Empirical formulas describing the Darcy-Weisbach friction factor remain indispensable for applications in sciences and engineering dealing with turbulent flows. Despite their practical significance, these formulas have remained without theoretical interpretation for many decades. To close this knowledge gap, much research has been devoted to the development of the so-called “spectral link” introduced in the early 2000s. Such a theory is entirely based on elegant phenomenological arguments that make no contact with equations describing turbulent wall flows. The spectral link spawned alternative approaches, now labeled “cospectral budget” (or CSB) models, that describe how turbulent eddies contribute to wall stresses. The CSB overcomes some of the shortcomings of the phenomenological approach and is here employed to provide a thorough clarification of the link between spectral properties of velocity fluctuations and the scaling of friction factors in turbulent pipe flows in the hydraulically smooth and fully rough regimes.

中文翻译：

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壁湍流的光谱特性与Darcy-Weisbach摩擦系数的标度之间的关系

描述Darcy-Weisbach摩擦因数的经验公式对于处理湍流的科学和工程应用仍然是必不可少的。尽管它们具有实际意义，但数十年来，这些公式一直没有进行理论解释。为了弥合这一知识鸿沟，许多研究致力于发展2000年代初引入的所谓“光谱链接”。这种理论完全基于优雅的现象学论点，这些论点与描述湍流壁流的方程式没有任何联系。频谱链接产生了替代方法，这些方法现在称为“共谱预算”（或CSB）模型，用于描述湍流涡流如何影响壁面应力。