Alternate Scales for Turbulent Boundary Layer on Transitional Rough Walls: Universal Log Laws

The present work deals with four new alternate transitional surface roughness scales for description of the turbulent boundary layer. The nondimensional roughness scale $ϕ$ is associated with the transitional roughness wall inner variable $ζ=Z+∕ϕ$⁠, the roughness friction Reynolds number $Rϕ=Rτ∕ϕ$⁠, and the roughness Reynolds number $Reϕ=Re∕ϕ$⁠. The two layer theory for turbulent boundary layers in the variables, mentioned above, is presented by method of matched asymptotic expansions for large Reynolds numbers. The matching in the overlap region is carried out by the Izakson–Millikan–Kolmogorov hypothesis, which gives the velocity profiles and skin friction universal log laws, explicitly independent of surface roughness, having the same constants as the smooth wall case. In these alternate variables, just above the wall roughness level, the mean velocity and Reynolds stresses are universal and do not depend on surface roughness. The extensive experimental data provide very good support to our universal relations. There is no universality of scalings in traditional variables and different expressions are needed for inflectional type roughness, monotonic Colebrook–Moody roughness, $k$-type roughness, $d$-type roughness, etc. In traditional variables, the velocity profile and skin friction predictions for the inflectional roughness, $k$-type roughness, and $d$-type roughness are supported well by the extensive experimental data. The pressure gradient effect from the matching conditions in the overlap region leads to the universal composite laws, which for weaker pressure gradients yields log laws and for strong adverse pressure gradients provides the half-power laws for universal velocity profiles and in traditional variables the additive terms in the two situations depend on the wall roughness.