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Drawing on an analogy to critical phenomena, it was shown that the Nikuradseturbulent friction factor ($f_t$) measurements in pipes of radius $R$ and wallroughness $r$ can be collapsed onto a one-dimensional curve expressed as aconveyance law $f_t Re^{1/4}=g_o(\chi)$, where $Re$ is a bulk Reynolds number,$\chi =Re^{3/4}\left({r}/{R}\right)$. The implicit function $g_o(.)$ wasconjectured based on matching two asymptotic limits of $f_t$. However, theconnection between $g_o(.)$ and the phenomenon it proclaims to represent -turbulent eddies - remains lacking. Using models for the wall-normal velocityspectrum and return-to-isotropy for pressure-strain effects to close aco-spectral density budget, a derivation of $g_o(.)$ is offered. The proposedmethod explicitly derives the solution of the conveyance law and provides aphysical interpretation of $\chi$ as a dimensionless length scale reflectingthe competition between viscous sublayer thickness and characteristic height ofroughness elements. The application of the proposed method to other publishedmeasurements spanning roughness and Reynolds numbers beyond the originalNikuradse range is further discussed.

Roughness-induced critical phenomenon analogy for turbulent friction factor explained by a co-spectral budget model