The effect of the wake of previous strokes on the aerodynamic forces of a flapping model insect wing is studied using the method of computational fluid dynamics. The wake effect is isolated by comparing the forces and flows of the starting stroke （when the wake has not developed） with those of a later stroke （when the wake has developed）. The following has been shown. （1） The wake effect may increase or decrease the lift and drag at the beginning of a half-stroke （downstroke or upstroke）, depending on the wing kinematics at stroke reversal. The reason for this is that at the beginning of the half-stroke, the wing “impinges” on the spanwise vorticity generated by the wing during stroke reversal and the distribution of the vorticity is sensitive to the wing kinematics at stroke reversal. （2） The wake effect decreases the lift and increases the drag in the rest part of the half-stroke. This is because the wing moves in a downwash field induced by previous half-stroke＇s starting vortex, tip vortices and attached leading edge vortex （these vortices form a downwash producing vortex ring）. （3） The wake effect decreases the mean lift by 6%-18% （depending on wing kinematics at stroke reversal） and slightly increases the mean drag. Therefore, it is detrimental to the aerodynamic performance of the flapping wing.
The aerodynamic forces and flow structure of a model insect wing is studied by solving the Navier-Stokes equations numerically.After an initial start from rest,the wing is made to execute an azimuthal rotation(sweeping)at a large angle of attack and constant angular velocity.The Reynolds number(Re)considered in the present note is 480(Re is based on the mean chord length of the wing and the speed at 60% wing length from the wing root).During the constant-speed sweeping motion,the stall is absent and large and approximately constant lift and drag coefficients can be maintained.The mechanism for the absence of the stall or the maintenance of large aerodynamic force coefficients is as follows.Soon after the initial start,a vortex ring,which consists of the leading-edge vortex(LEV),the starting vortex,and the two wing-tip vortices,is formed in the wake of the wing.During the subsequent motion of the wing,a base-to-tip spanwise flow converts the vorticity in the LEV to the wing tip and the LEV keeps an approximately constant strength.This prevents the LEV from shedding.As a result, the size of the vortex ring increases approximately linearly with time,resulting in an approximately constant time rate of the first moment of vorticity,or approximately constant lift and drag coefficients. The variation of the relative velocity along the wing span causes a pressure gradient along the wing- span.The base-to-tip spanwise flow is mainly maintained by the pressure-gradient force.
Flexible insect wings deform passively under the periodic loading during flapping flight. The wing flexibility is considered as one of the specific mechanisms on improving insect flight performance. The constitutive relation of the insect wing material plays a key role on the wing deformation, but has not been clearly understood yet. A viscoelastic constitutive relation model was established based on the stress relaxation ex- periment of a dragonfly wing （in vitro）. This model was examined by the finite element analysis of the dynamic deformation response for a model insect wing under the action of the periodical inertial force in flapping. It is revealed that the viscoelastic constitutive relation is rational to characterize the biomaterial property of insect wings in contrast to the elastic one. The amplitude and form of the passive viscoelastic deformation of the wing is evidently dependent on the viscous parameters in the constitutive relation.
The equations of motion of an insect with flapping wings are derived and then simplified to that of a flying body using the ＂rigid body＂ assumption. On the basis of the simplified equations of motion, the longitudinal dynamic flight stability of four insects （hoverfly, cranefly, dronefly and hawkmoth） in hovering flight is studied （the mass of the insects ranging from 11 to 1,648 mg and wingbeat frequency from 26 to 157Hz）. The method of computational fluid dynamics is used to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis are used to solve the equations of motion. The validity of the ＂rigid body＂ assumption is tested and how differences in size and wing kinematics influence the applicability of the ＂rigid body＂ assumption is investigated. The primary findings are： （1） For insects considered in the present study and those with relatively high wingbeat frequency （hoverfly, drone fly and bumblebee）, the ＂rigid body＂ assumption is reasonable, and for those with relatively low wingbeat frequency （cranefly and howkmoth）, the applicability of the ＂rigid body＂ assumption is questionable. （2） The same three natural modes of motion as those reported recently for a bumblebee are identified, i.e., one unstable oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode. （3） Approximate analytical expressions of the eigenvalues, which give physical insight into the genesis of the natural modes of motion, are derived. The expressions identify the speed derivative Mu （pitching moment produced by unit horizontal speed） as the primary source of the unstable oscillatory mode and the stable fast subsidence mode and Zw （vertical force produced by unit vertical speed） as the primary source of the stable slow subsidence mode.
