When the pipe surface is smooth (the "smooth pipe" curve in Figure 2), the friction factor's variation with Re can be modeled by the Kármán–Prandtl resistance equation for turbulent flow in smooth pipes with the parameters suitably adjusted
The numbers 1.930 and 0.537 are phenomenological; these specific values provide a fairly good fit to the data. The product  (called the "friction Reynolds number") can be considered, like the Reynolds number, to be a (dimensionless) parameter of the flow: at fixed values of , the friction factor is also fixed.
In the Kármán–Prandtl resistance equation,  can be expressed in closed form as an analytic function of  through the use of the Lambert  function:
In this flow regime, many small vortices are responsible for the transfer of momentum between the bulk of the fluid to the pipe wall. As the friction Reynolds number  increases, the profile of the fluid velocity approaches the wall asymptotically, thereby transferring more momentum to the pipe wall, as modeled in Blasius boundary layer theory.