三维非定常流中捕获所有LCS的自主动力系统
An Autonomous Dynamical System Captures all LCSs in Three-Dimensional Unsteady Flows
摘要：
拉格朗日拟序结构（LCS）是在具有任意时间依赖性的流动中形成有限时间示踪剂图案的物质表面。根据其变形特性，椭圆和双曲型LCS已被确定从不同的变分原理，解决不同的方程。在这里，我们观察到，在三维空间中，所有变分LCS的初始位置是不变流形相同的自治动力系统，产生的中间特征向量场，$ξ_{2}(x_{0})$，柯西-格林应变张量。该$ξ_{2}$系统允许通过经典方法（例如为自主动力系统开发的庞加莱映射）检测任何不稳定流中的LCS。作为例子，我们考虑稳定和时间非周期流，并使用其双$ξ_{2}$ -系统从一个单一的计算揭示双曲和椭圆LCS。
英文摘要：
Lagrangian coherent structures (LCSs) are material surfaces that shape finite-time tracer patterns in flows with arbitrary time dependence. Depending on their deformation properties, elliptic and hyperbolic LCSs have been identified from different variational principles, solving different equations. Here we observe that, in three dimensions, initial positions of all variational LCSs are invariant manifolds of the same autonomous dynamical system, generated by the intermediate eigenvector field, $ξ_{2}(x_{0})$, of the Cauchy-Green strain tensor. This $ξ_{2}$-system allows for the detection of LCSs in any unsteady flow by classic methods, such as Poincaré maps, developed for autonomous dynamical systems. As examples, we consider both steady and time-aperiodic flows, and use their dual $ξ_{2}$-system to uncover both hyperbolic and elliptic LCSs from a single computation.
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