Unveiling the Ties Between Nikuradse and Kolmogorov: How to Derive the Diagram From the Spectrum

We model the relation, first evinced by Nikuradse, among the Reynolds number, the relative roughness, and the friction coefficient ($f$) of turbulent flows in rough pipes. To that end, we identify the eddies that effect most of the momentum transfer between the viscous layer and the turbulent flow, and derive an expression for $f$ in terms of the characteristic velocity of those eddies, $u_s$. Then we use Kolmog\'orov's spectrum for the inertial range to determine $u_s$, and show that the resulting expression for $f$ gives a gradual transition between the scalings of Blasius and Strickler, but fails to give the hump or the bellies of Nikuradse's diagram. To obtain an expression for $f$ that also gives the bellies, we include an exponential spectrum for the dissipation range. Last, to obtain an expression for $f$ that also gives the hump, we include von K\'arm\'an's spectrum for the energy-containing range. This final expression for $f$ is in minute qualitative agreement with Nikuradse's diagram; it affords a way of interpreting successive portions of the diagram as manifestations of the varying habits of momentum transfer; and it reveals the existence of close ties between two milestones of experimental and theoretical turbulence.