overcome the above problems needs to be improved.How to generate a stable super-low-speed flow field that closely meets the flight speed of birds is another difficulty that needs to be overcome.(8) New kinematic designs and research on bird-like flapping wings need to be explored, especially for largescale bird-like models with high aspect ratios. Flapgliding flight may have the potential to become a new flight mode with low energy consumption. In addition,the conversion mechanism between flapping and gliding is another potential design point that can be further discussed.

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.