Stability of Underwater Periodic Locomotion

Most aquatic vertebrates swim by lateral flapping of their bodies and caudal fins. While much effort has been devoted to understanding the flapping kinematics and its influence on the swimming efficiency, little is known about the stability (or lack of) of periodic swimming. It is believed that stability limits maneuverability and body designs/flapping motions that are adapted for stable swimming are not suitable for high maneuverability and vice versa. In this paper, we consider a simplified model of a planar elliptic body undergoing prescribed periodic heaving and pitching in potential flow. We show that periodic locomotion can be achieved due to the resulting hydrodynamic forces, and its value depends on several parameters including the aspect ratio of the body, the amplitudes and phases of the prescribed flapping. We obtain closed-form solutions for the locomotion and efficiency for small flapping amplitudes, and numerical results for finite flapping amplitudes. We then study the stability of the (finite amplitude flapping) periodic locomotion using Floquet theory. We find that stability depends nonlinearly on all parameters. Interesting trends of switching between stable and unstable motions emerge and evolve as we continuously vary the parameter values. This suggests that, for live organisms that control their flapping motion, maneuverability and stability need not be thought of as disjoint properties, rather the organism may manipulate its motion in favor of one or the other depending on the task at hand.Comment: 15 pages, 15 figure

Global modes and nonlinear analysis of inverted-flag flapping

An inverted flag has its trailing edge clamped and exhibits dynamics distinct from that of a conventional flag, whose leading edge is restrained. We perform nonlinear simulations and a global stability analysis of the inverted-flag system for a range of Reynolds numbers, flag masses and stiffnesses. Our global stability analysis is based on a linearisation of the fully-coupled fluid-structure system of equations. The calculated equilibria are steady-state solutions of the fully-coupled nonlinear equations. By implementing this approach, we (i) explore the mechanisms that initiate flapping, (ii) study the role of vortex shedding and vortex-induced vibration (VIV) in large-amplitude flapping, and (iii) characterise the chaotic flapping regime. For point (i), we identify a deformed-equilibrium state and show through a global stability analysis that the onset of flapping is due to a supercritical Hopf bifurcation. For large-amplitude flapping, point (ii), we confirm the arguments of Sader et al. (2016) that for a range of parameters this regime is a VIV. We also show that there are other flow regimes for which large-amplitude flapping persists and is not a VIV. Specifically, flapping can occur at low Reynolds numbers ($<50$), albeit via a previously unexplored mechanism. Finally, with respect to point (iii), chaotic flapping has been observed experimentally for Reynolds numbers of $O(10^4)$, and here we show that chaos also persists at a moderate Reynolds number of 200. We characterise this chaotic regime and calculate its strange attractor, whose structure is controlled by the above-mentioned deformed equilibria and is similar to a Lorenz attractor. These results are contextualised with bifurcation diagrams that depict the different equilibria and various flapping regimes

Efficiency of Lift Production in Flapping and Gliding Flight of Swifts

Many flying animals use both flapping and gliding flight as part of their routine behaviour. These two kinematic patterns impose conflicting requirements on wing design for aerodynamic efficiency and, in the absence of extreme morphing, wings cannot be optimised for both flight modes. In gliding flight, the wing experiences uniform incident flow and the optimal shape is a high aspect ratio wing with an elliptical planform. In flapping flight, on the other hand, the wing tip travels faster than the root, creating a spanwise velocity gradient. To compensate, the optimal wing shape should taper towards the tip (reducing the local chord) and/or twist from root to tip (reducing local angle of attack). We hypothesised that, if a bird is limited in its ability to morph its wings and adapt its wing shape to suit both flight modes, then a preference towards flapping flight optimization will be expected since this is the most energetically demanding flight mode. We tested this by studying a well-known flap-gliding species, the common swift, by measuring the wakes generated by two birds, one in gliding and one in flapping flight in a wind tunnel. We calculated span efficiency, the efficiency of lift production, and found that the flapping swift had consistently higher span efficiency than the gliding swift. This supports our hypothesis and suggests that even though swifts have been shown previously to increase their lift-to-drag ratio substantially when gliding, the wing morphology is tuned to be more aerodynamically efficient in generating lift during flapping. Since body drag can be assumed to be similar for both flapping and gliding, it follows that the higher total drag in flapping flight compared with gliding flight is primarily a consequence of an increase in wing profile drag due to the flapping motion, exceeding the reduction in induced drag

Modal decomposition of fluid-structure interaction with application to flag flapping

Modal decompositions such as proper orthogonal decomposition (POD), dynamic mode decomposition (DMD) and their variants are regularly used to educe physical mechanisms of nonlinear flow phenomena that cannot be easily understood through direct inspection. In fluid-structure interaction (FSI) systems, fluid motion is coupled to vibration and/or deformation of an immersed structure. Despite this coupling, data analysis is often performed using only fluid or structure variables, rather than incorporating both. This approach does not provide information about the manner in which fluid and structure modes are correlated. We present a framework for performing POD and DMD where the fluid and structure are treated together. As part of this framework, we introduce a physically meaningful norm for FSI systems. We first use this combined fluid-structure formulation to identify correlated flow features and structural motions in limit-cycle flag flapping. We then investigate the transition from limit-cycle flapping to chaotic flapping, which can be initiated by increasing the flag mass. Our modal decomposition reveals that at the onset of chaos, the dominant flapping motion increases in amplitude and leads to a bluff-body wake instability. This new bluff-body mode interacts triadically with the dominant flapping motion to produce flapping at the non-integer harmonic frequencies previously reported by Connell & Yue (2007). While our formulation is presented for POD and DMD, there are natural extensions to other data-analysis techniques

A Dynamic Model Investigation of the Effect of a Sharp-Edge Vertical Gust on Blade Periodic Flapping Angles and Bending Moments of a Two-Blade Rotor

A two-blade rotor having a diameter of 4 feet and a solidity of 0.037 was subjected to sharp-edge vertical gusts while being operated at various forward speeds to study the effect of the gusts on the blade periodic bending moments and flapping angles. Variables studied included gust velocity, collective pitch angle, flapping hinge offset, and tip-speed ratio. Dimensionless coefficients are derived for the periodic components of the incremental changes in blade flapping angles and bending moments which arise when a rotor blade penetrates a sharp-edge gust. Mental changes in both the flapping angles and bending moments are essentially proportional to gust velocity, and the coefficients express the ratio of these increments to gust velccity. The results show that the flapping coefficient usually increases with an increase in collective pitch angle, is generally dependent on tip-speed ratio, and is essentially independent of the amount of flapping hinge offset. The bending-moment coefficient is also dependent on collective pitch angle and tip-speed ratio. Expected reductions in bending moments are realized by the use of flapping hinges, and further reductions in bending moments are achieved as the amount of flapping hinge offset is increased. Comparison of the experimental results of this investigation with limited available theoretical results shows substantial agreement but indicates that the assumption that the response of the rotor to a sharp-edge gust is independent of the collective pitch angle prior to gust entry is probably inadequate

Periodic and Chaotic Flapping of Insectile Wings

Insects use flight muscles attached at the base of the wings to produce impressive wing flapping frequencies. The maximum power output of these flight muscles is insufficient to maintain such wing oscillations unless there is good elastic storage of energy in the insect flight system. Here, we explore the intrinsic self-oscillatory behavior of an insectile wing model, consisting of two rigid wings connected at their base by an elastic torsional spring. We study the wings behavior as a function of the total energy and spring stiffness. Three types of behavior are identified: end-over-end rotation, chaotic motion, and periodic flapping. Interestingly, the region of periodic flapping decreases as energy increases but is favored as stiffness increases. These findings are consistent with the fact that insect wings and flight muscles are stiff. They further imply that, by adjusting their muscle stiffness to the desired energy level, insects can maintain periodic flapping mechanically for a range of operating conditions

Can scalable design of wings for flapping wing micro air vehicle be inspired by natural flyers?

Lift production is constantly a great challenge for flapping wing micro air vehicles (MAVs). Designing a workable wing, therefore, plays an essential role. Dimensional analysis is an effective and valuable tool in studying the biomechanics of flyers. In this paper, geometric similarity study is firstly presented. Then, the pw−AR ratio is defined and employed in wing performance estimation before the lumped parameter is induced and utilized in wing design. Comprehensive scaling laws on relation of wing performances for natural flyers are next investigated and developed via statistical analysis before being utilized to examine the wing design. Through geometric similarity study and statistical analysis, the results show that the aspect ratio and lumped parameter are independent on mass, and the lumped parameter is inversely proportional to the aspect ratio. The lumped parameters and aspect ratio of flapping wing MAVs correspond to the range of wing performances of natural flyers. Also, the wing performances of existing flapping wing MAVs are examined and follow the scaling laws. Last, the manufactured wings of the flapping wing MAVs are summarized. Our results will, therefore, provide a simple but powerful guideline for biologists and engineers who study the morphology of natural flyers and design flapping wing MAVs