Law of the wall
This law of the wall was first published in 1930 by Hungarian-American mathematician, aerospace engineer, and physicist Theodore von Kármán.
[1] 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.
[2] 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:[3] where From experiments, the von Kármán constant is found to be
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.
,[2] 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.
are given by:[2][5] Intermediate values are generally given by the empirically derived Nikuradse diagram,[2] though analytical methods for solving for this range have also been proposed.
[6] 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.
[7] 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.
[8][9] In 1996, Cipra submitted experimental evidence in support of these power-law descriptions.
[11] 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.
For scalars (most notably temperature), the self-similar logarithmic law of the wall has been theorized (first formulated by B.
[15][16][17][18] 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.
Below the region where the law of the wall is applicable, there are other estimations for friction velocity.
[19] In the region known as the viscous sublayer, below 5 wall units, the variation of
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 intersect, at
with an eddy viscosity formulation based on a near-wall turbulent kinetic energy
Comparisons with DNS data of fully developed turbulent channel flows for
