Harms Tanner D, Brunton Steven L, McKeon Beverley J
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91106, USA.
Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA.
R Soc Open Sci. 2024 Oct 30;11(10):240586. doi: 10.1098/rsos.240586. eCollection 2024 Oct.
Complex flows are often characterized using the theory of Lagrangian coherent structures (LCS), which leverages the motion of flow-embedded tracers to highlight features of interest. LCS are commonly employed to study fluid mechanical systems where flow tracers are readily observed, but they are broadly applicable to dynamical systems in general. A prevailing class of LCS analyses depends on reliable computation of flow gradients. The finite-time Lyapunov exponent (FTLE), for example, is derived from the Jacobian of the flow map, and the Lagrangian-averaged vorticity deviation (LAVD) relies on velocity gradients. Observational tracer data, however, are typically sparse (e.g. drifters in the ocean), making accurate computation of gradients difficult. While a variety of methods have been developed to address tracer sparsity, they do not provide the same information about the flow as gradient-based approaches. This work proposes a purely Lagrangian method, based on the data-driven machinery of regression, for computing instantaneous and finite-time flow gradients from sparse trajectories. The tool is demonstrated on a common analytical benchmark to provide intuition and demonstrate performance. The method is seen to effectively estimate gradients using data with sparsity representative of observable systems.
复杂流动通常使用拉格朗日相干结构（LCS）理论来表征，该理论利用嵌入流场的示踪剂的运动来突出感兴趣的特征。LCS通常用于研究易于观察流场示踪剂的流体力学系统，但它们通常广泛适用于一般的动力系统。一类流行的LCS分析依赖于流场梯度的可靠计算。例如，有限时间李雅普诺夫指数（FTLE）是从流场映射的雅可比矩阵导出的，而拉格朗日平均涡度偏差（LAVD）依赖于速度梯度。然而，观测示踪剂数据通常很稀疏（例如海洋中的漂流器），使得梯度的精确计算变得困难。虽然已经开发了多种方法来解决示踪剂稀疏问题，但它们提供的关于流场的信息与基于梯度的方法不同。这项工作提出了一种基于回归的数据驱动机制的纯拉格朗日方法，用于从稀疏轨迹计算瞬时和有限时间的流场梯度。该工具在一个常见的分析基准上进行了演示，以提供直观认识并展示性能。结果表明，该方法能够有效地利用具有可观测系统代表性稀疏性的数据来估计梯度。
