Power Law Turbulent Velocity Profile in Transitional Rough Pipes

Abstract

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.