Friction Factor Directly From Transitional Roughness in a Turbulent Pipe Flow

The friction factor data from transitional rough test pipes, from the measurements of Sletfjerding and Gudmundsson (2003, “Friction Factor Directly From Roughness Measurements,” J. Energy Resour. Technol. 125, pp. 126–130), have been analyzed in terms of directly measurable roughness parameters, $Ra$ the arithmetic mean roughness, $RZ$ the mean peak to valley heights roughness, $Rq$ the root mean square (rms) roughness, and $Rq∕H$ rms textured roughness (⁠$H$⁠, the Hurst exponent is a texture parameter), in addition to $h$ the equivalent sand grain roughness. The proposed friction factor $λ$⁠, in terms of new scaling parameter, viz., the roughness Reynolds number $Reϕ=Re∕ϕ$ (where $ϕ$ is a nondimensional roughness scale), is a universal relation for all kinds of surface roughness. This means that Prandtl’s smooth pipe friction factor relation would suffice provided that the traditional Reynolds number Re is replaced by the roughness Reynolds number $Reϕ$⁠. This universality is very well supported by the extensive rough pipe data of Sletfjerding and Gudmundsson, Shockling’s (2005, “Turbulence Flow in Rough Pipe,” MS thesis, Princeton University) machined honed pipe surface roughness data, and Nikuradse’s (1933, Laws of Flow in Rough Pipe, VDI, Forchungsheft No. 361) sand grain roughness data. The predictions for the roughness function $ΔU+$⁠, and the roughness scale $ϕ$ for inflectional roughness compare very well with the data of the above mentioned researchers. When surface roughness is present, there is no universality of scaling of the friction factor $λ$ with respect to the traditional Reynolds number Re, and different expressions are needed for various types of roughnesses, as suggested, for example, with inflectional roughness, monotonic roughness, etc. In traditional variables, the proposed friction factor prediction for inflectional roughness in the pipes, is supported very well by the experimental data of Sletfjerding and Gudmundsson, Shockling, and Nikuradse. In the present work, the predictions of friction factor as implicit relations, as well as approximate explicit relations, have also been proposed for various roughness scales.