Interaction of gap flow with flapping dynamics of two side-by-side elastic foils

Abstract

Two side-by-side elastic foils placed in an axial flow with the leading edges clamped lose their stability to exhibit in-phase or out-of-phase modes due to the proximity induced effects. Of particular, the passive out-of-phase flapping mode typically represents the clapping mechanism exhibited by biological organisms such as jellyfish and squid for swimming via jet propulsion. An impact of the viscous gap-flow dynamics on such passive flapping modes and vice versa is not well understood for the side-by-side elastic foil system. In the present work, we explore the mutual interaction of two side-by-side elastic foils performing flapping motion with the viscous gap-flow via a high-order finite element based fluid-elastic formulation with an exact tracking of fluid-foil interface. We show that the gap-flow exhibits pulsating flow with higher net drag for the passive out-of-phase coupled mode compared to the in-phase flapping where it exhibits uniform flow rate. Three distinct gap-flow velocity patterns are identified as functions of the coupled flapping modes: (i) unsteady symmetrical gap-flow with variable gap for the out-of-phase, (ii) unsteady alternating biased asymmetrical gap-flow with a uniform gap for the in-phase, and (iii) unsteady alternating biased asymmetrical gap-flow with variable gap for the mixed in-phase and out-of-phase. We examine the role of the gap-flow on the coupled fluid-elastic instability and the passive flapping modes. Two side-by-side elastic foils can experience significantly lower drag compared to their single foil counterpart and the two side-by-side rigid foils by undergoing static outward deformation. We utilize this phenomenon to understand the greater propensity of the flapping instability of the two side-by-side elastic foils in contrast to their single foil counterpart. We show that the coupled system does not exhibit the out-of-phase flapping if there is no gap-flow between the foils. We also find that two elastic foils when placed in proximity to each other always lose their stability to exhibit the out-of-phase coupling irrespective of whether the fully developed flapping exhibits in-phase or the out-of-phase flapping. The transition from the initial out-of-phase to the in-phase flapping is characterized by the loss of symmetry in the jet-like gap flow at the exit area of the side-by-side foils.