We propose an efficient Eulerian approach to compute the Lagrangian-averaged vorticity deviation (LAVD) of given flow fields. Traditional approaches need to solve [Formula: see text] ordinary differential equations (ODEs) for a [Formula: see text]-dimensional flow. Furthermore, if the velocity data are discrete, interpolation is required to obtain the velocity and vorticity data along the particle trace of any sampling point, which could be quite time-consuming and even affect the accuracy of the solutions. In contrast, our proposed Eulerian approach only needs to solve one single partial differential equation (PDE) in order to obtain the LAVD field and no interpolation is required. Based on the doubling technique, we also propose an efficient iterative Eulerian-type algorithm to compute the long-time LAVD for periodic flows. After that, relation between the long-time LAVD and the coherent ergodic partition is briefly discussed. Numerical examples will show the accuracy, efficiency and effectiveness of our proposed Eulerian approaches.

中文翻译：

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基于欧拉公式的拉格朗日平均涡度偏差快速计算

我们提出了一种有效的欧拉方法来计算给定流场的拉格朗日平均涡度偏差 (LAVD)。传统方法需要为 [公式：参见文本] 维流求解 [公式：参见文本] 常微分方程 (ODE)。此外，如果速度数据是离散的，则需要插值来获得沿任何采样点的粒子轨迹的速度和涡度数据，这可能非常耗时，甚至会影响求解的准确性。相比之下，我们提出的欧拉方法只需要求解一个偏微分方程 (PDE) 即可获得 LAVD 场，并且不需要插值。基于加倍技术，我们还提出了一种有效的迭代欧拉型算法来计算周期性流的长时间 LAVD。在那之后，简要讨论了长时间LAVD与相干遍历分区之间的关系。数值示例将显示我们提出的欧拉方法的准确性、效率和有效性。