Concluding Remarks

Flexible wings have been found to be beneficial for both natural and human-made flyers. Birds can flex their wings during upstroke to minimize the drag and can still maintain a smooth surface by sliding their feathers over each other. Bats, whose wings consist of membrane and arm bones, can only flex their wings a bit to avoid structure failure or flutter; however, they can enlarge the wing camber during the downstroke. Insects can bend the wing chordwise to generate camber while preventing bending in the spanwise direction.

Fixed, flexible wings can facilitate steadier, better controlled flight. In a gusty environment, a flexible wing can provide a more consistent lift-to-drag ratio than can a rigid wing by adaptively adjusting the camber in accordance with the instantaneous flow field. By responding to the aerodynamic loading variations, a membrane wing can also adaptively conduct passive camber control to delay stall. Moreover, a flexible wing can adjust the wing shape to its motion to mitigate the loss of lift due to the non-linear wing-wake interaction, which can be significant for a rigid wing in hover.

A membrane wing is found to exhibit flutter, whose frequency is about an order of magnitude higher than that of the vortex shedding frequency. The flutter exists even under a steady-state free-stream condition. Such intrinsic vibrations result from coupled aerodynamics and structural dynamics.

We also highlighted the impacts of the anisotropic nature of flexibility on resulting flight performance by considering the chordwise, spanwise, isotropic, and anisotropic wing flexibilities individually. Chordwise and spanwise flexibilities inter­act with surrounding air, and depending on the imposed kinematics and wing flexi­bility, they can enhance propulsive force. Studies on anisotropic wing structures are more recent and can provide insightful observations that can be applied to MAV development. The impact of structural flexibility on aerodynamics undergoing plung­ing motion in forward flight can be highlighted as follows:

1. The thrust generation consists of contributions due to both leading-edge suction

and the pressure projection of the chordwise deformed rear foil. When the rear

foil’s flexibility increases, the thrust of the teardrop element decreases, as does the effective angle of attack.

2. Within a certain range, as chordwise flexibility increases, even though the effec­tive angle of attack and the net aerodynamic force are reduced due to chordwise shape deformation, both mean and instantaneous thrust are enhanced due to the increase in the projected area normal to the flight trajectory.

3. For the spanwise flexible case, correlations of the motion from the root to the tip play a role. Within a suitably selected range of spanwise flexibility, the effective angle of attack and thrust forces of a plunging wing are enhanced due to the wing deformations.

Furthermore, we have demonstrated that as the Reynolds number drops to O(102)- O(103), which corresponds to flyers such as fruit flies and honey bees, the viscous effects are significant, and the flexibility of the wing structure can noticeably change the instantaneous, effective angle of attack via the large-scale vertical flows. The structural flexibility can also mitigate the induced downward jet via shape deforma­tion. Both effects can result in enhanced aerodynamic performance. Furthermore, lift generated on the flexible wing scales with the relative tip deformation parameter, whereas the optimal lift is obtained when the wing deformation synchronizes with the imposed translation, which we also observe in fruit flies and honey bees. Hence, under such modest Reynolds number, the observation that synchronized rotation is aerodynamically preferable for flexible wings is different from what we observe for the rigid wing. It is recalled that in Chapter 3, we have concluded that appropriate combinations of advanced rotation and dynamic stall associated with large AoAs can produce more favorable lift. These findings clearly highlight the effect of wing flexibility.

Since the various scaling parameters vary with the length and time scales in dif­ferent proportionality, the scaling invariance of both fluid dynamics and structural dynamics as the size changes is fundamentally difficult and challenging. It also seems that there is a desirable level of structural flexibility to support desirable aerodynam­ics. Significant work needs to be done to better understand the interaction between structural flexibility and aerodynamic performance under unpredictable wind gust conditions. Dimensional analysis and non-dimensionalization of the governing equa­tions for the fluid and the wing structure lead to a system of non-dimensional param­eters, such as Reynolds number (Re), reduced frequency (k), Strouhal number (St), aspect ratio (AR), effective stiffness (П1), effective AoA (ae), thickness ratio (h*), the density ratio (p*), and finally the force coefficients. Compared to the set of parameters in rigid flapping wing aerodynamics, three additional non-dimensional parameters were introduced: П1; p*, and h*. From the scaling arguments, the time – averaged force cofficient could be related to non-dimensional relative tip defor­mation. The tip deformation is an outcome of the interplay between the imposed kinematics and the response of the wing structure dictated by the wingtip amplitude and the phase lag. The amplitude of the maximum relative wingtip deformation, у, was obtained from the non-dimensional beam analysis and is only a function of the a priori known non-dimensional paramters. By considering the energy balance of the wing, the time-averaged force normalized by the effective stiffness was related to y . The time-averaged force can be related to the resultant force on the wing depending on the situation, such as fluid/inertial force, with/without free-stream, or thrust/lift/weight. These results enable us to estimate the order of magnitude of the time-averaged force generation for a flexible flapping wing using a priori known parameters. Furthermore, for propulsive efficiency it was seen that the opti­mal efficiency was obtained for the motion frequency that is lower than the natural frequency. The current scaling shows that smaller flyers need to flap faster from the efficiency point of view, but the relative payload capacity increases because their weight reduces at a much faster rate compared to larger flyers.

The study of hawkmoth hovering suggests that flexibilty plays a role in both the resulting wing kinematics and aerodynamic force generation. However, because of the limited number of studies regarding fluid-structure interaction in biological flyer-like models, further investigations are needed to derive general conclusions regarding the role of wing flexibility in their flights.