Analysis of passive flexion in propelling a plunging plate using a torsion spring model

Abstract

We mimic a flapping wing through a fluid–structure interaction (FSI) framework based upon a generalized lumped-torsional flexibility model. The developed fluid and structural solvers together determine the aerodynamic forces, wing deformation and self-propelled motion. A phenomenological solution to the linear single-spring structural dynamics equation is established to help offer insight and validate the computations under the limit of small deformation. The cruising velocity and power requirements are evaluated by varying the flapping Reynolds number ($20\leqslant Re_{f}\leqslant 100$), stiffness (represented by frequency ratio,$1\lesssim \unicode[STIX]{x1D714}^{\ast }\leqslant 10$) and the ratio of aerodynamic to structural inertia forces (represented by a dimensionless parameter$\unicode[STIX]{x1D713}$($0.1\leqslant \unicode[STIX]{x1D713}\leqslant 3$)). For structural inertia dominated flows ($\unicode[STIX]{x1D713}\ll 1$), pitching and plunging are shown to always remain in phase ($\unicode[STIX]{x1D719}\approx 0$) with the maximum wing deformation occurring at the end of the stroke. When aerodynamics dominates ($\unicode[STIX]{x1D713}>1$), a large phase difference is induced ($\unicode[STIX]{x1D719}\approx \unicode[STIX]{x03C0}/2$) and the maximum deformation occurs at mid-stroke. Lattice Boltzmann simulations show that there is an optimal$\unicode[STIX]{x1D714}^{\ast }$at which cruising velocity is maximized and the location of optimum shifts away from unit frequency ratio ($\unicode[STIX]{x1D714}^{\ast }=1$) as$\unicode[STIX]{x1D713}$increases. Furthermore, aerodynamics administered deformations exhibit better performance than those governed by structural inertia, quantified in terms of distance travelled per unit work input. Closer examination reveals that although maximum thrust transpires at unit frequency ratio, it is not transformed into the highest cruising velocity. Rather, the maximum velocity occurs at the condition when the relative tip displacement${\approx}\,0.3$.

Three-dimensional effects on the aerodynamic performance of flapping wings in tandem configuration

“…7b-c shows the time evolution of the horizontal and vertical forces exerted on the plate normalized by 1 2 ρV 2 C, where V = 2πf A is the maximum vertical velocity of the leading edge. It should be pointed out that results from Arora et al (2018) correspond to cycles 48 − 49, whereas the present results are for cycles 12 − 13, since no changes were observed with respect to previous cycles. Nevertheless, a good agreement is observed between the present simulations and those from Arora et al (2018), with relative differences in the peak-to-peak amplitudes of the forces of less than 3%, and an absolute difference of the maximum tip-deflection angle lower than 0.19 • .…”

“…In the next sections, two examples of the capabilities of the proposed methodology are presented. In §4.1 the study from Arora et al (2018) presented in §3.2 for validation is extended to a 3D configuration. In §4.2 the proposed algorithm is employed to model a deformable filament attached to a sphere, as an idealized model of the spider ballooning problem.…”

“…Most of the works which study the FSI of multi-body system derived the equations of motion specifically for the problem under consideration (Zhang et al 2013;Arora et al 2018;Suzuki et al 2019;Yao & Yeo 2019). On the contrary, few works are found in the literature which are able to compute the interaction problem of a generic multibody system with a surrounding flow.…”

“…The second validation case is taken from Arora et al (2018) and consists of a 2D self-propelling flexible plate. The plate is modelled using a lumped-torsional flexibility model as shown in Fig.…”

“…The bodies are discretized using a Lagrangian grid with equidistant points separated ∆x = 0.004C and the time step is ∆t = 5 • 10 −5 T (where T = f −1 ). On the other hand, simulations in Arora et al (2018) are performed using a Lattice-Boltzmann method with a computational domain 50C × 20C and spatial discretization ∆x = 0.005C. 7a depicts the time evolution during two cycles of the tip deflection angle (defined as the angle between the horizontal and the line that joins the leading edge and the trailing edge).…”