From Colebrook-White roughness to Nikuradse sand grains: A multiscale momentum-transfer model for turbulent friction over rough surfaces.

Abstract

Wall roughness plays a central role in determining turbulent friction in pipe flows, yet most predictive frameworks continue to characterize the wall by a single effective roughness scale. Classical datasets-notably Nikuradse's sand-grain experiments and the rough-pipe measurements of Colebrook and White-demonstrate that this description is incomplete: surfaces with comparable nominal roughness heights can exhibit markedly different friction curves, including branch splitting and multiregime behavior. Reexamining the Colebrook-White experiments, we identify clear evidence of multiscale additivity, whereby the contribution of a smaller roughness scale enters once it becomes dynamically accessible and increases approximately linearly with its areal coverage. Motivated by this observation, we develop a multiscale extension of the Gioia-Chakraborty momentum-transfer framework in which roughness elements at distinct geometric scales contribute additive increments to the total friction. In the model, each scale becomes dynamically active when its height exceeds a Reynolds-number-dependent viscous cutoff proportional to the Kolmogorov length, while its contribution is weighted by a scale-dependent coefficient representing the geometric prominence of that roughness scale. This construction generalizes the original scale-selection mechanism by allowing multiple roughness scales to participate concurrently in outer-layer momentum transfer. The resulting framework provides a unified and continuous description connecting the classical roughness-dominated Nikuradse-Strickler regime, Colebrook-White-type trends at intermediate roughness, and smooth-wall behavior. For the applications considered here, however, the model coefficients are not uniquely determined from surface properties and are selected in a data-guided manner; establishing a quantitative mapping from measured surface topography to these coefficients remains an open problem.