On the attached eddy hypothesis and the turbulent friction factor of pipe flows with rough wall.*

The characterization of turbulent friction factors in rough pipe flows holds significant importance in fluid dynamics research. Among the various equations employed by engineers, the classical Colebrook-White (CW) equation stands out as the most popular choice. However, despite its accuracy, the CW equation suffers from certain limitations. Being an implicit formulation, it provides limited insight into the near-wall momentum transfer mechanisms.

In this study, we propose an innovative approach by incorporating the attached eddies hypothesis to develop an alternative equation that describes the turbulent friction factor within rough wall pipe flows. Building upon the work of Anbarlooei et al. (H.R. Anbarlooei, D.O.A. Cruz, F. Ramos "New power-law scaling for the friction factor of extreme Reynolds number pipe flows," The Physics of Fluids 32(9):95121.), our formulation expands to include the effects of vortexes generated by the rough wall elements on the momentum transfer near the wall region.

The resulting new equation offers explicit representation and demonstrates accurate reproduction of experimental data, addressing the limitations of the classical CW equation. This advancement opens new avenues for understanding and optimizing fluid flow in rough-walled pipes, which has practical implications across various engineering applications.