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

Abstract

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.

The leading-edge vortex of yacht sails

“…A stable LEV grows in the direction in which the vorticity is extracted. The vorticity and circulation of the LEV can significantly increase the lift and thus it is exploited on both man-made and natural flyers [3,4,5,6]. Remarkably, it has been identified across a wide range of Re.…”

Three-dimensional flow structures and unsteady forces on pitching and surging revolving flat plates

“…Moreover, Lentink and Dickinson (2009) point to the rotational inertial mechanisms (i.e., centripetal and Coriolis accelerations) in combination with the spanwise flow (Ellington et al 1996), which are present in the case of revolving wings, as the responsible mechanism for stabilizing the LEV and thus augmenting force generation. Recently, Jardin and David (2014) showed that spanwise gradient of the local wing speed by itself leads to stabilization of the LEV but not to enhanced lift; the latter is observed when including the rotational inertial effects.…”

Coriolis effects enhance lift on revolving wings

“…T his burst, referred to as global burst (as opposed to tip burst), is not visible in case 0 [7] w hich indicates the occurrence o f an instability associated with rotational shear. Because o f this instability, it is here delicate to conclude w hether or not rotational shear prom otes LEV attachm ent, as observed for rectilinear shear [7], However, it is clear that rotational shear does not prom ote lift generation, w hich is very sim ilar to that obtained in the case o f rectilinear shear. T herefore, when com pared to the rectilinear shear case, it appears that the global burst induced by the rotational shear has only a w eak influence on lift generation.…”

“…T he w ing aspect ratio is set to . We add the lift coefficient obtained for a w ing em bedded in a rectilinear shear flow (w ithout source term s) and reproduced from Jardin and D avid [7]. In this particular case, referred to as case 0 (-), the spanw ise gradient in flow speed is equal in m agnitude to that im posed in cases A, B, C, and D but the flow is rectilinear rather than being rotational.…”

“…The vorticity produced at the leading edge is balanced and does not accumulate inside the LEV which would otherwise rapidly shed into the wake. Recently, Jardin and David [7] test this hypothesis by considering a wing embedded in a spanwise varying oncoming flow (rectilinear shear flow). The authors show that spanwise gradients in flow speed does tend to limit vortex growth via spanwise flow drainage but that this mechanism does not suffice in generating high sustained lift.…”