拉格朗日正交分解
Lagrangian Proper Orthogonal Decomposition
AI总结：
本文提出拉格朗日正交分解方法，通过粒子轨迹构建速度时间序列并分离波动，利用主成分分析提取特征空间，验证了该方法在湍流模拟中能有效重构轨迹特征，且需随轨迹长度增加而增加模态数量以提升重建精度。
摘要：
我们介绍了一种用于湍流拉格朗日轨迹的模态表示方法，称为拉格朗日正交分解（LPOD）。通过粒子轨迹集合构建速度时间序列，并对其进行归一化以分离波动。然后对所得数据集应用主成分分析，时间实例定义特征空间。该方法在直接数值模拟和三维粒子追踪实验中进行了测试，显示主导模态在两种情况下均表现出相似的结构和能量分布。通过组合模态和系数进行截断重建，重新缩放波动并进行时间积分。对于积分时间尺度量级的轨迹，使用有限数量的模态（约10个）可以准确再现单粒子扩散和曲率统计，而捕捉加速度分布尾部则需要更多模态（约30-60个）。更长的轨迹需要越来越多的模态以实现准确的重建。这些结果表明，通过随机采样模态拉格朗日动力学可能实现数据驱动的合成粒子轨迹生成。
英文摘要：
We introduce a modal representation for Lagrangian trajectories in turbulence, termed Lagrangian Proper Orthogonal Decomposition (LPOD). An ensemble of particle trajectories is used to construct velocity time series, which are normalized independently for each trajectory to isolate fluctuations. Principal Component Analysis is then applied to the resulting dataset, with temporal instances defining the feature space. The method is tested on trajectories from both direct numerical simulations of homogeneous isotropic turbulence and three-dimensional particle-tracking experiments, showing that the leading modes exhibit similar structures and energy distributions in both cases. Truncated reconstructions are obtained by combining modes and coefficients, rescaling the fluctuations, and integrating in time. For trajectories of the order of the integral time scale, single-particle dispersion and curvature statistics are accurately reproduced using a limited number of modes (c.a. 10), whereas capturing the tails of acceleration distributions requires a larger set (c.a. 30-60). Longer trajectories require progressively more modes for accurate reconstruction. These results suggest a possible route to data-driven generation of synthetic particle trajectories via stochastic sampling of the modal Lagrangian dynamics.
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