The logarithmic law of the wall is a self similar solution for the mean velocity parallel to the wall, and is valid for flows at high Reynolds numbers — in an overlap region with approximately constant shear stress and far enough from the wall for viscous effects to be negligible: where From experiments, the von Kármán constant is found to be and for a smooth wall. With dimensions, the logarithmic law of the wall can be written as: where y0 is the distance from the boundary at which the idealized velocity given by the law of the wall goes to zero. This is necessarily nonzero because the turbulent velocity profile defined by the law of the wall does not apply to the laminar sublayer. The distance from the wall at which it reaches zero is determined by comparing the thickness of the laminar sublayer with the roughness of the surface over which it is flowing. For a near-wall laminar sublayer of thickness and a characteristic roughness length-scale, 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: 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.
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.
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: where, This approximation can be used farther than 5 wall units, but by the error is more than 25%.
Buffer Layer
In the buffer layer, between 5 wall units and 30 wall units, neither law holds, such that: with the largest variation from either law occurring approximately where the two equations intercept, at. That is, before 11 wall units the linear approximation is more accurate and after 11 wall units the logarithmic approximation should be used, though neither are relatively accurate at 11 wall units. The mean streamwise velocity profile is improved for with an eddy viscosity formulation based on a near-wall turbulent kinetic energy function and the van Driest mixing length equation. Comparisons with DNS data of fully developed turbulent channel flows for showed good agreement.
