It has long been proposed that animals flying in the air and swimming in the water could extract energy from neighbour-induced flows. A large number of mechanisms have been proposed to explain whether, and if so how, animals can save energy by moving in two-dimensional (2D) formations—individuals swim in the horizontal plane. Seldom studies explore the mechanisms in three-dimensional (3D) formations—individuals swim in both horizontal and vertical planes, even though most animals perform 3D behaviour. In this letter, taking a pair of bio-inspired robotic fish as experimental physical models, we explore the energy cost of the follower when swimming close to a neighbour in 3D formations (mainly in the vertical plane). We found the cost of the follower is mainly affected by how it spatiotemporally interacts with the 3D vortices shed by the neighbour in 3D formations. A simple linear correlation was found between the spatial factor (the height difference) and temporal factor (the body phase difference) when the follower saves most energy compared to swimming alone. Preliminary flow visualisations and 3D computational fluid dynamic simulations show this is due to the structure of vortices along the span of the caudal fin's trailing edge. Our studies shed new light on the energy saving control of multiple artificial underwater robots in 3D formations.

中文翻译：

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三维形式的学校养鱼机器鱼的节能

长期以来，人们一直认为动物在空中飞行和在水中游泳可以从邻居感应的水流中提取能量。已经提出了大量的机制来解释动物是否以及如何（通过这种方式）可以通过以二维（2D）形式移动来节省能量，即个体在水平面内游泳。很少有研究探索三维（3D）形成的机制-即使大多数动物都执行3D行为，个人也会在水平面和垂直平面上游泳。在这封信中，以一对受生物启发的机器鱼为实验物理模型，我们研究了跟随者在3D编队中（主要在垂直平面中）靠近邻居游泳时的能量消耗。我们发现跟随者的成本主要受其时空与邻居在3D编队中喷出的3D涡旋相互作用的方式所影响。与单独游泳相比，追随者节省最多能量时，在空间因素（高度差）和时间因素（身体相位差）之间发现了简单的线性相关性。初步的流动可视化和3D计算的流体动力学模拟表明，这是由于沿尾鳍后缘跨度的旋涡结构所致。我们的研究为3D编队中的多个人工水下机器人的节能控制提供了新的思路。当追随者比单独游泳时节省最多能量时，在空间因素（高度差）和时间因素（身体相位差）之间发现简单的线性相关性。初步的流动可视化和3D计算的流体动力学模拟表明，这是由于沿尾鳍后缘跨度的旋涡结构所致。我们的研究为3D编队中的多个人工水下机器人的节能控制提供了新的思路。与单独游泳相比，追随者节省最多能量时，在空间因素（高度差）和时间因素（身体相位差）之间发现了简单的线性相关性。初步的流动可视化和3D计算的流体动力学模拟表明，这是由于沿尾鳍后缘跨度的旋涡结构所致。我们的研究为3D编队中的多个人工水下机器人的节能控制提供了新的思路。