On the incomplete similarity for turbulent velocity profiles in rough pipes

The possibility of extending Barenblatt's hypothesis of incomplete similarity for the velocity profile is analyzed for rough-pipe flows. The verification of this hypothesis, which leads to a power law in the intermediate region of the flow, is performed using Nikuradse's well-known data for pipes with homogeneous sand roughness. As detailed comparison between the log law and the power law would require much higher measurement resolution, no attempt is made here to assess the validity of one or the other law, and the power law profile is simply used as a working hypothesis. The power law fits well for a large part of the pipe section both for the transitionally rough regime and the fully rough regime. The coefficients of the power law show an interesting behavior as a function of the Reynolds number and the relative roughness. For the fully rough regime, it is possible to obtain simple expressions for the power law coefficients and the resistance law.