Relation between the spectral properties of wall turbulence and the scaling of the Darcy-Weisbach friction factor

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.

Figure 1

Scheme of the idealized dimensionless vertical velocity spectrum $E_{ww}/E_{\mathrm{Kol}}\left(K_a\right)$.

Figure 2

${\alpha }_1{\alpha }_2$ as a function of $d/R$ in the fully rough turbulent flow case.

Figure 3

${\alpha }_1{\alpha }_2^{3/4}$ as a function of Re in the smooth turbulent flow case.