Power Law Turbulent Velocity Profile in Transitional Rough Pipes

Alternate power law velocity profile $u+=Aζα$ in transitional rough pipe fully turbulent flow, has been proposed, in terms of new appropriate inner rough wall variables (⁠$ζ=Z+∕ϕ$⁠, $uϕ=u∕ϕ$⁠), and new parameters $Rϕ=Rτ∕ϕ$ termed as the roughness friction Reynolds number, $Reϕ=Re∕ϕ$ termed as the roughness Reynolds number and $ϕ$ termed as roughness scale (along with normal wall coordinate $Z=y+ϵr$ where $ϵr$ is the shift of the origin of boundary layer due to the rough wall, $Z+=Zuτ∕ν$ and $u+=u∕uτ$⁠). The envelope of the power law shows that the power law constants $α$ and $A$ depend on the parameter $Rϕ$ (i.e., $α=α(Rϕ)$ and $A=A(Rϕ)$⁠) but explicitly independent of the wall roughness parameter $h∕δ$ (roughness height $h$ in pipe of radius $δ$⁠). The roughness scale $ϕ$ has been related to the roughness function $ΔU+$ of Clauser representing the velocity shift caused by wall roughness. The present results of the velocity profile, just slightly above the wall roughness level $h$⁠, remain valid for all types of wall roughness. The data of Nikuradse for sand-grain roughness, in transitional and fully rough pipes, has been considered, which provides good support to the predictions of an alternate power law velocity profile, based on single parameter $Rϕ$⁠, the roughness friction Reynolds number.