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${\approx}\,0.3$.