Turbulent Flow in a Transitional Rough Pipe Universal Laws; Power Law Velocity With Equivalent Log Law Velocity in the Overlap Region: Commencing From Fully Smooth Pipe Flow

The turbulent transitional rough pipe universal power law and equivalent log law are, independent of wall roughness, without any closure model. The open Reynolds mean momentum equations are employed without closure models such as eddy viscosity or mixing length. That all components of Reynolds stress are of same order of the wall shear stress, $τw$⁠. The key parameters are wall roughness scale $ϕ$⁠, roughness friction Reynolds number $Rτϕ=Reτ/ϕ$⁠, and roughness average Reynolds number $Rbϕ=Reb/ϕ$⁠. The s three layers (inner, meso, and outer), with overlap region reveals dual solutions: power law and log law. The power law friction factor can be expressed as $λ=(CS,n,Re/ϕ)$⁠. The power law index n and prefactor $CS$ remain as fully smooth pipe power law constants and do not depend on the roughness friction Reynolds number $Reτ/ϕ$⁠. The power law velocity and friction factor exhibit envelopes where the tangent at a point $Reτ/ϕ=exp(α−1−κB)$ yields equivalent log laws. If outer layer is neglected, the power law friction factor simplifies to $λ=CS(Re/ϕ)−n$⁠. As an engineering approximation, the power laws fr are extrapolated within a $±$ 5 percent domain, a limited range of Reynolds numbers with experimental and direct numerical simulation (DNS) data. Additionally, log law theory for transitional rough pipe is extended to higher-order effects $(Reτ/ϕ)−p$⁠, where $p=1,2,…,∞$⁠. The power law and log law work comparison were made with turbulent transitional experimental and DNS data.