Aerodynamics of flexible wing in bees' hovering flight

The aerodynamics of 2-dimensional flexible wings in bees＇ normal hovering flight is studied. Four insect flapping flight coordinate systems, including a global system, a bodyfixed system, a rigid wing-fixed system and a flexible wingfixed system, are established to represent the insects＇ position, gesture, wing movement and wing deformation, respectively. Then the transformations among four coordinate systems are studied. It is found that the elliptic coordinate system can improve the computation accuracy and reduce the calculation complexity in a 2-dimensional rigid wing. The computation model of a 2-dimensional flexible wing is established, and the changes of the force, moment, and power are investigated. According to the computation results, the large lift and drag peaks at the beginning and end of the stroke can be explained by the superposition of the rapid translational acceleration, the fast pitching-up rotation and the Magnus effect; and the small force and drag peaks can be explained by the convex flow effect and the concave flow effect. Compared with the pressure force, pressure moment and translational power, the viscous force, viscous moment and rotational power are small and can be ignored.

Experimental investigation on the wing-wake interaction at the mid stroke in hovering flight of dragonfly

This paper focuses on flow structures of the wing-wake interaction between the hind wing and the wake of the forewing in hovering flight of a dragonfly since there are arguments whether the wing-wake interaction is useful or not.A mechanical flapping model with two tandem wings is used to study the interaction.In the device,two identical simplified model wings are mounted to the flapping model and they are both scaled up to keep the Reynolds number similar to those of dragonfly in hovering flight since our experiment is conducted in a water tank.The kinetic pattern of dragonfly(Aeschna juncea) is chosen because of its special interesting asymmetry.A multi-slice phase-locked stereo particle image velocimetry(PIV) system is used to record flow structures around the hind wing at the mid downstroke(t/T=0.25) and the mid upstroke(t/T=0.75).To make comparison of the flow field between with and without the influence of the wake,flow structures around a single flapping wing(hind wing without the existence of the forewing) at these two stroke phases are also recorded.A local vortex identification scheme called swirling strength is applied to determine the vortices around the wing and they are visualized with the iso-surface of swirling strength.This paper also presents contour lines of z at each spanwise position of the hind wing,the vortex core position of the leading edge vortex(LEV) of hind wing with respect to the upper surface of hind wing,the circulation of the hind wing LEV at each spanwise position and so on.Experimental results show that dimension and strength of the hind wing LEV are impaired at the mid stroke in comparison with the single wing LEV because of the downwash from the forewing.Our results also reveal that a wake vortex from the forewing traverses the upper surface of the hind wing at the mid downstroke and its distance to the upper surface is about 40% of the wing chord length.At the instant,the distance of the hind wing LEV to the upper surface is about 20% of the wing chord length.Thus,there must be a wing-wake interaction mechanism that makes the wake vortex become an additional LEV of the hind wing and it can partly compensate the hind wing for its lift loss caused by the downwash from the forewing.

Experimental Investigation of Flexible Hawkmoth-like Wings on the Wing-wake Interaction in Hovering Flight

To study wing-wake interaction for various wing flexibilities, force measurements and digital particle image velocimetry were carried out on flapping hawkmoth-like wings in a water tank. Wing thickness was employed as a design variable for the wing flexi- bility distributions. Abrupt flap-down and phase delay in flexible wings influenced the behaviors of the Leading-Edge Vortex （LEV） and Trailing-Edge Vortex （TEV）, generated by the previous stroke. While the rigid wing exhibited a detached LEV at the end of the stroke, wing with specific flexibilities obtained attached LEVs. The attached LEVs induced a relatively rapid flow toward the wing surface as a result of encountering the TEV, and the flow caused a higher lift peak. On the other hand, the wings with larger wing deformations generated distinctive changes in LEV and TEV behaviors. The flap-down helped the TEV form closer to the wing surface, and it thus caused a downwash rather than wing-wake interaction. Furthermore, the most flexible wing had a newly-formed pair of LEVs above the wing during the wing reversal, thereby being not able to generate the wing-wake interaction. These results help to understand the different vortex structures generated by flexible wings during the wing reversal and the corresponding effects of wing-wake interaction.

Numerical Study of Two-Winged Insect Hovering Flight

The two-winged insect hovering flight is investigated numerically using the lattice Boltzmann method(LBM).A virtual model of two elliptic foils with flapping motion is used to study the aerodynamic performance of the insect hovering flight with and without the effect of ground surface.Systematic studies have been carried out by changing some parameters of the wing kinematics,including the stroke amplitude,attack angle,and the Reynolds number for the insect hovering flight without ground effect,as well as the distance between the flapping foils and the ground surface when the ground effect is considered.The influence of the wing kinematic parameters and the effect of the ground surface on the unsteady forces and vortical structures are analyzed.The unsteady forces acting on the flapping foils are verified to be closely associated with the time evolution of the vortex structures,foil translational and rotational accelerations,and interaction between the flapping foils and the existed vortical flow.Typical unsteady mechanisms of lift production are identified by examining the vortical structures around the flapping foils.The results obtained in this study provide some physical insight into the understanding of the aerodynamics and flow structures for the insect hovering flight.

Control for going from hovering to small speed flight of a model insect

The longitudinal steady-state control for going from hovering to small speed flight of a model insect is studied, using the method of computational fluid dynamics to compute the aerodynamic derivatives and the techniques based on the linear theories of stability and control for determining the non-zero equilibrium points. Morphological and certain kinematical data of droneflies are used for the model insect. A change in the mean stroke angle （δФ） results in a horizontal forward or backward flight; a change in the stroke amplitude （δФ） or a equal change in the down- and upstroke angles of attack （δα1） results in a vertical climb or decent; a proper combination of δФ and δФ controls （or δФ and δα1 controls） can give a flight of any （small） speed in any desired direction.

Dynamic flight stability of hovering model insects:theory versus simulation using equations of motion coupled with Navier-Stokes equations

In the present paper, the longitudinal dynamic flight stability properties of two model insects are predicted by an approximate theory and computed by numerical sim- ulation. The theory is based on the averaged model （which assumes that the frequency of wingbeat is sufficiently higher than that of the body motion, so that the flapping wings＇ degrees of freedom relative to the body can be dropped and the wings can be replaced by wingbeat-cycle-average forces and moments）; the simulation solves the complete equations of motion coupled with the Navier-Stokes equations. Comparison between the theory and the simulation provides a test to the validity of the assumptions in the theory. One of the insects is a model dronefly which has relatively high wingbeat frequency （164 Hz） and the other is a model hawkmoth which has relatively low wingbeat frequency （26 Hz）. The results show that the averaged model is valid for the hawkmoth as well as for the dronefly. Since the wingbeat frequency of the hawkmoth is relatively low （the characteristic times of the natural modes of motion of the body divided by wingbeat period are relatively large） compared with many other insects, that the theory based on the averaged model is valid for the hawkmoth means that it could be valid for many insects.

Lateral dynamic flight stability of hovering insects: theory vs. numerical simulation

In the present paper, the lateral dynamic flight stability properties of two hovering model insects are predicted by an approximate theory based on the averaged model, and computed by numerical simulation that solves the complete equations of motion coupled with the Naviertokes equations. Comparison between the theoretical and simulational results provides a test to the validity of the assumptions made in the theory. One of the insects is a model dronefly which has relatively high wingbeat frequency （164Hz） and the other is a model hawkmoth which has relatively low wingbeat frequency （26 Hz）. The following conclusion has been drawn. The theory based on the averaged model works well for the lateral motion of the dronefly. For the hawkmoth, relatively large quantitative differences exist between theory and simulation. This is because the lateral non-dimensional eigenvalues of the hawkmoth are not very small compared with the non-dimensional flapping frequency （the largest lateral non-dimensional eigenvalue is only about 10% smaller than the non-dimensional flapping frequency）. Nevertheless, the theory can still correctly predict variational trends of the dynamic properties of the hawkmoth＇s lateral motion.

Stabilization control of a bumblebee in hovering and forward flight

Our previous study shows that the hovering and forward flight of a bumblebee do not have inherent stability （passive stability）. But the bumblebees are observed to fly stably. Stabilization control must have been applied. In this study, we investigate the longitudinal stabilization control of the bumblebee. The method of computational fluid dynamics is used to compute the control derivatives and the techniques of eigenvalue and eigenvector analysis and modal decomposition are used for solving the equations of motion. Controllability analysis shows that at all flight speeds considered, although inherently unstable, the flight is controllable. By feedbacking the state variables, i.e. vertical and horizontal velocities, pitching rate and pitch angle （which can be measured by the sensory system of the insect）, to produce changes in stroke angle and angle of attack of the wings, the flight can be stabilized, explaining why the bumblebees can fly stably even if they are passively unstable.

Lateral Flight Stability of Two Hovering Model Insects

The longitudinal disturbance motion of different insects at hovering flight has the same modal structure. Here, we consider the case of lateral motion. The lateral dynamic flight stability of two model insects, hoverfly and honeybee, at hovering flight is studied. The method of computational fluid dynamics is applied to compute the stability derivatives. The techniques of eigenvalue and eigenvector analysis are used to solve the equations of motion. Results show that the lateral disturbance motion of the hoverfly has three natural modes of motion： an unstable divergence mode, a stable oscillatory mode and a stable subsidence mode, and the flight is unstable; while the honeybee has a different modal structure （a stable slow subsidence mode, a stable fast subsidence mode, and a nearly neutrally stable oscillatory mode） and the flight is nearly neutrally stable. The change in modal structure between the two insects is due to their roll-moment/side-velocity derivative having different sign, and the sign difference is because that the hoverfly has a relatively small, but the honeybee has a relatively large, distance between the wing roots and the center of mass. Thus, unlike the case of longitudinal motion, for lateral motion, some insects have different modal structures and stability properties from others.

Flow Visualization of Rhinoceros Beetle （Trypoxylus dichotomus） in Free Flight

Aerodynamic characteristics of the beetle, Trypoxylus dichotomus, which has a pair of elytra （forewings） and flexible hind wings, are investigated. Visualization experiments were conducted for various flight conditions of a beetle, Trypoxylus di- chotomus： free, tethered, hovering, forward and climbing flights. Leading edge, trailing edge and tip vortices on both wings were observed clearly. The leading edge vortex was stable and remained on the top surface of the elytron for a wide interval during the downstroke of free forward flight. Hence, the elytron may have a considerable role in lift force generation of the beetle. In addition, we reveal a suction phenomenon between the gaps of the hind wing and the elytron in upstroke that may improve the positive lift force on the hind wing. We also found the reverse clap-fling mechanism of the T. dichotomus beetle in hovering flight. The hind wings touch together at the beginning of the upstroke. The vortex generation, shedding and interaction give a better understanding of the detailed aerodynamic mechanism of beetle flight.

Analysis of 1/2 sub-harmonic resonance in a maneuvering rotor system

This paper focuses on the 1/2 sub-harmonic resonance of an aircraft’s rotor system under hovering flight that can be modeled as a maneuver load G in the equations of motion.The effect on the rotor system is analyzed by using theoretical methods.It is shown that the sub-harmonic resonance may occur due to maneuvering flight conditions.The larger the eccentricity E and the maneuver load G,the greater the sub-harmonic resonance.The effects of nonlinear stiffness,damping of the system,maneuver load,and eccentricity on the sub-harmonic resonance region in parameter planes are also investigated.Bifurcation diagrams of the analytical solutions are in good agreement with that of the numerical simulation solutions.These results will contribute to the understanding of the nonlinear dynamic behaviors of maneuvering rotor systems.