Turbulent flow in smooth and rough pipes

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 400 × 10³ (R⁺ > 9 × 10³), a logarithmic region further out. New experiments on a rough pipe with a honed surface finish with k_rms/D= 19.4 × 10^-6, over a Reynolds number range of 57 × 10³-21 × 10⁶, 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 k _s⁺ ≤ 3.5, which supports the suggestion that the original smooth pipe was indeed hydraulically smooth for Re_D≤24 × 10⁶. The relationship between the velocity shift, ΔU//u_τ, and the roughness Reynolds number, k _s⁺, has been used to generalize the form of the transition from smooth to fully rough flow for an arbitrary relative roughness k _rms/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.