Experiments and numerical simulations on hovering three-dimensional flexible flapping wings.

In this paper, the applicability and accuracy of high-fidelity experimental and numerical approaches in the analysis of three-dimensional flapping (revolving and pitching) wings operating under hovering flight conditions, i.e. where unsteady and three-dimensional rotational effects are strong, are assessed. Numerical simulations are then used to explore the role of mass and frequency ratios on aerodynamic performance, wing dynamics and flow physics. It is shown that time-averaged lift increases with frequency ratio, up to a certain limit that depends on mass ratio and beyond which upward wing bending and flexibility induced phase lag between revolving an pitching motions at stroke reversal become strong and contribute to phases of negative lift that counterbalances the initial lift increase. This wing dynamics, which is dominated by spanwise bending, also affects wing-wake interactions and, in turn, leading edge vortex formation.

On the lift-optimal aspect ratio of a revolving wing at low Reynolds number.

Lentink & Dickinson (2009 J. Exp. Biol.212, 2705-2719. (doi:10.1242/jeb.022269)) showed that rotational acceleration stabilized the leading-edge vortex on revolving, low aspect ratio (AR) wings and hypothesized that a Rossby number of around 3, which is achieved during each half-stroke for a variety of hovering insects, seeds and birds, represents a convergent high-lift solution across a range of scales in nature. Subsequent work has verified that, in particular, the Coriolis acceleration plays a key role in LEV stabilization. Implicit in these results is that there exists an optimal AR for wings revolving about their root, because it is otherwise unclear why, apart from possible morphological reasons, the convergent solution would not occur for an even lower Rossby number. We perform direct numerical simulations of the flow past revolving wings where we vary the AR and Rossby numbers independently by displacing the wing root from the axis of rotation. We show that the optimal lift coefficient represents a compromise between competing trends with competing time scales where the coefficient of lift increases monotonically with AR, holding Rossby number constant, but decreases monotonically with Rossby number, when holding AR constant. For wings revolving about their root, this favours wings of AR between 3 and 4.

Coriolis effects enhance lift on revolving wings.

At high angles of attack, an aircraft wing stalls. This dreaded event is characterized by the development of a leading edge vortex on the upper surface of the wing, followed by its shedding which causes a drastic drop in the aerodynamic lift. At similar angles of attack, the leading edge vortex on an insect wing or an autorotating seed membrane remains robustly attached, ensuring high sustained lift. What are the mechanisms responsible for both leading edge vortex attachment and high lift generation on revolving wings? We review the three main hypotheses that attempt to explain this specificity and, using direct numerical simulations of the Navier-Stokes equations, we show that the latter originates in Coriolis effects.

Spanwise gradients in flow speed help stabilize leading-edge vortices on revolving wings.

While a leading-edge vortex on an infinite translating wing is shed after a short distance of travel, its counterpart on a finite span revolving insect wing or maple seed membrane exhibits robust attachment. The latter explains the aerodynamic lift generated by such biological species. Here we analyze the mechanisms responsible for leading-edge vortex attachment. We compute the Navier-Stokes solution of the flow past a finite span wing (i) embedded in a uniform oncoming flow, (ii) embedded in a spanwise varying oncoming flow, and (iii) revolving about its root. We show that over flapping amplitudes typical of insect flight (ϕ = 120°), the spanwise gradient of the local wing speed may suffice in maintaining leading-edge vortex attachment. We correlate this result with the development of spanwise flow, driven by the spanwise gradient of pressure, and we evaluate the sensitivity of such a mechanism to the Reynolds number. It is noted, however, that leading-edge vortex attachment through the spanwise gradient of the local wing speed does not promote large lift, which ultimately arises from centrifugal and Coriolis effects.