A unified friction factor formulation: Bridging laminar and turbulent friction factor with critical points analysis

The friction factor is widely recognized as a pivotal parameter in the analysis of fluid-boundary interactions; however, a comprehensive grasp of friction mechanics remains elusive. This investigation revisits measurements from the benchmark Nikuradse measurements, furnishing indirect evidence of two critical points in pipe turbulence. It underscores that friction factors within laminar and turbulent regimes are intimately interconnected, bearing significant relations to subcritical and critical phenomena. The two critical points directing the laminar-turbulent transition consist of a standard non-equilibrium phase transition and a fully matured turbulent regime, accompanied by an extensive crossover to its asymptotic scaling. Relying on a mathematical model, the scaled friction factor for rough pipes converges into a unified curve. New formula of friction factor of pipe flow is derived, and it was illustrated that it can be derived as a geometric weighted parameter, bridging the laminar and turbulent friction factors. Conclusively, the proffered model was juxtaposed with pipe experimental data from the antecedent study and more contemporaneous transitional pipe data to authenticate the aptness of the suggested model, and it united the friction factor in all three regimes.