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Abstract The inverted flag configuration is inspired by biological structures (e.g. leaves on a tree branch), showing rich dynamics associated with instabilities at lower flow speeds than the regular flag configuration. In the biological counterpart, the arrangement of leaves and twigs on foliage creates a complex interacting environment that promotes certain dynamic fluttering modes. While enabling a large amplitude response for reduced flow speeds is advantageous in emerging fields such as energy harvesting, still, little is known about the consequence of such interactions. In this work, we numerically study the canonical bio-inspired problem of the flow-structural interaction of a 2D inverted flag behind a cylindrical bluff body, mimicking a leaf behind a tree branch, to investigate its distinct fluttering regimes. The separation distance between the cylinder and flag is gradually modified to determine the effective distance beyond which small-amplitude or large-amplitude flapping occurs for different flow velocities. It is shown that the flag exhibits a periodic large amplitude−low frequency response mode when the cylinder is placed at a sufficiently large distance in front of the flag. At smaller distances, when the flag is within the immediate wake of the cylinder, the flag undergoes a high frequency−small amplitude response. Finally, the flag’s piezoelectric power harvesting capability is investigated numerically and experimentally for varying geometrical and electrical parameters associated with these two conditions. Two separate optimal response modes with the highest energy output have also been identified. 

Abstract Wind-induced stress is the primary mechanical cause of tree failures. Among different factors, the branching mechanism plays a central role in the stress distribution and stability of trees in windstorms. A recent study showed that Leonardo da Vinci’s original observation, stating that the total cross section of branches conserved across branching nodes is the optimal configuration for resisting wind-induced damage in rigid trees, is correct. However, the breaking risk and the optimal branching pattern of trees are also a function of their reconfiguration capabilities and the processes they employ to mitigate high wind-induced stress hotspots. In this study, using a numerical model of rigid and flexible branched trees, we explore the role of flexibility and branching patterns of trees in their reconfiguration and stress mitigation capabilities. We identify the robust optimal branching mechanism for an extensive range of tree flexibility. Our results show that the probability of a tree breaking at each branching level from the stem to terminal foliage strongly depends on the cross section changes in the branching nodes, the overall tree geometry, and the level of tree flexibility. Three response categories have been identified: the stress concentration in the main trunk, the uniform stress level through the tree’s height, and substantial stress localization in the terminal branches. The reconfigurability of the tree determines the dominant response mode. The results suggest a very similar optimal branching law for both flexible and rigid trees wherein uniform stress distribution occurs throughout the tree’s height. An exception is the very flexible branched plants in which the optimal branching pattern deviates from this prediction and is strongly affected by the reconfigurability of the tree. 

The bistable fluttering response of heavy inverted flags with different aspect ratios ( $AR$ ) is investigated to determine how the vortical structures affect the intermittent vibration response of the flag. A heavy inverted flag in a uniform flow may exhibit several response modes; amongst them are three major modes that occur over an extended velocity range: stationary, large-scale periodic oscillation and one-sided deflected modes. Significant hysteretic bistability is observed at the transition between these modes for all $AR$ , which is notably different from the conventional flag vibration with a fixed leading edge and free trailing edge where no hysteresis is observed at the lower $AR$ limit ( $AR<1$ ). The difference is associated with the distinct roles of vortices around the flag. Experiments with flags made of spring steel are conducted in a wind tunnel, where the flow speed is steadily increased and later decreased to obtain different oscillatory modes of the heavy inverted flags. The experimental results are used to validate the numerical model of the same problem. It is found that different critical velocities exist for increasing and decreasing flow velocities, and there is a sustained hysteresis for all $AR$ controlled by the initiation threshold and growth of the leading-edge and side-edge vortices. The effect of the vortices in the bistable oscillation regime is quantified by formulating a modal force partitioning approach. It is shown that $AR$ can significantly alter the static and dynamic vortex interaction with the flexible plate, thereby changing the flag's hysteresis behaviour and bistable response.