Relation of flow speed to wall distance
In fluid dynamics, the law of the wall (also known as the logarithmic law of the wall) states that the average velocity of a turbulent flow at a certain point is proportional to the logarithm of the distance from that point to the "wall", or the boundary of the fluid region. This law of the wall was first published in 1930 by Hungarian-American mathematician, aerospace engineer, and physicist Theodore von Kármán. It is only technically applicable to parts of the flow that are close to the wall ( ;20% of the height of the flow), though it is a good approximation for the entire velocity profile of natural streams.
General logarithmic formulation.
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 (direct) viscous effects to be negligible:
formula_1   with   formula_2   formula_3   and   formula_4
where 
From experiments, the von Kármán constant is found to be formula_5 and formula_6 for a smooth wall.
With dimensions, the logarithmic law of the wall can be written as:
formula_7
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 formula_8 and a characteristic roughness length-scale formula_9,
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, formula_10, where
formula_11
The flow is hydraulically smooth for formula_12, hydraulically rough for formula_13, and transitional for intermediate values.
Values for formula_14 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,
formula_15
where formula_16 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 formula_17 to formula_18 is approximately 1:1, such that:
For  formula_19
formula_20
where, 
This approximation can be used farther than 5 wall units, but by formula_21 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:
For  formula_22
formula_23
formula_24
with the largest variation from either law occurring approximately where the two equations intersect, at formula_25. 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 formula_17 is improved for formula_27 with an eddy viscosity formulation based on a near-wall turbulent kinetic energy formula_28 function and the van Driest mixing length equation. Comparisons with DNS data of fully developed turbulent channel flows for formula_29 showed good agreement.
Notes.
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