|} Intuitively, this means that if the roughness elements are hidden within the laminar sublayer, they have a much different effect on the turbulent law of the wall velocity profile than if they are sticking out into the main part of the flow. This is also often more formally formulated in terms of a boundary Reynolds number, , where The flow is hydraulically smooth for , hydraulically rough for , and transitional for intermediate values. Values for are given by: {| border="0" |- | || for hydraulically smooth flow |- | || for hydraulically rough flow. |} Intermediate values are generally given by the empirically derived Nikuradse diagram, though analytical methods for solving for this range have also been proposed. For channels with a granular boundary, such as natural river systems, where is the average diameter of the 84th largest percentile of the grains of the bed material. Power law solutions Works by Barenblatt and others have shown that besides the logarithmic law of the wall — the limit for infinite Reynolds numbers — there exist power-law solutions, which are dependent on the Reynolds number. In 1996, Cipra submitted experimental evidence in support of these power-law descriptions. This evidence itself has not been fully accepted by other experts. In 2001, Oberlack claimed to have derived both the logarithmic law of the wall, as well as power laws, directly from the Reynolds-averaged Navier–Stokes equations, exploiting the symmetries in a Lie group approach. However, in 2014, Frewer et al. refuted these results. For scalars For scalars (most notably temperature), the self-similar logarithmic law of the wall has been theorized (first formulated by B. A. Kader) and observed in experimental and computational studies. In many cases, extensions to the original law of the wall formulation (usually through integral transformations) are generally needed to account for compressibility, variable-property and real fluid effects. Near the wall Below the region where the law of the wall is applicable, there are other estimations for friction velocity. Viscous sublayer In the region known as the viscous sublayer, below 5 wall units, the variation of to is approximately 1:1, such that: For where, {| border="0" ||| is the wall coordinate: the distance y to the wall, made dimensionless with the friction velocity and kinematic viscosity , |- ||| is the dimensionless velocity: the velocity u parallel to the wall as a function of y (distance from the wall), divided