Law of the wall

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.

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 (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.

Below the region where the law of the wall is applicable, there are other estimations for friction velocity.