text: When the pipe surfaces roughness height  is significant (typically at high Reynolds number), the friction factor departs from the smooth pipe curve, ultimately approaching an asymptotic value ("rough pipe" regime). In this regime, the resistance to flow varies according to the square of the mean flow velocity and is insensitive to Reynolds number. Here, it is useful to employ yet another dimensionless parameter of the flow, the roughness Reynolds number'
  where the roughness height  is scaled to the pipe diameter .
  It is illustrative to plot the roughness function :
  Figure 3 shows  versus  for the rough pipe data of Nikuradse, Shockling, and Langelandsvik.
  In this view, the data at different roughness ratio  fall together when plotted against , demonstrating scaling in the variable . The following features are present:
  * When , then  is identically zero: flow is always in the smooth pipe regime. The data for these points lie to the left extreme of the abscissa and are not within the frame of the graph. 
  * When , the data lie on the line ;  flow is in the smooth pipe regime. 
  * When , the data asymptotically approach a horizontal line; they are independent of , , and .
  * The intermediate range of  constitutes a transition from one behavior to the other. The data depart from the line  very slowly, reach a maximum near , then fall to a constant value.
  Afzal's fit to these data in the transition from smooth pipe flow to rough pipe flow employs an exponential expression in  that ensures proper behavior for  (the transition from the smooth pipe regime to the rough pipe regime):
  and 
  This function shares the same values for its term in common with the Kármán–Prandtl resistance equation, plus one parameter 0.305 or 0.34 to fit the asymptotic behavior for  along with one further parameter, 11, to govern the transition from smooth to rough flow. It is exhibited in Figure 3.
  The  friction factor for another analogous roughness becomes
  and
  This function shares the same values for its term in common with the Kármán–Prandtl resistance equation, plus one parameter 0.305 or 0.34 to fit the asymptotic behavior for  along with one further parameter, 26, to govern the transition from smooth to rough flow. 
  The Colebrook–White relation fits the friction factor with a function of the form
  This relation has the correct behavior at extreme values of , as shown by the labeled curve in Figure 3: when  is small, it is consistent with smooth pipe flow, when large, it is consistent with rough pipe flow. However its performance in the transitional domain overestimates the friction factor by a substantial margin. Colebrook acknowledges the discrepancy with Nikuradze's data but argues that his relation is consistent with the measurements on commercial pipes. Indeed, such pipes are very different from those carefully prepared by Nikuradse: their surfaces are characterized by many different roughness heights and random spatial distribution of roughness points, while those of Nikuradse have surfaces with uniform roughness height, with the points extremely closely packed.