振荡流的可分辨极限和漂移的层次
Hierarchy of Distinguished Limits and Drifts for Oscillating Flows
摘要：
本文用双时间法研究了流体质点在随时间振荡的一般速度场中的拉格朗日运动。我们的目标是：㈠有系统地计算最一般和实际可用的渐近解; ㈡在不作任何假设的情况下计算平均运动，以检查双定时法的适用范围; ㈢对各种漂移运动进行分类，并找出其适用范围; ㈣对所审议的区域采用逻辑顺序;（v）为将同样的思想应用于研究更复杂的复杂系统打开大门。我们的方法来研究漂移是相当不寻常的：而不是解决轨道的常微分方程，我们认为一个双曲型偏微分方程的标量拉格朗日场$a(\vx,t)$，轨迹表示的特征曲线，这个偏微分方程。它使我们得到拉格朗日运动的纯欧拉描述，这大大简化了计算。在这个问题中有两个小的标度参数：两个时间尺度的比值和一个无量纲的振动振幅。它把我们引到一系列关于区别极限的问题。我们已经考虑了四种不同的极限，它们在慢时间的尺度上彼此不同。我们已经证明，每个独特的限制产生了无穷多个解决方案的$a(\vx,t)$。一个经典的漂移出现在一个纯粹的振荡流的情况下。与此同时，我们已经表明，漂移运动的概念是不够的拉格朗日动力学的描述和一些“扩散”条款确实出现。给出了漂移和“扩散”的五个不同选择的例子。
英文摘要：
Lagrangian motions of fluid particles in a general velocity field oscillating in time are studied with the use of the two-timing method. Our aims are: (i) to calculate systematically the most general and practically usable asymptotic solutions; (ii) to check the limits of applicability of the two-timing method by calculating the averaged motion without making any assumptions; (iii) to classify various drift motions and find their limits of applicability; (iv) to introduce a logical order into the area under consideration; (v) to open the gate for application of the same ideas to the studying more complex systems. Our approach to study a drift is rather unusual: instead of solving the ODE for trajectories we consider a hyperbolic PDE for a scalar lagrangian field $a(\vx,t)$, trajectories represent the characteristics curves for this PDE. It leads us to purely eulerian description of lagrangian motion, that greatly simplifies the calculations. There are two small scaling parameters in the problem: a ratio of two time-scales and a dimensionless vibrational amplitude. It leads us to the sequence of problems for distinguished limits. We have considered four distinguished limits which differ from each other by the scale of slow time. We have shown that each distinguished limit produces an infinite number of solutions for $a(\vx,t)$. A classical drift appears in the case of a purely oscillating flow. At the same time, we have shown that the concept of drift motion is not sufficient for the description of lagrangian dynamics and some `diffusion' terms do appear. Five examples of different options of drifts and `diffusion' are given.
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