Alternate Scales for Turbulent Flow in Transitional Rough Pipes: Universal Log Laws

In transitional rough pipes, the present work deals with alternate four new scales, the inner wall transitional roughness variable $ζ=Z+∕ϕ$⁠, associated with a particular roughness level, defined by roughness scale $ϕ$ connected with roughness function $▵U+$⁠, the roughness friction Reynolds number $Rϕ$ (based on roughness friction velocity), and roughness Reynolds number $Reϕ$ (based on roughness average velocity) where the mean turbulent flow, little above the roughness sublayer, does not depend on pipes transitional roughness. In these alternate variables, a two layer mean momentum theory is analyzed by the method of matched asymptotic expansions for large Reynolds numbers. The matching of the velocity profile and friction factor by Izakson-Millikan-Kolmogorov hypothesis gives universal log laws that are explicitly independent of pipe roughness. The data of the velocity profile and friction factor on transitional rough pipes provide strong support to universal log laws, having the same constants as for smooth walls. There is no universality of scalings in traditional variables and different expressions are needed for various types of roughness, as suggested, for example, with inflectional-type roughness, monotonic Colebrook-Moody roughness, etc. In traditional variables, the roughness scale, velocity profile, and friction factor prediction for inflectional pipes roughness are supported very well by experimental data.