Mean flow measurements are presented for fully developed turbulent pipe flow over a Reynolds number range of $57\,{\times}\,10^3$ to $21\,{\times}\,10^6$ where the flow exhibits hydraulically smooth, transitionally rough, and fully rough behaviours. The surface of the pipe was prepared with a honing tool, typical of many engineering applications, achieving a ratio of characteristic roughness height to pipe diameter of 1 : 17000. Results for the friction factor show that in the transitionally rough regime this surface follows a Nikuradse (1933)-type inflectional relationship rather than the monotonic Colebrook (1939) relationship used in the Moody diagram. This result supports previous suggestions that the Moody diagram in the transitional regime must be used with caution. Outer scaling of the mean velocity data shows excellent collapse and strong evidence for Townsend's outer layer similarity hypothesis for rough-walled flows. Finally, the pipe exhibited smooth behaviour for scaled roughness height $k_s^+ \,{\le}\, 3.5$, which supports the suggestion by Zagarola & Smits (1998) that their pipe was hydraulically smooth for $Re_D\,\,{\leq}\, 24\,{\times}\,10^6$.

Recent experiments at Princeton University have revealed aspects of smooth pipe flow behaviour that suggest a more complex scaling than previously noted. In particular, the pressure gradient results yield a new friction factor relationship for smooth pipes, and the velocity profiles indicate the presence of a power-law region near the wall and, for Reynolds numbers greater than about 400x103 (R+>9x103), a logarithmic region further out. New experiments on a rough pipe with a honed surface finish with krms/D=19.4x10-6, over a Reynolds number range of 57x103-21x106, show that in the transitionally rough regime this surface follows an inflectional friction factor relationship rather than the monotonic relationship given in the Moody diagram. Outer-layer scaling of the mean velocity data and streamwise turbulence intensities for the rough pipe show excellent collapse and provide strong support for Townsend's outer-layer similarity hypothesis for rough-walled flows. The streamwise rough-wall spectra also agree well with the corresponding smooth-wall data. The pipe exhibited smooth behaviour for ks+ < or =3.5, which supports the suggestion that the original smooth pipe was indeed hydraulically smooth for ReD< or =24x106. The relationship between the velocity shift, DeltaU/utau, and the roughness Reynolds number, ks+, has been used to generalize the form of the transition from smooth to fully rough flow for an arbitrary relative roughness krms/D. These predictions apply for honed pipes when the separation of pipe diameter to roughness height is large, and they differ significantly from the traditional Moody curves.

A method for evaluating a universal transitional resistance diagram for pipes that relates the pressure drop in the pipe to Reynolds number, as a function of relative surface roughness, is presented. The method assumes a universal wake function, a logarithmic overlap region and a power fit in the viscous and buffer layer. Estimates can be made of the friction factor–Reynolds number relationship for arbitrary relative roughness, based on a given surface geometry. The method is illustrated for a pipe with a honed surface finish and uses data of Shockling (“Turbulent flow in a rough pipe,” MSE dissertation, Princeton University, 2005). Honed roughness demonstrates an inflectional behavior in the transitionally rough regime, much like sand grain roughness [Laws of flow in rough pipes, VDI Forschungsh, 361 (1933), 1292 (NACA TM, 1950)], but the method proposed here can be applied to any given roughness behavior. It is suggested that the critical parameter that determines whether the resistance diagram shows inflectional characteristics is the ratio of roughness height to outer layer scale. Based on analysis of data from previous researchers it is suggested that if the relative surface roughness krms∕D&lt;0.0025, where krms is the rms amplitude of the roughness and D is the pipe diameter, inflectional relationships should be observed.

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