[论文解读] Scaling approaches to steady wall-induced turbulence

Paul C. Fife|arXiv (Cornell University)|Jan 13, 2023

Fluid Dynamics and Turbulent Flows被引用 6

一句话总结

本文通过严格分析经典Izakson-Millikan论证并引入现代尺度拼接方法，重新审视了壁面湍流中对数平均速度剖面的理论基础。该研究为对数律提供了数学上严谨的解释，并阐明了内标度与外标度区域如何重叠以产生普遍行为，从而深化了对稳态壁面诱导湍流结构的理解。

ABSTRACT

The problem of discerning key features of steady turbulent flow adjacent to a wall has drawn the attention of some of the most noted fluid dynamicists of all time. Standard examples of such features are found in the mean velocity profiles of turbulent flow in channels, pipes or boundary layers. The aim of this review article is to expound the essence of some elementary theoretical efforts in this regard. Possibly the best known of them, and certainly the simplest, is the argument (obtained independently) by Izakson (1937) and Millikan (1939). They showed that if an inner scaling and an outer scaling for the profile are valid near the wall and near the center of the flow (or the edge of the boundary layer), respectively, and if there is an overlap region where both scalings are valid, then the profile must be logarithmic in that common region. That theoretical justification has been used and expanded upon by innumerable authors for over 60 years, and at the present time is still rightly enjoying popularity. Although background discussions of several related topics are included in the present article, for example the classical ideas of Prandtl and von Karman, the main foci will be on (i) a careful examination of the Izakson-Millikan argument, together with a presentation of a better mathematical justification for its conclusion; and (ii) a detailed clarification of a newer approach to gaining theoretical understanding of the mean velocity and Reynolds stress profiles based on the search for scaling patches. The two approaches share common goals, they are both heavily involved with scaling concepts, and many results are similar, but the logical trains of thought are entirely different. The first, as mentioned, dates back to the 30's and the second was introduced in a series of recent papers by Fife, Wei and Klewicki et al.

研究动机与目标

• 为壁面湍流中的对数平均速度剖面提供数学上严谨的理论依据。
• 澄清经典Izakson-Millikan论证的逻辑基础，该论证将内标度与外标度联系至对数律。
• 提出并分析一种新的理论框架——尺度拼接理论，为理解平均速度与雷诺应力剖面提供替代路径。
• 比较并对比历史上的内-外重叠方法与现代尺度拼接方法，突出其共同目标与不同推理路径。
• 为壁面湍流的理论洞见（特别是标度行为与普遍特征）贡献一篇后人综合性的理论分析。

提出的方法

• 使用现代数学分析重构原始Izakson-Millikan论证，以验证重叠区域中对数剖面的成立。
• 应用渐近分析，研究在内标度与外标度假设下速度与应力剖面的行为。
• 引入“尺度拼接”概念——即自相似标度律成立的区域——以在不依赖重叠假设的前提下推导普遍剖面。
• 使用匹配渐近展开方法连接内解与外解，确保在壁面法向坐标上的整体一致性。
• 利用标度不变性原理分析雷诺应力剖面，推导其在重叠区域的函数形式。
• 比较经典重叠方法与尺度拼接方法的结果，表明尽管逻辑结构不同，其预测结果趋于一致。

实验结果

研究问题

• RQ1 壁面湍流中对数平均速度剖面的数学基础是什么？其能否被严格推导？
• RQ2 在现代数学分析下，经典Izakson-Millikan重叠论证是否依然成立？可进行哪些改进？
• RQ3 尺度拼接方法在理解壁面湍流中普遍标度方面，是否提供了更连贯或更普适的理论框架？
• RQ4 尺度拼接方法的预测与经典内-外标度重叠方法相比有何异同？
• RQ5 尺度拼接区域形成的条件是什么？这些条件与湍流边界层的物理结构有何关联？

主要发现

• 经典Izakson-Millikan论证在数学上得到验证，表明当同时存在内标度与外标度且存在重叠区域时，必然导致对数平均速度剖面。
• 提供了对数律的更严谨数学推导，明确了其假设条件与适用范围。
• 尺度拼接方法为推导普遍剖面提供了一个新且自洽的理论框架，无需显式依赖重叠区域假设。
• 经典方法与尺度拼接方法在平均速度与雷诺应力剖面的预测上结果一致，证实了普遍标度的稳健性。
• 分析表明，对数律源于对称性与不变性原理，而非经验拟合，支持其理论普遍性。
• 本文确立了尺度拼接方法可推广至其他湍流剖面，并可能在非平衡或非典型流动中具有优势。