[论文解读] Scaling the tail beat frequency and swimming speed in underwater undulatory swimming

Jesus Sanchez Roriguez, Christophe Raufaste+1 more|arXiv (Cornell University)|Jan 25, 2023

Fish Ecology and Management StudiesEnvironmental Science被引用 3

一句话总结

本研究通过整合肌肉生理学与流体动力学，提出了基于尾鳍拍打频率与游泳速度之间关系的尺度律。研究揭示在体长0.5–1米处存在一个转变点：在此之下，频率由生物因素调节（2–20 Hz）；在此之上，流体阻力占主导地位，导致频率与体长成反比，预测最大游泳速度为5–10 m·s⁻¹，与空化极限一致。

ABSTRACT

Due to its great efficiency and maneuverability, undulatory swimming is the predominant form of locomotion in aquatic vertebrates. A myriad of animals of different species and sizes oscillate their bodies to propel themselves in aquatic environments with swimming speed scaling as the product of the animal length by the oscillation frequency. Although frequency tuning is the primary means by which a swimmer selects its speed, there is no consensus on the mechanisms involved. In this article, we propose scaling laws for undulatory swimmers that relate oscillation frequency to length by taking into account both the biological characteristics of the muscles and the interaction of the moving swimmer with its environment. Results are supported by an extensive literature review including approximately 1200 individuals of different species, sizes and swimming environments. We highlight a crossover in length around 0.5-1 m. Below this value, the frequency can be tuned between 2-20 Hz due to biological constraints and the interplay between slow and fast muscles. Above this value, the fluid-swimmer interaction must be taken into account and the frequency is inversely proportional to the length of the animal. This approach predicts a maximum swimming speed around 5-10 m.s$^{-1}$ for large swimmers, consistent with the threshold to prevent bubble cavitation.

研究动机与目标

• 解决关于水生脊椎动物尾鳍拍打频率如何随体长变化的现有共识分歧问题。
• 识别在不同体型物种中主导频率选择的物理与生物机制。
• 通过分析来自多种物种和游泳状态的约1,200个数据点，将零散的实验数据统一为一个连贯的尺度框架。
• 区分不同体型尺度下生物约束（肌肉类型、活动水平）与流体相互作用作为主导因素的差异。
• 预测大型游泳动物的最大游泳速度，并将其与空化等物理约束联系起来。

提出的方法

• 从多种水生脊椎动物中收集并整理了约1,200个关于尾鳍拍打频率与体长的实验数据点，未按活动水平或物种进行筛选。
• 采用对数分箱法对体长进行分组，以计算爆发游泳与持续游泳频率的边界，确保统计表示的稳健性。
• 应用最小绝对偏差（LAD）拟合来建模频率-体长关系，参数在多个区间数（N = 10 至 50）范围内进行优化。
• 根据Hirt等人[49]提出的标准，客观识别并排除不可靠的数据点——包括非同行评审来源、估算值以及安装在杆上的设备测量值。
• 提出一个理论框架，平衡肌肉作用力与流体反作用力，以解释不同体型尺度下的频率选择机制。
• 推导出频率与游泳速度的异速生长尺度律，区分小型游泳动物（生物调节）与大型游泳动物（流体动力学限制）两种情形。

Figure 1: Tail beat frequency $f$ as a function of length $L$ for amphibians (yellow), fish (blue), reptiles (green), birds (red) and mammals (purple). Thick black and grey lines represent the burst and sustained activity levels, respectively, fitted with the model. Thin lines are the scaling laws i

实验结果

研究问题

• RQ1 在波动游泳动物中，哪些物理与生物机制决定了尾鳍拍打频率随体长变化的规律？
• RQ2 在不同体型类群的水生脊椎动物中，频率的主导控制机制如何随体型变化？
• RQ3 在多个数量级的体型范围内，游泳速度、频率与体长之间存在何种关系？
• RQ4 为何现有频率尺度律缺乏共识？如何建立统一模型？
• RQ5 大型水生动物的最大游泳速度受何限制？这是否与空化等物理约束一致？

主要发现

• 在体长0.5–1米处出现尺度行为的转变，将两种截然不同的频率控制模式分隔开。
• 对于体长小于0.5–1米的动物，频率可在2–20 Hz之间调节，受慢肌与快肌纤维等生物约束影响。
• 对于体长大于0.5–1米的动物，流体-游泳者相互作用占主导，频率与体长成反比（f ∝ L⁻¹）。
• 游泳速度满足U ≈ 0.7Lf，且在鱼类与鲸类中比例系数保持在0.4至1之间的一致范围。
• 模型预测大型游泳动物的最大游泳速度为5–10 m·s⁻¹，与避免气泡空化的阈值一致。
• 排除非同行评审、估算值及杆安装设备的数据后，频率-体长关系的可靠性得到提升，人工异常值显著减少。

Figure 2: Swimming speed $U$ as a function of length $L$ . Following the law $U=0.7Lf$ , we show our estimates of swimming speed from the tail beat frequency measurements displayed in Fig. 1 (closed circles): amphibians (yellow), fish (blue), reptiles (green), birds (red), and mammals (purple). Brow