标题： 层流和湍流流体流动的数学定义 显示英文标题

标题： On a mathematical definition of laminar and turbulent fluid flow

摘要： 如标题所述，本研究提出了层流和湍流的数学定义，即一种可用于构思和证明关于此类流动的数学定理的定义。 该定义基于人类长期已知的实验事实：每当重复一个给定的流动时，如果流动是湍流，结果将不会相同。 湍流流动并非严格可重复。 从这一基本事实可以得出关于湍流流动的更复杂的真理：通过大量重复结果求平均得到的平均流动并不是自然流动，即它是一种在任何实验中都无法自然出现的流动。 所提出的定义需要一些初步的数学概念，这些概念也在文中进行了介绍：函数之间的接近性，实现的集合，流动的平均方法，以及实现（物理流动）和平均值（平均流动）的不同特性。 引入了受限非线性的概念，并证明了层流只能在受限非线性条件下存在，而湍流是普遍非线性的结果。 研究了稳态湍流的特殊情况，并对集合平均和时间平均是否相等提出了疑问。 还提供了两个已解决的例子来说明定义所隐含的意义和方法：冯·卡门涡街和受强制白噪声扰动的层流。 显示英文摘要

摘要： As stated in the title, the present research proposes a mathematical definition of laminar and turbulent flows, i.e., a definition that may be used to conceive and prove mathematical theorems about such flows. The definition is based on an experimental truth long known to humans: Whenever one repeats a given flow, the results will not be the same if the flow is turbulent. Turbulent flows are not strictly repeatable. From this basic fact follows a more elaborate truth about turbulent flows: The mean flow obtained by averaging the results of a large number of repetitions is not a natural flow, that is, it is a flow that cannot occur naturally in any experiment. The proposed definition requires some preliminary mathematical notions, which are also introduced in the text: Proximity between functions, the ensemble of realisations, the method of averaging the flows, and the distinct properties of realisations (physical flows) and averages (mean flows). The notion of restricted nonlinearity is introduced and it is demonstrated that laminar flows can only exist in conditions of restricted nonlinearity, whereas turbulent flows are a consequence of general nonlinearity. The particular case of steady-state turbulent flow is studied, and an uncertainty is raised about the equality of ensemble average and time average. Two solved examples are also offered to illustrate the meaning and methods implied by the definitions: The von K\`arm\`an vortex street and a laminar flow with imposed white-noise perturbation.